
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.
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
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