Graph-based time-space trade-offs for approximate near neighbors
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We take a first step towards a rigorous asymptotic analysis of graph-based approaches for finding (approximate) nearest neighbors in high-dimensional spaces, by analyzing the complexity of (randomized) greedy walks on the approximate near neighbor graph. For random data sets of size n = 2^o(d) on the d-dimensional Euclidean unit sphere, using near neighbor graphs we can provably solve the approximate nearest neighbor problem with approximation factor c > 1 in query time n^ρ_q + o(1) and space n^1 + ρ_s + o(1), for arbitrary ρ_q, ρ_s ≥ 0 satisfying (2c^2 - 1) ρ_q + 2 c^2 (c^2 - 1) √(ρ_s (1 - ρ_s))≥ c^4. Graph-based near neighbor searching is especially competitive with hash-based methods for small c and near-linear memory, and in this regime the asymptotic scaling of a greedy graph-based search matches the recent optimal hash-based trade-offs of Andoni-Laarhoven-Razenshteyn-Waingarten [SODA'17]. We further study how the trade-offs scale when the data set is of size n = 2^Θ(d), and analyze asymptotic complexities when applying these results to lattice sieving.
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