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 data structures. The rotation distance between a pair of trees is the minimum number of rotations needed to convert one tree into the other. In this paper we establish a tight bound of 2 n − 6 2n - 6 on the maximum rotation distance between two n n -node trees for all large n n . The hard and novel part of the proof is the lower bound, which makes use of volumetric arguments in hyperbolic 3 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.
Highlights
A rotation in a binary tree is a local restructuring of the tree that changes it into another tree
The problem addressed in this paper is: what is the maximum rotation distance between any pair of n-node binary trees? We show that for all n ~ 11 this distance is at most 2n - 6 and that for all sufficiently large n this bound is tight
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Summary
A rotation in a binary tree is a local restructuring of the tree that changes it into another tree. Pallo [7] proposed a heuristic search algorithm to compute the rotation distance between two given trees Our interest in this problem stems from our attempt to solve the dynamic optimality conjecture concerning the performance of splaying [8, 10]. A system that is isomorphic to binary trees related by rotations is that of triangulations of a polygon related by the diagonal flip operation This is the operation that converts one triangulation of a polygon into another by removing a diagonal in the triangulation and adding the diagonal that subdivides the resulting quadrilateral in the opposite way. Our approach to solving the rotation distance problem is based on the observation that any sequence of diagonal flips converting one triangulation of a polygon into another gives a way to dissect (into tetrahedra) a polyhedron formed from the two triangulations. We construct particular polyhedra that require many tetrahedra to triangulate them. §4 contains remarks and some open problems
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