Embedding Stacked Polytopes on a Polynomial-Size Grid
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454_2017_9887_ReferencePDF.pdf
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Author(s)
Schulz, André
Demaine, Erik D
Date Issued
March 2017
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Demaine, Erik D., and André Schulz. “Embedding Stacked Polytopes on a Polynomial-Size Grid.” Discrete & Computational Geometry 57, no. 4 (March 21, 2017): 782–809.
Version
Author's final manuscript
Abstract
A stacking operation adds a d-simplex on top of a facet of a simplicial d-polytope while maintaining the convexity of the polytope. A stacked d-polytope is a polytope that is obtained from a d-simplex and a series of stacking operations. We show that for a fixed d every stacked d-polytope with n vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by O(n[superscript 2 log[subscript 2](2d)], except for one axis, where the coordinates are bounded by O(n[superscript 3 log[subscript 2](2d)]. The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00454-017-9887-6
