diff --git a/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb b/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb index 34be111c..9e165156 100644 --- a/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb +++ b/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb @@ -1,810 +1,765 @@ { - "cells": [ - { - "cell_type": "markdown", - "metadata": { - "id": "NGGk7Y6pFWqf" - }, - "source": [ - "[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/pinecone-io/examples/blob/master/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb) [![Open nbviewer](https://raw.githubusercontent.com/pinecone-io/examples/master/assets/nbviewer-shield.svg)](https://nbviewer.org/github/pinecone-io/examples/blob/master/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb)\n", - "\n", - "# Using LlamaIndex with Pinecone\n", - "\n", - "Accompanying notebook to [this documentation](https://docs.pinecone.io/docs/llamaindex).\n", - "Note this notebook requires a paid OpenAI plan to avoid rate limits." - ] - }, - { - "cell_type": "markdown", - "metadata": { - "id": "4-4KqRZKChu_" - }, - "source": [ - "# Set up environment\n" - ] - }, - { - "cell_type": "code", - "execution_count": 1, - "metadata": { - "id": "l_2XILBGBxNI" - }, - "outputs": [], - "source": [ - "!pip install -qU \\\n", - " llama-index==0.9.34 \\\n", - " \"pinecone-client[grpc]\"==3.0.0 \\\n", - " arxiv==2.1.0\n" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "lghB__9ECB7a", - "outputId": "d3cdffb6-5ef1-4e6d-fd91-13a15c845ee0" - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Enter your Pinecone API Key: \u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\n", - "Enter your OpenAI API Key: \u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\n" - ] - } - ], - "source": [ - "import os\n", - "from getpass import getpass\n", - "\n", - "pinecone_api_key = os.getenv(\"PINECONE_API_KEY\") or getpass(\"Enter your Pinecone API Key: \")\n", - "openai_api_key = os.getenv(\"OPENAI_API_KEY\") or getpass(\"Enter your OpenAI API Key: \")\n" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "kq0EB52hCYl5", - "outputId": "c6dfa7d1-2c02-4790-f1ac-02f41b797887" - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Python 3.10.12\n" - ] - } - ], - "source": [ - "# Notebook runs on Python version:\n", - "!python -V" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "id": "6n1FKszfCo6i" - }, - "source": [ - "# Load the data\n" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "LUHLfBHYCqPR", - "outputId": "20a957e1-562d-4920-c47d-7927c5bf5c7d" - }, - "outputs": [ - { - "data": { - "text/plain": [ - "Document(id_='8ace8b98-2057-4113-ac54-dcb474cbf89b', embedding=None, metadata={'page_label': '1', 'file_name': 'hnsw.pdf'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={}, hash='786d6f516cefedb35510cd9443b8bbef8256f606a479688b4ade59c548117475', text=\"IEEE TRANSACTIONS ON JOURNAL NAME, MANUS CRIPT ID 1 \\n Efficient and robust approximate nearest \\nneighbor search using Hierarchical Navigable \\nSmall World graphs \\nYu. A. Malkov, D. A. Yashunin \\nAbstract \u2014 We present a new approach for the approximate K -nearest neighbor search based on navigable small world \\ngraphs with controllable hierarchy (Hierarchical NSW , HNSW ). The proposed solution is fully graph -based, without any need for \\nadditional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. \\nHierarchical NSW incrementally builds a multi -layer structure consisting from hierarchical set of proximity graphs (layers) for \\nnested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an \\nexponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Sma ll \\nWorld (NSW) struc tures while additionally having the links separated by their characteristic distance scales. Starting search \\nfrom the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a \\nlogarithmic complexity scalin g. Additional employment of a heuristic for selecting proximity graph neighbors significantly \\nincreases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the \\nproposed general metric space search i ndex is able to strongly outperform previous opensource state -of-the-art vector -only \\napproaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation . \\nIndex Terms \u2014 Graph and tree search strategies, Artificial Intelligence, Information Search and Retrieval, Information Storage \\nand Retrieval, Information Technology and Systems, Search process, Graphs and networks, Data Structures, Nearest neighbor \\nsearch, Big data, Approximate search, Simil arity search \\n\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \\uf075 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \\n1 INTRODUCTION\\nonstantly growing amount of the available info r-\\nmation resources has led to high demand in scalable \\nand efficient similarity search data structures. One of the \\ngenerally used approaches for information search is the \\nK-Nearest Neighbor Search (K -NNS). The K -NNS a s-\\nsumes you have a defined distance function between the \\ndata elements and aims at finding the K elements from \\nthe dataset which minimize the distance to a given query. \\nSuch algorithms are used in many applications, such as \\nnon-parametric machine learn ing algorithms, image fe a-\\ntures matching in large scale databases [1] and semantic \\ndocument retrieval [2]. A na\u00efve approach to K -NNS is to \\ncompute the distances between the query and every el e-\\nment in the dataset and select the elements with minimal \\ndistance. Unfortunately , the complexity of the na\u00efve a p-\\nproach scales linearly with the number of stored elements \\nmaking it infeasible for large -scale datasets. This has led \\nto a high interest in development of fast and scalable K -\\nNNS algorithms. \\nExact solutions for K -NNS [3-5] may offer a substantial \\nsearch speedup only in case of relatively low dimensional \\ndata du e to \u201ccurse of dimensionality\u201d. To overcome this \\nproblem a concept of Approximate Nearest Neighbors \\nSearch (K -ANNS) was proposed, which relaxes the cond i-\\ntion of the exact search by allowing a small number of errors. The quality of an inexact search (the re call) is d e-\\nfined as the ratio between the number of found true nea r-\\nest neighbors and K. The m ost popular K -ANNS sol u-\\ntions are based on approximated versions of tree alg o-\\nrithms [6, 7], locality -sensitive hashing (LSH) [8, 9] and \\nproduct quantization (PQ) [10-17]. Proximity graph K -\\nANNS algorithms [10, 18 -26] have recently gained pop u-\\nlarity offering a better performance on high dimensional \\ndatasets. However, the power -law scaling of the proxim i-\\nty graph routing causes extreme performance degrad a-\\ntion in case o f low dimensional or clustered data. \\nIn this paper we propose the Hierarchical Navigable \\nSmall World (Hierarchical NSW, HNSW), a new fully \\ngraph based incremental K -ANNS structure, which can \\noffer a much better logarithmic complexity scaling. The \\nmain contributions are: explicit selection of the graph\u2019s \\nenter -point node, separation of links by different scales \\nand use of an advanced heuristic to select the neighbors. \\nAlternatively, Hierarchical NSW algorithm can be seen as \\nan extension of the probabilistic skip list structure [27] \\nwith proximity graphs instead of the linked lists. Perfo r-\\nmance e valuation has demonstrated that the proposed \\ngeneral metric space method is able to strongly outpe r-\\nform previous opensource state -of-the-art approaches \\nsuitable only for vector spaces. \\n\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \\n\\uf0b7 Y. M alkov is with the Federal state budgetary institution of science \\nInstitute of Applied Physics of the Russian Academy of Sciences, 46 \\nUl'yanov Street, 603950 Nizhny Novgorod, Russia . E-mail: \\nyurymalkov@mail.ru . \\n\\uf0b7 D. Y ashunin . Addres: 31-33 ul. Krasnozvezdnaya, 603104 Nizhny \\nNovgorod, Russia. E -mail: yashuninda@yandex.ru C \\nThis work has been submitted to the IEEE for possible publication. Copyright \\nmay be transferred without notice, after which this version may no longer be \\naccessible. \", start_char_idx=None, end_char_idx=None, text_template='{metadata_str}\\n\\n{content}', metadata_template='{key}: {value}', metadata_seperator='\\n')" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "import arxiv\n", - "from pathlib import Path\n", - "from llama_index import download_loader\n", - "\n", - "# Download paper to local file system (LFS)\n", - "# `id_list` contains 1 item that matches our PDF's arXiv ID\n", - "paper = next(arxiv.Client().results(arxiv.Search(id_list=[\"1603.09320\"])))\n", - "paper.download_pdf(filename=\"hnsw.pdf\")\n", - "\n", - "# Download and instantiate `PDFReader` from LlamaHub\n", - "PDFReader = download_loader(\"PDFReader\")\n", - "loader = PDFReader()\n", - "\n", - "# Load HNSW PDF from LFS\n", - "documents = loader.load_data(file=Path('./hnsw.pdf'))\n", - "\n", - "# Preview one of our documents\n", - "documents[0]" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "id": "lIMqktjNDC-o" - }, - "outputs": [], - "source": [ - "# Clean up our Documents' content\n", - "import re\n", - "\n", - "def clean_up_text(content: str) -> str:\n", - " \"\"\"\n", - " Remove unwanted characters and patterns in text input.\n", - "\n", - " :param content: Text input.\n", - "\n", - " :return: Cleaned version of original text input.\n", - " \"\"\"\n", - "\n", - " # Fix hyphenated words broken by newline\n", - " content = re.sub(r'(\\w+)-\\n(\\w+)', r'\\1\\2', content)\n", - "\n", - " # Remove specific unwanted patterns and characters\n", - " unwanted_patterns = [\n", - " \"\\\\n\", \" \u2014\", \"\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\", \"\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\", \"\u2014\u2014\u2014\u2014\u2014\",\n", - " r'\\\\u[\\dA-Fa-f]{4}', r'\\uf075', r'\\uf0b7'\n", - " ]\n", - " for pattern in unwanted_patterns:\n", - " content = re.sub(pattern, \"\", content)\n", - "\n", - " # Fix improperly spaced hyphenated words and normalize whitespace\n", - " content = re.sub(r'(\\w)\\s*-\\s*(\\w)', r'\\1-\\2', content)\n", - " content = re.sub(r'\\s+', ' ', content)\n", - "\n", - " return content\n", - "\n", - "# Call function\n", - "cleaned_docs = []\n", - "for d in documents:\n", - " cleaned_text = clean_up_text(d.text)\n", - " d.text = cleaned_text\n", - " cleaned_docs.append(d)" - ] - }, - { - "cell_type": "code", - "execution_count": 9, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/", - "height": 122 - }, - "id": "8MXyzzqMDTzb", - "outputId": "07162a89-57e0-47fd-bef8-39419c05f045" - }, - "outputs": [ - { - "data": { - "application/vnd.google.colaboratory.intrinsic+json": { - "type": "string" - }, - "text/plain": [ - "\"IEEE TRANSACTIONS ON JOURNAL NAME, MANUS CRIPT ID 1 Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs Yu. A. Malkov, D. A. Yashunin Abstract We present a new approach for the approximate K-nearest neighbor search based on navigable small world graphs with controllable hierarchy (Hierarchical NSW , HNSW ). The proposed solution is fully graph-based, without any need for additional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. Hierarchical NSW incrementally builds a multi-layer structure consisting from hierarchical set of proximity graphs (layers) for nested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an exponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Sma ll World (NSW) struc tures while additionally having the links separated by their characteristic distance scales. Starting search from the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a logarithmic complexity scalin g. Additional employment of a heuristic for selecting proximity graph neighbors significantly increases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the proposed general metric space search i ndex is able to strongly outperform previous opensource state-of-the-art vector-only approaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation . Index Terms Graph and tree search strategies, Artificial Intelligence, Information Search and Retrieval, Information Storage and Retrieval, Information Technology and Systems, Search process, Graphs and networks, Data Structures, Nearest neighbor search, Big data, Approximate search, Simil arity search 1 INTRODUCTIONonstantly growing amount of the available info rmation resources has led to high demand in scalable and efficient similarity search data structures. One of the generally used approaches for information search is the K-Nearest Neighbor Search (K-NNS). The K-NNS a ssumes you have a defined distance function between the data elements and aims at finding the K elements from the dataset which minimize the distance to a given query. Such algorithms are used in many applications, such as non-parametric machine learn ing algorithms, image fe atures matching in large scale databases [1] and semantic document retrieval [2]. A na\u00efve approach to K-NNS is to compute the distances between the query and every el ement in the dataset and select the elements with minimal distance. Unfortunately , the complexity of the na\u00efve a pproach scales linearly with the number of stored elements making it infeasible for large-scale datasets. This has led to a high interest in development of fast and scalable K-NNS algorithms. Exact solutions for K-NNS [3-5] may offer a substantial search speedup only in case of relatively low dimensional data du e to \u201ccurse of dimensionality\u201d. To overcome this problem a concept of Approximate Nearest Neighbors Search (K-ANNS) was proposed, which relaxes the cond ition of the exact search by allowing a small number of errors. The quality of an inexact search (the re call) is d efined as the ratio between the number of found true nea rest neighbors and K. The m ost popular K-ANNS sol utions are based on approximated versions of tree alg orithms [6, 7], locality-sensitive hashing (LSH) [8, 9] and product quantization (PQ) [10-17]. Proximity graph K-ANNS algorithms [10, 18-26] have recently gained pop ularity offering a better performance on high dimensional datasets. However, the power-law scaling of the proxim ity graph routing causes extreme performance degrad ation in case o f low dimensional or clustered data. In this paper we propose the Hierarchical Navigable Small World (Hierarchical NSW, HNSW), a new fully graph based incremental K-ANNS structure, which can offer a much better logarithmic complexity scaling. The main contributions are: explicit selection of the graph\u2019s enter-point node, separation of links by different scales and use of an advanced heuristic to select the neighbors. Alternatively, Hierarchical NSW algorithm can be seen as an extension of the probabilistic skip list structure [27] with proximity graphs instead of the linked lists. Perfo rmance e valuation has demonstrated that the proposed general metric space method is able to strongly outpe rform previous opensource state-of-the-art approaches suitable only for vector spaces. Y. M alkov is with the Federal state budgetary institution of science Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia . E-mail: yurymalkov@mail.ru . D. Y ashunin . Addres: 31-33 ul. Krasnozvezdnaya, 603104 Nizhny Novgorod, Russia. E-mail: yashuninda@yandex.ru C This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. \"" - ] - }, - "execution_count": 9, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "cleaned_docs[0].get_content()" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "id": "KJ4ypePGDb2y" - }, - "source": [ - "# Transform the data" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "id": "97rPsTCtExKz" - }, - "source": [ - "## Metadata" - ] - }, - { - "cell_type": "code", - "execution_count": 10, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "CBH8_dahDciC", - "outputId": "e992cb20-65dd-42c9-b304-b5bbd56dc705" - }, - "outputs": [ - { - "data": { - "text/plain": [ - "{'page_label': '1', 'file_name': 'hnsw.pdf'}" - ] - }, - "execution_count": 10, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Unhelpful metadata\n", - "cleaned_docs[0].metadata" - ] - }, - { - "cell_type": "code", - "execution_count": 11, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "VfEJ4rKwDhR5", - "outputId": "30bab882-62bd-4da5-8be7-f924c00ca7c5" - }, - "outputs": [ - { - "data": { - "text/plain": [ - "{'page_label': '1',\n", - " 'file_name': 'hnsw.pdf',\n", - " 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'],\n", - " 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}" - ] - }, - "execution_count": 11, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Iterate through `documents` and add our new key:value pairs\n", - "\n", - "# Helpful metadata additions\n", - "metadata_additions = {\"authors\": [\"Yu. A. Malkov\", \"D. A. Yashunin\"],\n", - " \"title\": \"Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs\"}\n", - "\n", - "[cd.metadata.update(metadata_additions) for cd in cleaned_docs] # Updates dict in place\n", - "\n", - "# Let's confirm everything worked:\n", - "cleaned_docs[0].metadata\n", - "\n", - "# Great!" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "id": "wbMepf_OEnEB" - }, - "source": [ - "## Ingestion pipeline" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "id": "pZMCyGA9En-8" - }, - "outputs": [], - "source": [ - "from llama_index.node_parser import SemanticSplitterNodeParser\n", - "from llama_index.embeddings import OpenAIEmbedding\n", - "from llama_index.ingestion import IngestionPipeline\n", - "\n", - "# This will be the model we use both for Node parsing and for vectorization\n", - "embed_model = OpenAIEmbedding(api_key=openai_api_key)\n", - "\n", - "# Define the initial pipeline\n", - "pipeline = IngestionPipeline(\n", - " transformations=[\n", - " SemanticSplitterNodeParser(\n", - " buffer_size=1,\n", - " breakpoint_percentile_threshold=95,\n", - " embed_model=embed_model,\n", - " ),\n", - " embed_model,\n", - " ],\n", - " )\n" - ] - }, - { - "cell_type": "markdown", - "metadata": { - "id": "KWk8cAV-Ezt-" - }, - "source": [ - "# Upsert the data" - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": { - "id": "UCvt6VtfE0N7" - }, - "outputs": [], - "source": [ - "from pinecone.grpc import PineconeGRPC\n", - "from pinecone import ServerlessSpec\n", - "\n", - "from llama_index.vector_stores import PineconeVectorStore\n", - "\n", - "# Initialize connection to Pinecone\n", - "pc = PineconeGRPC(api_key=pinecone_api_key)\n", - "index_name = \"llama-integration-example\"\n", - "\n", - "# Create your index (can skip this step if your index already exists)\n", - "pc.create_index(\n", - " index_name,\n", - " dimension=1536,\n", - " spec=ServerlessSpec(cloud=\"aws\", region=\"us-west-2\"),\n", - ")\n", - "\n", - "# Initialize your index\n", - "pinecone_index = pc.Index(index_name)\n", - "\n", - "# Initialize VectorStore\n", - "vector_store = PineconeVectorStore(pinecone_index=pinecone_index)\n" - ] - }, + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/pinecone-io/examples/blob/main/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb) [![Open nbviewer](https://raw.githubusercontent.com/pinecone-io/examples/main/assets/nbviewer-shield.svg)](https://nbviewer.org/github/pinecone-io/examples/blob/main/learn/generation/llama-index/using-llamaindex-with-pinecone.ipynb)\n", + "\n", + "# Using LlamaIndex with Pinecone\n", + "\n", + "Accompanying notebook to [this documentation](https://docs.pinecone.io/integrations/llamaindex). It uses LlamaIndex and Pinecone to perform semantic search and build a RAG pipeline, then evaluates the answers.\n", + "\n", + "This notebook calls the OpenAI API, which is a paid service." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Set up the environment" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:28.997570Z", + "iopub.status.busy": "2026-10-08T21:57:28.997467Z", + "iopub.status.idle": "2026-10-08T21:57:50.211183Z", + "shell.execute_reply": "2026-10-08T21:57:50.210557Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/", - "height": 1000, - "referenced_widgets": [ - "a704adf4487944888ac5204c8af01bb0", - "018284be942d4b1c80b786005d8e8c12", - "d0b95b4b3c3c4249a8a0949bbee03244", - "7bae5fa30a85469189d1d4cb2f1b8432", - "307b2cf43a6948b9bd74d77d5340bdc8", - "c02f8cabac564a02af0da144cfa7f18d", - "12fc582a4b41446aa7a0b7c994ca3da0", - "ed195d80d1bd4e87a1c21c2483ebc462", - "0844f0c5c3e141bf8216ca09248fc674", - "c90f1ae40f794847a051ad6b6e767129", - "151236c9274b4f5a9c74cd38a046e583" - ] - }, - "id": "Y6hoZsafE6Rg", - "outputId": "e7e4b1be-c4a2-4989-c04f-62c7aeb8d564" - }, - "outputs": [ - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "a704adf4487944888ac5204c8af01bb0", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "Upserted vectors: 0%| | 0/46 [00:00: RelatedNodeInfo(node_id='3f6962ed-6349-4fe4-bd5a-dee09f6d4b9e', node_type=, metadata={'page_label': '1', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='b96b4def9ce374e601c1ad43943a4730ed91e855fd9667b91adddb9a7736afbd'), : RelatedNodeInfo(node_id='b7cce664-4af1-4e43-a356-b27db7e2a717', node_type=, metadata={}, hash='36c04cc6d883a81c5da2c5dfc2d77828bc0601479f6700ea1e290536b85e5062')}, hash='a05ce131f4543ade9a9ef4b8c267c2524cc5821cb57b64f239c9c54f93032aa8', text='IEEE TRANSACTIONS ON JOURNAL NAME, MANUS CRIPT ID 1 Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs Yu. A. Malkov, D. A. Yashunin Abstract We present a new approach for the approximate K-nearest neighbor search based on navigable small world graphs with controllable hierarchy (Hierarchical NSW , HNSW ). The proposed solution is fully graph-based, without any need for additional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. Hierarchical NSW incrementally builds a multi-layer structure consisting from hierarchical set of proximity graphs (layers) for nested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an exponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Sma ll World (NSW) struc tures while additionally having the links separated by their characteristic distance scales. Starting search from the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a logarithmic complexity scalin g. Additional employment of a heuristic for selecting proximity graph neighbors significantly increases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the proposed general metric space search i ndex is able to strongly outperform previous opensource state-of-the-art vector-only approaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation . Index Terms Graph and tree search strategies, Artificial Intelligence, Information Search and Retrieval, Information Storage and Retrieval, Information Technology and Systems, Search process, Graphs and networks, Data Structures, Nearest neighbor search, Big data, Approximate search, Simil arity search 1 INTRODUCTIONonstantly growing amount of the available info rmation resources has led to high demand in scalable and efficient similarity search data structures. One of the generally used approaches for information search is the K-Nearest Neighbor Search (K-NNS). The K-NNS a ssumes you have a defined distance function between the data elements and aims at finding the K elements from the dataset which minimize the distance to a given query. Such algorithms are used in many applications, such as non-parametric machine learn ing algorithms, image fe atures matching in large scale databases [1] and semantic document retrieval [2]. A na\u00efve approach to K-NNS is to compute the distances between the query and every el ement in the dataset and select the elements with minimal distance. Unfortunately , the complexity of the na\u00efve a pproach scales linearly with the number of stored elements making it infeasible for large-scale datasets. This has led to a high interest in development of fast and scalable K-NNS algorithms. Exact solutions for K-NNS [3-5] may offer a substantial search speedup only in case of relatively low dimensional data du e to \u201ccurse of dimensionality\u201d. To overcome this problem a concept of Approximate Nearest Neighbors Search (K-ANNS) was proposed, which relaxes the cond ition of the exact search by allowing a small number of errors. The quality of an inexact search (the re call) is d efined as the ratio between the number of found true nea rest neighbors and K. The m ost popular K-ANNS sol utions are based on approximated versions of tree alg orithms [6, 7], locality-sensitive hashing (LSH) [8, 9] and product quantization (PQ) [10-17]. 