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

TL;DR

The paper tackles the exact interleaving distance between merge trees, a metric with strong theoretical guarantees but NP-hard computation. Building on the ε-good map framework, it introduces a fixed-parameter algorithm parameterized by the numbers of leaf nodes, η_f and η_g, which are independent of the candidate distance ε. The approach leverages a path-based construction along leaf-to-root paths, augmented/extended merge trees, and a gluing-based continuous map to verify ε-goodness, yielding a runtime of O(n^2 log n + η_g^{η_f}(η_f+η_g) n log n). This reparameterization yields a stable, significantly more practical complexity bound than prior τ-dependent FPT methods, with correctness formally established. The work paves the way for robust topological comparisons of scalar fields and suggests extensions to related descriptors like Reeb graphs and time-varying data.

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 $O(2^{2τ}(2τ)^{2τ+2}\cdot n^2\log^3n)$ 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.

Figures (11)

• Figure 1: An example of a merge tree of the height field on the graph $(x, f(x))$ of the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^4 - 4x^2 + x$. The corresponding merge tree is shown on the right, with function values increasing toward $\infty$ at the root node colored in blue. The coral (light red) and sky-blue colored nodes in the merge tree correspond to the local minima in the domain, where new connected components are born. The green node in the merge tree corresponds to the saddle where two components merge. The coral edge in the merge tree corresponds to the coral region in the domain, the sky-blue edge to the sky-blue region, the green edge to the green region and the blue edge to the blue region.
• Figure 2: Illustration of construction of $\varepsilon$-compatible maps between the (extended) merge trees $M_f$ and $M_g$. A blue node $u$ in $M_f$ is mapped to a red node $\alpha(u)$ in $M_g$, satisfying $\tilde{g}(\alpha(u)) = \tilde{f}(u) + \varepsilon$. Conversely, a red node $v$ in $M_g$ is mapped to a point $\beta(v)$ in $M_f$, satisfying $\tilde{f}(\beta(v)) = \tilde{g}(v) + \varepsilon$.
• Figure 3: Illustration of the Ancestor-Shift property in Definition \ref{['def:epsilon-goodmap']} for an $\varepsilon$-good map $\alpha^\varepsilon: M_f \rightarrow M_g$. Here $\alpha^\varepsilon(v_1) \succeq \alpha^\varepsilon(v_2)$, then it must hold that $i^{2\varepsilon}(v_1)\succeq i^{2\varepsilon}(v_2)$ for $v_1, v_2 \in M_f$.
• Figure 4: Illustration of the Ancestor-Closeness property in Definition \ref{['def:epsilon-goodmap']} for an $\varepsilon$-good map $\alpha^\varepsilon: M_f \rightarrow M_g$. The black region in $M_g$ represents the complement of $\mathrm{Im}(\alpha^{\varepsilon})$ in $M_g$. For a point $w \notin \mathrm{Im}(\alpha^{\varepsilon})$, let $w^a$ be its nearest ancestor such that $w^a \in \mathrm{Im}(\alpha^{\varepsilon})$. Then we must have $|\tilde{g}(w^a) - \tilde{g}(w)| \leq 2\varepsilon$.
• Figure 5: Illustration of the augmentation process. The red nodes denote newly inserted nodes, while the black nodes represent the already existing nodes of the merge trees $M_f$ and $M_g$. Consider a critical level $\tilde{f}^{-1}(c)$ in $M_f$, then there is a corresponding level $\tilde{g}^{-1}(c + \varepsilon)$ added in $M_g$, and two additional red nodes of degree 2 are introduced at this level. Similarly, for a critical level $\tilde{g}^{-1}(c')$ in $M_g$, a corresponding level $\tilde{f}^{-1}(c' - \varepsilon)$ is added in $M_f$ and two additional degree two nodes are introduced.

Theorems & Definitions (25)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.1: Candidate Set Generation agarwal_computing_2018
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3: Gluing Lemma armstrong1983basic
• Lemma 4.4