Finding Triconnected Components by Local Replacements

We present a parallel algorithm for nding triconnected components on a CRCW PRAM. The time complexity of our algorithm is O(logn) and the processor-time product is O((m + n) log logn) where n is the number of vertices, and m is the number of edges of the input graph. Our algorithm, like other parallel algorithms for this problem, is based on open ear decomposition but it employs a new technique, local replacement, to improve the complexity. Only the need to use the subroutines for connected components and integer sorting, for which no optimal parallel algorithm that runs in O(log n) time is known, prevents our algorithm from achieving optimality.