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='3f6962ed-6349-4fe4-bd5a-dee09f6d4b9e', node_type=, metadata={'page_label': '1', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='b96b4def9ce374e601c1ad43943a4730ed91e855fd9667b91adddb9a7736afbd'), : RelatedNodeInfo(node_id='1363c481-8016-4b69-bc55-7bdb99667a09', node_type=, metadata={'page_label': '1', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='0da740827557c82dd646e627033cc6ed256242861507322798efb3bcfe60ce90'), : RelatedNodeInfo(node_id='893d4caa-bc26-4f00-afbc-50a333bf3068', node_type=, metadata={}, hash='6824b16b4c7372b7e9ed77bfb4369bce8678a2575eee15c4a29d11ede3bf4a90')}, hash='a3e0e4ce0c25a84e7c66c8b6627ccf9b6f8a14551bb6e076103245a7e1f3ba20', text='Proximity graph K-ANNS algorithms [10, 18-26] have recently gained pop ularity offering a better performance on high dimensional datasets. However, the power-law scaling of the proxim ity graph routing causes extreme performance degrad ation in case o f low dimensional or clustered data. In this paper we propose the Hierarchical Navigable Small World (Hierarchical NSW, HNSW), a new fully graph based incremental K-ANNS structure, which can offer a much better logarithmic complexity scaling. The main contributions are: explicit selection of the graph\u2019s enter-point node, separation of links by different scales and use of an advanced heuristic to select the neighbors. Alternatively, Hierarchical NSW algorithm can be seen as an extension of the probabilistic skip list structure [27] with proximity graphs instead of the linked lists. Perfo rmance e valuation has demonstrated that the proposed general metric space method is able to strongly outpe rform previous opensource state-of-the-art approaches suitable only for vector spaces. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='2837a61fa167ae56612a9f1bfccac2dedb3d80a3ef623ff14e7f4f45436361c5'), : RelatedNodeInfo(node_id='4ccc877a-ac28-4d90-8c31-fad31303a384', node_type=, metadata={}, hash='f019be979fe677dabcac2ee3273fc5217f86e93819179b9f8f4a88095df4abbe')}, hash='f6052cf0b504adea83f8e2edc9258452d05be64f0bc9b2b70d9623c62434ce7f', text='For a given proximity graph, we start the search at some enter point (it can be random or supplied by a separate algorithm) and iteratively t raverse the graph. At each step of the traversal the algorithm examines the distances from a query to the neighbors of a current base node and then selects as the next base node the adjacent node that minimizes the distance, while co nstantly keeping track of the best discovered neighbors. 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='4e438db4-0331-442c-a948-ceffafe685b9', node_type=, metadata={'page_label': '2', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='6d68bfd69cc029b5a4c3bc418642838d7fbbfb5ffae8d888c91813515be74be3'), : RelatedNodeInfo(node_id='222db67c-4c76-43d4-ae7d-fdb097aebeb7', node_type=, metadata={'page_label': '2', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='f6f59e5a9c74f94461bc37678ae707814eb0f25ad7fb5b4ef4d0d2ba45938f46'), : RelatedNodeInfo(node_id='3edf4852-cc8a-40f9-8a1e-c6a769a04945', node_type=, metadata={}, hash='9ebd41883facac66642cda2109849cf55aa5ee0835f8af1a4f18ad0390f50048')}, hash='b9418844df768bc19995ab8c69b5e83d230e897cb0fc9fe09533f64582ed9455', text='The search is terminated when some stopping condition is met (e.g. the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques , such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approach es are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure\u2019s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges betwee n the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing . Const ruction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , however, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\\uf061. For \\uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \\uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg\u2019s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg\u2019s graphs suffers fr om polylogarit hmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg\u2019s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44] . The routing can be divided into two phases: \u201czoom-out\u201d and \u201czoom-in\u201d [32]. The greedy algorithm starts in the \u201czoom-out\u201d phase ', start_char_idx=673, end_char_idx=6175, text_template='{metadata_str}\\n\\n{content}', metadata_template='{key}: {value}', metadata_seperator='\\n'),\n", - " TextNode(id_='3edf4852-cc8a-40f9-8a1e-c6a769a04945', embedding=[0.0011280025355517864, 0.028634879738092422, -0.008988010697066784, -0.02151937037706375, -0.0123297069221735, 0.015772229060530663, -0.011479878798127174, 0.0005437458166852593, -0.029066994786262512, -0.05473468080163002, -0.0034317204263061285, 0.004033081699162722, -0.0004892811994068325, 0.023435084149241447, 0.004205928184092045, 0.009369713254272938, 0.025926953181624413, 9.220441825164016e-06, 0.0061144400388002396, 0.000755752669647336, -0.03776692971587181, 0.006161252502351999, -0.01920034922659397, -0.039668239653110504, 0.010946935974061489, 0.03886162117123604, 0.017227020114660263, -0.002436053240671754, -0.0033615014981478453, 0.000886737834662199, 0.029700765386223793, 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='ac87ce7f-3b99-4d1c-8b82-d9c7f059993e', node_type=, metadata={'page_label': '3', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='529b41ec4e69365c2c387267849757909c288ca7388d03abede03687d10f5cd5'), : RelatedNodeInfo(node_id='4ccc877a-ac28-4d90-8c31-fad31303a384', node_type=, metadata={'page_label': '2', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='f019be979fe677dabcac2ee3273fc5217f86e93819179b9f8f4a88095df4abbe'), : RelatedNodeInfo(node_id='f4506a9d-7fe6-4a42-bfac-011125776c59', node_type=, metadata={}, hash='94a3f25dcc8b82b9cd4c2f97e540d8ce12e0ba5794960fff70043ff5009a81b6')}, hash='e23e9157920914069525bbe1d5724c4f6e027e621c459ad715503afb5ca87fe6', text='AUTHOR ET AL.: TITL E 3 from a low degree node and traverses the graph simult aneously increasing the node\u2019s degree until the characteri stic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the \u201czoom-in\u201d phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber o f distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44] ; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such stru cture t he search starts from the upper laye r which has only the long est links (the \u201czoom-in\u201d phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illu stration) . After that, the search switches to the lower layer (which h as shorter links ), restarts from the element which was the local minimum in the previous layer and the pr ocess repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of rout ing in a navigable small world network . One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only \u201cshort\u201d links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer . In case we m erge connections from all layers, the stru cture becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insert ion-the stochasticity is achieved by using level random ization , thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure ). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='9ebd41883facac66642cda2109849cf55aa5ee0835f8af1a4f18ad0390f50048'), : RelatedNodeInfo(node_id='9f486445-5722-4e11-b707-e80d2d5ea0f5', node_type=, metadata={}, hash='c1e256ad162d3e063c953134567f3fcb536d84738e50932df060563449f2c532')}, hash='60eb2ef7bb8a1d1bab15180fce99679d756be94be9dda7b0c8a3777b49febfb6', text='The Hierarchical Layer=0Layer=1Layer=2Decreasing characteristic radius Cluster 2Inserted elemente2Cluster 1 Fig. 1. Illustration of the Hierarchical NSW idea. The search starts from an element from the top layer (shown red). Red arrows show direction of the greedy algorithm from the entry point to the query (shown green). 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'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='ac87ce7f-3b99-4d1c-8b82-d9c7f059993e', node_type=, metadata={'page_label': '3', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='529b41ec4e69365c2c387267849757909c288ca7388d03abede03687d10f5cd5'), : RelatedNodeInfo(node_id='f4506a9d-7fe6-4a42-bfac-011125776c59', node_type=, metadata={'page_label': '3', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='94a3f25dcc8b82b9cd4c2f97e540d8ce12e0ba5794960fff70043ff5009a81b6'), : RelatedNodeInfo(node_id='c0307639-48c0-491f-8764-96c569466731', node_type=, metadata={}, hash='3030b0a277e4b333ec1f851f06922b5927eefa4ec00da742c135a9dbea146e2d')}, hash='298d357c80738b234e86b5905b9408ce494469de3aea5463b42632baef826075', text='Fig. 2. Illustration of the heuristic used to select the graph neig hbors for two isolated clusters. A new ele ment is inserted on the boundary of Cluster 1. All of the closest neighbors of the element belong to the Cluster 1, thus missing the edges of Delaunay graph between the clusters. The heuristic, however, selects element e2 from Cluster 2, thus, maintaining the global connectivity in case the inserted element is the closest to e2 compared to any other element from Cluster 1. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='c1e256ad162d3e063c953134567f3fcb536d84738e50932df060563449f2c532'), : RelatedNodeInfo(node_id='8c21b77e-4f06-4882-ba1e-370d5130957a', node_type=, metadata={}, hash='93e2bd46be97d41d53efff8213292088cf3640063f9dfb9b053f597e1e67cebc')}, hash='bcea9e6d7481b9c82ddb57a20fc33d59e0f4b7bfdd70e9133b35d120e28d8ce3', text='4 IEEE TRANSACTIONS ON JOUR NAL NAME, MANUSCRIP T ID NSW approach thus can utilize the same methods for making the distributed approximate search/overlay structures [45]. For the selection of the proximity graph con nections during the element insertion we utilize a heuristic that takes into account the distances between the candidate elements to create diverse connections (a similar alg orithm was utilized in the spatial approximation tree [4] to select the tree child ren) instead of just selecting the closest neighbors. The heuristic examines the candidates starting from the nearest (with respect to the inserted element ) and creates a connection to a candidate only if it is closer to the base (inserted) element compared to any of the a lready connected candidates (see Section 4 for the details). When the number of candidates is large enough the heuristic allows getting the exact relative neighborhood graph [46] as a subgraph , a minimal subgraph of the D elaunay graph deducible by using only the distances b etween the nodes. The relative neighborhood graph allows easily keeping the global connected component, even in case of highly clustered data (see Fig. 2 for illustration). Note that the heuristic creates extra edges compared to the exact relative neighborhood graph s, allowing contro lling the number of the connections which is important for search performance . For the case of 1D data the heuristic allows getting the exact Delaunay subgraph (which in this c ase coincides with the relative neighborhood graph) by using only information about the distances between the elements, thus making a direct transition from Hiera rchical NSW to the 1D probabilistic skip list algorithm. Base variant of the Hierarchical NSW proximity graphs was also used in ref. [18] (called \u2018sparse neighbo rhood graphs\u2019) for proximity graph searching. Similar heuristic was also a focus of the FANNG algorithm [47] (published shortly after the first versions of the current manuscript were posted online) with a slightly different interpretation, based on the sparse neighborhood graph \u2019s property of the exact routing [18]. 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='f7cdbb96-f0d3-447d-8a76-6ed3c785d9cf', node_type=, metadata={'page_label': '4', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='d97f7a991825ab14e884476720832b329e6162b0c22aa7ce2dbbdbdf9e31e718'), : RelatedNodeInfo(node_id='c0307639-48c0-491f-8764-96c569466731', node_type=, metadata={'page_label': '4', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='3030b0a277e4b333ec1f851f06922b5927eefa4ec00da742c135a9dbea146e2d'), : RelatedNodeInfo(node_id='d02fb582-e6cb-401b-8ce7-958c303674a4', node_type=, metadata={}, hash='a740055013cf242d8aa467a2d63404ffd20dd60375c9779806741543d18f93c5')}, hash='e680203d844e9b24d2f4c9835669f2ba5a2a211c3565be30afa91a1eee8a4875', text='4 ALGORITHM DESCRIPTION Network construction algorithm (alg. 1) is organized via consecutive insertions of the stored elements into the graph structure. For every inserted element an integer maximum layer l is randomly selected with an expone ntially decaying probability distribution (normalized by the mL parameter , see line 4 in alg. 1 ). The first phase of the insertion process starts from the top layer by greedily traversing the graph in order to find the ef closest neighbors to the inserted element q in the layer. After that, the algorithm continues the search from the next layer using the found closest neighbors from the previous layer as enter points, and the process repeats. Closest neighbors at each layer are found by a variant of the greedy search algorithm described in alg. 2, which is an updated version of the algorithm from [26]. To obtain the approximate ef nearest neighbors in some layer l\u0441, a dynamic list W of ef close st found elements (initially filled with enter points) is kept during the search. The list is updated at each step by evaluating the neighborhood of the closest previously non-evaluated element in the list until the neighborhood of every element from the l ist is evaluated. Compared to limiting the number of distance calculations, Hierarchical NSW stop condition has a n advantage-it allows discarding candidat es for evalution that are further f rom the query than the furthest element in the list, thus avoid ing bloating of search structures . As in NSW, the list is emulated via two priority queues for better performance. The distinctions from NSW (along with some queue optimizations) are: 1) the enter point is a fixed parameter; 2) instead of changing the numb er of multi-searches, the quality of the search is controlled by a different parameter ef (which was set to K in NSW [26]). Algorithm 1 INSERT( hnsw, q, M, Mmax, efConstruction , mL) Input: multilayer graph hnsw, new element q, number of established connections M, maximum number of connections for each element per layer Mmax, size of the dynamic candidate list efConstruction , normalization factor for level gene ration mL Output: update hnsw inserting element q 1 W \u2190 \u2205 // list for the currently found nearest elements 2 ep \u2190 get enter point for hnsw 3 L \u2190 level of ep // top layer for hnsw 4 l \u2190 \u230a-ln(unif(0..1))\u2219mL\u230b // new element\u2019s level 5 for lc \u2190 L \u2026 l+1 6 W \u2190 SEARCH-LAYER(q, ep, ef=1, lc) 7 ep \u2190 get the nearest element from W to q 8 for lc \u2190 min(L, l) \u2026 0 9 W \u2190 SEARCH-LAYER(q, ep, efConstruction , lc) 10 neighbors \u2190 SELECT-NEIGHBORS( q, W, M, lc) // alg. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='93e2bd46be97d41d53efff8213292088cf3640063f9dfb9b053f597e1e67cebc'), : RelatedNodeInfo(node_id='0d8d2c21-d52f-4ac9-9ab5-45e9d047c92e', node_type=, metadata={}, hash='4d5386312d0a02d6465873f4184fe8ae5fe151b881466b3eee0d85b5d1b48d94')}, hash='3dfd4138ef425a6df921b228396a563e5c5bc216757741d51b6ecb6b4295e7dd', text='3 or alg. 4 11 add bidirectionall connectionts from neighbors to q at layer lc 12 for each e \u2208 neighbors // shrink connections if needed 13 eConn \u2190 neighbourhood (e) at layer lc 14 if \u2502eConn \u2502 > Mmax // shrink connections of e // if lc = 0 then Mmax = Mmax0 15 eNewConn \u2190 SELECT-NEIGHBORS( e, eConn, Mmax, lc) // alg. 3 or alg. 4 16 set neighbourhood (e) at layer lc to eNewConn 17 ep \u2190 W 18 if l > L 19 set enter point for hnsw to q Algorithm 2 SEARCH-LAYER(q, ep, ef, lc) Input: query element q, enter points ep, number of nearest to q elements to return ef, layer number lc Output: ef closest neighbors to q 1 v \u2190 ep // set of visited elements 2 C \u2190 ep // set of candidates 3 W \u2190 ep // dynamic list of found nearest neighbors 4 while \u2502C\u2502 > 0 5 c \u2190 extract nearest element from C to q 6 f \u2190 get furthest element from W to q 7 if distance(c, q) > distance(f, q) 8 break // all elements in W are evaluated 9 for each e \u2208 neighbourhood (c) at layer lc // update C and W 10 if e \u2209 v 11 v \u2190 v \u22c3 e 12 f \u2190 get furthest element from W to q 13 if distance(e, q) < distance(f, q) or \u2502W\u2502 < ef 14 C \u2190 C \u22c3 e 15 W \u2190 W \u22c3 e 16 if \u2502W\u2502 > ef 17 remove furthest element from W to q 18 return W ', start_char_idx=4723, end_char_idx=5899, text_template='{metadata_str}\\n\\n{content}', metadata_template='{key}: {value}', metadata_seperator='\\n'),\n", - " TextNode(id_='0d8d2c21-d52f-4ac9-9ab5-45e9d047c92e', 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='25f116c1-a2c2-4e7e-962a-03dfb10f692c', node_type=, metadata={'page_label': '5', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='e1663cfd3111fed6dbd50273ea61332f334365a4c919ba5ce45d3bf80b117d82'), : RelatedNodeInfo(node_id='d02fb582-e6cb-401b-8ce7-958c303674a4', node_type=, metadata={'page_label': '4', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='a740055013cf242d8aa467a2d63404ffd20dd60375c9779806741543d18f93c5'), : RelatedNodeInfo(node_id='3cf659af-f281-41a5-809d-edbf54df58fa', node_type=, metadata={}, hash='fb616510d1ae94278e654db3bc24b44e98539633164cac497fca6a6916d6cb2c')}, hash='75d43226f95e60c44e726355a8f1bb202acd6884a7862a9eab3f342fec32eaf4', text='AUTHOR ET AL.: TITL E 5 Algorithm 3 SELECT-NEIGHBORS-SIMPLE( q, C, M) Input: base element q, candidate elements C, number of neighbors to return M Output: M nearest elements to q return M nearest elements from C to q Algorithm 5 K-NN-SEARCH (hnsw, q, K, ef) Input: multilayer graph hnsw, query element q, number of nearest neighbors to return K, size of the dynamic candidate list ef Output: K nearest elements to q 1 W \u2190 \u2205 // set for the current nearest elements 2 ep \u2190 get enter point for hnsw 3 L \u2190 level of ep // top layer for hnsw 4 for lc \u2190 L \u2026 1 5 W \u2190 SEARCH-LAYER(q, ep, ef=1, lc) 6 ep \u2190 get nearest element from W to q 7 W \u2190 SEARCH-LAYER(q, ep, ef, lc =0) 8 return K nearest elements from W to q During the first phase of the search the ef parameter is set to 1 (simple greedy search) to avoid introduction of add itional parameters. 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='25f116c1-a2c2-4e7e-962a-03dfb10f692c', node_type=, metadata={'page_label': '5', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='e1663cfd3111fed6dbd50273ea61332f334365a4c919ba5ce45d3bf80b117d82'), : RelatedNodeInfo(node_id='0d8d2c21-d52f-4ac9-9ab5-45e9d047c92e', node_type=, metadata={'page_label': '5', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='4d5386312d0a02d6465873f4184fe8ae5fe151b881466b3eee0d85b5d1b48d94'), : RelatedNodeInfo(node_id='d6c4aa0c-20dd-4e5e-8c63-e49f7e9b325a', node_type=, metadata={}, hash='d9d05b637eee7d4fbe52c40810ced13e8afa49c846a992102ff89ebf0375349c')}, hash='8ce7fd27e42d7243135a91799b7ae34b52e3ae17830f122679bddb49537cc7d3', text='When the search reaches the layer that is equal or less than l, the second phase of the construction algorithm is initiated . The second phase differs in two points: 1) the ef parameter is increased from 1 to efConstruction in order to control the recall of the greedy search procedure; 2) the found closest neighbors on each layer are also used as candidates for the connections of the inserted element. Two methods for the selection of M neighbors from the candidates were considered: simple connection to the closest elements (alg. 3) and the heuristic that accounts for the distances between the candidate elements to create connections in diverse direction s (alg. 4), described in the Section 3. The heuristic has two additional parameters: extendCandidates (set to false by default) which extends the candidate set and useful only for extremely clustered data, and keepPrunedConnections which allows getting fixed number of connection per element . The maximum number of connections that an element can have per layer is defined by the parameter Mmax for every layer higher than zero (a special parameter Mmax0 is used for the ground layer sep arately). If a node is already full at the moment of making of a new connection, then its extended connection list gets shrunk by the same algorithm that used for the neighbors selection (algs. 3 or 4). The insertion procedure terminates when the conne ctions of the inserted elements are established on the zero layer. The K-ANNS search algorithm used in Hierarchical NSW is presented in alg. 5. It is roughly equivalent to the insertion algorithm for an item with layer l=0. The diffe rence is that the closest neighbors found at the ground layer which are used as candidates for the connections are now returned as the search result. The quality of the search is controlled by the ef parameter (corresponding to efConstruction in the construction algorithm). 