Scheduling tree dags on parallel architectures

Abstract

We provide optimal within a constant explicit upper bounds on the makespan of schedules for tree-structured programs on mesh arrays of processors, and provide polynomial-time algorithms to find schedules with makespan matching these bounds. In particular, we show how to find, in polynomial time, a (nonpreemptive) schedule for a binary tree dag withn unit execution time tasks and heighth on ad-dimensional mesh array withm processors and links of unit bandwidth and unit propagation delay whose makespan isO(n/m+n ¹/(d+1)+h), i.e., optimal within a constant factor. Further, we extend these schedules to bounded degree forest dags with arbitrary positive integer execution time tasks and to meshes when the propagation delay of all the links is an arbitrary positive integer. Thus, we provide a polynomial-time approximation algorithm for an NP-hard problem, with a performance ratio that is a constant.

We also show how to schedule tree dags on any parallel architecture that satisfies certain natural, not very restrictive, conditions that are satisfied by most parallel architectures used in practice. Letε be a fixed positive real number. We provide polynomial time computable schedules for binary tree dags withn unit execution time tasks and heighth ∉ (g(n)n ^−ɛ,g(n) logn) on any parallel architecture satisfying those conditions, with unit bandwidth and unit propagation delay links, with optimal up to a constant makespanO(g(n)+ft), whereg is a function that depends only on that architecture. The number of processors used is optimal within a constant factor ifh ≤g(n)n ^−ɛ, and is optimal within anO(logn) factor ifh≥g(n)logn. As an example, for hypercube and complete binary tree architectures, we achieve optimal within a constant makespanO(h) whenh=Ω(log² n), using an optimal within anO(logn) factor number of processors. Further, we extend these schedules to the case of bounded-degree forest dags with tasks of arbitrary positive integer execution times and architectures when the propagation delay of all the links is a given arbitrary positive integer.