Abstract: "Assignment of tasks of a parallel program onto processors of a distributed-memory architecture determines the completion time of the program. The problem is to determine an assignment which minimizes communication overhead leading to minimal completion time. This paper presents assignment heuristics by taking into account two major factors that determine interprocessor communication overhead, temporal link contention and unequal distances between processors on a given topology. We use a new Temporal Communication Graph (TCG) to model task graphs. Using this model, we identify communication steps that conflict both temporally and spatially. Further, this model is strong enough to identify critical communication edges which make the completion time longer. Assignment heuristics are developed to reduce contention on critical communication edges. An iterative heuristic derives optimal assignment by pair-wise exchanging of processors, involved with the critical communication edge, within d-hops. The goodness of our heuristics are verified through simulation experiments by applying them to random task graphs with a variety of topologies (hypercubes and meshes) using wormhole e-cube routing. The heuristics are general and can be applied to programs with regular/irregular communication and any host system exhibiting message transfer over minimal distance."