4.1 In fluence of the construction parameters Algorithm construction parameters mL and Mmax0 are r esponsible for maintaining the small world navigability in the constructed graphs. Setting mL to zero (this corr esponds to a single layer in the graph) and Mmax0 to M leads to production of directed k-NN graphs with a po wer-law search complexity well studied before [21, 29] (assuming using the alg. 3 for neighbor selection). Setting mL to zero and Mmax0 to infinity leads to production of NSW graphs with polylogarithmi c complexity [25, 26] . Finally, setting mL to some non-zero value leads to emergence of controllable hierarchy graphs w hich allow logarithmic search complexity by introduction of layers (see the Se ction 3). To achieve the optimum performance advantage of the controllable hierarchy, the overlap between neighbors on different layers (i.e. percent of element neighbors that are also belong to other layers) has to be small. In order to decrease the overlap we need to decrease the mL. Howe ver, at the same time, decreas ing mL leads to an increase of average hop number during a greedy search on each la yer, which negatively affects the performance. This leads to existence of the optimal value for the mL parameter . A simple choice for the optimal mL is 1/ln( M), this co rresponds to the skip list parameter p=1/M with an ave rage single element overlap between the layers. Simul ations done on an Intel Core i7 5930K CPU show that the proposed selection of mL is a reasonable choice (see Fig. ', start_char_idx=843, end_char_idx=4280, text_template='{metadata_str}\\n\\n{content}', metadata_template='{key}: {value}', metadata_seperator='\\n'),\n", - " TextNode(id_='d6c4aa0c-20dd-4e5e-8c63-e49f7e9b325a', embedding=[-0.010554810985922813, 0.03364754468202591, -0.006250801496207714, -0.007888867519795895, 0.004859716631472111, 0.03228188678622246, 0.002233726205304265, -0.008455471135675907, -0.03620452806353569, -0.05654414743185043, 0.0010596580104902387, 0.026092829182744026, -0.011230376549065113, 0.011455565690994263, -0.013307924382388592, 7.990565791260451e-05, 0.01760830171406269, 0.006991744972765446, 0.015676036477088928, -0.005873065907508135, -0.021821508184075356, -0.003259787568822503, -0.005727782845497131, -0.04166716709733009, 0.011099622584879398, 0.03600113093852997, 0.019714904949069023, -0.011397452093660831, -0.012763112783432007, 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It is hard to expect the same behavior for high dime nsional data since in this case the k-NN graph already has Algorithm 4 SELECT-NEIGHBORS-HEURISTIC( q, C, M, lc, extendCandidates, kee pPrunedConnections ) Input: base element q, candidate elements C, number of neighbors to return M, layer number lc, flag indicating whether or not to extend candidate list extendCandidates , flag indicating whether or not to add discarded elements keepPrunedConnections Output: M elements selected by the heuristic 1 R \u2190 \u2205 2 W \u2190 C // working queue for the candidates 3 if extendCandidates // extend candidates by their neighbors 4 for each e \u2208 C 5 for each eadj \u2208 neighbourhood (e) at layer l c 6 if eadj \u2209 W 7 W \u2190 W \u22c3 eadj 8 Wd \u2190 \u2205 // queue for the discarded candidates 9 while \u2502W\u2502 > 0 and \u2502R\u2502< M 10 e \u2190 extract nearest element from W to q 11 if e is closer to q compared to any element from R 12 R \u2190 R \u22c3 e 13 else 14 Wd \u2190 Wd \u22c3 e 15 if keepPrunedConnections // add some of the discarded // connections from Wd 16 while \u2502Wd\u2502> 0 and \u2502R\u2502< M 17 R \u2190 R \u22c3 extract nearest element from Wd to q 18 return R ', start_char_idx=4280, end_char_idx=5597, text_template='{metadata_str}\\n\\n{content}', metadata_template='{key}: {value}', metadata_seperator='\\n'),\n", - " TextNode(id_='662fadd5-2b76-4e69-9af6-b811ae9b9d76', embedding=[-0.01618100143969059, 0.037960369139909744, 0.011161034926772118, -0.018665991723537445, 0.004548674914985895, 0.015709709376096725, 0.006330298725515604, -0.0036917817778885365, -0.044701263308525085, -0.05846868082880974, -0.0005105655873194337, 0.015609738416969776, 0.0016129229916259646, 0.018466049805283546, -0.009954242967069149, 0.012746287509799004, 0.01795191317796707, 0.002231492893770337, 0.02097960375249386, -0.011011078022420406, -0.02110813744366169, 0.010875403881072998, -0.019065875560045242, -0.03581813722848892, 0.010961093008518219, 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='f8322ab2-464d-41fc-925e-3a4c7cea9500', node_type=, metadata={'page_label': '6', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='d232911f8873acf038cf99a4b08b6ce7397002ea53087d8d527a294338919e6f'), : RelatedNodeInfo(node_id='d6c4aa0c-20dd-4e5e-8c63-e49f7e9b325a', node_type=, metadata={'page_label': '5', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='d9d05b637eee7d4fbe52c40810ced13e8afa49c846a992102ff89ebf0375349c'), : RelatedNodeInfo(node_id='5637c528-af81-4f94-a32c-65d197cd11fa', node_type=, metadata={}, hash='add6d4be54548cdeaccd267a3d3515a6866595bd7e009ff745e62d08a5254c22')}, hash='15f7b3d178c216f76fe36a8708cfbcb27edf81c5057977fc49ad7204d63f3d12', text='6 IEEE TRANSACTIONS ON JOUR NAL NAME, MANUSCRIP T ID very short greedy algorithm paths [28]. Surprisingly, i ncreasing the mL from zero leads to a measurable increase in speed on very high dimensional data (100k dense ra ndom d=1024 vectors, see plot in F ig. 4), and does not i ntroduce any penalty for the Hierarchical NSW approach. For real data such as SIFT vectors [1] (which have co mplex mixed structure ), the performance improvement by increasing the mL is higher, but less prominent at current settings compared to improvement from the heuristic (see Fig. 5 for 1-NN search performance on 5 million 128-dimensional SIFT vectors from the learning set of BI GANN [13]). Selection of the Mmax0 (the maximum number of co nnections that an element can have in the zero layer) also has a strong in fluence on the search performance, esp ecially in case of high quality (high recall) search. Simul ations show that setting Mmax0 to M (this corresponds to k-NN graphs on each layer if the neighbors selection heuri stic is not used) leads to a very strong per formance penalty at high recall. Simulations also suggest that 2\u2219 M is a good choice for Mmax0: setting the parameter higher leads to performance degradation and excessive memory usage . In Fig. 6 there are presented results of search performance for the 5M SIFT learn dataset depending on the Mmax0 parameter (done on an Intel Core i5 2400 CPU) . The su ggested value gives performance close to optimal at diffe rent recalls. In all of the considered cases, use of the heuristic for proximity graph neighbors selection (alg. 4) leads to a higher or similar search performance compared to the na\u00efve connection to the nearest neighbors (alg. 3). The effect is the most prominent for low dimensional data, at high recall for mid-dimensional data and for the case of highly clust ered data (ideologically discontinuity can be regarded as a local low dimensional feature), see the comparison in Fig. 7 (Core i5 2400 CPU) . When using the closest neighbors as connections for the proximity graph, the Hierarchical NSW algorithm fails to ac hieve a high recall for clustered data because the search stucks at the clusters boundaries. Contrary, when the heuristic is used (together with candidates \u2019 extension, line 3 in Alg. 4) , clustering leads to even higher performance. For uniform and very high dimensional data there is a little difference between the neighbors selecting methods (see Fig. 4) , possibly due to the fact that in this case almost all of the nearest neighbors are s elected by the heuristic. The only meaningful construction parameter left for the user is M. A reasonable range of M is from 5 to 48. Simulations show that smaller M generally produces be tter results for lower recalls and/or lower dimensional data, while b igger M is better for high recall and/or high dimensional data (see Fig. 8 for illustration , Core i5 2400 CPU ). The parameter also defines the memory consum ption of the algorithm (which is proportional to M), so it should be selected with care. Selection o f the efConstruction parameter is straigh tforward. As it was suggested in [26] it has to be large enough to produce K-ANNS recall close to unity during the construction process (0.95 is enough for the most use-cases). And just like in [26], this parameter can possibly 0,0 0,5 1,0 1,5 2,00,010,020,030,04Query time, msmL10M d=4 random vectors,M=6, Mmax0=12Recall 0.9, 1-NN Simple neighbors HeuristicAutoselect 0,0 0,5 1,0 1,5 2,014,114,214,314,414,5Query time, msmL100k random vectors, d=1024M=20, Mmax0=40,Recall=0.9, 1-NN Simple neighbors HeuristicAutoselect 0,0 0,5 1,0 1,5 2,00,090,100,110,120,130,140,15Query time, msmL5M SIFT, d=128,M=16, Mmax0=32,Recall=0.9, 1-NN Simple Neighbors HeuristicAutoselect Fig. 3. Plots for query time vs mL parameter for 10M random vectors with d=4. The a utoselected value 1/ln( M) for mL is shown by an arrow. 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='f8322ab2-464d-41fc-925e-3a4c7cea9500', node_type=, metadata={'page_label': '6', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='d232911f8873acf038cf99a4b08b6ce7397002ea53087d8d527a294338919e6f'), : RelatedNodeInfo(node_id='662fadd5-2b76-4e69-9af6-b811ae9b9d76', node_type=, metadata={'page_label': '6', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='3236fe12a2d7914bc719666a8449ddbdf1366959993b519a71e15f08f06cebbf'), : RelatedNodeInfo(node_id='a1d30860-4f7b-4fd0-9321-f56d44e6db45', node_type=, metadata={}, hash='9251afed1768787e3b0c983c89aa42e1b5c7296700bd6a767b47d022ba8f5f48')}, hash='5f830ff9be5e1f5cdad5cfa85daae73f01480a502e0b26e1f823b1df34868898', text='Fig. 4. Plots for query time vs mL param eter for 100k random vectors with d=1024. The autoselected value 1/ln( M) for mL is shown by an arrow. Fig. 5. Plots for query time vs mL parameter for 5M SIFT learn dataset. The autoselected value 1/ln( M) for mL is shown by an arrow. 0 20 40 60 80 100 120 140 160 180 2000.11Query time, msMmax05M SIFT, d=128, M=20,mL=0.33, 10-NN Recall=0.4 Recall=0.8 Recall=0.94Autoselect 0,20,40,60,81,00,01 0,1Query time, msRecall10M random vectors, d=10 M=16, 10-NN baseline-no clusters heuristic-no clusters baseline-100 clusters heuristic-100 clusters 10-310-210-11000,01 0,1 1Query time, msRecall error (1-recall)5M SIFT, d=128,10-NN M=2 M=3 M=6 M=12 M=20 M=40 Fig. 6. Plots for query time vs Mmax0 parameter for 5M SIFT learn dataset. The autoselected value 2\u2219M for Mmax0 is shown by an arrow. 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'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='8f94e813-f60d-4e9d-8ab1-fd4dbe0d2806', node_type=, metadata={'page_label': '7', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='7fbb867180683b87b30353e1d2934dc5adcd9ec1fa7656efc613598ec012c796'), : RelatedNodeInfo(node_id='07c7720d-a275-4c64-be9c-c36acb2f979b', node_type=, metadata={'page_label': '6', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='53ca6ad88b4c6df8598233252e4cb82ee64803f95b6dcc7d15bb325cbd95d7b9'), : RelatedNodeInfo(node_id='cbde7997-67b1-47e7-9200-f557ee4ebd6f', node_type=, metadata={}, hash='80a4266695ea779451c19b54df1ebe512e19ec7da1541d85ebaf473b4a6e4855')}, hash='5a8a85203f927def05de0cc174a51735691e1d8f555d7bfefadedcf83dde79ff', text='AUTHOR ET AL.: TITL E 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index q uality tradeoff is co ntrolled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruct ion=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complex ity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer . One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial position s of the data elements and , thus , when we tra verse the graph there is a fixed probability p=exp( -mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the p robability of not reaching the ta rget on s-th step is bounded by exp( -s\u00b7 mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp( -mL)), which is independent of the dataset size . If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C\u00b7 S, independent ly of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional dataset s. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbor s per element . 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='8f94e813-f60d-4e9d-8ab1-fd4dbe0d2806', node_type=, metadata={'page_label': '7', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='7fbb867180683b87b30353e1d2934dc5adcd9ec1fa7656efc613598ec012c796'), : RelatedNodeInfo(node_id='cfefe5cf-3a32-476b-a287-6ea83ba44e5e', node_type=, metadata={'page_label': '7', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='840f6cfb576f47d00f18e9b3bee05bb1714510a5de7ecd8c387b32a569b4c2d7'), : RelatedNodeInfo(node_id='1a88d60d-a80a-4fb7-8f18-0b92a1897131', node_type=, metadata={}, hash='9ec6a656506c1153ef6daf48ab647292bbe3c7eae6fba1ec8d2c38228b59eef6')}, hash='ab6dbcfe9ba8a664e2922627bb87d5f41697fa21d16b95b1885fa9312dcd3a9c', text='Thus, t o avoid stucking into a l ocal minimum the greedy search algo rithm em ploy s a backtracking procedure on the zero layer. Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via t he minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling . Such empirical investigation of the Delaunay graph approximation resilience require s having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e datasets, mak ing such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a sequence of K-ANN-searches at different layers with a su bsequent use of heuristic (which has fixed complexity at fixed efConstruction ). The average number of lay ers for an element to be added in is a constant that depends on mL: \\uf05b\\uf05d\\uf05b \\uf05d 1 ln( (0,1)) 1 1LL E l E unif m m\\uf02b \\uf03d \\uf02d \\uf02b \\uf03d \\uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N\u2219log( N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16 , efConstruction =100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='9ec6a656506c1153ef6daf48ab647292bbe3c7eae6fba1ec8d2c38228b59eef6'), : RelatedNodeInfo(node_id='0acfda72-f820-47c8-84ea-5afeab7d8a6a', node_type=, metadata={}, hash='babce99b4e9b7522bb0b2e1eea50733d61e51ca846dd2bb307fb828f7ab6b1b2')}, hash='28dd45decf5e33f4770f1feae8ad46ba108199e008759ec7caa06718e9ce2caf', text='8 IEEE TRANSACTIONS ON JOUR NAL NAME, MANUSCRIP T ID 10-510-410-310-210-1100100100010000100000Distance computationsRecall error (1-recall)(a) 10M random vectors, d=4M=6, 10-NN NSW Hierarchical NSW102103104105106107108020040060080010001200Distance computationsDataset size(b) Random vectors, d=8M=6, Recall=0.95, 10-NN NSW Hierarchical NSW1021031041051061071080,00,51,01,52,02,5Query time, msDataset size(c) Random vectors, d=8M=6, Recall=0.95, 10-NN NSW Hierarchical NSW Fig. 12. Comparison between NSW and Hierarchical NSW: (a) distance calculation number vs accuracy tradeoff for a 10 million 4-dimensional random vectors dataset; (b-c) performance scaling in t erms of number of distance calculations (b) and raw query(c) time on a 8-dimensional random vectors dataset. 4.2.3 Memory cost The memory consumption of the Hierarchical NSW is mostly defined by the storage of graph connections. The number of connection s per element is Mmax0 for the zero layer and Mmax for all other l ayers. Thus, the average memory consumption per element is (Mmax0+mL \u2219Mmax)\u2219bytes_per_link . If we limit the maximum total number of elements by approximately four billions, we can use four-byte unsigned integers to store the co nnections. Tests suggest that typical close to optimal M values usually lie in a range between 6 and 48. This means that the typical memory requirements for the i ndex (excluding the size of the data) are about 60-450 bytes per object, which is in a good agreement with the simul ations. 5 PERFORMANCE EVALUATIO N The Hierarchical NSW algorithm was implemented in C++ on top of the Non Metric Space Library (nmslib) [49]1, which already had a functional NSW i mplementation (under name \u201csw-graph\u201d). D ue to several limitations posed by the library, to achieve a better pe rformance, the Hierarchical NSW implementation uses custom distance functions together with C-style memory management , which avoid s unnecessary implicit addres sing and allows efficient hardware and software prefetc hing during the graph traversal . Comparing the performance of K-ANNS algorithms is a nontrivial task since the state-of-the-art is constantly changing as new algorithms and implementations are emerging. In this work we concentr ated on comparison with the best algorithms in Euclid spaces that have open source implementations. An implementation of the Hie rarchical NSW algorithm presented in this paper is also distributed as a part of the open source nmslib library1 together with a n external C++ memory-efficient header-only version with support for incremental index constru ction2. The comparison section consists of four parts: compa rison to the baseline NSW (5.1), comparison to the state-ofthe-art algorithms in Euclid spaces (5.2) , rerun of the su b-1 https://github.com/searchivarius/nmslib 2 https://github.c om/nmslib/hnsw set of tests [34] in general metric spaces in which NSW failed (5.3) and comparison to state-of-the-art PQ-algorithms on a large 200M SIFT dataset (5.4). 5.1 Comparison with baseline NSW For the baseline NSW algorithm implementation, we u sed the \u201csw-graph\u201d from nmslib 1.1 (which is slightly upda ted compared to the implementation tested in [33, 34] ) to demonstrate the improvements in speed and algorithmic complexity (measured by the number of distance comp utations). Fig. 12(a) presents a co mparison of Hierarchical NSW to the basic NSW algorithm for d=4 random hypercube data made on a Core i5 2400 CPU (10-NN search). Hiera rchical NSW uses much less distance computations during a search on the dataset , especially at high recalls . The scalings of the algorithms on a d=8 random hype rcube dataset for a 10-NN search with a fixed recall of 0.95 are presented in Fig. 12 (b). It clearly demostrates that H ierarchical NSW has a complexity scaling for this setting not worse than logarithmic and outperforms NSW at any dataset size. 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The testing system utilize s python bindings of the algorithms \u2013 it consequentially runs the K-ANN search for one thousand queries (ra ndomly extracted from the initial dataset) with preset alg orithm parameters producing an output containing recall and average time of a single search. The considered alg orithms are: 1. Baseline NSW algorithm from nmslib 1.1 (\u201csw-graph\u201d). 2. FLANN 1.8.4 [6]. A popular library4 containing several algorithms, built-in in Op enCV5. We used the available auto-tuning procedure with several reruns to infer the best parameters. 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Hierarchical NSW was compiled using the GCC 5.3 with-Ofast optimization flag. The parameters and description of the used datasets are outlined in Table 1. For all of the datasets except GloVe we used the L2 distance. For GloVe we used the cosine similarity which is equivalent to L2 after vector normalization. The brute-force (BF) t ime is measured by the nmslib library. 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='ae1c1db7-3b98-4dd3-a16a-8a4830697078', node_type=, metadata={'page_label': '9', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='fe190817b0f2ba13237ad1733b1745b49119ef0fae7916a922b4361f7b85b60c'), : RelatedNodeInfo(node_id='9954c592-e1a0-4cf6-abcf-735a44ae4d7c', node_type=, metadata={'page_label': '9', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='f12c56f6194cfcb957185b18cb47df49b0c4ce1133c0cf8f7ae6fc24d0a96356'), : RelatedNodeInfo(node_id='b9f51f04-61b6-440f-a2b8-a79c76671d62', node_type=, metadata={}, hash='68c24941e4e55e8b975a04fd9252f0106b5c489f45f596a60c614bbc2dfdb83f')}, hash='d6462c7fa5a7e5cb302c579cfc4a07faa16a642b91efee9ac96bd178c1ed9163', text='Results for the vector data are presented in Fig. 13. For SIFT, GloVE, DEEP and CoPhIR datasets Hierarchical NSW clearly outperforms the rivals by a large margin. For low dimensional data (d=4) Hiera rchical NSW is 7 https://github.com/FALCONN-LIB/FALCONN slightly faster at high recall compared to the Annoy while strongly outperforms the other algorithms. 5.3 Comparison in general spaces A recent comparison of algorithms [34] in general spaces (i.e. non-symmetric or with violation of triangle inequal ity) showed that the baseline NSW algorithm has severe problems on low dimensional datasets. To test the pe rformance of the Hierarchical NSW algorithm we have repeated a subset of tests from [34] on which NSW pe rformed poorly or suboptimal . For that purpose we used a built-in nmslib testing system which had scripts to run tests from [34]. The evaluated algorithms included the VP-tree, permutation techniques (NAPP and bruteforce filtering) [49, 55-57], the basic NSW algorithm and NNDescent-produced proximity graphs [29] (both in pair with the NSW graph search algorithm). As in the original tests, for every dataset the test includes the results of e ither NSW or NNDescent, depending on which structure performed better. No custom distance functions or special TABLE 1 Parameters of the used datasets on vector spaces benchmark. Dataset Description Size d BF time Space SIFT Image feature vectors [13] 1M 128 94 ms L2 GloVe Word embeddings trained on tweets [52] 1.2M 100 95 ms cosine CoPhIR MPEG-7 features extracted from the images [53] 2M 272 370 ms L2 Random vectors Random vectors in hypercube 30M 4 590 ms L2 DEEP One million subset of the billion deep image features dataset [14] 1M 96 60 ms L2 MNIST Handwritten digit images [54] 60k 784 22 ms L2 TABLE 2. Used datasets for repetition of the Non-Metric data tests subset. Dataset Description Size d BF time Distance Wiki-sparse TF-IDF (term frequency \u2013inverse document frequency) vectors (created via GENSIM [58]) 4M 105 5.9 s Sparse cosine Wiki-8 Topic histograms created from sparse TF-IDF vectors of the wiki-sparse dataset (created via GENSIM [58]) 2M 8-Jensen\u2013Shannon (JS) divergence Wiki-128 Topic histograms created from sparse TF-IDF vectors of the wiki-sparse dataset (created via GENSIM [58]) 2M 128 1.17 s Jensen\u2013Shannon (JS) divergence ImageNet Signatures extracted from LSVRC-2014 with SQFD (signature quadratic form) distance [59] 1M 272 18.3 s SQFD DNA DNA (deoxyribonucleic acid) dataset sampled from the Human Genome 5 [34]. 