Chain Rotations: a New Look at Tree Distance

TL;DR

This paper introduces chain rotation, a new operation on binary trees, and analyzes the chain distance, providing bounds and comparisons to the traditional rotation distance.

Contribution

It defines chain rotation and chain distance, establishes bounds, and compares these to existing rotation distance bounds, offering new insights into tree transformation metrics.

Findings

Upper bound of chain distance: C(S,T) <= n-1

Existence of tree pairs where the bound is tight

Comparison with rotation distance bounds (2n-6 and 5n/3-4)

Abstract

As well known the rotation distance D(S,T) between two binary trees S, T of n vertices is the minimum number of rotations of pairs of vertices to transform S into T. We introduce the new operation of chain rotation on a tree, involving two chains of vertices, that requires changing exactly three pointers in the data structure as for a standard rotation, and define the corresponding chain distance C(S,T). As for D(S,T), no polynomial time algorithm to compute C(S,T) is known. We prove a constructive upper bound and an analytical lower bound on C(S,T) based on the number of maximal chains in the two trees. In terms of n we prove the general upper bound C(S,T)<= n-1 and we show that there are pairs of trees for which this bound is tight. No similar result is known for D(S,T) where the best upper and lower bounds are 2n-6 and 5n/3-4 respectively.