Locally Correct Interleavings Between Merge Trees

Abstract

Merge trees are typically used as a topological summary of scalar fields. To analyze e.g. a time-varying scalar field via its merge tree representation, one needs a suitable method to match and compare two merge trees. We consider the interleaving distance as a method to match and compare merge trees. An interleaving between two merge trees consists of two maps, one in each direction. These maps must satisfy ancestor relations and hence introduce a "shift" between points and their image. An optimal interleaving minimizes the maximum shift; the interleaving distance is the value of this shift. However, to study the evolution of merge trees, we need not only a number but also a meaningful matching between the two trees. The two maps of an optimal interleaving induce a matching, but due to the bottleneck nature of the interleaving distance, this matching fails to capture local similarities between the trees. In this paper we hence propose a notion of local optimality for interleavings. To do so, we define the residual interleaving distance, a generalization of the interleaving distance that allows additional constraints on the maps. This allows us to define locally correct interleavings, which use a range of shifts across the two merge trees that reflect the local similarity well. We give a constructive proof that a locally correct interleaving always exists.