On constant factor approximation for earth mover distance over doubling metrics

Given a metric space (X, dX), the earth mover distance between two distributions over X is defined as the minimum cost of a bipartite matching between the two distributions. The doubling dimension of a metric (X, dX) is the smallest value α such that every ball in X can be covered by 2 ball of half the radius. A metric (or a sequence of metrics) is called doubling precisely if its doubling dimension is bounded. We study the efficient algorithms for approximating earth mover distance over doubling precisely metrics. Our first result is a near linear time (in the size of the X) algorithm for estimating EMD over doubling metric X , with a O(αX) approximation ratio, where αX is the doubling dimension of X . Given a metric (X, dX), we can use Õ(n ) preprocessing time to create a data structure of size Õ(n), such that subsequent EMD queries can be answered in Õ(n) time, with approximation ratio O(αX/ǫ). Our second result is an encoding scheme, which is a weaker form of sketching. In an encoding scheme, distributions are encoded, such that the EMD between two distributions can be estimated in sub linear time, given the encodings of the two distributions. In particular, given (X, dX), by using Õ(n ) preprocessing time, every subsequent distribution μ can be encoded into F (μ) in Õ(n) time. The query for EMD between μ and ν can be answered in Õ(n) time, with approximation ratio O(αX/ǫ), given the two encodings F (μ) and F (ν). The encoding scheme has immediate applications. In a 2-player game where 1 player knows μ and the other knows ν, there is a communication protocol with small communication complexity, through which the two players can approximate the EMD between μ and ν. Another application is distance oracle, where we are given a metric (X, dX) and s distributions μ1, μ2, · · · , μs overX , we can use Õ(n+sn) preprocessing time, creating a data structure of size Õ(n + sn), such the query for EMD between μi and μj can be answered in Õ(n ) time, with approximation ratio O(αX/ǫ).