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 $O(2^{2τ}(2τ)^{2τ+2}\cdot n^2\log^3n)$ where $τ$ is the degree-bound parameter and $n$ is the total number of nodes in both the merge trees. Their algorithm exhibits exponential complexity in $τ$, 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, $η_f$ and $η_g$, 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 $O\!\left(n^2\log n+η_g^{η_f}(η_f+η_g)\, n \log n \right)$. 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 $η_f$ and $η_g$ 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.

翻译：合并树是标量数据分析中用于捕捉子水平集（类似地，超水平集）拓扑的基本拓扑结构。交错距离是一种理论上严谨、稳定的度量，用于比较合并树。然而，精确计算该距离是NP难的。首个用于其精确计算的固定参数可解（FPT）算法引入了两个合并树间$\varepsilon$-良好映射的概念，其中$\varepsilon$是交错距离的候选值。该算法的复杂度为$O(2^{2τ}(2τ)^{2τ+2}\cdot n^2\log^3n)$，其中$τ$是度约束参数，$n$是两个合并树的总节点数。该算法的复杂度在$τ$上呈指数级，且随$\varepsilon$值增大而增加。在本文中，我们提出了一种改进的FPT算法，用于计算两个合并树间的$\varepsilon$-良好映射。我们的算法引入了两个新参数$η_f$和$η_g$，分别对应于合并树$M_f$和$M_g$中的叶节点数量。这种参数化的动机源于观察到合并树可以分解为一组唯一的从叶到根的路径。所提算法实现了$O\!\left(n^2\log n+η_g^{η_f}(η_f+η_g)\, n \log n \right)$的复杂度。为获得此降低的复杂度，我们假设从$M_f$到$M_g$的可能$\varepsilon$-良好映射数量不超过从$M_g$到$M_f$的数量。值得注意的是，参数$η_f$和$η_g$与$\varepsilon$的选择无关。与他们的算法相比，我们的方法显著减少了计算最优$\varepsilon$-良好映射的搜索空间。我们还为所提算法提供了形式化的正确性证明。