Summary: A parallel randomized algorithm for finding the connected components of an undirected graph is presented. The algorithm has an expected running time of \(T=O(\log(n))\) with \(P=O((m+n)/\log(n))\) processors, where \(m\) is the number of edges and \(n\) is the number of vertices. The algorithm is optimal in the sense that the product \(P\cdot T\) is a linear function of the input size. The algorithm requires \(O(m+n)\) space, which is the input size, so it is optimal in space as well.