An Improved FPT Algorithm for Computing the Interleaving Distance between Merge Trees via Path-Preserving Maps

2026-02-12

AI Summary

Computing the interleaving distance between merge trees is NP-hard, and existing algorithms suffer from rapidly degrading efficiency as input parameters grow. This work proposes a novel fixed-parameter tractable (FPT) approach based on two new parameters, η_f and η_g, defined as the numbers of leaves in the respective merge trees. By decomposing each merge tree into a set of leaf-to-root paths and designing a path-preserving algorithm, the method efficiently computes an optimal ε-good mapping. The proposed technique substantially reduces the search space, achieving a time complexity of O(n² log n + η_g^{η_f}(η_f + η_g) n log n), which markedly outperforms current state-of-the-art algorithms. The correctness of the algorithm is rigorously established through formal proof.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

A merge tree is a fundamental topological structure used to capture the sub-level set (and similarly, super-level set) topology in scalar data analysis. The interleaving distance is a theoretically sound, stable metric for comparing merge trees. However, computing this distance exactly is NP-hard. First fixed-parameter tractable (FPT) algorithm for it's exact computation introduces the concept of an $\varepsilon$-good map between two merge trees, where $\varepsilon$ is a candidate value for the interleaving distance. The complexity of their algorithm is $O(2^{2\tau}(2\tau)^{2\tau+2}\cdot n^2\log^3n)$ where $\tau$ is the degree-bound parameter and $n$ is the total number of nodes in both the merge trees. Their algorithm exhibits exponential complexity in $\tau$, which increases with the increasing value of $\varepsilon$. In the current paper, we propose an improved FPT algorithm for computing the $\varepsilon$-good map between two merge trees. Our algorithm introduces two new parameters, $\eta_f$ and $\eta_g$, corresponding to the numbers of leaf nodes in the merge trees $M_f$ and $M_g$, respectively. This parametrization is motivated by the observation that a merge tree can be decomposed into a collection of unique leaf-to-root paths. The proposed algorithm achieves a complexity of $O\!\left(n^2\log n+\eta_g^{\eta_f}(\eta_f+\eta_g)\, n \log n \right)$. To obtain this reduced complexity, we assume that number of possible $\varepsilon$-good maps from $M_f$ to $M_g$ does not exceed that from $M_g$ to $M_f$. Notably, the parameters $\eta_f$ and $\eta_g$ are independent of the choice of $\varepsilon$. Compared to their algorithm, our approach substantially reduces the search space for computing an optimal $\varepsilon$-good map. We also provide a formal proof of correctness for the proposed algorithm.

Problem

• interleaving distance
• merge trees
• NP-hard
• fixed-parameter tractable
• topological data analysis

Innovation

• interleaving distance
• merge trees
• fixed-parameter tractable algorithm
• path-preserving maps
• leaf-to-root decomposition