The Reeb Graph Edit Distance is Universal

TL;DR

This paper introduces a stable and universal edit distance for Reeb graphs of piecewise linear functions, establishing it as an upper bound for all other stable distances, unlike the interleaving and functional distortion distances.

Contribution

The paper defines a new edit distance for Reeb graphs that is proven to be both stable and universal, surpassing existing distances in generality.

Findings

The edit distance is stable under small function perturbations.

The edit distance is universal, bounding all other stable distances.

Interleaving and functional distortion distances are not universal.

Abstract

We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.