Labeled Interleaving Distance for Reeb Graphs

TL;DR

This paper introduces a labeled interleaving distance for Reeb graphs, proving its relation to the ordinary interleaving distance and providing algorithms for its computation, thereby advancing topological data analysis tools.

Contribution

It extends the labeled interleaving distance concept from merge trees to Reeb graphs, establishes its properties, and offers efficient algorithms for contour trees.

Findings

Labeled interleaving distance equals the minimum over labelings of the ordinary interleaving distance.

Efficient algorithms are provided for computing the labeled interleaving distance for contour trees.

Counterexamples show the ordinary interleaving distance is not intrinsic on unlabeled Reeb graphs.

Abstract

Merge trees, contour trees, and Reeb graphs are graph-based topological descriptors that capture topological changes of (sub)level sets of scalar fields. Comparing scalar fields using their topological descriptors has many applications in topological data analysis and visualization of scientific data. Recently, Munch and Stefanou introduced a labeled interleaving distance for comparing two labeled merge trees, which enjoys a number of theoretical and algorithmic properties. In particular, the labeled interleaving distance between merge trees can be computed in polynomial time. In this work, we define the labeled interleaving distance for labeled Reeb graphs. We then prove that the (ordinary) interleaving distance between Reeb graphs equals the minimum of the labeled interleaving distance over all labelings. We also provide an efficient algorithm for computing the labeled interleaving distance between two labeled contour trees (which are special types of Reeb graphs that arise from simply-connected domains). In the case of merge trees, the notion of the labeled interleaving distance was used by Gasparovic et al. to prove that the (ordinary) interleaving distance on the set of (unlabeled) merge trees is intrinsic. As our final contribution, we present counterexamples showing that, on the contrary, the (ordinary) interleaving distance on (unlabeled) Reeb graphs (and contour trees) is not intrinsic. It turns out that, under mild conditions on the labelings, the labeled interleaving distance is a metric on isomorphism classes of Reeb graphs, analogous to the ordinary interleaving distance. This provides new metrics on large classes of Reeb graphs.