1M-2.4 s Levenshtein 0,00,20,40,60,81,00,01 0,1 1 10 100Query time, msRecall1M SIFT, 10-NNBF: 94 ms Hierarchical NSW NSW Annoy VP-tree FLANN0,00,20,40,60,81,00,01 0,1 1 10 100Query time, msRecall1.2M GloVe, 10-NNBF: 95 ms Hierarchical NSW NSW Annoy VP-tree FLANN FALCONN0,01 0,1 1 10 1000,00,20,40,60,81,0RecallQuery time, ms2M CoPhIR, 10-NNBF: 370 ms Hierarchical NSW NSW Annoy VP-tree FLANN0,01 0,1 10,20,40,60,81,0RecallQuery time, ms30M random d=4,10-NNBF: 590 ms Hierachical NSW NSW Annoy VP-tree FLANN0,1 1 100,00,20,40,60,81,0RecallQuery time, ms60k MNIST, 10-NNBF: 22 ms Hierarchical NSW NSW Annoy VP-tree FLANN0,01 0,1 1 10 1000,00,20,40,60,81,0RecallQuery time, ms1M Deep,10-NNBF: 60 ms Hierarchical NSW NSW Annoy VP-tree FLANN Fig. 13. Results of the comparison of Hierarchical NSW with open source implementations of K-ANNS algorithms on five datasets for 10-NN searches. The time of a brute-force search is denoted as the BF. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='f649ce611b6774a4cd9f66da8321d5b30f0788c45132f9089f6e5ad2654ea481'), : RelatedNodeInfo(node_id='9cdcb748-dfc4-475c-84c3-2994870e1b39', node_type=, metadata={}, hash='f98911041081d3f5f85ddf94980da2bd7d016a01b8cff04f88a2661de6cc4aee')}, hash='eacc717f0c4e3b021bf7b463d51f88c06799d84e82980688d067fa0414b9fb3e', text='The results are presented in Fig. 14. Hierarchical NSW significantly improves the performance of NSW and is a leader for any of the tested datasets. The strongest e nhancement over NSW, almost by 3 orders of magnitude is observed for the dataset with the lo west dimensionality, the wiki-8 with JS-divergence. This is an important result that demonstrates the robustness of Hierarchical NSW , as for the original NSW this dataset was a stumbling block. Note that for the wiki-8 to nullify the effect of impleme ntation results are presented for the distance computations number instead of the CPU time. 5.4 Comparison with product quantization based algorithms. Product quantization K-ANNS algorithms [10-17] are considered as the state-of-the-art on billion scale dataset s since they can efficiently compress stored data , allowing modest RAM usage while achieving millisecond search times on modern CPUs. To compare the performance of Hierarchical NSW against PQ algorithms we used the facebook Faiss library8 as the baseline (a new library with state-of-the-art PQ algorithms [12, 15] implementations, released after the current manuscript was submitted) compiled with the OpenBLAS backend. The tests where done for a 200M subset of 1B SIFT dataset [13] on a 4X Xeon E5-4650 v2 server with 128Gb of RAM. The ann-benchmark testbed was not feasible for these experiments because of its rel iance on 32-bit floating point format (requiring more than 100 Gb just to store the data). To get the results for Faiss PQ algorithms we have utilized built-in scripts with the parameters from Faiss wiki9. For the Hierarchical NSW algorithm we used a special build outside of the nmslib with a small memory footprint, simple non-vectorized 8 https://github.com/facebookresearch/faiss 2017 May build. From 2018 Faiss library has its own implementation of Hierarchical NSW. 9 https://github.com/facebookresearch/faiss/wiki/Indexing-1G-vectors TABLE 3. Parameters for comparison between Hierarchical NSW and Faiss on a 200M subset of 1B SIFT dataset. Algorithm Build time Peak memory (runtime) Parameters Hierarchical NSW 5.6 hours 64 Gb M=16, efConstruction =500 (1) Hierarchical NSW 42 minutes 64 Gb M=16, efConstruction =40 (2) Faiss 12 hours 30 Gb OPQ64, IMI2x14, PQ64 (1) Faiss 11 hours 23.5 Gb OPQ32, IMI2x14, PQ32 (2) 0,00,20,40,60,81,00,1 1 10 100Query time, msRecall200M SIFT, 1-NN Faiss 1 Faiss 2 HNSW 1 HNSW 21 10 1000,10,20,30,40,50,6Query time, msDataset size, millions Fig. 15 Results of comparison with Faiss library on the 200M SIFT dataset from [13]. The inset shows the scaling of the query time vs the dataset size for Hierarchical NSW. 0,20,40,60,81,00,1 1 10 100Query time, msRecall2M Wiki-128 (JS-div), 10-NNBF: 1.17 s Hierarchical NSW NSW VP-tree NAPP0,800,850,900,951,0010 100 1000Query time, msRecall1M ImageNet, 10-NNBF: 18.3 s Hierarchical NSW NSW VP-tree NAPP bruto-force filtering0,60,70,80,91,0100 1000 10000 100000Number of distance computationsRecall2M Wiki-8 (JS-div), 10-NN Hierarchical NSW Old kNN(NNDescent) VP-tree NAPP0,60,81,010 100 1000Query time, msRecall1M DNA (edit dist), 10-NNBF: 2.4 s Hierarchical NSW Old kNN (NNDescent) VP-tree NAPP brute-force filt. bin.0,00,20,40,60,81,01 10 100 1000Query time, msRecall4M Wiki-sparse, 10-NNBF: 5.9 s Hierarchical NSW NSW NAPP Fig. 14. Results of the comparison of Hierarchical NSW with general space K-ANNS algorithms from the Non Metric Space Library on five datasets for 10-NN searches. The time of a brute-force search is denoted as the BF. 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A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={: RelatedNodeInfo(node_id='f5200883-2089-4dda-a775-3fce21c08c8d', node_type=, metadata={'page_label': '11', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='9155c451a9f664cbd27fcf49ad70bf51e356de7e889d0adfeecd6f65d39a7df9'), : RelatedNodeInfo(node_id='6fb9afa9-f8a9-41c4-b881-a0f5bacb1f55', node_type=, metadata={'page_label': '10', 'file_name': 'hnsw.pdf', 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='6adbd248e806e55f1c580547ff3950fdbf775d51e647ba5e25716cc0047db79d'), : RelatedNodeInfo(node_id='9bb1f7ca-b5fc-402b-bc58-d377bb8d1942', node_type=, metadata={}, hash='4e8ce1f0e8f3b2a4c6acf1abd7b45bf5d852d17d2b4ba6fbf7d2ce6f8f95252d')}, hash='3400db742aff02668206a7f3bc4aaef1589c3a372881bac4eaf1027e02adb697', text='AUTHOR ET AL.: TITL E 11 integer distance functions and support for incremental index const ruction10. The results are presented in Fig. 15 with summariz ation of the parameters in Table 3. The peak memory co nsumption was measured by using linux \u201ctime \u2013v\u201d tool in separate test runs after index construction for both of the algorithms. Even though Hierarchical NSW requires si gnificantly more RAM , it can achieve much higher accur acy, while offering a massive advance in search speed and much faster index construction. The inset in Fig. 15 presents the scaling of the query time vs the dataset size for Hierarchical NSW. Note that the scaling deviates from the pure logarithm , possibly due to relatively high dimensionality of the dataset. 6 DISCUSSION By using structure decomposition of navigable small world graphs together with the smart neighbor selectio n heuristic the proposed Hierarchical NSW approach ove rcomes several important problems of the basic NSW structure advancing the state-of\u2013the-art in K-ANN search. Hierarchical NSW offers an excellent performance and is a clear leader on a large variety of the datasets, surpa ssing the opensource rivals by a large margin in case of high dimensional data. Even for the datasets where the previous algorithm (NSW) has lost by orders of magn itude, Hierarchical NSW was able to come first. Hiera rchical NSW supports continuous incremental indexing and can also be used as an efficient method for getting approxima tions of the k-NN and relative neighborhood graphs, which are byproducts of the index construction. Robustness of the approach is a strong feature which makes it very attractive for practical applications. The algorithm is applicable in generalized metric spaces pe rforming the best on any of the datasets tested in this p aper, and thus eliminating the need for complicated sele ction of the best algorithm for a spe cific problem. We stress the importance of the algorithm\u2019s robustness since the data may have a complex structure with different effe ctive dimensionality across the scales. For instance, a d ataset can consist of points lying on a curve that randomly fills a high dimensional cube, thus being high dimensio nal at large scale and low dimensional at small scale . In order to perform efficient search in such datasets an a pproximate nearest neighbor algorithm has to work well for both cases of high and low dimensio nality. There are several ways to further increase the efficie ncy and applicability of the Hierarchical NSW approach. There is still one meaningful parameter left which stron gly affects the construction of the index \u2013 the number of added connections per la yer M. Potentially , this param eter can be inferred directly by using different heuri stics [4]. It would also be interesting to compare H ierarchical NSW on the full 1B SIFT and 1B DEEP datasets [10-14] and add support for element updates and removal. One of the apparent shortcomings of the proposed a p-10 https://github.com/nmslib/hnsw proach compared to the basic NSW is the loss of the po ssibility of distributed search. The search in the Hiera rchical NSW structure always starts from the top layer, thus the structure cannot be made distributed by using the same techniques as described in [26] due to cogne stion of the higher layer elements. Simple workarounds can be used to distribute the structure, such as partitio ning the data across cluster nodes studied in [6], however in this case, the total parallel throughput of the system does not scale well with the number of computer nodes. Still, there are other possible known ways to make this particular structure distributed. Hierarchical NSW is ideologically very similar to the well-known one-dimensional exact search probabilistic skip list st ructure, and thus can use the same techniques to make the stru cture distributed [45]. Potentially this can lead to even be tter distributed performance compared to the base NSW due to logarithmic scalability and ideally uniform load on the nodes. 7 ACKNOWL EDGEMENTS We thank Leonid Boytsov for many helpful discussions, assistance with Non-Metric Space Library integration and comments on the manuscript. We thank Seth Hoffert and Azat Davletshin for the suggestions on the manuscript and the algorithm and fellows who contributed to the algorithm on the github repository. We also thank Valery Kalyagin for support of this work. The reported study was funded by RFBR, according to the research project No. 16-31-60104 mol_\u0430_dk. 8 REFERENCES [1] D. G. Lowe, \"Distinctive image features from scale-invariant keypoints,\" International journal of computer vision, vol. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='c9ca3468dc963e2b6dc4c5199556fd6348deec4e8132065bbd7fecf8cd0e3f9a'), : RelatedNodeInfo(node_id='1f5869fe-3b2e-4025-908d-e92d46faa944', node_type=, metadata={}, hash='f2c98cc1dcedcc4f7a8354ac9d711661ca03c7ef887abbd4dd13135130120e6a')}, hash='5985e02dfffca85d682d346eb9fd00f3db7905a93c2eab7b8c7829e0b53d94be', text='AUTHOR ET AL.: TITL E 13 [49] L. Boytsov and B. Naidan, \"Engineering Efficient and Effe ctive Non-metric Space Library,\" in Similarity Search and Applications : Springer, 2013, pp. 280-293. [50] L. Boytsov and B. Naidan, \"Learning to prune in metric and non-metric spaces,\" in Advances in Neural Information Processing Systems , 2013, pp. 1574-1582. [51] A. Andoni and I. 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Yashunin'], 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}, hash='f2c98cc1dcedcc4f7a8354ac9d711661ca03c7ef887abbd4dd13135130120e6a')}, hash='cea0a7fa5bb6f9d314841410692d8b5edb14e5e8556220d856c952256e53fb35', text='30, no. 9, pp. 1647-1658, 2008. [57] E. S. Tellez, E. Ch\u00e1vez, and G. Navarro, \"Succinct nearest neighbor search,\" Information Systems, vol. 38, no. 7, pp. 1019-1030, 2013. [58] P. Sojka, \"Software fram ework for topic modelling with large corpora,\" in In Proceedings of the LREC 2010 Workshop on New Challenges for NLP Frameworks , 2010: Citeseer. [59] C. Beecks, \"Distance-based similarity models for content-based multimedia retrieval,\" Hochschulbibliothek der R heinisch-Westf\u00e4lischen Technischen Hochschule Aachen, 2013. Yury A. Malkov received a Master\u2019s degree in physics from Nizhny Novgorod State University in 2009, and a PhD degree in laser phy sics from the Institute of Applied Physics RAS in 2015. He is author of 20+ papers on physics and computer science . Yury currently occupies a position of a Project Leader in Samsung AI Center in Moscow . His current research interests include deep lear ning, scalable sim ilarity search, biological and artificial neural networks. Dmitry A. Yashunin received a Master\u2019s degree in physics from Nizhny Novgorod State University in 2009, and a PhD degree in laser physics from the Institute of Applied Physics RAS in 2015. From 2008 to 2012 he was working in Mera Networks . He is author of 10+ papers on physics. Dmitry currently woks at Intelli-Vision in the pos ition of a leading research engineer. His current research interests include scalable similarity search, computer vision and deep lear ning. ', start_char_idx=1426, end_char_idx=2885, text_template='{metadata_str}\\n\\n{content}', metadata_template='{key}: {value}', metadata_seperator='\\n')]" - ] - }, - "execution_count": 12, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# Our pipeline with the addition of our PineconeVectorStore\n", - "pipeline = IngestionPipeline(\n", - " transformations=[\n", - " SemanticSplitterNodeParser(\n", - " buffer_size=1,\n", - " breakpoint_percentile_threshold=95,\n", - " embed_model=embed_model,\n", - " ),\n", - " embed_model,\n", - " ],\n", - " vector_store=vector_store # Our new addition\n", - " )\n", - "\n", - "# Now we run our pipeline!\n", - "pipeline.run(documents=cleaned_docs)\n" - ] - }, + "name": "stdout", + "output_type": "stream", + "text": [ + "Note: you may need to restart the kernel to use updated packages.\n" + ] + } + ], + "source": [ + "%pip install -qU \\\n", + " pinecone==9.1.0 \\\n", + " llama-index==0.14.25 \\\n", + " llama-index-vector-stores-pinecone==0.9.0 \\\n", + " llama-index-readers-file==0.7.0 \\\n", + " arxiv==4.0.1\n", + "\n", + "import os\n", + "import re\n", + "from getpass import getpass\n", + "from pathlib import Path\n", + "from urllib.request import urlretrieve\n", + "\n", + "import arxiv\n", + "import nest_asyncio\n", + "from llama_index.core import Settings, VectorStoreIndex\n", + "from llama_index.core.evaluation import RelevancyEvaluator\n", + "from llama_index.core.ingestion import IngestionPipeline\n", + "from llama_index.core.node_parser import SemanticSplitterNodeParser\n", + "from llama_index.core.query_engine import RetrieverQueryEngine\n", + "from llama_index.core.retrievers import VectorIndexRetriever\n", + "from llama_index.embeddings.openai import OpenAIEmbedding\n", + "from llama_index.llms.openai import OpenAI\n", + "from llama_index.readers.file import PDFReader\n", + "from llama_index.vector_stores.pinecone import PineconeVectorStore\n", + "from pinecone import ServerlessSpec\n", + "from pinecone.grpc import PineconeGRPC" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:50.213273Z", + "iopub.status.busy": "2026-10-08T21:57:50.213081Z", + "iopub.status.idle": "2026-10-08T21:57:50.215566Z", + "shell.execute_reply": "2026-10-08T21:57:50.215207Z" + } + }, + "outputs": [], + "source": [ + "# Get a Pinecone API key from app.pinecone.io and an OpenAI API key from platform.openai.com/api-keys\n", + "if not os.getenv(\"PINECONE_API_KEY\"):\n", + " os.environ[\"PINECONE_API_KEY\"] = getpass(\"Enter your Pinecone API key: \")\n", + "if not os.getenv(\"OPENAI_API_KEY\"):\n", + " os.environ[\"OPENAI_API_KEY\"] = getpass(\"Enter your OpenAI API key: \")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Load the data\n", + "\n", + "This notebook uses the [canonical HNSW paper](https://arxiv.org/pdf/1603.09320) by Yuri Malkov as the sample dataset. Download the PDF from arXiv.org and load it with LlamaIndex's `PDFReader`." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:50.216721Z", + "iopub.status.busy": "2026-10-08T21:57:50.216641Z", + "iopub.status.idle": "2026-10-08T21:57:51.989445Z", + "shell.execute_reply": "2026-10-08T21:57:51.988984Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 15, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "IJmHDNzUFPkK", - "outputId": "675db08a-ac2d-4803-9d30-2a4a7767ef63" - }, - "outputs": [ - { - "data": { - "text/plain": [ - "{'dimension': 1536,\n", - " 'index_fullness': 0.0,\n", - " 'namespaces': {'': {'vector_count': 46}},\n", - " 'total_vector_count': 46}" - ] - }, - "execution_count": 15, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "pinecone_index.describe_index_stats()" + "data": { + "text/plain": [ + "Document(id_='7b2e8f21-29b4-446a-b324-f40afad7bac4', embedding=None, metadata={'page_label': '1', 'file_name': 'hnsw.pdf'}, excluded_embed_metadata_keys=[], excluded_llm_metadata_keys=[], relationships={}, metadata_template='{key}: {value}', metadata_separator='\\n', text_resource=MediaResource(embeddings=None, data=None, text=\"IEEE TRANSACTIONS ON JOURNAL NAME, MANUSCRIPT ID 1 \\n \\nEfficient and robust approximate nearest \\nneighbor search using Hierarchical Navigable \\nSmall World graphs \\nYu. A. Malkov, D. A. Yashunin \\nAbstract — We present a new approach for the approximate K -nearest neighbor search based on navigable small world \\ngraphs with controllable hierarchy (Hierarchical NSW, HNSW). The proposed solution is fully graph-based, without any need for \\nadditional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. \\nHierarchical NSW incrementally builds a multi-layer structure consisting from hierarchical set of proximity graphs (layers) for \\nnested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an \\nexponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Sma ll \\nWorld (NSW) struc tures while additionally having the links separated by their characteristic distance scales. Starting search \\nfrom the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a \\nlogarithmic complexity scalin g. Additional employment of a heuristic for selecting proximity graph neighbors significantly \\nincreases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the \\nproposed general metric space search i ndex is able to strongly outperform previous opensource state-of-the-art vector-only \\napproaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation. \\nIndex Terms — Graph and tree search strategies, Artificial Intelligence, Information Search and Retrieval, Information Storage \\nand Retrieval, Information Technology and Systems, Search process, Graphs and networks, Data Structures, Nearest neighbor \\nsearch, Big data, Approximate search, Similarity search \\n—————————— \\uf075 —————————— \\n1 INTRODUCTION\\nonstantly growing amount of the available info r-\\nmation resources has led to high demand in scalable \\nand efficient similarity search data structures. One of the \\ngenerally used approaches for information search is the \\nK-Nearest Neighbor Search (K -NNS). The K -NNS a s-\\nsumes you have a defined distance function between the \\ndata elements and aims at finding the K elements from \\nthe dataset which minimize the distance to a given query. \\nSuch algorithms are used in many applications, such as \\nnon-parametric machine learn ing algorithms, image fe a-\\ntures matching in large scale databases [1] and semantic \\ndocument retrieval [2]. A naïve approach to K -NNS is to \\ncompute the distances between the query and every el e-\\nment in the dataset and select the elements with minimal \\ndistance. Unfortunately, the complexity of the naïve a p-\\nproach scales linearly with the number of stored elements \\nmaking it infeasible for large-scale datasets. This has led \\nto a high interest in development of fast and scalable K -\\nNNS algorithms. \\nExact solutions for K-NNS [3-5] may offer a substantial \\nsearch speedup only in case of relatively low dimensional \\ndata du e to “curse of dimensionality”. To overcome this \\nproblem a concept of Approximate Nearest Neighbors \\nSearch (K-ANNS) was proposed, which relaxes the cond i-\\ntion of the exact search by allowing a small number of \\nerrors. The quality of an inexact search (the re call) is d e-\\nfined as the ratio between the number of found true nea r-\\nest neighbors and K. The m ost popular K -ANNS sol u-\\ntions are based on approximated versions of tree alg o-\\nrithms [6, 7], locality -sensitive hashing (LSH) [8, 9] and \\nproduct quantization (PQ) [10-17]. Proximity graph K -\\nANNS algorithms [10, 18-26] have recently gained pop u-\\nlarity offering a better performance on high dimensional \\ndatasets. However, the power -law scaling of the proxim i-\\nty graph routing causes extreme performance degrad a-\\ntion in case of low dimensional or clustered data. \\nIn this paper we propose the Hierarchical Navigable \\nSmall World (Hierarchical NSW, HNSW), a new fully \\ngraph based incremental K -ANNS structure, which can \\noffer a much better logarithmic complexity scaling. The \\nmain contributions are: explicit selection of the graph’s \\nenter-point node, separation of links by different scales \\nand use of an advanced heuristic to select the neighbors. \\nAlternatively, Hierarchical NSW algorithm can be seen as \\nan extension of the probabilistic skip list structure [27] \\nwith proximity graphs instead of the linked lists. Perfo r-\\nmance e valuation has demonstrated that the proposed \\ngeneral metric space method is able to strongly outpe r-\\nform previous opensource state-of-the-art approaches \\nsuitable only for vector spaces. \\n———————————————— \\n\\uf0b7 Y. Malkov is with the Federal state budgetary institution of science \\nInstitute of Applied Physics of the Russian Academy of Sciences, 46 \\nUl'yanov Street, 603950 Nizhny Novgorod, Russia. E-mail: \\nyurymalkov@mail.ru. \\n\\uf0b7 D. Yashunin. Addres: 31-33 ul. Krasnozvezdnaya, 603104 Nizhny \\nNovgorod, Russia. E-mail: yashuninda@yandex.ru \\nC \\nThis work has been submitted to the IEEE for possible publication. Copyright \\nmay be transferred without notice, after which this version may no longer be \\naccessible. \", path=None, url=None, mimetype=None), image_resource=None, audio_resource=None, video_resource=None, text_template='{metadata_str}\\n\\n{content}')" ] - }, + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Download paper to local file system (LFS)\n", + "# `id_list` contains 1 item that matches our PDF's arXiv ID\n", + "paper = next(arxiv.Client().results(arxiv.Search(id_list=[\"1603.09320\"])))\n", + "urlretrieve(paper.pdf_url, \"hnsw.pdf\")\n", + "\n", + "# Instantiate `PDFReader` from LlamaHub\n", + "loader = PDFReader()\n", + "\n", + "# Load HNSW PDF from LFS\n", + "documents = loader.load_data(file=Path(\"./hnsw.pdf\"))\n", + "\n", + "# Preview one of our documents\n", + "documents[0]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Clean up the text: remove leftover `\\n` characters and rejoin words that were hyphenated across line breaks." + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:51.990810Z", + "iopub.status.busy": "2026-10-08T21:57:51.990744Z", + "iopub.status.idle": "2026-10-08T21:57:51.997640Z", + "shell.execute_reply": "2026-10-08T21:57:51.997159Z" + } + }, + "outputs": [ { - "cell_type": "markdown", - "metadata": { - "id": "cqMV456NH9ae" - }, - "source": [ - "# Query the data" + "data": { + "text/plain": [ + "\"IEEE TRANSACTIONS ON JOURNAL NAME, MANUSCRIPT ID 1 Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs Yu. A. Malkov, D. A. Yashunin Abstract — We present a new approach for the approximate K-nearest neighbor search based on navigable small world graphs with controllable hierarchy (Hierarchical NSW, HNSW). The proposed solution is fully graph-based, without any need for additional search structures, which are typically used at the coarse search stage of the most proximity graph techniques. Hierarchical NSW incrementally builds a multi-layer structure consisting from hierarchical set of proximity graphs (layers) for nested subsets of the stored elements. The maximum layer in which an element is present is selected randomly with an exponentially decaying probability distribution. This allows producing graphs similar to the previously studied Navigable Sma ll World (NSW) struc tures while additionally having the links separated by their characteristic distance scales. Starting search from the upper layer together with utilizing the scale separation boosts the performance compared to NSW and allows a logarithmic complexity scalin g. Additional employment of a heuristic for selecting proximity graph neighbors significantly increases performance at high recall and in case of highly clustered data. Performance evaluation has demonstrated that the proposed general metric space search i ndex is able to strongly outperform previous opensource state-of-the-art vector-only approaches. Similarity of the algorithm to the skip list structure allows straightforward balanced distributed implementation. Index Terms — Graph and tree search strategies, Artificial Intelligence, Information Search and Retrieval, Information Storage and Retrieval, Information Technology and Systems, Search process, Graphs and networks, Data Structures, Nearest neighbor search, Big data, Approximate search, Similarity search 1 INTRODUCTIONonstantly growing amount of the available info rmation resources has led to high demand in scalable and efficient similarity search data structures. One of the generally used approaches for information search is the K-Nearest Neighbor Search (K-NNS). The K-NNS a ssumes you have a defined distance function between the data elements and aims at finding the K elements from the dataset which minimize the distance to a given query. Such algorithms are used in many applications, such as non-parametric machine learn ing algorithms, image fe atures matching in large scale databases [1] and semantic document retrieval [2]. A naïve approach to K-NNS is to compute the distances between the query and every el ement in the dataset and select the elements with minimal distance. Unfortunately, the complexity of the naïve a pproach scales linearly with the number of stored elements making it infeasible for large-scale datasets. This has led to a high interest in development of fast and scalable K-NNS algorithms. Exact solutions for K-NNS [3-5] may offer a substantial search speedup only in case of relatively low dimensional data du e to “curse of dimensionality”. To overcome this problem a concept of Approximate Nearest Neighbors Search (K-ANNS) was proposed, which relaxes the cond ition of the exact search by allowing a small number of errors. The quality of an inexact search (the re call) is d efined as the ratio between the number of found true nea rest neighbors and K. The m ost popular K-ANNS sol utions are based on approximated versions of tree alg orithms [6, 7], locality-sensitive hashing (LSH) [8, 9] and product quantization (PQ) [10-17]. Proximity graph K-ANNS algorithms [10, 18-26] have recently gained pop ularity offering a better performance on high dimensional datasets. However, the power-law scaling of the proxim ity graph routing causes extreme performance degrad ation in case of low dimensional or clustered data. In this paper we propose the Hierarchical Navigable Small World (Hierarchical NSW, HNSW), a new fully graph based incremental K-ANNS structure, which can offer a much better logarithmic complexity scaling. The main contributions are: explicit selection of the graph’s enter-point node, separation of links by different scales and use of an advanced heuristic to select the neighbors. Alternatively, Hierarchical NSW algorithm can be seen as an extension of the probabilistic skip list structure [27] with proximity graphs instead of the linked lists. Perfo rmance e valuation has demonstrated that the proposed general metric space method is able to strongly outpe rform previous opensource state-of-the-art approaches suitable only for vector spaces. — Y. Malkov is with the Federal state budgetary institution of science Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov Street, 603950 Nizhny Novgorod, Russia. E-mail: yurymalkov@mail.ru. D. Yashunin. Addres: 31-33 ul. Krasnozvezdnaya, 603104 Nizhny Novgorod, Russia. E-mail: yashuninda@yandex.ru C This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. \"" ] - }, + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def clean_up_text(content: str) -> str:\n", + " \"\"\"\n", + " Remove unwanted characters and patterns in text input.\n", + "\n", + " :param content: Text input.\n", + "\n", + " :return: Cleaned version of original text input.\n", + " \"\"\"\n", + "\n", + " # Fix hyphenated words broken by newline\n", + " content = re.sub(r\"(\\w+)-\\n(\\w+)\", r\"\\1\\2\", content)\n", + "\n", + " # Remove specific unwanted patterns and characters\n", + " unwanted_patterns = [\n", + " \"\\\\n\",\n", + " \" —\",\n", + " \"——————————\",\n", + " \"—————————\",\n", + " \"—————\",\n", + " r\"\\\\u[\\dA-Fa-f]{4}\",\n", + " r\"\\uf075\",\n", + " r\"\\uf0b7\",\n", + " ]\n", + " for pattern in unwanted_patterns:\n", + " content = re.sub(pattern, \"\", content)\n", + "\n", + " # Fix improperly spaced hyphenated words and normalize whitespace\n", + " content = re.sub(r\"(\\w)\\s*-\\s*(\\w)\", r\"\\1-\\2\", content)\n", + " content = re.sub(r\"\\s+\", \" \", content)\n", + "\n", + " return content\n", + "\n", + "\n", + "# Call function\n", + "cleaned_docs = []\n", + "for d in documents:\n", + " cleaned_text = clean_up_text(d.text)\n", + " d.set_content(cleaned_text)\n", + " cleaned_docs.append(d)\n", + "\n", + "# Inspect output\n", + "cleaned_docs[0].get_content()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Transform the data" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Metadata\n", + "\n", + "The default metadata isn't very useful, so add the authors' names and the paper's title." + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:51.998669Z", + "iopub.status.busy": "2026-10-08T21:57:51.998617Z", + "iopub.status.idle": "2026-10-08T21:57:52.000377Z", + "shell.execute_reply": "2026-10-08T21:57:52.000063Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 17, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "WUAD0kIFH_fc", - "outputId": "61ed62f7-0673-4860-fbd7-63483dfe077d" - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "['AUTHOR ET AL.: TITL E 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index q uality tradeoff is co ntrolled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruct ion=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complex ity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer . One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial position s of the data elements and , thus , when we tra verse the graph there is a fixed probability p=exp( -mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the p robability of not reaching the ta rget on s-th step is bounded by exp( -s\u00b7 mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp( -mL)), which is independent of the dataset size . If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C\u00b7 S, independent ly of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional dataset s. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbor s per element . ', 'Thus, t o avoid stucking into a l ocal minimum the greedy search algo rithm em ploy s a backtracking procedure on the zero layer. Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via t he minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling . Such empirical investigation of the Delaunay graph approximation resilience require s having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e datasets, mak ing such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a sequence of K-ANN-searches at different layers with a su bsequent use of heuristic (which has fixed complexity at fixed efConstruction ). The average number of lay ers for an element to be added in is a constant that depends on mL: \\uf05b\\uf05d\\uf05b \\uf05d 1 ln( (0,1)) 1 1LL E l E unif m m\\uf02b \\uf03d \\uf02d \\uf02b \\uf03d \\uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N\u2219log( N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16 , efConstruction =100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. ', 'The search is terminated when some stopping condition is met (e.g. the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques , such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approach es are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure\u2019s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges betwee n the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing . Const ruction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , however, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\\uf061. For \\uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \\uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg\u2019s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg\u2019s graphs suffers fr om polylogarit hmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg\u2019s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44] . The routing can be divided into two phases: \u201czoom-out\u201d and \u201czoom-in\u201d [32]. The greedy algorithm starts in the \u201czoom-out\u201d phase ', 'AUTHOR ET AL.: TITL E 3 from a low degree node and traverses the graph simult aneously increasing the node\u2019s degree until the characteri stic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the \u201czoom-in\u201d phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber o f distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44] ; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such stru cture t he search starts from the upper laye r which has only the long est links (the \u201czoom-in\u201d phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illu stration) . After that, the search switches to the lower layer (which h as shorter links ), restarts from the element which was the local minimum in the previous layer and the pr ocess repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of rout ing in a navigable small world network . One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only \u201cshort\u201d links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer . In case we m erge connections from all layers, the stru cture becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insert ion-the stochasticity is achieved by using level random ization , thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure ). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. ', 'When the search reaches the layer that is equal or less than l, the second phase of the construction algorithm is initiated . The second phase differs in two points: 1) the ef parameter is increased from 1 to efConstruction in order to control the recall of the greedy search procedure; 2) the found closest neighbors on each layer are also used as candidates for the connections of the inserted element. Two methods for the selection of M neighbors from the candidates were considered: simple connection to the closest elements (alg. 3) and the heuristic that accounts for the distances between the candidate elements to create connections in diverse direction s (alg. 4), described in the Section 3. The heuristic has two additional parameters: extendCandidates (set to false by default) which extends the candidate set and useful only for extremely clustered data, and keepPrunedConnections which allows getting fixed number of connection per element . The maximum number of connections that an element can have per layer is defined by the parameter Mmax for every layer higher than zero (a special parameter Mmax0 is used for the ground layer sep arately). If a node is already full at the moment of making of a new connection, then its extended connection list gets shrunk by the same algorithm that used for the neighbors selection (algs. 3 or 4). The insertion procedure terminates when the conne ctions of the inserted elements are established on the zero layer. The K-ANNS search algorithm used in Hierarchical NSW is presented in alg. 5. It is roughly equivalent to the insertion algorithm for an item with layer l=0. The diffe rence is that the closest neighbors found at the ground layer which are used as candidates for the connections are now returned as the search result. The quality of the search is controlled by the ef parameter (corresponding to efConstruction in the construction algorithm). 4.1 In fluence of the construction parameters Algorithm construction parameters mL and Mmax0 are r esponsible for maintaining the small world navigability in the constructed graphs. Setting mL to zero (this corr esponds to a single layer in the graph) and Mmax0 to M leads to production of directed k-NN graphs with a po wer-law search complexity well studied before [21, 29] (assuming using the alg. 3 for neighbor selection). Setting mL to zero and Mmax0 to infinity leads to production of NSW graphs with polylogarithmi c complexity [25, 26] . Finally, setting mL to some non-zero value leads to emergence of controllable hierarchy graphs w hich allow logarithmic search complexity by introduction of layers (see the Se ction 3). To achieve the optimum performance advantage of the controllable hierarchy, the overlap between neighbors on different layers (i.e. percent of element neighbors that are also belong to other layers) has to be small. In order to decrease the overlap we need to decrease the mL. Howe ver, at the same time, decreas ing mL leads to an increase of average hop number during a greedy search on each la yer, which negatively affects the performance. This leads to existence of the optimal value for the mL parameter . A simple choice for the optimal mL is 1/ln( M), this co rresponds to the skip list parameter p=1/M with an ave rage single element overlap between the layers. Simul ations done on an Intel Core i7 5930K CPU show that the proposed selection of mL is a reasonable choice (see Fig. ']\n" - ] - } - ], - "source": [ - "from llama_index import VectorStoreIndex\n", - "from llama_index.retrievers import VectorIndexRetriever\n", - "\n", - "# Due to how LlamaIndex works here, if your Open AI API key was\n", - "# not set to an environment variable before, you have to set it at this point\n", - "if not os.getenv('OPENAI_API_KEY'):\n", - " os.environ['OPENAI_API_KEY'] = openai_api_key\n", - "\n", - "# Instantiate VectorStoreIndex object from our vector_store object\n", - "vector_index = VectorStoreIndex.from_vector_store(vector_store=vector_store)\n", - "\n", - "# Grab 5 search results\n", - "retriever = VectorIndexRetriever(index=vector_index, similarity_top_k=5)\n", - "\n", - "# # Query vector DB\n", - "answer = retriever.retrieve('How does logarithmic complexity affect graph construction?')\n", - "\n", - "# # Inspect results\n", - "print([i.get_content() for i in answer])" + "data": { + "text/plain": [ + "{'page_label': '1', 'file_name': 'hnsw.pdf'}" ] - }, + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "cleaned_docs[0].metadata" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:52.001200Z", + "iopub.status.busy": "2026-10-08T21:57:52.001156Z", + "iopub.status.idle": "2026-10-08T21:57:52.002942Z", + "shell.execute_reply": "2026-10-08T21:57:52.002634Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 18, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "lstYKSRxcT06", - "outputId": "c7bbcd40-8eda-4257-c9af-ae360cba0ead" - }, - "outputs": [ - { - "data": { - "text/plain": [ - "5" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# See how many Nodes we we retrieved (should be 5, because we made similarity_top_k=5)\n", - "\n", - "len([i.get_content() for i in answer])" + "data": { + "text/plain": [ + "{'page_label': '1',\n", + " 'file_name': 'hnsw.pdf',\n", + " 'authors': ['Yu. A. Malkov', 'D. A. Yashunin'],\n", + " 'title': 'Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs'}" ] - }, + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Iterate through `documents` and add our new key:value pairs\n", + "metadata_additions = {\n", + " \"authors\": [\"Yu. A. Malkov\", \"D. A. Yashunin\"],\n", + " \"title\": \"Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs\",\n", + "}\n", + "\n", + "# Update dict in place\n", + "[cd.metadata.update(metadata_additions) for cd in cleaned_docs]\n", + "\n", + "# Let's confirm everything worked:\n", + "cleaned_docs[0].metadata" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Ingestion pipeline\n", + "\n", + "The pipeline uses `SemanticSplitterNodeParser` with OpenAI's `text-embedding-3-small` model to split the Documents into semantically coherent Nodes, then embeds each Node. Don't run it yet. You'll add a vector store to it in the next section." + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:52.003802Z", + "iopub.status.busy": "2026-10-08T21:57:52.003761Z", + "iopub.status.idle": "2026-10-08T21:57:52.005466Z", + "shell.execute_reply": "2026-10-08T21:57:52.005079Z" + } + }, + "outputs": [], + "source": [ + "# This will be the model we use both for Node parsing and for vectorization\n", + "embed_model = OpenAIEmbedding(\n", + " model=\"text-embedding-3-small\",\n", + " api_key=os.environ[\"OPENAI_API_KEY\"],\n", + ")\n", + "\n", + "# Define the initial pipeline\n", + "pipeline = IngestionPipeline(\n", + " transformations=[\n", + " SemanticSplitterNodeParser(\n", + " buffer_size=1,\n", + " breakpoint_percentile_threshold=95,\n", + " embed_model=embed_model,\n", + " ),\n", + " embed_model,\n", + " ],\n", + ")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Upsert the data\n", + "\n", + "Create a Pinecone index and wrap it in a LlamaIndex `PineconeVectorStore`." + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:52.006314Z", + "iopub.status.busy": "2026-10-08T21:57:52.006250Z", + "iopub.status.idle": "2026-10-08T21:57:57.678220Z", + "shell.execute_reply": "2026-10-08T21:57:57.677820Z" + } + }, + "outputs": [], + "source": [ + "# Initialize connection to Pinecone\n", + "pc = PineconeGRPC(api_key=os.environ[\"PINECONE_API_KEY\"])\n", + "index_name = \"llama-integration-example\"\n", + "\n", + "# Create your index (can skip this step if your index already exists)\n", + "pc.create_index(\n", + " name=index_name,\n", + " dimension=1536,\n", + " metric=\"cosine\",\n", + " spec=ServerlessSpec(cloud=\"aws\", region=\"us-east-1\"),\n", + ")\n", + "\n", + "# Initialize your index\n", + "pinecone_index = pc.Index(index_name)\n", + "\n", + "# Initialize VectorStore\n", + "vector_store = PineconeVectorStore(pinecone_index=pinecone_index)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Add the vector store to the pipeline and run it." + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:57:57.679452Z", + "iopub.status.busy": "2026-10-08T21:57:57.679395Z", + "iopub.status.idle": "2026-10-08T21:58:07.693491Z", + "shell.execute_reply": "2026-10-08T21:58:07.692614Z" + } + }, + "outputs": [], + "source": [ + "# Our pipeline with the addition of our PineconeVectorStore\n", + "pipeline = IngestionPipeline(\n", + " transformations=[\n", + " SemanticSplitterNodeParser(\n", + " buffer_size=1,\n", + " breakpoint_percentile_threshold=95,\n", + " embed_model=embed_model,\n", + " ),\n", + " embed_model,\n", + " ],\n", + " vector_store=vector_store, # Our new addition\n", + ")\n", + "\n", + "# Now we run our pipeline!\n", + "nodes = pipeline.run(documents=cleaned_docs)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The vector count shows how many Nodes the splitter produced." + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:07.696283Z", + "iopub.status.busy": "2026-10-08T21:58:07.696111Z", + "iopub.status.idle": "2026-10-08T21:58:07.828880Z", + "shell.execute_reply": "2026-10-08T21:58:07.827957Z" + } + }, + "outputs": [ { - "cell_type": "markdown", - "metadata": { - "id": "oio_1WkcJ1fd" - }, - "source": [ - "# Build a RAG app with the data" + "data": { + "text/plain": [ + "DescribeIndexStatsResponse(dimension=1536, total_vector_count=46, metric='cosine', namespaces=1)" ] - }, + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "pinecone_index.describe_index_stats()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Query the data" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:07.831464Z", + "iopub.status.busy": "2026-10-08T21:58:07.831290Z", + "iopub.status.idle": "2026-10-08T21:58:08.319061Z", + "shell.execute_reply": "2026-10-08T21:58:08.318366Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 19, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/", - "height": 122 - }, - "id": "2e1MoWDLIDea", - "outputId": "a87744a3-6bf4-4f30-9992-5de7832d4266" - }, - "outputs": [ - { - "data": { - "application/vnd.google.colaboratory.intrinsic+json": { - "type": "string" - }, - "text/plain": [ - "'Logarithmic complexity in graph construction allows for efficient and scalable routing in the graph. It achieves this by organizing the graph into different layers based on their length scale. The search procedure starts from the top layer, which contains the longest links, and greedily traverses through the elements until a local minimum is reached. Then, the search switches to the lower layer with shorter links and repeats the process. By separating the links into layers, the maximum number of connections per element in all layers can be kept constant, resulting in a logarithmic complexity scaling for routing in the graph. This logarithmic complexity is achieved by selecting an integer level for each element, which determines the maximum layer it belongs to. The construction algorithm incrementally builds a proximity graph for each layer, and the search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer. Overall, logarithmic complexity in graph construction allows for efficient and robust approximate nearest neighbor search.'" - ] - }, - "execution_count": 19, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "from llama_index.query_engine import RetrieverQueryEngine\n", - "\n", - "# Pass in your retriever from above, which is configured to return the top 5 results\n", - "query_engine = RetrieverQueryEngine(retriever=retriever)\n", - "\n", - "# Now you query:\n", - "llm_query = query_engine.query('How does logarithmic complexity affect graph construction?')