Efficient Heuristic Algorithms for Interleaving Distance between Merge Trees

2025-09-19

AI Summary

Computing the interleaving distance between merge trees remains computationally challenging—particularly when trees are unlabelled or possess non-overlapping label sets. To address this, we propose a heuristic algorithm based on leaf-label assignment that leverages the hierarchical structure of merge trees to infer structural correspondences among nodes. Our method combines a greedy optimization strategy with label propagation to enable efficient approximation of the interleaving distance in polynomial time—significantly outperforming existing exact algorithms. It accommodates labelled trees with partial or disjoint label sets, as well as fully unlabelled trees. Experiments on synthetic time-varying electron density data demonstrate substantial improvements in computational efficiency while preserving acceptable accuracy. To our knowledge, this is the first work to introduce label-driven structural matching for interleaving distance approximation, thereby broadening the practical applicability of topological data analysis in scientific visualization and comparative analysis of tree-structured data.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Heuristic Search
• Machine Learning: Structured Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Merge trees are fundamental structures in topological data analysis. Interleaving distance is a widely accepted metric for comparing merge trees, with applications in visualization and scientific computing. While a greedy algorithm exists for finding the interleaving distance between labeled merge trees with overlapping labels, computing the interleaving distance between unlabeled trees or labeled trees with disjoint labels remains a significant challenge. In this work, we introduce a novel heuristic algorithm for approximating the interleaving distance between labeled merge trees with partial agreement and disagreement. Our method strategically assigns labels primarily to the leaves of the trees to infer structural correspondence. We also introduce an enhanced version of a previous algorithm that offers improved performance. Both algorithms run in polynomial time and provide practical, efficient alternatives for comparing merge trees, particularly in cases involving unlabeled or structurally diverse data. This work contributes a new direction for merge tree analysis and offers promising tools for real-world applications. We demonstrate this application on the simulation of time-varying electron density.

Problem

• Computing interleaving distance between unlabeled merge trees
• Approximating distance for trees with disjoint label sets
• Providing efficient polynomial-time algorithms for comparison

Innovation

• Heuristic algorithm for interleaving distance approximation
• Strategic leaf label assignment for structural correspondence
• Enhanced polynomial-time algorithm for unlabeled trees