Sparse geometric graphs with small dilation

Given a set S of n points in the plane, 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, and dilation O(n / (k + 1)) can be computed in time O(n log n). We also construct n–point sets for which any geometric graph with n – 1 + k edges has dilation O(n / (k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position. This work was supported by LG Electronics and NUS research grant R-252-000-166-112. Research by B.A. has been supported in part by NSF ITR Grant CCR-00-81964 and by a grant from US-Israel Binational Science Foundation. Part of the work was carried out while B.A. was visiting TU/e in February 2004 and in the summer of 2005. MdB was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301