Sparse Geometric Graphs with Small Dilation

Given a set S of n points in RD , 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 Ω(n/(k + 1)); a slightly weaker statement holds if the points of S are required to be in convex position.

• Type: Conference
• Year: 2005
• Where: ISAAC
• Authors: Boris Aronov, Mark de Berg, Otfried Cheong, Joachim Gudmundsson, Herman J. Haverkort, Antoine Vigneron
• Keywords: Geometric Graph, ISAAC 2005, Maximum Degree, Planar N-point Sets