Locally Correct Interleavings between Merge Trees

2025-12-18

AI Summary

The global interleaving distance for merge trees—constrained by the bottleneck property—fails to capture local structural similarity. To address this limitation in time-varying terrain analysis, we introduce *locally correct interleaving*, a refined notion of interleaving that induces a *residual interleaving distance*. Unlike conventional approaches enforcing a single global shift, our method constructs mappings that preserve ancestor relationships, enabling fine-grained, geometrically interpretable matching of local topological structures in merge trees—while guaranteeing theoretical existence. We prove the existence of locally correct interleavings and demonstrate empirically that the residual interleaving distance effectively identifies both locally isomorphic and divergent regions in evolving terrains. This yields a new metric for topological time-series analysis that balances mathematical rigor with practical utility. (132 words)

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Intelligent Robots: Localization, Mapping, and Navigation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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.

Problem

• Proposes locally optimal interleavings for merge trees.
• Generalizes interleaving distance with residual constraints.
• Ensures matching captures local similarities in trees.

Innovation

• Introduces locally correct interleavings for merge trees
• Defines residual interleaving distance with additional constraints
• Constructively proves existence of locally optimal matchings