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

TL;DR

This paper presents an improved fixed-parameter tractable algorithm for exactly computing the interleaving distance between merge trees, reducing complexity by leveraging leaf-path decompositions and new parameters.

Contribution

The paper introduces a novel FPT algorithm that uses leaf-node path parameters to significantly reduce the complexity of computing the interleaving distance between merge trees.

Findings

Reduced algorithm complexity to $O(n^2 \, \log n + \eta_g^{\eta_f}(\eta_f + \eta_g) \, n \log n)$

Proved correctness of the new algorithm

Achieved more efficient computation of the interleaving distance

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.