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 measure 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 does 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 measure for ordered merge trees. Analogous 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. The labelling-based definition fairly directly leads to an efficient algorithm for computing the monotone interleaving distance, but unfortunately it computes only an approximation thereof. Instead, we discover a surprising connection between the monotone interleaving distance of ordered merge trees and the Fr\'{e}chet distance of 1D curves. As a result, the monotone interleaving distance between two ordered merge trees of total complexity $n$ can be computed exactly in $\tilde O(n^2)$ time. The connection between the monotone interleaving distance and the Fr\'{e}chet distance establishes a new bridge between the fields of computational topology/topological data analysis, where interleaving distances are studied extensively, and computational geometry, where Fr\'{e}chet distances are studied extensively.

翻译：合并树是描述具有层次结构数据（如地形和标量场）的常见拓扑描述符。交错距离则是比较合并树的常用距离度量。然而，合并树的交错距离仅基于层次结构，忽略了底层数据中可能存在的任何其他几何或拓扑属性。此外，计算交错距离是NP难的。本文引入了一种能够捕捉数据中固有顺序的有序合并树形式。我们进一步定义了交错距离的一种自然变体——单调交错距离，这是一种用于有序合并树的保序距离度量。与常规合并树的交错距离类似，我们证明单调变体具有三种等价定义：基于两个映射、单个映射或标签。基于标签的定义直接导出了计算单调交错距离的高效算法，但不幸的是，该算法仅能计算近似值。相反，我们发现有序合并树的单调交错距离与一维曲线的弗雷歇距离之间存在惊人联系。因此，总复杂度为$n$的两个有序合并树之间的单调交错距离可在$\tilde O(n^2)$时间内精确计算。单调交错距离与弗雷歇距离之间的联系，在计算拓扑学/拓扑数据分析（广泛研究交错距离）与计算几何（广泛研究弗雷歇距离）领域之间架起了一座新桥梁。