Graph-Based Time-Space Trade-Offs for Approximate Near Neighbors

Thijs Laarhoven, Marc Herbstritt
2018 International Symposium on Computational Geometry  
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
more » ... o(1) , for arbitrary ρ q , ρ s ≥ 0 satisfying (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.
doi:10.4230/lipics.socg.2018.57 dblp:conf/compgeom/Laarhoven18 fatcat:ybdrcak45vbirce44fbnwu6hyq