Graph-based time-space trade-offs for approximate near neighbors

We take a first step towards a rigorous asymptotic analysis of graph-based methods for finding (approximate) nearest neighbors in high-dimensional spaces, by analyzing the complexity of randomized greedy walks on the approximate nearest 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 ^ρq+o(1) and space n ^1+ρs+o(1), for arbitrary ρ _q,ρ _s ≥ 0 satisfying (2c ² - 1)ρq + 2c ²(c ² - 1)√ρ _s(1 - ρ _s) ≥ c ⁴. (1) 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 optimal hash-based trade-offs of Andoni-Laarhoven-Razenshteyn-Waingarten [5]. 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.