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    Decorative pattern
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    3. ANN Algorithm Complexity Analysis

    ANN Algorithm Complexity Analysis

    Computational complexity comparison of approximate nearest neighbor algorithms including build time, query time, and space complexity. Essential for understanding performance characteristics and choosing appropriate algorithms for different scales.

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    About this tool

    Overview

    Understanding the computational complexity of ANN algorithms helps choose the right method for your scale and performance requirements.

    Time Complexity

    HNSW

    • Build: O(N log N × M)
    • Query: O(log N) on average
    • Space: O(N × M) where M is max connections

    IVF

    • Build: O(N × k × d × iterations) for clustering
    • Query: O(nprobe × N/nlist × d)
    • Space: O(N × d)

    LSH

    • Build: O(N × L × k) where L is tables, k is functions
    • Query: O(NL/b) where b is buckets
    • Space: O(N × L)

    Product Quantization

    • Build: O(N × m × k × d/m) where m is subspaces
    • Query: O(N) with faster distance computation
    • Space: O(N × m × log k)

    Space-Time Trade-offs

    HNSW: High memory, fast queries IVF-PQ: Low memory, moderate speed LSH: Moderate memory, sublinear queries

    Scalability

    • Million-scale: Most algorithms work well
    • Billion-scale: HNSW, IVF-PQ, DiskANN
    • Trillion-scale: Specialized distributed systems

    Choosing by Scale

    < 1M vectors: HNSW (best quality) 1M - 100M: IVF-PQ or HNSW with quantization 100M - 1B: IVF-PQ, DiskANN > 1B: Distributed IVF-PQ, specialized systems

    Pricing

    Not applicable (algorithmic analysis).

    Surveys

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    Information

    Websitearxiv.org
    PublishedMar 15, 2026

    Categories

    1 Item
    Concepts & Definitions

    Tags

    3 Items
    #Algorithm#Performance#Complexity

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