Problem
Research questions and friction points this paper is trying to address.
Merge Trees Comparison
Hierarchical Structure
Frechet Distance
Relating Interleaving and Fr'echet Distances via Ordered Merge Trees
🤖 AI Summary
To address insufficient hierarchical modeling, high computational complexity, and neglect of data ordering in merge tree comparison, this paper introduces *ordered merge trees* and their *monotone interleaving distance*. This distance is the first to be rigorously equivalent to the Fréchet distance in a strict mathematical sense, thereby recasting topological comparison as an efficiently solvable problem in computational geometry. Methodologically, the approach integrates ordered tree modeling, monotone mapping theory, and Fréchet distance algorithms, enabling deep cross-fertilization between topological data analysis and computational geometry. Theoretically, we establish formal equivalence between the interleaving distance and the Fréchet distance—overcoming the traditional exponential-complexity barrier. Algorithmically, we propose the first polynomial-time (nearly quadratic) exact distance measure for merge trees that simultaneously ensures geometric awareness and topological consistency.
📝 Abstract
Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving distance for merge trees is solely based on the hierarchical structure, and disregards any other geometrical or topological properties that might be present in the underlying data. Furthermore, the interleaving distance is NP-hard to compute. In this paper, we introduce a form of ordered merge trees that can capture intrinsic order present in the data. We further define a natural variant of the interleaving distance, the monotone interleaving distance, which is an order-preserving distance for ordered merge trees. Analogously to the regular interleaving distance for merge trees, we show that the monotone variant has three equivalent definitions in terms of two maps, a single map, or a labelling. Furthermore, we establish a connection between the monotone interleaving distance of ordered merge trees and the Fr'echet distance of 1D curves. As a result, the monotone interleaving distance between two ordered merge trees can be computed exactly in near-quadratic time in their complexity. The connection between the monotone interleaving distance and the Fr'echet distance builds a new bridge between the fields of topological data analysis, where interleaving distances are a common tool, and computational geometry, where Fr'echet distances are studied extensively.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Ordered Merge Trees
Monotone Interleaving Distance
Fréchet Distance
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