An Explicit Formulation of the Earth Movers Distance with Continuous Road Map Distances

Abstract

The Earth movers distance (EMD) is a measure of distance between probability distributions which is at the heart of mass transportation theory. Recent research has shown that the EMD plays a crucial role in studying the potential impact of Demand-Responsive Transportation (DRT) and Mobility-on-Demand (MoD) systems, which are growing paradigms for one-way vehicle sharing where people drive (or are driven by) shared vehicles from a point of origin to a point of destination. While the ubiquitous physical transportation setting is the road network, characterized by systems of roads connected together by interchanges, most analytical works about vehicle sharing represent distances between points in a plane using the simple Euclidean metric. Instead, we consider the EMD when the ground metric is taken from a class of one-dimensional, continuous metric spaces, reminiscent of road networks. We produce an explicit formulation of the Earth movers distance given any finite road network R. The result generalizes the EMD with a Euclidean R1 ground metric, which had remained one of the only known non-discrete cases with an explicit formula. Our formulation casts the EMD as the optimal value of a finite-dimensional, real-valued optimization problem, with a convex objective function and linear constraints. In the special case that the input distributions have piece-wise uniform (constant) density, the problem reduces to one whose objective function is convex quadratic. Both forms are amenable to modern mathematical programming techniques.