\n", - "\n", - "llm_query.response\n" - ] - }, + "name": "stdout", + "output_type": "stream", + "text": [ + "['AUTHOR ET AL.: TITLE 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index quality tradeoff is controlled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruction=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complexity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer. One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial positions of the data elements and , thus, when we tra verse the graph there is a fixed probability p=exp(-mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the probability of not reaching the ta rget on s-th step is bounded by exp(-s· mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp(-mL)), which is independent of the dataset size. If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C· S, independently of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional datasets. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbors per element. Thus, to avoid stucking into a l ocal minimum the greedy search algo rithm em ploys a backtracking procedure on the zero layer. ', 'AUTHOR ET AL.: TITLE 3 from a low degree node and traverses the graph simultaneously increasing the node’s degree until the characteristic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the “zoom-in” phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber of distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44]; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such structure the search starts from the upper layer which has only the longest links (the “zoom-in” phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illustration). After that, the search switches to the lower layer (which has shorter links), restarts from the element which was the local minimum in the previous layer and the process repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of routing in a navigable small world network. One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only “short” links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer. In case we merge connections from all layers, the structure becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insertion-the stochasticity is achieved by using level randomization, thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. The Hierarchical Layer=0Layer=1Layer=2Decreasing characteristic radius Cluster 2Inserted elemente2Cluster 1 Fig. ', 'the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques, such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approaches are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure’s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges between the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing. Construction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , h owever, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\\uf061. For \\uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \\uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg’s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg’s graphs suffers fr om polylogarithmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg’s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44]. The routing can be divided into two phases: “zoom-out” and “zoom-in” [32]. ', 'Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via the minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling. Such empirical investigation of the Delaunay graph approximation resilience requires having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e d atasets, making such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a s equence of K-ANN-searches at different layers with a subsequent use of heuristic (which has fixed complexity at fixed efConstruction). The average number of lay ers for an element to be added in is a constant that depends on mL: \\uf05b \\uf05d \\uf05b \\uf05d1 ln( (0,1)) 1 1 LLE l E unif m m\\uf02b \\uf03d \\uf02d \\uf02b \\uf03d \\uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N∙log(N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16, efConstruction=100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. Fig. ', '4 IEEE TRANSACTIONS ON JOURNAL NAME, MANUSCRIPT ID NSW approach thus can utilize the same methods for making the distributed approximate search/overlay structures [45]. For the selection of the proximity graph con nections during the element insertion we utilize a heuristic that takes into account the distances between the candidate elements to create diverse connections (a similar alg orithm was utilized in the spatial approximation tree [4] to select the tree children) instead of just selecting the closest neighbors. The heuristic examines the candidates starting from the nearest (with respect to the inserted element ) and creates a connection to a candidate only if it is closer to the base (inserted) element compared to any of the a lready connected candidates (see Section 4 for the details). When the number of candidates is large enough the heuristic allows getting the exact relative neighborhood graph [46] as a subgraph , a minimal subgraph of the D elaunay graph deducible by using only the distances b etween the nodes. The relative neighborhood graph allows easily keeping the global connected component, even in case of highly clustered data (see Fig. 2 for illustration). Note that the heuristic creates extra edges compared to the exact relative neighborhood graph s, allowing controlling the number of the connections which is important for search performance. For the case of 1D data the heuristic allows getting the exact Delaunay subgraph (which in this case coincides with the relative neighborhood graph) by using only information about the distances between the elements, thus making a direct transition from Hiera rchical NSW to the 1D probabilistic skip list algorithm. Base variant of the Hierarchical NSW proximity graphs was also used in ref. [18] (called ‘sparse neighborhood graphs’) for proximity graph searching. Similar heuristic was also a focus of the FANNG algorithm [47] (published shortly after the first versions of the current manuscript were posted online) with a slightly different interpretation, based on the sparse neighborhood graph’s property of the exact routing [18]. 4 ALGORITHM DESCRIPTION Network construction algorithm (alg. 1) is organized via consecutive insertions of the stored elements into the graph structure. For every inserted element an integer maximum layer l is randomly selected with an expone ntially decaying probability distribution (normalized by the mL parameter, see line 4 in alg. 1). The first phase of the insertion process starts from the top layer by greedily traversing the graph in order to find the ef closest neighbors to the inserted element q in the layer. After that, the algorithm continues the search from the next layer using the found closest neighbors from the previous layer as enter points, and the process repeats. Closest neighbors at each layer are found by a variant of the greedy search algorithm described in alg. 2, which is an updated version of the algorithm from [26]. To obtain the approximate ef nearest neighbors in some layer lс, a dynamic list W of ef closest found elements (initially filled with enter points) is kept during the search. The list is updated at each step by evaluating the neighborhood of the closest previously non-evaluated element in the list until the neighborhood of every element from the l ist is evaluated. Compared to limiting the number of distance calculations, Hierarchical NSW stop condition has a n advantage-it allows discarding candidates for evalution that are further from the query than the furthest element in the list, thus avoiding bloating of search structures. As in NSW, the list is emulated via two priority queues for better performance. ']\n" + ] + } + ], + "source": [ + "# Instantiate VectorStoreIndex object from your vector_store object\n", + "vector_index = VectorStoreIndex.from_vector_store(\n", + " vector_store=vector_store,\n", + " embed_model=embed_model,\n", + ")\n", + "\n", + "# Grab 5 search results\n", + "retriever = VectorIndexRetriever(index=vector_index, similarity_top_k=5)\n", + "\n", + "# Query vector DB\n", + "answer = retriever.retrieve(\n", + " \"How does logarithmic complexity affect graph construction?\"\n", + ")\n", + "\n", + "# Inspect results\n", + "print([i.get_content() for i in answer])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Build a RAG app with the data\n", + "\n", + "`RetrieverQueryEngine` uses the retriever you created above, so the LLM gets the top 5 search results as context." + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:08.321760Z", + "iopub.status.busy": "2026-10-08T21:58:08.321564Z", + "iopub.status.idle": "2026-10-08T21:58:10.243363Z", + "shell.execute_reply": "2026-10-08T21:58:10.242743Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 61, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "p1JitDo8NgSz", - "outputId": "1225b685-f781-449c-98f3-a746d9e55094" - }, - "outputs": [ - { - "data": { - "text/plain": [ - "['AUTHOR ET AL.: TITL E 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index q uality tradeoff is co ntrolled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruct ion=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complex ity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer . One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial position s of the data elements and , thus , when we tra verse the graph there is a fixed probability p=exp( -mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the p robability of not reaching the ta rget on s-th step is bounded by exp( -s\u00b7 mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp( -mL)), which is independent of the dataset size . If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C\u00b7 S, independent ly of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional dataset s. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbor s per element . ',\n", - " 'Thus, t o avoid stucking into a l ocal minimum the greedy search algo rithm em ploy s a backtracking procedure on the zero layer. Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via t he minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling . Such empirical investigation of the Delaunay graph approximation resilience require s having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e datasets, mak ing such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a sequence of K-ANN-searches at different layers with a su bsequent use of heuristic (which has fixed complexity at fixed efConstruction ). The average number of lay ers for an element to be added in is a constant that depends on mL: \\uf05b\\uf05d\\uf05b \\uf05d 1 ln( (0,1)) 1 1LL E l E unif m m\\uf02b \\uf03d \\uf02d \\uf02b \\uf03d \\uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N\u2219log( N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16 , efConstruction =100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. ',\n", - " 'The search is terminated when some stopping condition is met (e.g. the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques , such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approach es are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure\u2019s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges betwee n the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing . Const ruction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , however, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\\uf061. For \\uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \\uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg\u2019s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg\u2019s graphs suffers fr om polylogarit hmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg\u2019s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44] . The routing can be divided into two phases: \u201czoom-out\u201d and \u201czoom-in\u201d [32]. The greedy algorithm starts in the \u201czoom-out\u201d phase ',\n", - " 'AUTHOR ET AL.: TITL E 3 from a low degree node and traverses the graph simult aneously increasing the node\u2019s degree until the characteri stic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the \u201czoom-in\u201d phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber o f distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44] ; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such stru cture t he search starts from the upper laye r which has only the long est links (the \u201czoom-in\u201d phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illu stration) . After that, the search switches to the lower layer (which h as shorter links ), restarts from the element which was the local minimum in the previous layer and the pr ocess repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of rout ing in a navigable small world network . One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only \u201cshort\u201d links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer . In case we m erge connections from all layers, the stru cture becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insert ion-the stochasticity is achieved by using level random ization , thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure ). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. ',\n", - " 'When the search reaches the layer that is equal or less than l, the second phase of the construction algorithm is initiated . The second phase differs in two points: 1) the ef parameter is increased from 1 to efConstruction in order to control the recall of the greedy search procedure; 2) the found closest neighbors on each layer are also used as candidates for the connections of the inserted element. Two methods for the selection of M neighbors from the candidates were considered: simple connection to the closest elements (alg. 3) and the heuristic that accounts for the distances between the candidate elements to create connections in diverse direction s (alg. 4), described in the Section 3. The heuristic has two additional parameters: extendCandidates (set to false by default) which extends the candidate set and useful only for extremely clustered data, and keepPrunedConnections which allows getting fixed number of connection per element . The maximum number of connections that an element can have per layer is defined by the parameter Mmax for every layer higher than zero (a special parameter Mmax0 is used for the ground layer sep arately). If a node is already full at the moment of making of a new connection, then its extended connection list gets shrunk by the same algorithm that used for the neighbors selection (algs. 3 or 4). The insertion procedure terminates when the conne ctions of the inserted elements are established on the zero layer. The K-ANNS search algorithm used in Hierarchical NSW is presented in alg. 5. It is roughly equivalent to the insertion algorithm for an item with layer l=0. The diffe rence is that the closest neighbors found at the ground layer which are used as candidates for the connections are now returned as the search result. The quality of the search is controlled by the ef parameter (corresponding to efConstruction in the construction algorithm). 4.1 In fluence of the construction parameters Algorithm construction parameters mL and Mmax0 are r esponsible for maintaining the small world navigability in the constructed graphs. Setting mL to zero (this corr esponds to a single layer in the graph) and Mmax0 to M leads to production of directed k-NN graphs with a po wer-law search complexity well studied before [21, 29] (assuming using the alg. 3 for neighbor selection). Setting mL to zero and Mmax0 to infinity leads to production of NSW graphs with polylogarithmi c complexity [25, 26] . Finally, setting mL to some non-zero value leads to emergence of controllable hierarchy graphs w hich allow logarithmic search complexity by introduction of layers (see the Se ction 3). To achieve the optimum performance advantage of the controllable hierarchy, the overlap between neighbors on different layers (i.e. percent of element neighbors that are also belong to other layers) has to be small. In order to decrease the overlap we need to decrease the mL. Howe ver, at the same time, decreas ing mL leads to an increase of average hop number during a greedy search on each la yer, which negatively affects the performance. This leads to existence of the optimal value for the mL parameter . A simple choice for the optimal mL is 1/ln( M), this co rresponds to the skip list parameter p=1/M with an ave rage single element overlap between the layers. Simul ations done on an Intel Core i7 5930K CPU show that the proposed selection of mL is a reasonable choice (see Fig. ']" - ] - }, - "execution_count": 61, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "# To get the context (Nodes) again, we can now use the `.source_nodes` attribute.\n", - "# Let's inspect the 1st Node:\n", - "\n", - "llm_response_source_nodes = [i.get_content() for i in llm_query.source_nodes]\n", - "llm_response_source_nodes" + "data": { + "text/plain": [ + "'Logarithmic complexity in graph construction allows for efficient scaling as the dataset size increases. Specifically, it ensures that the time required to construct the graph grows at a manageable rate, proportional to the logarithm of the number of elements being processed. This means that even as the dataset expands significantly, the construction time remains relatively short, making it feasible to handle large datasets without a corresponding exponential increase in processing time. The construction process can be parallelized effectively, further enhancing performance while maintaining the quality of the index. Overall, logarithmic complexity contributes to the robustness and efficiency of the graph construction process, particularly in high-dimensional spaces.'" ] - }, + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Set the LLM that generates answers (and evaluates them later)\n", + "Settings.llm = OpenAI(model=\"gpt-4o-mini\")\n", + "\n", + "# Pass in your retriever from above, which is configured to return the top 5 results\n", + "query_engine = RetrieverQueryEngine(retriever=retriever)\n", + "\n", + "# Now you query:\n", + "llm_query = query_engine.query(\n", + " \"How does logarithmic complexity affect graph construction?\"\n", + ")\n", + "\n", + "llm_query.response" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Inspect the context (Nodes) that informed the answer with the `.source_nodes` attribute." + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:10.245402Z", + "iopub.status.busy": "2026-10-08T21:58:10.245287Z", + "iopub.status.idle": "2026-10-08T21:58:10.248391Z", + "shell.execute_reply": "2026-10-08T21:58:10.247776Z" + } + }, + "outputs": [ { - "cell_type": "markdown", - "metadata": { - "id": "ZbyrG7JgKATd" - }, - "source": [ - "# Evaluate the data" + "data": { + "text/plain": [ + "['AUTHOR ET AL.: TITLE 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index quality tradeoff is controlled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruction=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complexity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer. One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial positions of the data elements and , thus, when we tra verse the graph there is a fixed probability p=exp(-mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the probability of not reaching the ta rget on s-th step is bounded by exp(-s· mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp(-mL)), which is independent of the dataset size. If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C· S, independently of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional datasets. