Topology-induced Enhancement of Mappings

Abstract

In this paper we propose a new method to enhance a mapping $\mu(\cdot)$ of a parallel application's computational tasks to the processing elements (PEs) of a parallel computer. The idea behind our method \mswap 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 $\mu^{-1}(\cdot)$ and extending them to be unique on $V_a$, we can enhance $\mu(\cdot)$ by swapping labels of $V_a$ in a new way. Pairs of swapped labels are local \wrt the PEs, but not \wrt $G_a$. Moreover, permutations of the bitvectors' entries give rise to a plethora of hierarchies on the PEs. Through these hierarchies we turn \mswap into a hierarchical method for improving $\mu(\cdot)$ that is complementary to state-of-the-art methods for computing $\mu(\cdot)$ in the first place. In our experiments we use \mswap 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.