A direct comparison of three algorithms for reducing profile and wavefront

Abstract

The effectiveness of the reverse Cuthill-McKee, Gibbs-King and Sloan ordering algorithms for reducing the profile and wavefront of a sparse matrix with a symmetric pattern of zeros is assessed by applying them to each of Everstine's 30 test problems and comparing the results with those obtained by Armstrong's simulated annealing algorithm. This last procedure, although much too slow for practical use, is believed to give profiles which are close to the minimum and thus provides useful benchmark results for comparing possible procedures. For the test problems of Everstine, the reverse Cuthill-McKee procedure furnishes profiles which, on average, are 31% above those produced by the simulated annealing method. In the worst case, the reverse Cuthill-McKee algorithm yields a profile which is 114% in excess of the corresponding simulated annealing profile. The Gibbs-King scheme is generally more reliable than the reverse Cuthill-McKee scheme, but still yields an average profile which is 22% above that produced by the simulated annealing algorithm. Its worst-case performance gives a profile which is 67% in excess of the simulated annealing profile. The Sloan algorithm appears to be a substantial improvement on both the reverse Cuthill-McKee and Gibbs-King methods when measured by its average and worst-case profiles which are, respectively, 10 and 24% above those of the simulated annealing method. The root-mean-square wavefront reductions for the three algorithms mimic those of the profile reductions except that, in general, they are slightly further from the optimum values. In addition to those for Everstine's cases, some test results for a set of rectangular grid problems are also given. These examples correspond to meshes of 2D linear and quadratic finite elements which are used frequently in practice. The relative performance of the various algorithms is generally similar to that for Everstine's test problems although the spread of results is reduced significantly.