Space-Efficient approximation scheme for circular earth mover distance
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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We show that any $L_1$ embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid $\{0,1,\ldots,n\}^2 \subseteq \mathbb{R}^2$ incurs distortion $\Omega \left(\sqrt{\log n}\right)$. We also use Fourier analytic techniques to construct a simple $L_1$ embedding of this space which has distortion $O(\log n)$.