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Sparse geometric graphs with small dilation

Boris Aronov·Mark de Berg·Otfried Cheong·Joachim Gudmundsson·Herman Haverkort·Michiel Smid·Antoine Vigneron
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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.

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Boris Aronov, Mark de Berg, Otfried Cheong, Joachim Gudmundsson, Herman Haverkort, Michiel Smid, Antoine Vigneron. 2007-06-26. Sparse geometric graphs with small dilation. https://arxiv.org/abs/cs/0702080

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