A shortest cycle for each vertex of a graph

We present an algorithm that finds, for each vertex of an undirected graph, a shortest cycle containing it. While for directed graphs this problem reduces to the All-Pairs Shortest Paths problem, this is not known to be the case for undirected graphs. We present a truly sub-cubic randomized algorithm for the undirected case. Given an undirected graph with n vertices and integer weights in 1, . . . ,M , it runs in Õ( √ Mn) time where ω < 2.376 is the exponent of matrix multiplication. As a by-product, our algorithm can be used to determine which vertices lie on cycles of length at most t in Õ(Mnt) time. For the case of bounded real edge weights, a variant of our algorithm solves the problem up to an additive error of in Õ(n) time.