Locally Correct Interleavings between Merge Trees

TL;DR

This paper introduces locally correct interleavings for merge trees, enabling more meaningful local comparisons of terrains over time by generalizing the interleaving distance with residual constraints.

Contribution

It proposes a new notion of local optimality for interleavings, defining residual interleaving distance to better capture local similarities between merge trees.

Findings

Locally correct interleavings always exist.

Residual interleaving distance generalizes the standard interleaving distance.

Enhanced matching captures local similarities more effectively.

Abstract

Temporal sequences of terrains arise in various application areas. To analyze them efficiently, one generally needs a suitable abstraction of the data as well as a method to compare and match them over time. In this paper we consider merge trees as a topological descriptor for terrains and the interleaving distance as a method to match and compare them. 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 over time, 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.