The weak gap property in metric spaces of bounded doubling dimension

Abstract

We introduce the weak gap property for directed graphs whose vertex set S is a metric space of size n. We prove that, if the doubling dimension of S is a constant, any directed graph satisfying the weak gap property has O(n) edges and total weight, where denotes the weight of a minimum spanning tree of S. We show that 2-optimal TSP tours and greedy spanners satisfy the weak gap property. © 2009 Springer Berlin Heidelberg.