The time complexity of popular Approximate Nearest Neighbor (ANN) search algorithms varies depending on the method, but most aim for sub-linear query time relative to dataset size. For example, k-d trees and RKD forests have construction times of (O(n \log n)) and search times of (O(\log n)) in low dimensions, but their performance degrades to nearly (O(n)) in high-dimensional spaces due to the "curse of dimensionality." Hierarchical Navigable Small World (HNSW) graphs achieve (O(\log n)) search time by creating layered proximity graphs, while Locality-Sensitive Hashing (LSH) reduces search complexity to roughly (O(1)) per hash table (with multiple tables required for accuracy). Inverted File Index (IVF) methods partition data into clusters, leading to search times like (O(\sqrt{n})) when optimally configured. These complexities assume idealized conditions, but they provide a baseline for comparing scalability.
In practice, the theoretical time complexity directly influences how search speed degrades as datasets grow. Algorithms with logarithmic scaling, like HNSW, maintain relatively stable query times even as data increases exponentially. For instance, a dataset growing from 1M to 10M points might see search times increase by a factor of 2–3 (due to (O(\log n))), whereas linear scans would become 10x slower. LSH’s near-constant time per hash table allows predictable latency, but the need for multiple tables (e.g., 50–100) to maintain accuracy increases memory and computational overhead. IVF-based methods, while efficient for moderate-sized datasets, may require retuning the number of clusters as data grows to avoid performance drops. Real-world benchmarks often show HNSW outperforming others at scale due to its balance of speed and recall, even with high-dimensional data.
However, practical performance depends on factors beyond theoretical complexity. Memory access patterns, cache efficiency, and parallelization significantly impact speed. For example, graph-based methods like HNSW benefit from localized memory accesses during graph traversal, while LSH’s random hash table lookups can cause cache misses. Dimensionality also plays a role: algorithms like k-d trees degrade in high dimensions, while HNSW and IVF-PQ (Product Quantization) mitigate this through graph hierarchies or compressed representations. Implementation optimizations (e.g., SIMD instructions in FAISS) further affect real-world speed. Thus, while complexity provides a guideline, the choice of ANN algorithm often hinges on dataset properties (size, dimensionality), hardware constraints, and acceptable trade-offs between accuracy and latency.