Paper by Martin L. Demaine

Reference:

Gill Barequet, Nadia Benbernou, David Charlton, Erik D. Demaine, Martin L. Demaine, Mashhood Ishaque, Anna Lubiw, André Schulz, Diane L. Souvaine, Godfried T. Toussaint, and Andrew Winslow, “Bounded-Degree Polyhedronization of Point Sets”, Computational Geometry: Theory and Applications, volume 46, number 2, February 2013, pages 917–928.

Abstract:

In 1994 Grünbaum showed that, given a point set S in ℝ³, it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n log⁶ n) expected time that constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al.

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