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbors per element. Thus, to avoid stucking into a l ocal minimum the greedy search algo rithm em ploys a backtracking procedure on the zero layer. ',\n", + " 'AUTHOR ET AL.: TITLE 3 from a low degree node and traverses the graph simultaneously increasing the node’s degree until the characteristic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the “zoom-in” phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber of distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44]; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such structure the search starts from the upper layer which has only the longest links (the “zoom-in” phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illustration). After that, the search switches to the lower layer (which has shorter links), restarts from the element which was the local minimum in the previous layer and the process repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of routing in a navigable small world network. One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only “short” links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer. In case we merge connections from all layers, the structure becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insertion-the stochasticity is achieved by using level randomization, thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. The Hierarchical Layer=0Layer=1Layer=2Decreasing characteristic radius Cluster 2Inserted elemente2Cluster 1 Fig. ',\n", + " 'the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques, such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approaches are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure’s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges between the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing. Construction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , h owever, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\\uf061. For \\uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \\uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg’s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg’s graphs suffers fr om polylogarithmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg’s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44]. The routing can be divided into two phases: “zoom-out” and “zoom-in” [32]. ',\n", + " 'Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via the minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling. Such empirical investigation of the Delaunay graph approximation resilience requires having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e d atasets, making such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a s equence of K-ANN-searches at different layers with a subsequent use of heuristic (which has fixed complexity at fixed efConstruction). The average number of lay ers for an element to be added in is a constant that depends on mL: \\uf05b \\uf05d \\uf05b \\uf05d1 ln( (0,1)) 1 1 LLE l E unif m m\\uf02b \\uf03d \\uf02d \\uf02b \\uf03d \\uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N∙log(N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16, efConstruction=100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. Fig. ',\n", + " '4 IEEE TRANSACTIONS ON JOURNAL NAME, MANUSCRIPT ID NSW approach thus can utilize the same methods for making the distributed approximate search/overlay structures [45]. For the selection of the proximity graph con nections during the element insertion we utilize a heuristic that takes into account the distances between the candidate elements to create diverse connections (a similar alg orithm was utilized in the spatial approximation tree [4] to select the tree children) instead of just selecting the closest neighbors. The heuristic examines the candidates starting from the nearest (with respect to the inserted element ) and creates a connection to a candidate only if it is closer to the base (inserted) element compared to any of the a lready connected candidates (see Section 4 for the details). When the number of candidates is large enough the heuristic allows getting the exact relative neighborhood graph [46] as a subgraph , a minimal subgraph of the D elaunay graph deducible by using only the distances b etween the nodes. The relative neighborhood graph allows easily keeping the global connected component, even in case of highly clustered data (see Fig. 2 for illustration). Note that the heuristic creates extra edges compared to the exact relative neighborhood graph s, allowing controlling the number of the connections which is important for search performance. For the case of 1D data the heuristic allows getting the exact Delaunay subgraph (which in this case coincides with the relative neighborhood graph) by using only information about the distances between the elements, thus making a direct transition from Hiera rchical NSW to the 1D probabilistic skip list algorithm. Base variant of the Hierarchical NSW proximity graphs was also used in ref. [18] (called ‘sparse neighborhood graphs’) for proximity graph searching. Similar heuristic was also a focus of the FANNG algorithm [47] (published shortly after the first versions of the current manuscript were posted online) with a slightly different interpretation, based on the sparse neighborhood graph’s property of the exact routing [18]. 4 ALGORITHM DESCRIPTION Network construction algorithm (alg. 1) is organized via consecutive insertions of the stored elements into the graph structure. For every inserted element an integer maximum layer l is randomly selected with an expone ntially decaying probability distribution (normalized by the mL parameter, see line 4 in alg. 1). The first phase of the insertion process starts from the top layer by greedily traversing the graph in order to find the ef closest neighbors to the inserted element q in the layer. After that, the algorithm continues the search from the next layer using the found closest neighbors from the previous layer as enter points, and the process repeats. Closest neighbors at each layer are found by a variant of the greedy search algorithm described in alg. 2, which is an updated version of the algorithm from [26]. To obtain the approximate ef nearest neighbors in some layer lс, a dynamic list W of ef closest found elements (initially filled with enter points) is kept during the search. The list is updated at each step by evaluating the neighborhood of the closest previously non-evaluated element in the list until the neighborhood of every element from the l ist is evaluated. Compared to limiting the number of distance calculations, Hierarchical NSW stop condition has a n advantage-it allows discarding candidates for evalution that are further from the query than the furthest element in the list, thus avoiding bloating of search structures. As in NSW, the list is emulated via two priority queues for better performance. ']" ] - }, + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "llm_response_source_nodes = [i.get_content() for i in llm_query.source_nodes]\n", + "\n", + "llm_response_source_nodes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Evaluate the data\n", + "\n", + "`RelevancyEvaluator` checks whether the LLM's response is relevant to the context retrieved from Pinecone. It doesn't need ground truth data." + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:10.250170Z", + "iopub.status.busy": "2026-10-08T21:58:10.250063Z", + "iopub.status.idle": "2026-10-08T21:58:12.987984Z", + "shell.execute_reply": "2026-10-08T21:58:12.987061Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 57, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "ONqJhkXyJ2IU", - "outputId": "de6ee71f-9ac9-4b24-d91f-916b8e7c7928" - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n", - "Given the 5 chunks of content (below), is your LLM's response relevant? True\n", - " \n", - " ----Contexts----- \n", - " \n", - "['AUTHOR ET AL.: TITL E 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index q uality tradeoff is co ntrolled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruct ion=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complex ity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer . One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial position s of the data elements and , thus , when we tra verse the graph there is a fixed probability p=exp( -mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the p robability of not reaching the ta rget on s-th step is bounded by exp( -s\u00b7 mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp( -mL)), which is independent of the dataset size . If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C\u00b7 S, independent ly of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional dataset s. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbor s per element . ', 'Thus, t o avoid stucking into a l ocal minimum the greedy search algo rithm em ploy s a backtracking procedure on the zero layer. Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via t he minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling . Such empirical investigation of the Delaunay graph approximation resilience require s having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e datasets, mak ing such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a sequence of K-ANN-searches at different layers with a su bsequent use of heuristic (which has fixed complexity at fixed efConstruction ). The average number of lay ers for an element to be added in is a constant that depends on mL: \\uf05b\\uf05d\\uf05b \\uf05d 1 ln( (0,1)) 1 1LL E l E unif m m\\uf02b \\uf03d \\uf02d \\uf02b \\uf03d \\uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N\u2219log( N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16 , efConstruction =100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. ', 'The search is terminated when some stopping condition is met (e.g. the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques , such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approach es are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure\u2019s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges betwee n the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing . Const ruction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , however, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\\uf061. For \\uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \\uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg\u2019s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg\u2019s graphs suffers fr om polylogarit hmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg\u2019s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44] . The routing can be divided into two phases: \u201czoom-out\u201d and \u201czoom-in\u201d [32]. The greedy algorithm starts in the \u201czoom-out\u201d phase ', 'AUTHOR ET AL.: TITL E 3 from a low degree node and traverses the graph simult aneously increasing the node\u2019s degree until the characteri stic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the \u201czoom-in\u201d phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber o f distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44] ; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such stru cture t he search starts from the upper laye r which has only the long est links (the \u201czoom-in\u201d phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illu stration) . After that, the search switches to the lower layer (which h as shorter links ), restarts from the element which was the local minimum in the previous layer and the pr ocess repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of rout ing in a navigable small world network . One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only \u201cshort\u201d links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer . In case we m erge connections from all layers, the stru cture becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insert ion-the stochasticity is achieved by using level random ization , thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure ). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. ', 'When the search reaches the layer that is equal or less than l, the second phase of the construction algorithm is initiated . The second phase differs in two points: 1) the ef parameter is increased from 1 to efConstruction in order to control the recall of the greedy search procedure; 2) the found closest neighbors on each layer are also used as candidates for the connections of the inserted element. Two methods for the selection of M neighbors from the candidates were considered: simple connection to the closest elements (alg. 3) and the heuristic that accounts for the distances between the candidate elements to create connections in diverse direction s (alg. 4), described in the Section 3. The heuristic has two additional parameters: extendCandidates (set to false by default) which extends the candidate set and useful only for extremely clustered data, and keepPrunedConnections which allows getting fixed number of connection per element . The maximum number of connections that an element can have per layer is defined by the parameter Mmax for every layer higher than zero (a special parameter Mmax0 is used for the ground layer sep arately). If a node is already full at the moment of making of a new connection, then its extended connection list gets shrunk by the same algorithm that used for the neighbors selection (algs. 3 or 4). The insertion procedure terminates when the conne ctions of the inserted elements are established on the zero layer. The K-ANNS search algorithm used in Hierarchical NSW is presented in alg. 5. It is roughly equivalent to the insertion algorithm for an item with layer l=0. The diffe rence is that the closest neighbors found at the ground layer which are used as candidates for the connections are now returned as the search result. The quality of the search is controlled by the ef parameter (corresponding to efConstruction in the construction algorithm). 4.1 In fluence of the construction parameters Algorithm construction parameters mL and Mmax0 are r esponsible for maintaining the small world navigability in the constructed graphs. Setting mL to zero (this corr esponds to a single layer in the graph) and Mmax0 to M leads to production of directed k-NN graphs with a po wer-law search complexity well studied before [21, 29] (assuming using the alg. 3 for neighbor selection). Setting mL to zero and Mmax0 to infinity leads to production of NSW graphs with polylogarithmi c complexity [25, 26] . Finally, setting mL to some non-zero value leads to emergence of controllable hierarchy graphs w hich allow logarithmic search complexity by introduction of layers (see the Se ction 3). To achieve the optimum performance advantage of the controllable hierarchy, the overlap between neighbors on different layers (i.e. percent of element neighbors that are also belong to other layers) has to be small. In order to decrease the overlap we need to decrease the mL. Howe ver, at the same time, decreas ing mL leads to an increase of average hop number during a greedy search on each la yer, which negatively affects the performance. This leads to existence of the optimal value for the mL parameter . A simple choice for the optimal mL is 1/ln( M), this co rresponds to the skip list parameter p=1/M with an ave rage single element overlap between the layers. Simul ations done on an Intel Core i7 5930K CPU show that the proposed selection of mL is a reasonable choice (see Fig. ']\n" - ] - } - ], - "source": [ - "from llama_index.evaluation import RelevancyEvaluator\n", - "\n", - "# (Need to avoid peripheral asyncio issues)\n", - "import nest_asyncio\n", - "nest_asyncio.apply()\n", - "\n", - "# Define evaluator\n", - "evaluator = RelevancyEvaluator()\n", - "\n", - "# Issue query\n", - "llm_response = query_engine.query(\n", - " \"How does logarithmic complexity affect graph construction?\"\n", - ")\n", - "\n", - "# Grab context used in answer query & make it pretty\n", - "llm_response_source_nodes = [i.get_content() for i in llm_response.source_nodes]\n", - "\n", - "# # Take your previous question and pass in the response youwe got above\n", - "eval_result = evaluator.evaluate_response(query=\"How does logarithmic complexity affect graph construction?\", response=llm_response)\n", - "\n", - "print(f'\\nGiven the {len(llm_response_source_nodes)} chunks of content (below), is your LLM\\'s response relevant? {eval_result.passing}\\n \\\n", - " \\n ----Contexts----- \\n \\\n", - " \\n{llm_response_source_nodes}')\n" - ] - }, + "name": "stdout", + "output_type": "stream", + "text": [ + "\n", + "Given the 5 chunks of content (below), is your LLM's response relevant? True\n", + "\n", + " ----Contexts----- \n", + "\n", + "['AUTHOR ET AL.: TITLE 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index quality tradeoff is controlled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruction=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complexity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer. One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial positions of the data elements and , thus, when we tra verse the graph there is a fixed probability p=exp(-mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the probability of not reaching the ta rget on s-th step is bounded by exp(-s· mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp(-mL)), which is independent of the dataset size. If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C· S, independently of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional datasets. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbors per element. Thus, to avoid stucking into a l ocal minimum the greedy search algo rithm em ploys a backtracking procedure on the zero layer. ', 'AUTHOR ET AL.: TITLE 3 from a low degree node and traverses the graph simultaneously increasing the node’s degree until the characteristic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the “zoom-in” phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber of distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44]; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such structure the search starts from the upper layer which has only the longest links (the “zoom-in” phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illustration). After that, the search switches to the lower layer (which has shorter links), restarts from the element which was the local minimum in the previous layer and the process repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of routing in a navigable small world network. One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only “short” links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer. In case we merge connections from all layers, the structure becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insertion-the stochasticity is achieved by using level randomization, thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. The Hierarchical Layer=0Layer=1Layer=2Decreasing characteristic radius Cluster 2Inserted elemente2Cluster 1 Fig. ', 'the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques, such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approaches are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure’s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges between the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing. Construction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , h owever, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\\uf061. For \\uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \\uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg’s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg’s graphs suffers fr om polylogarithmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg’s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44]. The routing can be divided into two phases: “zoom-out” and “zoom-in” [32]. ', 'Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via the minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling. Such empirical investigation of the Delaunay graph approximation resilience requires having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e d atasets, making such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a s equence of K-ANN-searches at different layers with a subsequent use of heuristic (which has fixed complexity at fixed efConstruction). The average number of lay ers for an element to be added in is a constant that depends on mL: \\uf05b \\uf05d \\uf05b \\uf05d1 ln( (0,1)) 1 1 LLE l E unif m m\\uf02b \\uf03d \\uf02d \\uf02b \\uf03d \\uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N∙log(N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16, efConstruction=100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. Fig. ', '4 IEEE TRANSACTIONS ON JOURNAL NAME, MANUSCRIPT ID NSW approach thus can utilize the same methods for making the distributed approximate search/overlay structures [45]. For the selection of the proximity graph con nections during the element insertion we utilize a heuristic that takes into account the distances between the candidate elements to create diverse connections (a similar alg orithm was utilized in the spatial approximation tree [4] to select the tree children) instead of just selecting the closest neighbors. The heuristic examines the candidates starting from the nearest (with respect to the inserted element ) and creates a connection to a candidate only if it is closer to the base (inserted) element compared to any of the a lready connected candidates (see Section 4 for the details). When the number of candidates is large enough the heuristic allows getting the exact relative neighborhood graph [46] as a subgraph , a minimal subgraph of the D elaunay graph deducible by using only the distances b etween the nodes. The relative neighborhood graph allows easily keeping the global connected component, even in case of highly clustered data (see Fig. 2 for illustration). Note that the heuristic creates extra edges compared to the exact relative neighborhood graph s, allowing controlling the number of the connections which is important for search performance. For the case of 1D data the heuristic allows getting the exact Delaunay subgraph (which in this case coincides with the relative neighborhood graph) by using only information about the distances between the elements, thus making a direct transition from Hiera rchical NSW to the 1D probabilistic skip list algorithm. Base variant of the Hierarchical NSW proximity graphs was also used in ref. [18] (called ‘sparse neighborhood graphs’) for proximity graph searching. Similar heuristic was also a focus of the FANNG algorithm [47] (published shortly after the first versions of the current manuscript were posted online) with a slightly different interpretation, based on the sparse neighborhood graph’s property of the exact routing [18]. 4 ALGORITHM DESCRIPTION Network construction algorithm (alg. 1) is organized via consecutive insertions of the stored elements into the graph structure. For every inserted element an integer maximum layer l is randomly selected with an expone ntially decaying probability distribution (normalized by the mL parameter, see line 4 in alg. 1). The first phase of the insertion process starts from the top layer by greedily traversing the graph in order to find the ef closest neighbors to the inserted element q in the layer. After that, the algorithm continues the search from the next layer using the found closest neighbors from the previous layer as enter points, and the process repeats. Closest neighbors at each layer are found by a variant of the greedy search algorithm described in alg. 2, which is an updated version of the algorithm from [26]. To obtain the approximate ef nearest neighbors in some layer lс, a dynamic list W of ef closest found elements (initially filled with enter points) is kept during the search. The list is updated at each step by evaluating the neighborhood of the closest previously non-evaluated element in the list until the neighborhood of every element from the l ist is evaluated. Compared to limiting the number of distance calculations, Hierarchical NSW stop condition has a n advantage-it allows discarding candidates for evalution that are further from the query than the furthest element in the list, thus avoiding bloating of search structures. As in NSW, the list is emulated via two priority queues for better performance. ']\n" + ] + } + ], + "source": [ + "nest_asyncio.apply()\n", + "\n", + "# Define evaluator\n", + "evaluator = RelevancyEvaluator()\n", + "\n", + "# Issue query\n", + "llm_response = query_engine.query(\n", + " \"How does logarithmic complexity affect graph construction?