Parallel graph algorithms for hypercube computers

Abstract This paper presents several parallel algorithms on unweighted graphs for hypercube computers. The algorithms are for checking bipartiteness and for finding a spanning forest, the connected components, a fundamental cycle set, and the bridges of a graph. The algorithm for finding spanning forest is based on a strategy of successive elimination of non-forest edges. The input graph is partitioned equally among processors, which repeatedly eliminate non-forest edges and merge their results to finally construct the desired forest of the entire graph. In all the algorithms, low communication overhead is achieved by restricting the message-flow to only between the neighboring processors. The spanning-forest algorithm is used as a subroutine to design the remaining algortihms. Except for the bridge-finding algorithm, all others achieve optimal speedups for dense as well as sparse graphs, and each algorithm is optimally scalable up to a large number of processors depending upon the density of the input graph. For a graph of n vertices and m edges, the time complexity of the spanning-forest algorithm, using p processors, is O( m / p + n log p ), which corresponds to an optimal speedup for p ⩽( m / n )/(1+log( m / n )).

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