Faster Deterministic and Las Vegas Algorithms for Offline Approximate Nearest Neighbors in High Dimensions

We present a deterministic, truly subquadratic algorithm for offline (1 + epsilon)-approximate nearest or farthest neighbor search (in particular, the closest pair or diameter problem) in Hamming space in any dimension d <= n(delta), for a sufficiently small constant delta > 0. The running time of the algorithm is roughly n(2-epsilon 1/2+O(delta)) for nearest neighbors, or n(2-Omega(root epsilon / log(i/epsilon))) for farthest. The algorithm follows from a simple combination of expander walks, Chebyshev polynomials, and rectangular matrix multiplication. We also show how to eliminate errors in the previous Monte Carlo randomized algorithm of Alman, Chan, and Williams [FOCS'16] for offline approximate nearest or farthest neighbors, and obtain a Las Vegas randomized algorithm with expected running time n(2-Omega(root epsilon / log(i/epsilon))) . Finally, we note a simplification of Alman, Chan, and Williams' method and obtain a slightly improved Monte Carlo randomized algorithm with running time n(2-Omega(epsilon 1/3/ log2/3(1/epsilon))). As one application, we obtain improved deterministic and randomized (1+)-approximation algorithms for MAX-SAT.