Minimum average distance triangulations

  • Authors:

  • László Kozma

  • Affiliations:

  • Universität des Saarlandes, Saarbrücken, Germany

  • Venue:
  • ESA'12 Proceedings of the 20th Annual European conference on Algorithms
  • Year:
  • 2012

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Abstract

We study the problem of finding a triangulation T of a planar point set S such as to minimize the expected distance between two points x and y chosen uniformly at random from S. By distance we mean the length of the shortest path between x and y along edges of T, with edge weights given as part of the problem. In a different variant of the problem, the points are vertices of a simple polygon and we look for a triangulation of the interior of the polygon that is optimal in the same sense. We prove that a general formulation of the problem in which the weights are arbitrary positive numbers is strongly NP-complete. For the case when all weights are equal we give polynomial-time algorithms. In the end we mention several open problems.