Time-Optimal and Conflict-Free Mappings of Uniform Dependence Algorithms into Lower Dimensional Processor Arrays

Most existing methods of mapping algorithms into processor arrays are restricted to the case where n-dimensional algorithms or algorithms with n nested loops are mapped into (n—l)-dimensional arrays. However, in practice, it is interesting to map n-dimensional algorithms into (k —l)-dimensional arrays where k\u3c.n. For example, many algorithms at bit-level are at least 4-dimensional (matrix multiplication, convolution, LU decomposition, etc.) and most existing bit level processor arrays are 2-dimensional. A computational conflict occurs if two or more computations of an algorithm are mapped into the same processor and the same execution time. In this paper, necessary and sufficient conditions are derived to identify all mappings without computational conflicts, based on the Hermite normal form of the mapping matrix. These conditions are used to propose methods of mapping any n-dimensional algorithm into (k— l)-dimensional arrays, kn—3, optimality of the mapping is guaranteed

Piece-wise scheduling of composite task graphs onto distributed memory parallel computers

Heuristics for static scheduling of task graphs using list scheduling techniques have continued to improve by adding real-world factors such as processor speed, network transmission speed, interconnection topology, and link contention considerations to the basic task graph model. Yet, the resulting schedules do not fully model program loops and branches, startup costs for both process creation and message initiation, and a number of interesting parallel processing patterns such as meshes, tress, and supervisor/workers. In fact, improvements in the schedule may be obtained when the task graph is regular as when it contains repeated or replicated tasks, divide-and-conquer patterns of communication, or a mesh-structured pattern of computation. In this paper we describe a limited approach to scheduling composite task graphs that considers process and message startup costs, and three regular patterns : replicated, tree, and mesh. The approach is to model programs with such regular patterns as a composite task graph, where each regular structure is a decomposable sub-task node in the task graph. Then, we compute an optimal schedule for each sub-task. graph, piece the sub-tasks together, and perform an ordinary static scheduling heuristic on the pieces, to produce an overall schedule. We define a composite task graph as a hierarchical task graph containing regular-structured sub-task graphs as components. At the top level of this hierarchy, each graph node represents either a simple task or a hierarchically decomposable sub-task graph. We propose a piece-wise scheduling algorithm that simply allocates processors to sub-task graphs according to closed-form expressions which give determine the optimal number of processors, and then uses a list scheduling algorithm to schedule the flattened graph onto these processors. We do not address the pressing problem of loops and branches in the task graph representation, but we speculate that the technique of piece-wise scheduling introduced here can be adapted to a hybrid form of scheduling that may accommodate branches and loops. Piece-wise scheduling is not guaranteed to yield the best global schedule. Rather, it pieces together locally optimum sub-schedules. Finding globally optimum schedules for composite task graphs remains an open problem. We present an heuristic approach that has been experimentally used to schedule small parallel programs with encouraging results. More empirical evidence is needed to determine the usefulness of this technique, but early indications are encouraging