Abstract
The greedy triangulation of a finite planar point set is obtained by repeatedly inserting a shortest diagonal that doesn't intersect those already in the plane. We show that the problem of constructing the greedy triangulation of a finite set of points with integer coordinates in the plane is P-complete. This is the first known geometric P-complete problem where the input is given as a set of points. On the other hand, we provide general NC-methods for testing whether a given triangulation of a set of points and/or line segments can be built by inserting the diagonals in a given partial order, and for constructing such triangulations for simple polygons. As corollaries, we obtain NC-algorithms for testing whether a triangulation is respectively the greedy triangulation or the so called sweep-line triangulation, and for constructing respectively the greedy triangulation or the sweep-line triangulation of a simple polygon. The latter result solves the open problem posed by Atallah et al.
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Levcopoulos, C., Lingas, A., Wang, C. (1995). On parallel complexity of planar triangulations. In: Thiagarajan, P.S. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1995. Lecture Notes in Computer Science, vol 1026. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60692-0_64
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DOI: https://doi.org/10.1007/3-540-60692-0_64
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