Empty pseudo-triangles in point sets

  • Authors:

  • Hee-Kap Ahn;Sang Won Bae;Marc van Kreveld;Iris Reinbacher;Bettina Speckmann

  • Affiliations:

  • Department of Computer Science and Engineering, POSTECH, South Korea;Department of Computer Science, Kyonggi University, South Korea;Department of Information and Computing Sciences, Utrecht University, The Netherlands;Department of Computer Science, TU Braunschweig, Germany;Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2011

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Abstract

We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between @Q(n^2) and @Q(n^3) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between @Q(n^3) and @Q(n^6). If we count only star-shaped pseudo-triangles, the bounds are @Q(n^2) and @Q(n^5). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n^3) time. In the general case, the running times are O(n^6) for the maximization problems and O(nlogn) for the minimization problems.