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Counting Straight-Edge Tringulation of Planar Point Sets


Ray,  Saurabh
Algorithms and Complexity, MPI for Informatics, Max Planck Society;
International Max Planck Research School, MPI for Informatics, Max Planck Society;

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Ray, S. (2005). Counting Straight-Edge Tringulation of Planar Point Sets. Master Thesis, Universität des Saarlandes, Saarbrücken.

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A triangulation of a finite set S of points in R2 is a maximal set of line segments with disjoint interiors whose end points are in S. A set of points in the plane can have many triangulations and it is known that a set of n points always has more than (2.33n)[7] and fewer than 59n-(log(n)) [4] triangulation. However, these bounds are not tight. Also, counting the number of triangulation of a given a set of points efficiently remains an open problem. The fastest method so far is based on the so called t-path method [5] and it was the first algorithm having a running time sublinear on the number of triangulations counted. In this thesis, we consider a slightly different approach to counting the number of triangulations. Although we are unable to prove any non-trivial result about our algorithm yet, empirical results show that the running time of our algorithm for a set of n points is o(nlog2nT(n)) where T(n) is the number of triangulations counted, and in practice it performs much better than the earlier algorithm.