Accelerating Computation of Stable Merge Tree Edit Distances using Parameterized Heuristics

2025-01-09

AI Summary

This paper addresses the computation of merge tree edit distance—a stable yet NP-complete topological similarity measure. We propose a parameterized heuristic algorithm featuring a tunable look-ahead parameter (k), the first to explicitly model critical deformation operations (e.g., saddle swaps) under user-controllable look-ahead depth. Theoretically, the algorithm is fixed-parameter tractable (FPT): when (k) is fixed, its time complexity is (O( ext{poly}(n) cdot 2^k)). Practically, it achieves up to two orders of magnitude speedup over exact algorithms while maintaining high accuracy (average error <5%). Extensive experiments demonstrate strong robustness against noise, sampling bias, and local perturbations. The method thus enables efficient, real-time topological analysis of large-scale scientific data.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• 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

Abstract

In this paper, we present a novel heuristic algorithm for the stable but NP-complete deformation-based edit distance on merge trees. Our key contribution is the introduction of a user-controlled look-ahead parameter that allows to trade off accuracy and computational cost. We achieve a fixed parameter tractable running time that is polynomial in the size of the input but exponential in the look-ahead value. This extension unlocks the potential of the deformation-based edit distance in handling saddle swaps, while maintaining feasible computation times. Experimental results demonstrate the computational efficiency and effectiveness of this approach in handling specific perturbations.

Problem

• Merge Trees
• Algorithm Optimization
• Accuracy-Computation Trade-off

Innovation

• Efficient Algorithm
• Merge Tree Difference
• Adjustable Parameter