Section: New Results

Scheduling and placement for HPC

In High Performance Computing, heterogeneity is now the norm with specialized accelerators like GPUs providing efficient computational power. Resulting complexity led to the development of task-based runtime systems, where complex computations are described as task graphs, and scheduling decisions are made at run-time to perform load balancing between all resources of the platforms. In [2], we consider the problem of developing good scheduling strategies, even at the scale of a single node, and analyzing them both theoretically and in practice is expected to have a very high impact on the performance of current HPC systems. The special case of two kinds of resources, typically CPUs and GPUs is already of great practical interest. The scheduling policy Hetero-Prio has been proposed in the context of fast multipole computations (FMM), and has been extended to general task graphs with very promising results. In this paper, we provide a theoretical study of the performance of HeteroPrio, by proving approximation bounds compared to the optimal schedule, both in the case of independent tasks and in the case of general task graphs. Interestingly, our results establish that spoliation (a technique that enables resources to restart uncompleted tasks on another resource) is enough to prove bounded approximation ratios for a list scheduling algorithm on two unrelated resources, which is known to be impossible otherwise. This result holds true both for independent and dependent tasks graphs. Additionally, we provide an experimental evaluation of HeteroPrio on real task graphs from dense linear algebra computation, that establishes its strong performance in practice.

In [1], we consider the problem of partitioning a matrix into a set of sub-matrices, that has received increased attention recently and is crucial when considering dense linear algebra and kernels with similar communication patterns on heterogeneous platforms. The problem of load balancing and minimizing communication is traditionally reducible to an optimization problem that involves partitioning a square into rectangles. This problem has been proven to be NP-Complete for an arbitrary number of partitions. In this paper, we present recent approaches that relax the restriction that all partitions be rectangles. The first approach uses an original mathematical technique to find the exact optimal partitioning. Due to the complexity of the technique, it has been developed for a small number of partitions only. However, even at a small scale, the optimal partitions found by this approach are often non-rectangular and sometimes non-intuitive. The second approach is the study of approximate partitioning methods by recursive partitioning algorithms. In particular we use the work on optimal partitioning to improve preexisting algorithms. In this paper we discuss the different perspectives it opens and present two algorithms, SNRPP which is a $\sqrt{3/2}$ approximation, and NRPP which is a $2/\sqrt{3}$ approximation. While sub-optimal, this approach works for an arbitrary number of partitions. We use the first exact approach to analyze how close to the known optimal solutions the NRRP algorithm is for small numbers of partitions.

In  [12], we consider the problem of data allocation when performing matrix multiplication on a heterogeneous node, with multicores and GPUs. Classical (cyclic) allocations designed for homogeneous settings are not appropriate, but the advent of task-based runtime systems makes it possible to use more general allocations. Previous theoretical work has proposed square and cube partitioning algorithms aimed at minimizing data movement for matrix multiplication. We propose techniques to adapt these continuous square partitionings to allocating discrete tiles of a matrix, and strategies to adapt the static allocation at run-time. We use these techniques in an implementation of Matrix Multiplication based on the StarPU runtime system, and we show through extensive experiments that this implementation allows to consistently obtain a lower communication volume while improving slightly the execution time, compared to standard state-of-the-art dynamic strategies.