The ability to provide uniform shared-memory access to a significant number of processors in a single SMP node brings us much closer to the ideal PRAM parallel computer. In this paper, we develop new techniques for designing a uniform shared-memory algorithm from a PRAM algorithm and present the results of an extensive experimental study demonstrating that the resulting programs scale nearly linearly across a significant range of processors (from 1 to 64) and across the entire range of instance sizes tested. This linear speedup with the number of processors is, to our knowledge, the first ever attained in practice for intricate combinatorial problems. The example we present in detail here is a graph decomposition algorithm that also requires the computation of a spanning tree; this problem is not only of interest in its own right, but is representative of a large class of irregular combinatorial problems that have simple and efficient sequential implementations and fast PRAM algorithms, but have no known efficient parallel implementations. Our results thus offer promise for bridging the gap between the theory and practice of shared-memory parallel algorithms.

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The rigidity of squares of graphs in three-space has important applications to the study of flexibility in molecules. The Molecular Conjecture, posed in 1984 by T-S. Tay and W. Whiteley, states that the square G 2 of a graph G of minimum degree at least two is rigid if and only if G has six spanning trees which cover each edge of G at most five times. We give a lower bound on the degrees of freedom of G 2 in terms of forest covers of G. This provides a self-contained proof that the existence of the above six spanning trees is a necessary condition for the rigidity of G2. In addition, we prove that the truth of the Molecular Conjecture would imply that our lower bound is tight, and would also imply that a conjecture of Jacobs on ‘independent ’ squares is valid. 1