Flow in Planar Graphs with Multiple Sources and Sinks

Abstract

The problem of maximum flow in planar graphs has always been investigated under the assumption that there is only one source and one sink. Here we consider the case where there are many sources and sinks (single commodity) in a directed planar graph. An algorithm for the case when the demands of the sources and sinks are fixed and given in advance is presented. It can be implemented efficiently sequentially and in parallel and its complexity is dominated by the complexity of computing all shortest paths from a single source in a planar graph. If the demands are not known, an algorithm for computing the maximum flow is presented for the case where the number of faces that contain sources and sinks is bounded by a slowly growing function. Our result places the problem of computing a perfect matching in a planar bipartite graph in NC and it improves a previous parallel algorithm for the case of a single source, single sink in a planar directed1nd undirected) graph, both in terms of processor bounds and in its simple presentation.