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. For example, the channel networks formed by braided rivers carry intrinsic orders induced by the relative position of channels: from one bank to another, or from upstream to downstream. In this paper, we introduce a form of ordered merge trees that does capture intrinsic order present in the data. Furthermore, we define 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. There is no efficient constant factor approximation known to compute the interleaving distance. In contrast, we describe an $O(n^2)$ time algorithm that computes a 2-approximation of the monotone interleaving distance with an additive term $G$ that captures the maximum height differences of leaves of the input merge trees. In the real world setting of river network analysis, all leaves are at height 0; hence $G$ equals 0, and our algorithm is a proper 2-approximation.

翻译：合并树是用于描述具有层次结构数据（例如地形和标量场）的常见拓扑描述符，而交错距离则是比较合并树的一种常用距离度量。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。例如，辫状河道形成的河道网络具有由河道相对位置（从一岸到另一岸，或从上游到下游）所诱导的内在顺序。本文引入了一种能够捕捉数据中内在顺序的有序合并树形式，并定义了单调交错距离——一种针对有序合并树的保序距离度量。类似于普通合并树的交错距离，我们证明该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。目前尚无已知的高效常数因子近似算法来计算交错距离。相比之下，我们描述了一种运行时间为$O(n^2)$的算法，该算法可在加法项$G$（捕捉输入合并树中叶节点的最大高度差）下计算单调交错距离的2-近似。在河道网络分析的实际场景中，所有叶节点高度均为0，因此$G=0$，此时我们的算法是一个精确的2-近似。