Scheduling on Two Unbounded Resources with Communication Costs

Abstract

Heterogeneous computing systems became a popular and powerful platform, containing several heterogeneous computing elements (e.g. CPU+GPU). In this paper, we consider that we have two platforms, each with an unbounded number of processors. We want to execute an application represented as a Directed acyclic Graph (DAG) using these two platforms. Each task of the application has two possible execution times, depending on the platform it is executed on. Also, there is a cost to transfer data from one platform to another between successive tasks. The goal here is to minimize the finish execution time of the last task of the application (usually called makespan). We show that the problem is NP-complete for graphs of depth at least 3 but polynomial for graphs of depth at most 2. Finally, we focus on particular classes of graphs, by providing polynomial-time algorithms for bi-partite graphs, trees and 2-series-parallel graphs with different assumptions on communication delays.Résumé : Les systèmes de calculs hétérogènes (par exemple CPU+GPU) sont des plateformes populaires. Dans ce travail, nous considérons une machine avec deux plateformes homogènes de calcul, chacune contenant un nombre illimité de ressources de calcul. Nous cherchons à exécuter une application représentée par un graphe de dépendance dirigé et acyclique sur ces plateformes. Chaque tâche de l'application a deux possible modèle d'exécution en fonction de la plateforme sur laquelles elles sont exécutées. En plus nous considérons un coût de communication entre deux tâches successives si elles ne sont pas exécutées sur la même plateforme. Nous travaillons à minimiser le temps d'exécution de l'application.Nous montrons que le problème est NP-complet pour les graphes de profondeur au moins trois, mais polynomial pour les graphes de profondeur au plus deux. En plus, nous montrons qu'il est possible de calculer des solutions optimales en temps polynomial pour certaines classes de graphes définies récursivement (arbres, graphes série-parallèles).

A Fixed-Parameter Algorithm for Scheduling Unit Dependent Tasks with Unit Communication Delays

“…Figure 3 is the multistage graph associated with the precedence graph of Figure 1a and C = 6. We observe that the path (p 0 , p 1 1 , p 1 2 , p 1 3 , p 1 4 ) corresponds to the schedule shown in Figure 1b. On the same way, the path (p 0 , p 0 1 , p 0 2 , p 0 3 , p 0 4 ) corresponds to the schedule shown by Figure 4.…”

“…Shimada et al [19] developed two heuristic based methods to consider both malleable tasks and communications delays for executing a program on an homogeneous multi-core computing system. Ait-Aba et al [1] provided complexity results for an extension of the basic communication model for scheduling problems on an heterogeneous computing systems with two different resources.…”

Full Version -- Server Cloud Scheduling

“…In the conclusion of this paper a model very similar to the one studied in this work is mentioned as an interesting research direction. For a detailed discussion of related models, we refer to the preprint version of the above work [2]. The present model is closely related to the classical problem of makespan minimization on parallel machines with precedence constraints, where a set of jobs with processing times, a precedence relation on the jobs (or a task graph), and a set of m machines are given.…”