Consider a digraph with n vertices. For any fixed value k, we present linear-and almostlinear-time algorithms to find a cycle of length ≥ k, if one exists. We also find a cycle that has length ≥ log n/ log log n in polynomial time, if one exists. Under an appropriate complexity assumption it is known to be impossible to improve this guarantee by more than a log log n factor. Our approach is based on depth-first search.

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