Flow in planar graphs with multiple sources and sinks

Given a planar network with many sources and sinks, the problem of computing the maximum flow from the sources to the sinks is investigated. An algorithm that runs in O(log²n ) time using O(n^1.5) processors on an exclusive-read-exclusive-write parallel random-access machine (EREW PRAM) is obtained, when the amount of flow (demand) at each source and sink is assumed as input. When the demands are unknown, the problem remains open. However, in the special case in which the sources and sinks are all on one face (and the demands unknown), an algorithm that computes the maximum flow with time complexity O(log³n log log n) using O(n^1.5) processors is given. The results also hold for more general networks, namely, when the edge capacities have both lower and upper bounds