On approximation behavior and implementation of the greedy triangulation for convex planar point sets

• Authors: A Lingas
• Venue: SCG '86 Proceedings of the second annual symposium on Computational geometry
• Year: 1986

Manacher and Zorbrist conjectured that the greedy triangulation heuristic for minimum weight triangulation of n-point planar point sets yields solutions within an &Ogr;(n&egr;), &egr; n) bound on the approximation factor of the Delauney triangulation heuristic which holds for convex, planar n-point sets. To support the conjecture of Manacher and Zorbrist, we also show that the greedy triangulation heuristic for minimum weight triangulation of a (non-necessarily convex) polygon yields solutions at most h times longer than the optimum where h is the diameter of the tree dual to the produced greedy triangulation of the polygon. On the other hand, we present an implementation of the greedy triangulation heuristic for an n-vertex convex point set or a convex polygon taking &Ogr;(n2) time and &Ogr;(n) space which improves Gilbert's &Ogr;(n2logn)-time and &Ogr;(n2)-space bound in this case. To derive the latter result, we show that given a convex polygon P, one can find for all vertices v of P a shortest diagonal of P incident to v in linear time.