Relating Interleaving and Fréchet Distances via Ordered Merge Trees

TL;DR

This work introduces ordered merge trees to capture intrinsic data order alongside the hierarchical structure captured by traditional merge trees. It defines the monotone interleaving distance $d_{MI}$, an order-preserving analogue of the interleaving distance, and proves three equivalent formulations (two maps, a single map, or a labelling). A central result shows that $d_{MI}$ equals the Fréchet distance $d_F$ between the induced $1$D curves from ordered merge trees, enabling exact computation in near-quadratic time and bridging topological data analysis with computational geometry. The paper also presents two alternative definitions of $d_{MI}$ (monotone $ ext{δ}$-good interleaving and monotone label interleaving) and establishes a bijection between ordered merge trees and leaf-ordered merge trees, providing a robust, geometry-aware framework for comparing hierarchical data such as braided river networks.

Abstract

Merge trees are a common topological descriptor for data with a hierarchical component, such as terrains and scalar fields. The interleaving distance, in turn, is a common distance for comparing merge trees. However, the interleaving distance for merge trees is solely based on the hierarchical structure, and disregards any other geometrical or topological properties that might be present in the underlying data. Furthermore, the interleaving distance is NP-hard to compute. In this paper, we introduce a form of ordered merge trees that can capture intrinsic order present in the data. We further define a natural variant of the interleaving distance, the monotone interleaving distance, which is an order-preserving distance for ordered merge trees. Analogously to the regular interleaving distance for merge trees, we show that the monotone variant has three equivalent definitions in terms of two maps, a single map, or a labelling. Furthermore, we establish a connection between the monotone interleaving distance of ordered merge trees and the Fréchet distance of 1D curves. As a result, the monotone interleaving distance between two ordered merge trees can be computed exactly in near-quadratic time in their complexity. The connection between the monotone interleaving distance and the Fréchet distance builds a new bridge between the fields of topological data analysis, where interleaving distances are a common tool, and computational geometry, where Fréchet distances are studied extensively.

Figures (6)

• Figure 1: The Waimakariri River in New Zealand is a braided river. Photo by Greg O'Beirne obeirne2017waimakariri.
• Figure 2: A schematic river, at different levels of the terrain. We use a merge tree to represent the bars: leaves correspond to local maxima, and internal vertices correspond to two bars merging. We draw the tree upside down, following the convention that merge trees are rooted at positive infinity.
• Figure 3: Two merge trees and part of a $\delta$-interleaving. Mapping a point $x$ from $T$ to $T'$ through $\alpha$ (in blue), and mapping it back to $T$ via $\beta$ (in red) gives the unique ancestor $x^{2\delta}$ of $x$.
• Figure 4: (a) Visualisation of an ordered merge tree. (b) Lemma \ref{['lem:total-order-prop']}: if $x_1 <_h x_2$, then any descendants $\bar{x_1} \preceq x_1$ and $\bar{x_2} \preceq x_2$ at the same height $\bar{h}$ also satisfy $\bar{x_1} <_{\bar{h}} \bar{x_2}$.
• Figure 5: Visualisations of (a) a leaf-ordered merge tree and (b) the separating subtrees property of a leaf-order.

Theorems & Definitions (16)

• Definition 1
• Definition 2: Morozov, Beketayev and Weber morozov2013interleaving
• Definition 3
• Lemma 1
• Definition 4
• Lemma 2
• Lemma 3
• Corollary 4
• Lemma 4
• Lemma 5