Locally Correct Interleavings between Merge Trees

TL;DR

<3-5 sentence high-level summary> This work addresses the problem of comparing and tracking time-varying terrains represented by merge trees. It introduces the residual interleaving distance to allow constraints on interleavings and defines locally correct interleavings to capture local similarities, providing a constructive proof of their existence. The framework extends interleavings with partial maps, critical values, and bottleneck concepts, and outlines a practical approach via a modified dynamic program to compute locally correct interleavings. Overall, the paper strengthens the interpretability of merge-tree comparisons and enables more meaningful local matching in time-evolving terrains.

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.

Figures (11)

• Figure 1: A terrain and its merge tree; leaves and internal vertices represent minima and saddles.
• Figure 2: Two optimal interleavings between merge trees $T_1$ and $T_2$. Intuitively, the right interleaving induces a tighter and more meaningful matching than the left interleaving.
• Figure 3: Left: a merge tree $T$; the point $x|^{h}$ is the ancestor of $x$ at height $h$. Right: parts of a pair of $\delta$-compatible maps.
• Figure 4: Left: a partial up-map $\varphi$ that extends another partial up-map $\varphi'$. Middle: a partial up-map $\varphi$ and the corresponding map $\varphi^\uparrow[\delta]$; each non-dashed arrow has shift at least $\delta$. Right: a partial interleaving with shift $\delta$; note that $\psi(y)$ is an ancestor of both $x_1$ and $x_2$.
• Figure 6: Illustration of the residual shift. The shifts of arrows within the fan $F[(x, y)]$ (represented by the shaded area) are disregarded, so the $\varphi$-shift of $\alpha_2$ is less than the $\varphi$-shift of $\alpha_1$.

Theorems & Definitions (24)

• Definition 2.1: Morozov et al. morozov2013interleaving
• Lemma 2.2: agarwal2018computingtouli2022fpt
• Lemma 3.1
• Lemma 3.2
• Lemma 3.4
• Lemma 3.4
• Lemma 3.5
• Theorem 3.6
• Definition 3.8
• Lemma 3.8