Partitioning Tree-Shaped Task Graphs for Distributed Platforms With Limited Memory

Abstract

Scientific applications are commonly modeled as the processing of directed acyclic graphs of tasks, and for some of them, the graph takes the special form of a rooted tree. This tree expresses both the computational dependencies between tasks and their storage requirements. The problem of scheduling/traversing such a tree on a single processor to minimize its memory footprint has already been widely studied. The present paper considers the parallel processing of such a tree and study how to partition it for a homogeneous multiprocessor platform, where each processor is equipped with its own memory. We formally state the problem of partitioning the tree into connected parts such that each part can be processed on a single processor and the total resulting processing time is minimized. We prove that the problem is NP-complete, and we design polynomial-time heuristics to address it. An extensive set of simulations demonstrates the usefulness of these heuristics.

Mapping Tree-Shaped Workflows on Memory-Heterogeneous Architectures

“…Note that the problem of makespan minimization of a tree of tasks, by partitioning the tree so that each part fits (memory-wise) onto a processor, has already been tackled in the case of homogeneous processors [10]. As pointed out in Section 1, recent work by He et al has attempted to extend this approach to heterogeneous architectures [11].…”

“…One common form of such DAGs is a rooted directed tree, which we consider in this paper. These tree-shaped workflows occur in a variety of applications, for example in sparse matrix factorizations and computational physics [16,10].…”

“…To this end, we assume the workflow and its properties to be known before scheduling. Previous work [10] for completely homogeneous clusters (or other homogeneous platforms) showed the corresponding scheduling problem to be NP-complete and proposed several variants of a successful three-step heuristic. In summary, these steps are (i) to partition the tree into subtrees, minimizing the makespan and not taking the memory limit into account, (ii) to further partition subtrees too big for the memory limit, and finally (iii) to ensure that the number of subtrees is less than or equal to the number of processors.…”

“…To this end, after discussing related work (Section 2), we formalize the scheduling problem in Section 3. The algorithmic contribution, described in Section 4, consists of a three-step heuristic that builds upon the work by Gou et al [10] for the homogeneous case. We adapt two of these steps: (i) the assignment of tasks to processors, which now considers the different memory sizes, and (ii) when splitting or merging subtrees, we take the availability of suitable processors into account, i.e., those with sufficient mem-ory.…”

“…To evaluate the benefit from exploiting heterogeneity with HetPart, we employ simulations on real-world as well as randomly generated trees in Section 5. As standard of reference, we choose the state-of-the-art algorithm by Gou et al [10]. For a fair comparison, we use the same heterogeneous target architecture and let the reference algorithm work on it in different scenarios regarding hardware consumption.…”