# Improved Girth Approximation and Roundtrip Spanners

In this paper we provide improved algorithms for approximating the girth and producing roundtrip spanners of n-node m-edge directed graphs with non-negative edge lengths. First, for any integer k > 1, we provide a deterministic Õ(m^1+ 1/k) time algorithm which computes a O(k n) multiplicative approximation of the girth and a O(k n) multiplicative roundtrip spanner with Õ(n^1+1/k) edges. Second, we provide a randomized Õ(m√(n)) time algorithm that with high probability computes a 3-multiplicative approximation to the girth. Third, we show how to combine these algorithms to obtain for any integer k > 1 a randomized algorithm which in Õ(m^1+ 1/k) time computes a O(k k) multiplicative approximation of the girth and O(k k) multiplicative roundtrip spanner with high probability. The previous fastest algorithms for these problems either ran in All-Pairs Shortest Paths (APSP) time, i.e. Õ(mn), or were due Pachocki et al (SODA 2018) which provided a randomized algorithm that for any integer k > 1 in time Õ(m^1+1/k) computed with high probability a O(k n) multiplicative approximation of the girth and a O(k n) multiplicative roundtrip spanners with Õ(n^1+1/k) edges. Our first algorithm removes the need for randomness and improves the approximation factor in Pachocki et al (SODA 2018), our second is constitutes the first sub-APSP-time algorithm for approximating the girth to constant accuracy with high probability, and our third is the first time versus quality trade-offs for obtaining constant approximations.

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