Girth of a Planar Digraph with Real Edge Weights in O(n(log n)^3) Time

Abstract: The girth of a graph is the length of its shortest cycle. We give an algorithm that computes in O(n(log n)^3) time and O(n) space the (weighted) girth of an n-vertex planar digraph with arbitrary real edge weights. This is an improvement of a previous time bound of O(n^3/2), a bound which was only valid for non-negative edge-weights. Our algorithm can be modified to output a shortest cycle within the same time and space bounds if such a cycle exists.