8,670 research outputs found

    Max flow vitality in general and stst-planar graphs

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    The \emph{vitality} of an arc/node of a graph with respect to the maximum flow between two fixed nodes ss and tt is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only 2(n1)2(n-1) max flow instances and, In stst-planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in O(n)O(n) worst-case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in time proportional to the size of the set.Comment: 12 pages, 3 figure

    Single Source - All Sinks Max Flows in Planar Digraphs

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    Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201

    Faster Shortest Paths in Dense Distance Graphs, with Applications

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    We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the linear-time shortest-path algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph GG into regions of at most rr vertices each, for some parameter r<nr < n. The vertex set of the DDG is the set of Θ(n/r)\Theta(n/\sqrt r) vertices of GG that belong to more than one region (boundary vertices). The DDG has Θ(n)\Theta(n) arcs, such that distances in the DDG are equal to the distances in GG. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG (nicknamed FR-Dijkstra) runs in O(nlog(n)r1/2logr)O(n\log(n) r^{-1/2} \log r) time, and is a key component in many state-of-the-art planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the logn\log n dependency in the running time of the shortest-path algorithm, making it O(nr1/2log2r)O(n r^{-1/2} \log^2r). This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly linear-time, algorithms for problems such as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in O(nlogp)O(n \log p) time, where pp is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with rr vertices and kk boundary vertices in O(rlogk)O(r \log k) time, which is faster than has been previously known for small values of kk

    Maximum st-flow in directed planar graphs via shortest paths

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    Minimum cuts have been closely related to shortest paths in planar graphs via planar duality - so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason - so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no constraints on the source and sink. We believe this a promising avenue for developing algorithms that are more practical than the current asymptotically best algorithms for maximum st-flow.Comment: 20 pages, 4 figures. Short version to be published in proceedings of IWOCA'1
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