Parallel scheduling of task trees with limited memory

TL;DR

This paper explores parallel scheduling of tree-shaped task graphs with limited memory, aiming to optimize execution order for minimal memory use and processing time, especially in matrix factorization applications.

Contribution

It extends existing sequential memory minimization algorithms to parallel processing, analyzing complexity and proposing heuristics for trade-offs between memory and time.

Findings

Heuristics effectively balance memory and makespan.

Some heuristics keep memory within specified limits.

Experimental results validate heuristic performance on realistic trees.

Abstract

This paper investigates the execution of tree-shaped task graphs using multiple processors. Each edge of such a tree represents some large data. A task can only be executed if all input and output data fit into memory, and a data can only be removed from memory after the completion of the task that uses it as an input data. Such trees arise, for instance, in the multifrontal method of sparse matrix factorization. The peak memory needed for the processing of the entire tree depends on the execution order of the tasks. With one processor the objective of the tree traversal is to minimize the required memory. This problem was well studied and optimal polynomial algorithms were proposed. Here, we extend the problem by considering multiple processors, which is of obvious interest in the application area of matrix factorization. With multiple processors comes the additional objective to minimize the time needed to traverse the tree, i.e., to minimize the makespan. Not surprisingly, this problem proves to be much harder than the sequential one. We study the computational complexity of this problem and provide inapproximability results even for unit weight trees. We design a series of practical heuristics achieving different trade-offs between the minimization of peak memory usage and makespan. Some of these heuristics are able to process a tree while keeping the memory usage under a given memory limit. The different heuristics are evaluated in an extensive experimental evaluation using realistic trees.