The Reeb Graph Edit Distance Is Universal Ulrich Bauer Department of Mathematics, Technical University of Munich (TUM), Germany
[email protected] Claudia Landi Dipartimento di Scienze e Metodi dell’Ingegneria, Università degli Studi di Modena e Reggio Emilia, Reggio Emilia, Italy
[email protected] Facundo Mémoli Department of Mathematics, The Ohio State University, Columbus, OH, USA
[email protected] 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. 2012 ACM Subject Classification Theory of computation → Computational geometry; Mathematics of computing → Algebraic topology Keywords and phrases Reeb graphs, topological descriptors, edit distance, interleaving distance Digital Object Identifier 10.4230/LIPIcs.SoCG.2020.15 Funding This research has been partially supported by FAR 2014 (UniMORE), ARCES (University of Bologna), and the DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”. Acknowledgements We thank Barbara Di Fabio and Yusu Wang for valuable discussions. 1 Introduction The concept of a Reeb graph of a Morse function first appeared in [13] and has subsequently been applied to problems in shape analysis in [14, 10]. The literature on Reeb graphs in the computational geometry and computational topology is ever growing (see, e.g., [2, 3] for a discussion and references).