Optimal scheduling on parallel machines for a new order class
Introduction
An extensively studied problem in deterministic scheduling theory is that of scheduling a set of n partially ordered tasks on m identical machines in order to minimize the overall finishing time, or makespan. In nonpreemptive scheduling problem, it is assumed that once a processor is assigned to a task, it must work continuously on this task until it has been completed. When all tasks have the same execution time, the problem is referred to in the literature as the UET nonpreemptive scheduling problem and is designated by P|prec,pj=1|Cmax.
Let (V,<) be a UET system where V={1,…,n} is a set of n tasks to be executed on m identical machines, and < is a partial order on V that specifies the precedence constraints among tasks. For i,j∈V, if i<j then i must be completed before j can be begun. A successor (resp. predecessor) of a task i is a task j such that i<j (resp. j<i). For every task (resp. Γ−(i)) denotes the set of the successors (resp. predecessors) of i. Two tasks i,j∈V are comparable in (V,<), if i<j or j<i. Otherwise, they are said to be incomparable and denoted by i∥j. A subset S⊂V is linearly ordered if S does not contain any incomparable tasks. The number of arcs in the transitive closure of (V,<) is denoted by e.
For general m, determining an optimal schedule for UET systems is NP-complete [9], [12]. Although, the question of whether or not optimal scheduling algorithms exist which are polynomial for UET systems for m⩾3 remains open. Many papers have been devoted to solving various subcases [10]. Efficient algorithms are known if the number of machines is limited to 2 even for arbitrary precedence constraints [4], [1]. Hu [8] gives a polynomial-time algorithm to solve P|tree,pj=1|Cmax. For precedence constraints for which the longest path has at most h arcs, Dolev and Warmuth [2] give an O(nh(m−1)+1) algorithm. They also show in [3] that level orders can be solved in O(nm−1) time. For interval orders, Papadimitriou and Yannakakis [11] give a list scheduling algorithm which can be implemented in linear time.
In this paper we give a polynomial-time algorithm for a rich order class which contains properly interval orders and a subclass of the class of series parallel orders.
In Section 2 we recall the most important definitions of series parallel orders, level orders and interval orders. Then, we introduce the class of quasi-interval orders. In Section 3, we exhibit an efficient algorithm for finding optimal schedules for this new order class.
Section snippets
A new order class, quasi-interval orders
The first polynomial algorithm was developed by Hu for trees [8] which are a subclass of series-parallel orders.
A series-parallel order is defined recursively as follows [13].
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The partial order having single element and no arcs is a series-parallel order,
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If G1=(V1,E1) and G2=(V2,E2) are two series-parallel orders, so are the partial orders constructed by each of the following operations:
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parallel composition: Gp=(Vp,Ep) where Vp=V1∪V2 and Ep=E1∪E2,
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series composition: Gs=(Vs,Es) where Vs=V1∪V2 and
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The algorithm
The algorithm given by Papadimitriou and Yannakakis in [11] to produce an optimal schedule when the precedence graph is an interval order is as follows. Algorithm PY. Step 1. Sort the tasks in (V,<) on a list L=(l1,…,ln) in a nonincreasing order of their successor number. That is for all . Step 2. Schedule the tasks by always scheduling the first available task on list L. Theorem 5 PY algorithm solves the UET nonpreemptive scheduling problem for quasi-interval orders in O(n+e) time. Proof. It has been shown in [11] that
References (13)
- et al.
Scheduling precedence graphs of bounded height
J. Algorithms
(1984) NP-complete scheduling problems
J. Comput. Systems Sci.
(1975)- et al.
Optimal scheduling for two-processor systems
Acta Inform.
(1972) - et al.
Profile scheduling of opposing forests and level orders
SIAM J. Algebraic Discrete Methods
(1985) - M. Fujii, T. Kasami, K. Ninomiya, Optimal sequencing of two equivalent processors, SIAM J. Appl. Math. 17 (1969)...
Bounds on multiprocessing anomalies
Bell System Tech. J.
(1966)
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