Constant girth approximation for directed graphs in subquadratic time

In this paper we provide a Õ(m√n) time algorithm that computes a 3-multiplicative approximation of the girth of a n-node m-edge directed graph with non-negative edge lengths. This is the first algorithm which approximates the girth of a directed graph up to a constant multiplicative factor faster than All-Pairs Shortest Paths (APSP) time, i.e. O(mn). Additionally, for any integer k ≥ 1, we provide a deterministic algorithm for a O(kloglogn)-multiplicative approximation to the girth in directed graphs in Õ(m1+1/k) time. Combining the techniques from these two results gives us an algorithm for a O(klogk)-multiplicative approximation to the girth in directed graphs in Õ(m1+1/k) time. Our results naturally also provide algorithms for improved constructions of roundtrip spanners, the analog of spanners in directed graphs. The previous fastest algorithms for these problems either ran in All-Pairs Shortest Paths (APSP) time, i.e. O(mn), or were due Pachocki  which provided a randomized algorithm that for any integer k ≥ 1 in time Õ(m1+1/k) computed with high probability a O(klogn) multiplicative approximation of the girth. Our first algorithm constitutes the first sub-APSP-time algorithm for approximating the girth to constant accuracy, our second removes the need for randomness and improves the approximation factor in Pachocki , and our third is the first time versus quality trade-off for obtaining constant approximations.