Efficiently Constructing Sparse Navigable Graphs

Graph-based nearest neighbor search methods have seen a surge of popularity in recent years, offering state-of-the-art performance across a wide variety of applications. Central to these methods is the task of constructing a sparse navigable search graph for a given dataset endowed with a distance function. Unfortunately, doing so is computationally expensive, so heuristics are universally used in practice. In this work, we initiate the study of fast algorithms with provable guarantees for search graph construction. For a dataset with $n$ data points, the problem of constructing an optimally sparse navigable graph can be framed as $n$ separate but highly correlated minimum set cover instances. This yields a naive $O(n^3)$ time greedy algorithm that returns a navigable graph whose sparsity is at most $O(\log n)$ higher than optimal. We improve significantly on this baseline, taking advantage of correlation between the set cover instances to leverage techniques from streaming and sublinear-time set cover algorithms. Combined with problem-specific pre-processing techniques, we present an $\tilde{O}(n^2)$ time algorithm for constructing an $O(\log n)$-approximate sparsest navigable graph under any distance function. The runtime of our method is optimal up to logarithmic factors under the Strong Exponential Time Hypothesis via a reduction from Monochromatic Closest Pair. Moreover, we prove that, as with general set cover, obtaining better than an $O(\log n)$-approximation is NP-hard, despite the significant additional structure present in the navigable graph problem. Finally, we show that our techniques can also beat cubic time for the closely related and practically important problems of constructing $\alpha$-shortcut reachable and $\tau$-monotonic graphs, which are also used for nearest neighbor search. For such graphs, we obtain $\tilde{O}(n^{2.5})$ time or better algorithms.

Results of the Big ANN: NeurIPS'23 competition

The 2023 Big ANN Challenge, held at NeurIPS 2023, focused on advancing the state-of-the-art in indexing data structures and search algorithms for practical variants of Approximate Nearest Neighbor (ANN) search that reflect the growing complexity and diversity of workloads. Unlike prior challenges that emphasized scaling up classical ANN search ~\cite{DBLP:conf/nips/SimhadriWADBBCH21}, this competition addressed filtered search, out-of-distribution data, sparse and streaming variants of ANNS. Participants developed and submitted innovative solutions that were evaluated on new standard datasets with constrained computational resources. The results showcased significant improvements in search accuracy and efficiency over industry-standard baselines, with notable contributions from both academic and industrial teams. This paper summarizes the competition tracks, datasets, evaluation metrics, and the innovative approaches of the top-performing submissions, providing insights into the current advancements and future directions in the field of approximate nearest neighbor search.