Scheduling Associative Reductions with Homogeneous Costs when Overlapping Communications and Computations

Reduction is a core operation in parallel computing. Optimizing its cost has a high potential impact on the application execution time, particularly in MPI and MapReduce computations. In this paper, we propose an optimal algorithm for scheduling associative reductions. We focus on the case where communications and computations can be overlapped to fully exploit resources. Our algorithm greedily builds a spanning tree by starting from the sink and by adding a parent at each iteration. Bounds on the completion time of optimal schedules are then characterized. To show the algorithm extensibility, we adapt it to model variations in which either communication or computation resources are limited. Moreover, we study two specific spanning trees: while the binomial tree is optimal when there is either no transfer or no computation, the Fibonacci tree is optimal when the transfer cost is equal to the computation cost. Finally, approximation ratios of strategies that are derived from those trees are drawn.