Scheduling malleable tasks with precedence constraints

Abstract

In this paper we propose an approximation algorithm for scheduling malleable tasks with precedence constraints. Based on an interesting model for malleable tasks with continuous processor allotments by Prasanna and Musicus [22,23,24], we define two natural assumptions for malleable tasks: the processing time of any malleable task is non-increasing in the number of processors allotted, and the speedup is concave in the number of processors. We show that under these assumptions the work function of any malleable task is nondecreasing in the number of processors and is convex in the processing time.Furthermore, we propose a two-phase approximation algorithm for the scheduling problem. In the first phase we solve a linear program to obtain a fractional allotment for all tasks. By rounding the fractional solution, each malleable task is assigned a number of processors. In the second phase a variant of the list scheduling algorithm is employed. In the phases we use two parameters µ ∈ {1, . . . (m + 1)/2 } and ρ ∈ [0, 1] for the allotment and the rounding, respectively, where m is the number of processors. By choosing appropriate values of the parameters, we show (via a nonlinear program) that the approximation ratio of our algorithm is at most 100/63+100( √ 6469+13)/5481 ≈ 3.291919. We also show that our result is very close to the best asymptotic one.

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“…GRAD achieves O(1)-competitiveness with respect to makespan for job sets with arbitrary release times, and O(1)-competitiveness with respect to mean response time for batched job sets where all jobs are released simultaneously. Unlike many previous results, which either assume clairvoyance [24,19,25] or use instantaneous parallelism [11,9], GRAD removes these restrictive assumptions. Moreover, because the quantum length can be adjusted to amortize the cost of context-switching during processor reallocation, GRAD provides effective control over the scheduling overhead and ensures efficient utilization of processors.…”

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“…GRAD achieves O(1)-competitiveness with respect to makespan for job sets with arbitrary release times, and O(1)-competitiveness with respect to mean response time for batched job sets where all jobs are released simultaneously. Unlike many previous results, which either assume clairvoyance [29,21,31] or use instantaneous parallelism [10,6], GRAD removes these restrictive assumptions. Moreover, because the quantum length can be adjusted to amortize the cost of context-switching during processor reallocation, GRAD provides effective control over the scheduling overhead and ensures efficient utilization of processors.…”