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New results for the minimum weight triangulation problem

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Abstract

Given a finite set of points in a plane, a triangulation is a maximal set of nonintersecting line segments connecting the points. The weight of a triangulation is the sum of the Euclidean lengths of its line segments. No polynomial-time algorithm is known to find a triangulation of minimum weight, nor is the minimum weight triangulation problem known to be NP-hard. This paper proposes a new heuristic algorithm that triangulates a set ofn points inO(n 3) time and that never produces a triangulation whose weight is greater than that of a greedy triangulation. The algorithm produces an optimal triangulation if the points are the vertices of a convex polygon. Experimental results indicate that this algorithm rarely produces a nonoptimal triangulation and performs much better than a seemingly similar heuristic of Lingas. In the direction of showing the minimum weight triangulation problem is NP-hard, two generalizations that are quite close to the minimum weight triangulation problem are shown to be NP-hard.

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Authors and Affiliations

  1. Department of Computer Science, Virginia Polytechnic Institute and State University, 24061, Blacksburg, IA, USA

    L. S. Heath

  2. Department of Computer Science, University of Iowa, 52242, Iowa City, IA, USA

    S. V. Pemmaraju

Authors
  1. L. S. Heath
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  2. S. V. Pemmaraju
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Additional information

Communicated by D. T. Lee.

This research was done while the second author was with the Department of Computer Science, Virginia Polytechnic Institute and State University.

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Heath, L.S., Pemmaraju, S.V. New results for the minimum weight triangulation problem. Algorithmica 12, 533–552 (1994). https://doi.org/10.1007/BF01188718

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  • DOI: https://doi.org/10.1007/BF01188718

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