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**1 - 1**of**1**### Improved Approximation Algorithms for Earth-Mover Distance in Data Streams

, 2014

"... For two multisets S and T of points in [∆]2, 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) = minpi:S→T a∈S ||a−pi(a)||1, where pi ranges over all one-to-one mappin ..."

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For two multisets S and T of points in [∆]2, 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) = minpi:S→T a∈S ||a−pi(a)||1, where pi 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 = logO(1)(∆n). Our first algorithm gives a (1 + )-approximation using O(k−2 log4 n) space and works only in the insertion-only model. The second algorithm gives a O(min(k3, log∆))-approximation using O(log3 ∆ · log log ∆ · logn)-space in the turnstile model.