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We give a new characterization of the planar separator theorem in terms of mutually non-containing closed Jordan Curves. using this, we develop an O(n √n logn) maximum flow algorithm for directed planar networks (hence for any planar network).
Algorithms are given that compute maximum flows in planar directed networks either in O((logn)3) parallel time using O(n4) processors or O((logn)2) parallel time using O(n6) processors. The resource consumption of these algorithms is dominated by the cost of finding the value of a maximum flow. When such a value is given, or when the computation is on an undirected network, the bound is O((logn)2)...
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