Embedding Stacked Polytopes on a Polynomial-Size Grid
Author(s)Schulz, André; Demaine, Erik D
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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.
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Discrete & Computational Geometry
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.
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