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Rotation Distance, Triangulations, and Hyperbolic Geometry

Daniel D. Sleator, Robert E. Tarjan and William P. Thurston
Journal of the American Mathematical Society
Vol. 1, No. 3 (Jul., 1988), pp. 647-681
DOI: 10.2307/1990951
Stable URL: http://www.jstor.org/stable/1990951
Page Count: 35
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Rotation Distance, Triangulations, and Hyperbolic Geometry
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Abstract

A rotation in a binary tree is a local restructuring that changes the tree into another tree. Rotations are useful in the design of tree-based structures. The rotation distance between a pair of trees in the minimum number of rotations needed to convert one tree into the other. In this paper we establish a tight bound of 2n - 6 on the maximum rotation distance between two n-node trees for all large n. The hard and novel part of the proof is the lower bound, which makes use of volumetric arguments in hyperbolic 3-space. Our proof also gives a tight bound on the minimum number of tetrahedra needed to dissect a polyhedron in the worst case and reveals connections among binary trees, triangulations, polyhedra, and hyperbolic geometry.

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