Liang, H.-C.H.-C.LiangHSUEH-I LU2020-05-042020-05-042017https://www.scopus.com/inward/record.uri?eid=2-s2.0-85018691610&doi=10.1137%2f16M1057152&partnerID=40&md5=6582f4255bf3c69601bc065db978b91cLet G be an n-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of G with minimum weight and (2) a cycle of G with minimum weight. The best previously known algorithm for the former problem, running in O(n log3 n) time, can be obtained from the algorithm of Laçki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the O(n log3 n)-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in planar graphs, we solve both problems in O(n log n log log n) time via a divide-and-conquer algorithm that finds a shortest nondegenerate cycle. The kernel of our result is an O(n log log n)-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirected plane graph. © 2017 Society for Industrial and Applied Mathematics.Girth; Minimum cut; Planar graph; Shortest cycleDirected graphs; Graphic methods; Problem solving; Divide-and-conquer algorithm; Girth; Minimum cut; Minimum weight; Nondegenerate; Planar graph; Shortest cycle; Time algorithms; Graph theoryjournal article10.1137/16M10571522-s2.0-85018691610