Rotation distance using flows

TL;DR

This paper presents an elementary proof for the maximum rotation distance between binary trees with n internal nodes, utilizing a novel flow-based method inspired by potential functions and max-flow min-cut theorem.

Contribution

It introduces a new, elementary proof technique for rotation distance bounds using flow problems, simplifying previous complex arguments.

Findings

Maximum rotation distance is exactly 2n-6 for trees with n internal nodes.

The proof employs a flow problem and max-flow min-cut theorem, offering a new methodological approach.

The approach simplifies understanding of rotation distances compared to previous proofs.

Abstract

Splay trees are a simple and efficient dynamic data structure, invented by Sleator and Tarjan. The basic primitive for transforming a binary tree in this scheme is a rotation. Sleator, Tarjan, and Thurston proved that the maximum rotation distance between trees with n internal nodes is exactly 2n-6 for trees with n internal nodes (where n is larger than some constant). The proof of the upper bound is easy but the proof of the lower bound, remarkably, uses sophisticated arguments based on calculating hyperbolic volumes. We give an elementary proof of the same result. The main interest of the paper lies in the method, which is new. It basically relies on a potential function argument, similar to many amortized analyses. However, the potential of a tree is not defined explicitly, but by constructing an instance of a flow problem and using the max-flow min-cut theorem.