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.
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.
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Aronov, B., de Berg, M., Cheong, O., Gudmundsson, J., Haverkort, H., Vigneron, A. (2005). Sparse Geometric Graphs with Small Dilation. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_7
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DOI: https://doi.org/10.1007/11602613_7
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