ANN Algorithm Complexity Analysis

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).