Splay Top Trees

TL;DR

This paper introduces a new, simpler proof for the existence of top trees in dynamic graph algorithms, inspired by splay trees, which enables more straightforward implementations.

Contribution

It provides a direct proof of top trees' existence using splay tree ideas, offering new structural insights and simplifying their implementation.

Findings

New direct proof of top trees' existence

Simpler and more direct implementation methods

Enhanced understanding of top tree structure

Abstract

The top tree data structure is an important and fundamental tool in dynamic graph algorithms. Top trees have existed for decades, and today serve as an ingredient in many state-of-the-art algorithms for dynamic graphs. In this work, we give a new direct proof of the existence of top trees, facilitating simpler and more direct implementations of top trees, based on ideas from splay trees. This result hinges on new insights into the structure of top trees, and in particular the structure of each root path in a top tree.