A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two such geometric graphs is a challenging problem in pattern recognition. We study two notions of distance measures for geometric graphs, called the geometric edit distance (GED) and geometric graph distance (GGD). While the former is based on the idea of editing one graph to transform it into the other graph, the latter is inspired by inexact matching of the graphs. For decades, both notions have been lending themselves well as measures of similarity between attributed graphs. If used without any modification, however, they fail to provide a meaningful distance measure for geometric graphs—even cease to be a metric. We have curated their associated cost functions for the context of geometric graphs. Alongside studying the metric properties of GED and GGD, we investigate how the two notions compare. We further our understanding of the computational aspects of GGD by showing that the distance is $\mathcal{NP}$-hard to compute, even if the graphs are planar and arbitrary cost coefficients are allowed.

As a computationally tractable alternative, we propose in this paper the Graph Mover's Distance (GMD), which has been formulated as an instance of the earth mover's distance. The computation of the GMD between two geometric graphs with at most n vertices takes only $O(n^3)$-time. The GMD demonstrates extremely promising empirical evidence at recognizing letter drawings.

几何图是一种组合图，具有从其嵌入欧几里得空间中继承的几何形状。在两个这样的几何图的组合和几何结构中制定有意义的（不）相似性度量是模式识别中的一个具有挑战性的问题。我们研究几何图距离度量的两种概念，称为几何编辑距离（GED）和几何图距离（GGD）。前者基于编辑一个图以将其转换为另一图的想法，而后者则受到图的不精确匹配的启发。几十年来，这两个概念一直被用来衡量属性图之间的相似性。然而，如果不进行任何修改就使用它们，它们就无法为几何图形提供有意义的距离度量，甚至不再是一种度量。我们针对几何图的背景策划了相关的成本函数。除了研究 GED 和 GGD 的度量属性之外，我们还研究了这两个概念的比较。通过证明距离为，我们进一步理解了 GGD 的计算方面$\mathcal{NP}$- 难以计算，即使图形是平面的并且允许任意成本系数。

作为一种计算上易于处理的替代方案，我们在本文中提出了图移动器距离（GMD），它已被公式化为地球移动器距离的一个实例。两个最多有n 个顶点的几何图之间的 GMD 计算只需要$氧（n^3）$-时间。GMD 在识别字母图画方面展示了极有前景的经验证据。