Small Manhattan Networks and Algorithmic Applications for the Earth Mover’s Distance

Given a set S of n points in the plane, a Manhattan network on S is a (not necessarily planar) rectilinear network G with the property that for every pair of points in S the network G contains a path between them whose length is equal to the Manhattan distance between the points. A Manhattan network on S can be thought of as a graph G = (V,E) where the vertex set V corresponds to the points of S and a set of Steiner points S′. The edges in E correspond to horizontal and vertical line segments connecting points in S ∪ S′. A Manhattan network can also be thought of as a 1-spanner (for the L1-metric) for the points in S. We will show that there is a Manhattan network on S with O(n log n) vertices and edges which can be constructed in O(n log n) time. This allows us to to compute the L1-Earth Mover’s Distance on weighted planar point sets in O(n log n) time, which improves the currently best known result of O(n log n). At the expense of a slightly higher time and space complexity we are able to extend our approach to any dimension d ≥ 3. We will further show that our construction is optimal in the sense that there are point sets in the plane where every Manhattan network needs Ω(n log n) vertices and edges.