\"\n", + ")\n", + "\n", + "# Grab context used in answer query & make it pretty\n", + "llm_response_source_nodes = [i.get_content() for i in llm_response.source_nodes]\n", + "\n", + "# Take your previous question and pass in the response you got above\n", + "eval_result = evaluator.evaluate_response(\n", + " query=\"How does logarithmic complexity affect graph construction?\",\n", + " response=llm_response,\n", + ")\n", + "\n", + "# Print response\n", + "print(\n", + " f\"\\nGiven the {len(llm_response_source_nodes)} chunks of content (below), is your \"\n", + " f\"LLM's response relevant? {eval_result.passing}\\n\"\n", + " f\"\\n ----Contexts----- \\n\"\n", + " f\"\\n{llm_response_source_nodes}\"\n", + ")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Next, send an out-of-scope query that the RAG app can't answer from what's in the index." + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:12.990368Z", + "iopub.status.busy": "2026-10-08T21:58:12.990188Z", + "iopub.status.idle": "2026-10-08T21:58:13.892621Z", + "shell.execute_reply": "2026-10-08T21:58:13.891759Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": 58, - "metadata": { - "colab": { - "base_uri": "https://localhost:8080/" - }, - "id": "vl5glxDNj2Gz", - "outputId": "46b3cc7d-236a-4c71-de71-e2abfe784aea" - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n", - "AUTHOR ET AL.: TITL E 7 be auto-configured by using sample data. The construction process can be easily and efficiently parallelized with only few synchronization points (as demonstrated in Fig. 9) and no measurable effect on index quality. Construction speed/index q uality tradeoff is co ntrolled via the efConstruction parameter. The tradeoff between the search time and the index construction time is presented in Fig. 10 for a 10M SIFT dataset and shows that a reasonable quality index can be constructed for efConstruct ion=100 on a 4X 2.4 GHz 10-core Xeon E5-4650 v2 CPU server in just 3 minutes. Further increase of the efConstruction leads to little extra performance but in exchange of significantly longer construction time. 4.2 Complexity analysis 4.2.1 Search complex ity The complexity scaling of a single search can be strictly analyzed under the assumption that we build exact D elaunay graphs instead of the approximate ones. Suppose we have found the closest element on some layer (this is guaranteed by having the Delaunay graph) and then d escended to the next layer . One can show that the average number of steps before we find the closest element in the layer is bounded by a constant. Indeed, the layers are not correlated with the spatial position s of the data elements and , thus , when we tra verse the graph there is a fixed probability p=exp( -mL) that the next node belongs to the upper layer. However, the search on the layer always terminates before it reaches the element which belongs to the higher layer (otherwise the search on the upper layer would have stopped on a different element), so the p robability of not reaching the ta rget on s-th step is bounded by exp( -s\u00b7 mL). Thus the expected number of steps in a layer is bounded by a sum of geometric progression S =1/(1-exp( -mL)), which is independent of the dataset size . If we assume that the average degree of a node in the Delaunay graph is capped by a constant C in the limit of the large dataset (this is the case for random Euclid d ata [48], but can be in principle violated in exotic spaces), then the overall average number of distance evaluations in a layer is bounded by a constant C\u00b7 S, independent ly of the dataset size. And since the expectation of the maximum layer index by the construction scales as O(log(N)), the overall co mplexity scaling is O(log(N)), in agreement with the simulations on low dimensional dataset s. The inital assumption of having the exact Delaunay graph violates in Hierarchical NSW due to usage of a pproximate edge selection heuristic with a fixed number of neighbor s per element . \n", - "\n", - "Thus, t o avoid stucking into a l ocal minimum the greedy search algo rithm em ploy s a backtracking procedure on the zero layer. Simulations show that at least for low dimensional data (Fig. 11, d=4) the dependence of the required ef parameter (which determines the complexity via t he minimal number of hops during the backtracking) to get a fixed recall saturates with the rise of the dataset size . The b acktracking co mplexity is an additive term in respect to the final comple xity, thus, as follows from the empirical data, inaccuracies of the Delauna y graph approximation do not alter the scaling . Such empirical investigation of the Delaunay graph approximation resilience require s having the average number of Delaunay graph edges independent of the d ataset to evidence how well the edges are approximated with a constant number of connections in Hierarchical NSW. However, the average degree of Delaunay graph scales exponentially with the dimensionality [39]), thus for high dimensional data (e.g. d=128) the aforeme ntioned condition requires having extremely larg e datasets, mak ing such empricial investigation unfeasible. Further analitical evidence is required to confirm whether the resilience of Delaunay graph aproximations genera lizes to higher dimensional spaces. 4.2.2 Construction complexity The construction is done by iterative insertions of all el ements, while the insertion of an element is merely a sequence of K-ANN-searches at different layers with a su bsequent use of heuristic (which has fixed complexity at fixed efConstruction ). The average number of lay ers for an element to be added in is a constant that depends on mL: \uf05b\uf05d\uf05b \uf05d 1 ln( (0,1)) 1 1LL E l E unif m m\uf02b \uf03d \uf02d \uf02b \uf03d \uf02b (1) Thus, the insertion complexiy scaling is the same as the one for the search, meaning that at least for relatively low dimensional datasets the construction time scales as O(N\u2219log( N)). 0 5 10 15 20 25 30 35 400102030405060708090Build time, minutesThread count10M SIFT, d=128, M=16 , efConstruction =100 4X Xeon E5-4650 v2 (4x10 cores) Core i7-6850K (6 cores+HT) 0 2 4 6 8 10 120.20.30.40.50.60.7Query time, msBuild time, minutes10M SIFT, d=128, M=16,Recall 0.9, 1-NN 100101102103104105106107108109024681012Average ef to reach target recallDataset sizeRandom vectors, d=4M=6, Mmax0=12, 1-NN Recall=0.9 Recall=0.99 Recall=0.999 Fig. 9. Construction time for Hierarchical NSW on 10M SIFT dataset for different numbers of threads on two CPUs. Fig. 10. Plots of t he query time vs constru ction time tradeoff for Hierarchical NSW on 10M SIFT dataset. \n", - "\n", - "The search is terminated when some stopping condition is met (e.g. the number of distance calculations). Links to the closest neighbors in a k-NN graph serve as a simple approximation of the Delaunay graph [25, 26] ( a graph which guranties that the result of a basic greedy graph traversal is always the nearest neighbor). Unfortunately, Delaunay graph cannot be efficiently constructed without prior information about the structure of a space [4], but its approximation by the nearest neighbors can be done by using only distances between the stored elements . It was shown that proximity graph approaches with such approximation perform competitive to other k-ANNS thechniques , such as kd-trees or LSH [18-26]. The main drawbacks of the k-NN graph approach es are: 1) the power law scaling of the number of steps with the dataset size during the routing process [28, 29] ; 2) a possible loss of global connectivity which leads to poor search results on clusetered data . To overcome these problems many hybrid approaches have been proposed that use auxiliary algorithms applicable only for vector data (such as kd-trees [18, 19] and product quantization [10]) to find better candidates for the enter nodes by doing a coarse search. In [25, 26, 30] authors proposed a proximity graph K-ANNS algorithm called Navigable Small World (NSW, also known as Metricized Small World, MSW), which utilized navigable graphs, i.e. graphs with logarithmic or polylogarithmic scaling of the number of hops during the greedy traversal with the respect of the network size [31, 32]. The NSW graph is constructed via consec utive insertion of elements in random order by bidirectio nally connecting them to the M closest neighbors from the previously inserted elements. The M closest neighbors are found using the structure\u2019s search procedure (a variant of a greedy search from multipl e random enter nodes ). Links to the closest neighbors of the elements inserted in the beginning of the construction later become bridges betwee n the network hubs that keep the overall graph connectivity and allow the logarithmic scaling of the number of hops during greedy routing . Const ruction phase of the NSW structure can be eff iciently parallelized without global synchronization and without mesuarable effect on accuracy [26] , being a good choice for distributed search systems. The NSW approach delivered the state-of-the-art performance on some d atasets [33, 34] , however, due to the overall polylogarit hmic complexity scaling , the algorithm was still prone to severe performance degradation on low dimensional d atasets (on which NSW could lose to tree-based algorithms by several orders of magnitude [34]). 2.2 Navigable small world models Networks with logarithmic or polylogarithmic scaling of the greedy graph r outing are known as the navigable small world networks [31, 32] . Such networks are an i mportant topic of complex network theory aiming at u nderstanding of underlying mechanisms of real-life ne tworks formation in order to apply them for appli cations of scalable routing [32, 35, 36] and distributed similarity search [25, 26, 30, 37-40]. The first works to consider spatial models of navigable networks were done by J. Kleinberg [31, 41] as social ne twork models for the famous Milgram experiment [42]. Kleinberg studied a variant of random Watts-Strogatz networks [43], using a regular lattice graph in d-dimensional vector space together with augmentation of long-range links following a specific long link length di stribution r-\uf061. For \uf061=d the number of hops to get to the target by greedy routing scales polylogarithmically (i nstead of a power law for any other value of \uf061). This idea has inspired development of many K-NNS and K-ANNS algorithms based on the navigation effect [37-40]. But even though the Kleinberg\u2019s navigability criterion in principle can be extended for more general spaces, in o rder to build such a navigable network one has to know the data distribution beforehand . In addition, g reedy routing in Kleinberg\u2019s graphs suffers fr om polylogarit hmic complexity scalability at best. Another well-known class of navigable networks are the scale-free models [32, 35, 36] , which can reproduce several features of real-life networks and advertised for routing applications [35]. However, n etworks produced by such models have even worse power law complexity scaling of the greedy search [44] and, just like the Klei nberg\u2019s model, scale-free models require global knowledge of the data distribution, making them unusable for search applications. The above-described NSW algorithm uses a simpler, previously unknown model of navigable networks, a llowing decentralized graph construction and suitable for data in arbitrary spaces. It was suggested [44] that the NSW network formation mechanism may be respon sible for navigability of large-scale biological neural networks (presence of which is disputable): similar models were able to describe growth of small brain networks, while the model predicts several high-level features observed in large scale neural net works. However, the NSW model also suffers from the polylogarithmic search complexity of the routing process. 3 MOTIVATION The ways of improving the NSW search complexity can be identified through the analysis of the routing process, which was studied in detail in [32, 44] . The routing can be divided into two phases: \u201czoom-out\u201d and \u201czoom-in\u201d [32]. The greedy algorithm starts in the \u201czoom-out\u201d phase \n", - "\n", - "AUTHOR ET AL.: TITL E 3 from a low degree node and traverses the graph simult aneously increasing the node\u2019s degree until the characteri stic radius of the node links length reaches the scale of the distance to the query. Before the latter happens, the ave rage degree of a node can stay relatively small, which leads to an increased probabilit y of being stuck in a di stant false local minimum. One can avoid the described problem in NSW by star ting the search from a node with the maximum degree (good candidates are the first nodes inserted in the NSW structure [44]), directly going to the \u201czoom-in\u201d phase of the search. Tests show that setting hubs as starting points substantially increases probability of successful routing in the structure and provides significantly better perfo rmance at low dimensional data. However, it still has only a polyloga rithmic complexity scalability of a single greedy search at best , and performs worse on high d imensional data compared to Hierarchical NSW. The reason for the polylogarithmic complexity scaling of a single greedy search in NSW is that the overall nu mber o f distance computations is roughly proportional to a product of the average number of greedy algorithm hops by the average degree of the nodes on the greedy path. The average number of hops scales logarithmically [26, 44], while the average degree of the nodes on the greedy path also scales logarithmically due to the facts that: 1) the greedy search tends to go through the same hubs as the network grows [32, 44] ; 2) the average number of hub connections grow s logarithmically with an increase of the network size. Thus we get an overall po lylogarithmic dependence of the resulting complexity. The idea of Hierarchical NSW algorithm is to separate the links according to their length scale into different layers and then search in a multilayer graph. In this case we can evaluate only a needed fixed portion of the conne ctions for each element independently of the networks size, thus allowing a logarithmic scalability. In such stru cture t he search starts from the upper laye r which has only the long est links (the \u201czoom-in\u201d phase) . The algorithm greedily traverses through the elements from the upper layer until a local minimum is reached (see Fig. 1 for illu stration) . After that, the search switches to the lower layer (which h as shorter links ), restarts from the element which was the local minimum in the previous layer and the pr ocess repeats . The maximum number of connections per element in all layers can be made constant , thus allowing a logarithmic complexity scaling of rout ing in a navigable small world network . One way to form such a layered structure is to explici tly set links with different length scales by introducing layers. For every element we select an integer level l which defines the maximum layer for which the element belongs to. For all elements in a layer a proximity graph (i.e. graph containing only \u201cshort\u201d links that approximate Delaunay graph) is built incrementally. If we set an exp onentially decaying probabil ity of l (i.e. following a ge ometric distribution) we get a logarithmic scaling of the expected number of layers in the structure. The search procedure is an iterative greedy search starting from the top layer and finishing at the zero layer . In case we m erge connections from all layers, the stru cture becomes similar to the NSW graph (in this case the l can be put in correspondence to the node degree in NSW). In contrast to NSW, Hierarchical NSW constru ction algorithm does not require the elements to be shuffled before the insert ion-the stochasticity is achieved by using level random ization , thus allowing truly increme ntal indexing even in case of temporaril y alterating data distribution ( though changing the order of the inser tion slightly alters the performace due to only partially determenistic construction procedure ). The Hierarchical NSW idea is also very similar to a well-known 1D probabilistic skip list structure [27] and can be described using its terms. The major difference to skip list is that we generalize the structure by replacing the linked list with proximity graphs. \n", - "\n", - "When the search reaches the layer that is equal or less than l, the second phase of the construction algorithm is initiated . The second phase differs in two points: 1) the ef parameter is increased from 1 to efConstruction in order to control the recall of the greedy search procedure; 2) the found closest neighbors on each layer are also used as candidates for the connections of the inserted element. Two methods for the selection of M neighbors from the candidates were considered: simple connection to the closest elements (alg. 3) and the heuristic that accounts for the distances between the candidate elements to create connections in diverse direction s (alg. 4), described in the Section 3. The heuristic has two additional parameters: extendCandidates (set to false by default) which extends the candidate set and useful only for extremely clustered data, and keepPrunedConnections which allows getting fixed number of connection per element . The maximum number of connections that an element can have per layer is defined by the parameter Mmax for every layer higher than zero (a special parameter Mmax0 is used for the ground layer sep arately). If a node is already full at the moment of making of a new connection, then its extended connection list gets shrunk by the same algorithm that used for the neighbors selection (algs. 3 or 4). The insertion procedure terminates when the conne ctions of the inserted elements are established on the zero layer. The K-ANNS search algorithm used in Hierarchical NSW is presented in alg. 5. It is roughly equivalent to the insertion algorithm for an item with layer l=0. The diffe rence is that the closest neighbors found at the ground layer which are used as candidates for the connections are now returned as the search result. The quality of the search is controlled by the ef parameter (corresponding to efConstruction in the construction algorithm). 4.1 In fluence of the construction parameters Algorithm construction parameters mL and Mmax0 are r esponsible for maintaining the small world navigability in the constructed graphs. Setting mL to zero (this corr esponds to a single layer in the graph) and Mmax0 to M leads to production of directed k-NN graphs with a po wer-law search complexity well studied before [21, 29] (assuming using the alg. 3 for neighbor selection). Setting mL to zero and Mmax0 to infinity leads to production of NSW graphs with polylogarithmi c complexity [25, 26] . Finally, setting mL to some non-zero value leads to emergence of controllable hierarchy graphs w hich allow logarithmic search complexity by introduction of layers (see the Se ction 3). To achieve the optimum performance advantage of the controllable hierarchy, the overlap between neighbors on different layers (i.e. percent of element neighbors that are also belong to other layers) has to be small. In order to decrease the overlap we need to decrease the mL. Howe ver, at the same time, decreas ing mL leads to an increase of average hop number during a greedy search on each la yer, which negatively affects the performance. This leads to existence of the optimal value for the mL parameter . A simple choice for the optimal mL is 1/ln( M), this co rresponds to the skip list parameter p=1/M with an ave rage single element overlap between the layers. Simul ations done on an Intel Core i7 5930K CPU show that the proposed selection of mL is a reasonable choice (see Fig. \n" - ] - } - ], - "source": [ - "# For good measure, let's print out the context so that we can see it more clearly:\n", - "\n", - "for l in llm_response_source_nodes:\n", - " print(f'\\n{l}')" - ] - }, + "name": "stdout", + "output_type": "stream", + "text": [ + "To get to the other side.\n" + ] + } + ], + "source": [ + "query = \"Why did the chicken cross the road?\"\n", + "response = query_engine.query(query)\n", + "\n", + "print(response.response)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Then evaluate the response. The evaluator returns `False`, because the answer doesn't come from the retrieved context." + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:13.894566Z", + "iopub.status.busy": "2026-10-08T21:58:13.894409Z", + "iopub.status.idle": "2026-10-08T21:58:15.020678Z", + "shell.execute_reply": "2026-10-08T21:58:15.019351Z" + } + }, + "outputs": [ { - "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "82lyvQEXj78t" - }, - "outputs": [], - "source": [] + "name": "stdout", + "output_type": "stream", + "text": [ + "False\n" + ] } - ], - "metadata": { - "colab": { - "provenance": [] - }, - "kernelspec": { - "display_name": "Python 3 (ipykernel)", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.11.5" + ], + "source": [ + "eval_result = evaluator.evaluate_response(query=query, response=response)\n", + "\n", + "print(str(eval_result.passing))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Clean up\n", + "\n", + "When you no longer need the index, delete it." + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "execution": { + "iopub.execute_input": "2026-10-08T21:58:15.022767Z", + "iopub.status.busy": "2026-10-08T21:58:15.022606Z", + "iopub.status.idle": "2026-10-08T21:58:20.295156Z", + "shell.execute_reply": "2026-10-08T21:58:20.293861Z" } + }, + "outputs": [], + "source": [ + "pc.delete_index(name=index_name)" + ] + } + ], + "metadata": { + "colab": { + "provenance": [] + }, + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" }, - "nbformat": 4, - "nbformat_minor": 4 -} \ No newline at end of file + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.13" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +}