Minimum weight triangulation by cutting out triangles

Authors:
Magdalene Grantson;Christian Borgelt;Christos Levcopoulos
Affiliations:
Department of Computer Science, Lund University, Lund, Sweden;Department of Knowledge Processing and Language Engineering, University of Magdeburg, Magdeburg, Germany;Department of Computer Science, Lund University, Lund, Sweden
Venue:
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Year:
2005

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Abstract

We describe a fixed parameter algorithm for computing the minimum weight triangulation (MWT) of a simple polygon with (n–k) vertices on the perimeter and k hole vertices in the interior, that is, for a total of n vertices. Our algorithm is based on cutting out empty triangles (that is, triangles not containing any holes) from the polygon and processing the parts or the rest of the polygon recursively. We show that with our algorithm a minimum weight triangulation can be found in time at most O(n3k ! k), and thus in O(n3) if k is constant. We also note that k! can actually be replaced by bk for some constant b. We implemented our algorithm in Java and report experiments backing our analysis.