Sparse geometric graphs with small dilation

Abstract

Given a set S of n points in R(D), and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n/(k + 1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n - 1 + k edges has dilation Omega (n/(k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position. (c) 2007 Elsevier B.V. All rights reserved