Many scientific simulations, using the Message Passing Interface (MPI) programming model, are sensitive to the performance and scalability of reduction collective operations such as MPI_Allreduce and MPI_Reduce. These operations are the most widely used abstractions to perform mathematical operations over all processes that are part of the simulation. In this work, we propose a hierarchical design to implement the reduction operations on multicore systems. This design aims to improve the efficiency of reductions by 1) tailoring the algorithms and customizing the implementations for various communication mechanisms in the system 2) providing the ability to configure the depth of hierarchy to match the system architecture, and 3) providing the ability to independently progress each of this hierarchy. Using this design, we implement MPI_Allreduce and MPI_Reduce operations (and its nonblocking variants MPI_Iallreduce and MPI_Ireduce) for all message sizes, and evaluate on multiple architectures including InfiniBand and Cray XT5. We leverage and enhance our existing infrastructure, Cheetah, which is a framework for implementing hierarchical collective operations to implement these reductions. The experimental results show that the Cheetah reduction operations outperform the production-grade MPI implementations such as Open MPI default, Cray MPI, and MVAPICH2, demonstrating its efficiency, flexibility and portability. On Infini-Band systems, with a microbenchmark, a 512-process Cheetah nonblocking Allreduce and Reduce achieves a speedup of 23x and 10x, respectively, compared to the default Open MPI reductions. The blocking variants of the reduction operations also show similar performance benefits. A 512-process nonblocking Cheetah Allreduce achieves a speedup of 3x, compared to the default MVAPICH2 Allreduce implementation. On a Cray XT5 system, a 6144-process Cheetah Allreduce outperforms the Cray MPI by 145%. The evaluation with an application kernel, Conjugate Gradie- t solver, shows that the Cheetah reductions speeds up total time to solution by 195%, demonstrating the potential benefits for scientific simulations.
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