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12.6.3 Parallel Cluster Algorithms

As with the percolation models upon which the cluster algorithms are based, the phase transition in a spin model occurs when the clusters of bonded spins become large enough to span the entire lattice. Thus, near criticality (which in most cases is where we want to perform the simulation), clusters come in all sizes, from order N (where N is the number of sites in the lattice) right down to a single site. The highly irregular and nonlocal nature of the clusters means that cluster update algorithms do not vectorize well and hence give poor performance on vector machines. On this problem, a CRAY X-MP  is only about ten times faster than a Sun 4 workstation. The irregularity of the clusters also means that SIMD machines are not well suited to this problem [Apostolakis:92a;93a], [Baillie:91a], [Brower:91a], whereas for the Metropolis  type algorithms, they are perhaps the best machines available. It therefore appears that the optimum performance for this type of problem will come from MIMD parallel computers.

A parallel cluster algorithm involves distributing the lattice onto an array of processors using the usual domain decomposition. Clearly, a sequential algorithm can be used to label the clusters on each processor, but we need a procedure for converting these labels to their correct global values. We need to be able to tell many processors, which may be any distance apart, that some of their clusters are actually the same, to agree on which of the many different local labels for a given cluster should be assigned to be the global cluster label, and to pass this label to all the processors containing a part of that cluster. We have implemented two such algorithms, ``self-labelling''  and ``global equivalencing'' [Baillie:91a], [Coddington:90a].

Guy Robinson
Wed Mar 1 10:19:35 EST 1995