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Optimization for First Order Delaunay Triangulations

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Algorithms and Data Structures (WADS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4619))

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Abstract

This paper discusses optimization of quality measures over first order Delaunay triangulations. Unlike most previous work, our measures relate to edge-adjacent or vertex-adjacent triangles instead of only to single triangles. We give efficient algorithms to optimize certain measures, whereas other measures are shown to be NP-hard. For two of the NP-hard maximization problems we provide for any constant ε> 0, factor (1 − ε) approximation algorithms that run in 2O(1/ε)·n and \(2^{O(1/\varepsilon^2)}\cdot n\) time (when the Delaunay triangulation is given). For a third NP-hard problem the NP-hardness proof provides an inapproximability result. Our results are presented for the class of first-order Delaunay triangulations, but also apply to triangulations where every triangle has at most one flippable edge. One of the approximation results is also extended to k-th order Delaunay triangulations.

This research has been partially funded by the Netherlands Organisation for Scientific Research (NWO) under FOCUS/BRICKS grant number 642.065.503 (GADGET) and under the project GOGO.

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Authors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, 3508TB Utrecht, The Netherlands

    Marc van Kreveld, Maarten Löffler & Rodrigo I. Silveira

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  1. Marc van Kreveld
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  2. Maarten Löffler
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  3. Rodrigo I. Silveira
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Frank Dehne Jörg-Rüdiger Sack Norbert Zeh

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van Kreveld, M., Löffler, M., Silveira, R.I. (2007). Optimization for First Order Delaunay Triangulations. In: Dehne, F., Sack, JR., Zeh, N. (eds) Algorithms and Data Structures. WADS 2007. Lecture Notes in Computer Science, vol 4619. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73951-7_16

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  • DOI: https://doi.org/10.1007/978-3-540-73951-7_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73948-7

  • Online ISBN: 978-3-540-73951-7

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