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We present and analyze a portable, high-performance algorithm for finding connected components on modern distributed memory multiprocessors. The algorithm is a hybrid of the classic DFS on the subgraph local to each processor and a variant of the Shiloach-Vishkin PRAM algorithm on the global collection of subgraphs. We implement the algorithm in Split-C and measure performance on the the Cray T3D, the Meiko CS-2, and the Thinking Machines CM-5 using a class of graphs derived from cluster dynamics methods in computational physics. On a 256 processor Cray T3D, the implementation outperforms all previous solutions by an order of magnitude. A characterization of graph parameters allows us to select graphs that highlight key performance features. We study the effects of these parameters and machine characteristics on the balance of time between the local and global phases of the algorithm and find that edge density, surface-to-volume ratio, and relative communication cost dominate performance. By understanding the effect of machine characteristics on performance, the study sheds light on the impact of improvements in computational and/or communication performance on this challenging problem.