Publications (5)0 Total impact
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ABSTRACT: In this paper, we introduce a class of indicators that enable to compute
efficiently optimal transport plans associated to arbitrary distributions of N
demands and M supplies in R in the case where the cost function is concave. The
computational cost of these indicators is small and independent of N. A
hierarchical use of them enables to obtain an efficient algorithm.
02/2011;
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ABSTRACT: Consider a real line equipped with a (not necessarily intrinsic) distance. We
deal with the minimum-weight perfect matching problem for a complete graph
whose points are located on the line and whose edges have weights equal to
distances along the line. This problem is closely related to one-dimensional
Monge-Kantorovich trasnport optimization. The main result of the present note
is a "bottom-up" recursion relation for weights of partial minimum-weight
matchings.
02/2011;
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ABSTRACT: Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations. For the case of $N$ real-valued point masses we present an O(N |log epsilon|) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations. Comment: Added affiliation for the third author in arXiv metadata; no change in the source. AMS-LaTeX, 20 pages, 5 figures (pgf/TiKZ and embedded PostScript). Article accepted to SIAM J. Applied Math
02/2009;
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ABSTRACT: In this note, we introduce a class of indicators that enable to compute efficiently optimal transport plans associated to arbitrary distributions of $N$ demands and $N$ supplies in $\mathbf{R}$ in the case where the cost function is concave. The computational cost of these indicators is small and independent of $N$. A hierarchical use of them enables to obtain an efficient algorithm. oui
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ABSTRACT: In this note, we introduce a class of indicators that enable to compute efficiently optimal transport plans associated to arbitrary distributions of $N$ demands and $N$ supplies in $\mathbf{R}$ in the case where the cost function is concave. The cost of these indicators is small and independent of $N$. Using them recursively according to a particular algorithm allows to find an optimal transport plan in less than $N^2$ evaluations of the cost function.
