Tight quasi-universality of Reeb graph distances

TL;DR

This paper proves tight bounds showing that several distances between Reeb graphs are essentially equivalent up to a constant factor, establishing a form of universality for these metrics.

Contribution

The paper introduces a new distance called the functional contortion distance and proves tight bounds for various Reeb graph distances, including universality results for contour and merge trees.

Findings

Bi-Lipschitz bounds for Reeb graph distances

Introduction of the functional contortion distance

Universality of distances for contour and merge trees

Abstract

We establish tight bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.