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(m 1+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(m 1+1/k ) time. Our results naturally also provide algorithms for improved constructions of roundtrip spanners, the analog of spanners in directed graphs.

In this paper we propose a new segmentation algorithm which combines patch-based information with edge cues under a probabilistic framework. We use a mixture of multiple Gaussians for building the statistical model with color and spatial features, and we incorporate edge information based on texture, color and brightness differences into the EM algorithm. We evaluate our results qualitatively and quantitatively on a large data-set of natural images and compare our results to other state-of-the-art methods.