Sparse geometric graphs with small dilation

Authors:
Boris Aronov;Mark de Berg;Otfried Cheong;Joachim Gudmundsson;Herman Haverkort;Antoine Vigneron
Affiliations:
Department of Computer and Information Science, Polytechnic University, Brooklyn, New York;Department of Mathematics and Computing Science, TU Eindhoven, Eindhoven, The Netherlands;Division of Computer Science, KAIST, Daejeon, South Korea;IMAGEN Program, National ICT Australia Ltd, Australia;Department of Mathematics and Computing Science, TU Eindhoven, Eindhoven, The Netherlands;Department of Computer Science, National University of Singapore, Singapore
Venue:
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Year:
2005

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Abstract

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