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The Earth movers distance (EMD) is a measure of distance between probabilitydistributions which is at the heart of mass transportation theory. Recentresearch has shown that the EMD plays a crucial role in studying the potentialimpact of Demand-Responsive Transportation (DRT) and Mobility-on-Demand (MoD)systems, which are growing paradigms for one-way vehicle sharing where peopledrive (or are driven by) shared vehicles from a point of origin to a point ofdestination. While the ubiquitous physical transportation setting is the roadnetwork, characterized by systems of roads connected together by interchanges,most analytical works about vehicle sharing represent distances between pointsin a plane using the simple Euclidean metric. Instead, we consider the EMD whenthe ground metric is taken from a class of one-dimensional, continuous metricspaces, reminiscent of road networks. We produce an explicit formulation of theEarth movers distance given any finite road network R. The result generalizesthe EMD with a Euclidean R1 ground metric, which had remained one of the onlyknown non-discrete cases with an explicit formula. Our formulation casts theEMD as the optimal value of a finite-dimensional, real-valued optimizationproblem, with a convex objective function and linear constraints. In thespecial case that the input distributions have piece-wise uniform (constant)density, the problem reduces to one whose objective function is convexquadratic. Both forms are amenable to modern mathematical programmingtechniques.

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