### Eigenanalysis-based task mapping on parallel computers with cellular networks

Peng Zhang, Yuxiang Gao, Janet Fierson, Yuefan Deng
2013 Mathematics of Computation
Through eigenanalysis of communication matrices, we develop a new objective function formulation for mapping tasks to parallel computers with cellular networks. This new formulation significantly speeds up the solution process through consideration of the symmetries in the supply matrix of a network and a transformation of the demand matrix of any application. The extent of the speedup is not easily obtainable through analytical means for most production networks. However, numerical experiments
more » ... of mapping wave equations on 2D mesh onto 3D torus networks by simulated annealing demonstrate a far superior convergence rate and quicker escape from local minima with our new formulation than with the standard graph theory-based one. of popular networks. In Section 5, we validate and assess the value of our new formulation through analytical and numerical approaches by exploring the symmetries of networks. Conclusions are given in Section 6. Graph theory-based mapping formulation The problem of mapping n tasks of a parallel application onto p nodes of a networked system is well studied      14, 16, 17] . It is equivalent to the fundamental quadratic assignment problem (QAP)  when n = p. The basic task mapping graph formulation can be reformulated to express the hop-bytes metric in terms of the network supply matrix and the application demand matrix. A network topology graph is a directed graph G (P , L), where P and L are the sets of nodes and links. An element p i ∈ P represents a node and p = |P | is the total number of nodes. An element l ij ∈ L represents the connectedness between the node pair p i and p j , where l ij = 1 if there exists a link from p i to p j and l ij = 0 otherwise. |L| / |P | is the average node degree. The adjacency matrix of such a graph, a binary matrix L = [l ij ] ∈ R p×p , characterizes the connection topology of the network.