Improved Approximation Algorithms for Earth-Mover Distance in Data Streams

For two multisets S and T of points in [∆], such that |S| = |T | = n, the earth-mover distance (EMD) between S and T is the minimum cost of a perfect bipartite matching with edges between points in S and T , i.e., EMD(S, T ) = minπ:S→T ∑ a∈S ||a−π(a)||1, where π ranges over all one-to-one mappings. The sketching complexity of approximating earth-mover distance in the two-dimensional grid is mentioned as one of the open problems in [16, 11]. We give two algorithms for computing EMD between two multi-sets when the number of distinct points in one set is a small value k = log(∆n). Our first algorithm gives a (1 + )-approximation using O(k −2 log n) space and works only in the insertion-only model. The second algorithm gives a O(min(k, log∆))approximation using O(log∆ · log log∆ · logn)-space in the turnstile model.