Rotation Distance for Rank Bounded Trees

Abstract

AbstractComputing the rotation distance between two binary trees with n internal nodes efficiently (in \(\textsf {poly}(n)\) time) is a long standing open question in the study of height balancing in tree data structures. In this paper, we initiate the study of this problem bounding the rank of the trees given at the input (defined in [6] in the context of decision trees).We define the rank-bounded rotation distance between two given binary trees \(T_1\) and \(T_2\) (with n internal nodes) of rank at most r, denoted by \(d_r(T_1,T_2)\), as the length of the shortest sequence of rotations that transforms \(T_1\) to \(T_2\) with the restriction that the intermediate trees must be of rank at most r. We show that the rotation distance problem reduces in polynomial time to the rank bounded rotation distance problem. This motivates the study of the problem in the combinatorial and algorithmic frontiers. Observing that trees with rank 1 coincide exactly with skew trees (binary trees where every internal node has at least one leaf as a child), we show the following results in this frontier: We present an \(O(n^2)\) time algorithm for computing \(d_1(T_1,T_2)\). That is, when the given trees are skew trees (we call this variant as skew rotation distance problem) - where the intermediate trees are restricted to be skew as well. In particular, our techniques imply that for any two skew trees \(d(T_1,T_2) \le n^2\). We show the following upper bound: for any two trees \(T_1\) and \(T_2\) of rank at most \(r_1\) and \(r_2\) respectively, we have that: \(d_r(T_1,T_2) \le n^2 (1+(2n+1)(r_1+r_2-2))\) where \(r = \max \{r_1,r_2\}\). This bound is asymptotically tight for \(r=1\). En route our proof of the above theorems, we associate binary trees to permutations and relate the rotation operation on trees to transpositions in the corresponding permutations. We give exact combinatorial characterizations of permutations that correspond to binary trees and skew binary trees under this association. We also precisely characterize the transpositions that correspond to the set of rotations in binary trees.