Scheduling Trees of Malleable Tasks for Sparse Linear Algebra

Scientific workloads are often described as directed acyclic task graphs. In this paper, we focus on the multifrontal factorization of sparse matrices, whose task graph is structured as a tree of parallel tasks. Among the existing models for parallel tasks, the concept of \emph{malleable} tasks is especially powerful as it allows each task to be processed on a time-varying number of processors. Following the model advocated by Prasanna and Musicus~\cite{prasmus,prasmus2} for matrix computations, we consider malleable tasks whose speedup is $p^\alpha$, where $p$ is the fractional share of processors on which a task executes, and $\alpha$ ($0 < \alpha \leq 1$) is a parameter which does not depend on the task. We first motivate the relevance of this model for our application with actual experiments on multicore platforms. Then, we study the optimal allocation proposed by Prasanna and Musicus for makespan minimization using optimal control theory. We largely simplify their proofs by resorting only to pure scheduling arguments. Building on the insight gained thanks to these new proofs, we extend the study to distributed multicore platforms. There, a task cannot be distributed among several distributed nodes. In such a distributed setting (homogeneous or heterogeneous), we prove the NP-completeness of the corresponding scheduling problem, and propose some approximation algorithms. We finally assess the relevance of our approach by simulations on realistic trees. We show that the average performance gain of our allocations with respect to existing solutions (that are thus unaware of the actual speedup functions) is up to 16\% for $\alpha=0.9$ (the value observed in the real experiments)