Sublinear Time Approximation of the Cost of a Metric [math]-Nearest Neighbor Graph

Abstract

SIAM Journal on Computing, Volume 53, Issue 2, Page 524-571, April 2024.
Abstract. Let [math] be an [math]-point metric space. We assume that [math] is given in the distance oracle model, that is, [math] and for every pair of points [math] from [math] we can query their distance [math] in constant time. A [math]-nearest neighbor ([math]-NN) graph for [math] is a directed graph [math] that has an edge to each of [math]’s [math] nearest neighbors. We use [math] to denote the sum of edge weights of [math]. In this paper, we study the problem of approximating [math] in sublinear time when we are given oracle access to the metric space [math] that defines [math]. Our goal is to develop an algorithm that solves this problem faster than the time required to compute [math]. We first present an algorithm that in [math] time with probability at least [math] approximates [math] to within a factor of [math]. Next, we present a more elaborate sublinear algorithm that in time [math] computes an estimate [math] of [math] that satisfies with probability at least [math] [math], where [math] denotes the cost of the minimum spanning tree of [math]. Further, we complement these results with near matching lower bounds. We show that any algorithm that for a given metric space [math] of size [math], with probability at least [math], estimates [math] to within a [math] factor requires [math] time. Similarly, any algorithm that with probability at least [math] estimates [math] to within an additive error term [math] requires [math] time.