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.

翻译：合并树、等高线树及Reeb图是基于图的拓扑描述符，用于捕获标量场（子）水平集的拓扑变化。利用拓扑描述符比较标量场在拓扑数据分析与科学数据可视化中具有广泛应用。近期，Munch与Stefanou引入了一种标号交错距离用于比较两个标号合并树，该距离具备多项理论与算法性质，特别是合并树间的标号交错距离可在多项式时间内计算。本文定义了标号Reeb图的标号交错距离，并证明Reeb图之间的（普通）交错距离等于所有标号下标号交错距离的最小值。我们还提出了一种高效算法，用于计算两个标号等高线树（源于单连通域的一种特殊Reeb图）之间的标号交错距离。在合并树情形下，Gasparovic等人利用标号交错距离的概念证明了（无标号）合并树集合上的（普通）交错距离是内蕴的。作为最终贡献，我们给出反例表明：相反地，（无标号）Reeb图（及等高线树）上的（普通）交错距离并非内蕴。进一步发现，在标号的温和条件下，标号交错距离是Reeb图同构类上的度量，这为大规模Reeb图类提供了新度量。