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Conference Paper

Linear-Time Triangulation of a Simple Polygon Made Easier Via Randomization


Ramos,  Edgar A.
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Amato, N. M., Goodrich, M. T., & Ramos, E. A. (2000). Linear-Time Triangulation of a Simple Polygon Made Easier Via Randomization. In Proceedings of the 16th Annual Symposium on Computational Geometry (SCG-00) (pp. 201-212). New York, USA: ACM Press.

Cite as:
We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of non-optimal randomized algorithms due to Clarkson {\it et al.} (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottom-up preprocessing phase previous to the top-down construction phase.