Embedding Stacked Polytopes on a Polynomial-Size Grid
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Demaine_Embedding stacked.pdf
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Author(s)
Demaine, Erik D.
Schulz, Andre
Date Issued
January 2011
Journal
Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society of Industrial and Applied Mathematics (SIAM)
Citation
Erik D. Demaine and André Schulz, “Embedding Stacked Polytopes on a Polynomial-Size Grid”, in Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), San Francisco, California, USA, January 22–25, 2011.
Version
Author's final manuscript
Abstract
We show how to realize a stacked 3D polytope (formed
by repeatedly stacking a tetrahedron onto a triangular
face) by a strictly convex embedding with its n vertices
on an integer grid of size O(n4) x O(n4) x O(n18). We
use a perturbation technique to construct an integral 2D
embedding that lifts to a small 3D polytope, all in linear
time. This result solves a question posed by G unter M.
Ziegler, and is the rst nontrivial subexponential upper
bound on the long-standing open question of the
grid size necessary to embed arbitrary convex polyhedra,
that is, about effcient versions of Steinitz's 1916
theorem. An immediate consequence of our result is
that O(log n)-bit coordinates suffice for a greedy routing
strategy in planar 3-trees.
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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DOI of Published Version
http://www.siam.org/proceedings/soda/2011/SODA11_088_demainee.pdf
