  Let $G$ 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 $O(n\log^3 n)$
time, can be obtained from the algorithm of \Lacki, Nussbaum, Sankowski, and
Wulff-Nilsen for single-source all-sinks maximum flows. The best previously
known result for the latter problem is the $O(n\log^3 n)$-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 non-degenerate 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.
