Once again: Finding simple cycles in graphs (original) (raw)

We present a randomized algorithm that computes a simple cycle of length k in general graphs, where k is a xed integer, in O(maxfm; n logng) expected time. This algorithm can be derandomized with only a small loss in e ciency, yielding a deterministic algorithm for this task which runs in O(maxfm log n; n log ng) worst-case time. We show that the randomized algorithm may be parallelized. These algorithms improve upon previous results of many authors. Furthermore, we answer the question of AYZ 94], whether deciding if a given graph contains a triangle is as di cult as boolean multiplication of two n by n matrices, in the negative.