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Refining a triangulation of a planar straight-line graph to eliminate large angles

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1 Author(s)
Mitchell, S.A. ; Dept. of Appl. & Numerical Math., Sandia Nat. Labs., Albuquerque, NM, USA

We show that any planar straight line graph (PSLG) with v vertices can be triangulated with no angle larger than 7π/8 by adding O(v2log v) Steiner points in O(v2log2 v) time. We first triangulate the PSLG with an arbitrary constrained triangulation and then refine that triangulation by adding additional vertices and edges. We follow a lazy strategy of starting from an obtuse angle and exploring the triangulation in search of a sequence of Steiner points that will satisfy a local angle condition. Explorations may either terminate successfully (for example at a triangle vertex), or merge. Some PSLGs require Ω(v2) Steiner points in any triangulation achieving any largest angle bound less than π. Hence the number of Steiner points added by our algorithm is within a log v factor of worst case optimal. For most inputs the number of Steiner points and running time would be considerably smaller than in the worst case

Published in:

Foundations of Computer Science, 1993. Proceedings., 34th Annual Symposium on

Date of Conference:

3-5 Nov 1993