Finding All Useless Arcs in Directed Planar Graphs

We present a linear-time algorithm for simplifying flow networks on directed planar graphs: Given a directed planar graph on n vertices, a source vertex s and a sink vertex t, our algorithm removes all the arcs that do not participate in any simple s,t-path in linear-time. The output graph produced by our algorithm satisfies the prerequisite needed by the O(nlog n)-time algorithm of Weihe [FOCS'94 & JCSS'97] for computing maximum s,t-flow in directed planar graphs. Previously, Weihe's algorithm could not run in O(nlog n)-time due to the absence of the preprocessing step; all the preceding algorithms run in Ω̃(n^2)-time [Misiolek-Chen, COCOON'05 & IPL'06; Biedl, Brejová and Vinar, MFCS'00]. Consequently, this provides an alternative O(nlog n)-time algorithm for computing maximum s,t-flow in directed planar graphs in addition to the known O(nlog n)-time algorithms [Borradaile-Klein, SODA'06 & J.ACM'09; Erickson, SODA'10]. Our algorithm can be seen as a (truly) linear-time s,t-flow sparsifier for directed planar graphs, which runs faster than any maximum s,t-flow algorithm (which can also be seen of as a sparsifier). The simplified structures of the resulting graph might be useful in future developments of maximum s,t-flow algorithms in both directed and undirected planar graphs.