Connected components algorithms for mesh-connected parallel computers

We present a new CREW PRAM algorithm for nding connected components. For a graphG with n vertices andm edges, algorithmA0 requires at mostO(logn) parallel steps and performsO((n+m) logn) work in the worst case. The advantage our algorithm has over others in the literature is that it can be adapted to a 2-D mesh-connected communication model in which all CREW operations are replaced byO(logn) parallel row and column operations without increasing the time complexity. We present the mapping of A0 to a mesh-connected computer and describe two implementations, A1 and A2. Algorithm A1, which uses an adjacency matrix to represent the graph, performs O(n logn) work. Hence, it only achieveswork e ciency on dense graphs. The second implementation,A2, uses a sparse representation of the adjacency matrix and again performs O(logn) row and column operations but reduces the work to O((m + n) logn) on all graphs. We report MasPar MP-1 performance gures for implementations of the algorithms described. The implementations are exercised on a variety of parametrically generated graphs, di ering in structure and connectivity. These graphs are generated externally and read in as input for the algorithms, permitting comparison of di erent implementations on identical graphs.