Approximating All-Points Furthest Pairs and Maximum Spanning Trees in Metric Spaces

Abstract

Consider the problem of finding a point furthest from [Formula: see text] for each point [Formula: see text] in a metric space [Formula: see text], where [Formula: see text]. We prove this problem to have a deterministic [Formula: see text]-time [Formula: see text]-approximation algorithm. As a corollary, the maximum spanning tree problem in metric spaces has a deterministic [Formula: see text]-time [Formula: see text]-approximation algorithm. We also give a Monte Carlo [Formula: see text]-time algorithm outputting, for each [Formula: see text], a point [Formula: see text] satisfying [Formula: see text], where [Formula: see text]. As a corollary, we have a Monte Carlo [Formula: see text]-time algorithm for finding a spanning tree of weight at least [Formula: see text] in [Formula: see text].