Graph Mover's Distance: An Efficiently Computable Distance Measure for Geometric Graphs

Many applications in pattern recognition represent patterns as a geometric graph. The geometric graph distance (GGD) has recently been studied as a meaningful measure of similarity between two geometric graphs. Since computing the GGD is known to be $\mathcal{NP}$-hard, the distance measure proves an impractical choice for applications. 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. Alongside studying the metric properties of the GMD, we investigate the stability of the GGD and GMD. The GMD also demonstrates extremely promising empirical evidence at recognizing letter drawings from the {\tt LETTER} dataset \cite{da_vitoria_lobo_iam_2008}.

Distance Measures for Geometric Graphs

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.