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Matching point sets with respect to the Earth Mover's Distance
Cabello, Sergio ...
Let ▫$A$▫ and ▫$B$▫ be two sets of ▫$m$▫ resp. ▫$n$▫ weighted points in the plane, with ▫$m \leq n$▫. We present ▫$(1+\epsilon)$▫ and ▫$(2+\epsilon)$▫-approximation algorithms for the minimum ...Euclidean Earth Mover's Distance between ▫$A$▫ and ▫$B$▫ under translations and rigid motions respectively. In the general case where the sets have unequal total weights the algorithms run in ▫$O((n^3 m/\epsilon^4) \log^2 (n/\epsilon))$▫ time for translations and ▫$O((n^4 m^2/\epsilon^{4}) \log^{2}(n/\epsilon))$▫ time for rigid motions. When the sets have equal total weights, the respective running times decrease to ▫$O((n^2 /\epsilon^{4}) \log^{2}(n/\epsilon))$▫ and ▫$O((n^3 m/\epsilon^{4}) \log^{2}(n/\epsilon))$▫. We also show how to compute a ▫$(1+\epsilon)$▫ and ▫$(2+\epsilon)$▫-approximation of the minimum cost Euclidean bipartite matching under translations and rigid motions in ▫$O((n^{3/2}/\epsilon^{7/2}) \log^{5}n)$▫ and ▫$O((n/\epsilon)^{7/2} \log^{5}n)$▫ time respectively.
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