Topology-induced Enhancement of Mappings
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In this paper we propose a new method to enhance a mapping μ(·) of a parallel application's computational tasks to the processing elements (PEs) of a parallel computer. The idea behind our method is to enhance such a mapping by drawing on the observation that many topologies take the form of a partial cube. This class of graphs includes all rectangular and cubic meshes, any such torus with even extensions in each dimension, all hypercubes, and all trees. Following previous work, we represent the parallel application and the parallel computer by graphs G_a = (V_a, E_a) and G_p = (V_p, E_p). G_p being a partial cube allows us to label its vertices, the PEs, by bitvectors such that the cost of exchanging one unit of information between two vertices u_p and v_p of G_p amounts to the Hamming distance between the labels of u_p and v_p. By transferring these bitvectors from V_p to V_a via μ^-1(·) and extending them to be unique on V_a, we can enhance μ(·) by swapping labels of V_a in a new way. Pairs of swapped labels are local the PEs, but not G_a. Moreover, permutations of the bitvectors' entries give rise to a plethora of hierarchies on the PEs. Through these hierarchies we turn into a hierarchical method for improving μ(·) that is complementary to state-of-the-art methods for computing μ(·) in the first place. In our experiments we use to enhance mappings of complex networks onto rectangular meshes and tori with 256 and 512 nodes, as well as hypercubes with 256 nodes. It turns out that common quality measures of mappings derived from state-of-the-art algorithms can be improved considerably.
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