An efficient and fast parallel-connected component algorithm

A parallel algorithm for computing the connected components of undirected graphs is presented. Shared memory computation models are assumed. For a graph of <italic>e</italic> edges and <italic>n</italic> nodes, the time complexity of the algorithm is &Ogr;(<italic>e/p</italic> + (<italic>n</italic> log <italic>n</italic>)/<italic>p</italic> + log<supscrpt>2</supscrpt><italic>n</italic>) with <italic>p</italic> processors. The algorithm can be further refined to yield time complexity &Ogr;(<italic>H</italic>(<italic>e</italic>, <italic>n</italic>, <italic>p</italic>)/<italic>p</italic> + (<italic>n</italic> log <italic>n</italic>)/(<italic>p</italic> log(<italic>n</italic>/<italic>p</italic>)) + log<supscrpt>2</supscrpt><italic>n</italic>), where <italic>H</italic>(<italic>e, n, p</italic>) is very close to &Ogr;(<italic>e</italic>). These results show that linear speedup can be obtained for up to <italic>p</italic> ≤ <italic>e</italic>/log<supscrpt>2</supscrpt><italic>n</italic> processors when <italic>e</italic> ≥ <italic>n</italic> log <italic>n</italic>. Linear speedup can still be achieved with up to <italic>p</italic> ≤ <italic>n</italic>ε processors, 0 ≤ ε < 1, for graphs satisfying <italic>e</italic> ≥ <italic>n</italic> log<supscrpt>(*)</supscrpt><italic>n</italic>. Our results can be further improved if a more efficient integer sorting algorithm is available.

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