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Graph-based approaches to nearest neighbor search are popular and powerfultools for handling large datasets in practice, but they have limitedtheoretical guarantees. We study the worst-case performance of recentgraph-based approximate nearest neighbor search algorithms, such as HNSW, NSGand DiskANN. For DiskANN, we show that its "slow preprocessing" versionprovably supports approximate nearest neighbor search query with constantapproximation ratio and poly-logarithmic query time, on data sets with bounded"intrinsic" dimension. For the other data structure variants studied, includingDiskANN with "fast preprocessing", HNSW and NSG, we present a family ofinstances on which the empirical query time required to achieve a "reasonable"accuracy is linear in instance size. For example, for DiskANN, we show that thequery procedure can take at least $0.1 n$ steps on instances of size $n$ beforeit encounters any of the $5$ nearest neighbors of the query.

Worst-case Performance of Popular Approximate Nearest Neighbor Search Implementations: Guarantees and Limitations