E.G. Coffman Jr. and R.L. Graham, "Optimal Scheduling for two Processor Systems", Acta Informatica, 1, 1972, pp. 200-213.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... list scheduling algorithm of [18] For the time being, all we assume about the list of tasks is that, whenever t follows t in this list, then N(t ) N(t) We start by showing that there are subsets of tasks which have properties very similar to the blocks studied by Coffman and Graham in [1] in their algorithm for the 2 processor scheduling problem. Theorem 1 Let (T; OE) be an interval order, and let opt be the minimal length of a schedule for (T; OE) on m processors. Then there exist pairwise disjoint subsets (called blocks) i T , i = 1; r ( for some r) with 1. t 2 i ; ....

E. Coffman, Jr. and R. Graham. Optimal scheduling for two processor systems. Acta Inf., 1:200--213, 1972.

.... So [8] 1jp i = 1; prec(l ij = 1) r i jLmax P jp i = 1; intree(l ij = l)jLmax P jp i = 1; outtree(l ij = l) r i jCmax P jp i = 1; tree(l ij = l)jCmax Hu [23] P jp i = p; treejCmax P2jp i = 1; precjLmax Bernstein and Gertner [5] 1jp i = 1; prec(l ij 2 f0; 1g)jCmax Co man and Graham [9] 1jp i = 1; prec(l ij = 1)jCmax P2jp i = 1; precjCmax Fig. 5. Specializing our algorithm to known results. Our algorithm can solve all problems listed. Arrows denote generalization of results. The running times of the algorithm can be improved to O(n log n ne) in all cases using the ....

....arbitrary precedence constraints, individual processing times p i 1 and zero or unit latencies; the work in [10] proves this fact for unit latencies only. 2) Our algorithm specializes to most of the previously known cases such as the classical two processor scheduling due to Co man and Graham [9], again via the uni cation in [33] 3.2 NP Hardness The generic TCPS problem is NP complete and subsumes many simpler problems which are also known to be NP complete. Figure 6 summarizes these results. Hennessy and Gross [21] are the rst to show that precedence constrained scheduling with ....

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Coffman, E. G., and Graham, R. L. Optimal scheduling for two-processor systems. Acta Informatica 1 (1972), 200-213.

....static schedule of TT tasks, we can go on to perform the global scheduling and analysis of the whole application. 3. 2 Static schedule construction and holistic analysis For the construction of the cyclic static schedule for TT tasks and ST messages, we use a list scheduling based algorithm [5]. Assuming that in our application we have N time triggered task graphs G 1 ,G 2 , G N , the static schedule will be computed over a period T SS = LCM(T 1 ,T 2 , T N ) The input to the list scheduling algorithm is a graph consisting of n i instances of each G i , where n i =T SS T i . ....

E.G. Coffman Jr., R.L. Graham, "Optimal Scheduling for two Processor Systems", Acta Informatica, 1, 1972.

....connected processors. Many of these subproblems have been shown to be NP complete [15] Only for very restricted problems efficient algorithms are known: the execution lengths of all tasks are equal, the communication delays for information exchange are neglected, the number of processors is two [4], or special classes of precedence graphs are considered [9, 13, 16] In parallel architectures large delays occur before the result of the execution of a task on one processor can be used by a task on another processor. If these communication delays are not neglected, the scheduling problems ....

E.G. Coffman, Jr. and R.L. Graham. Optimal scheduling for two-processor systems. Acta Informatica, 1:200--213, 1972.

....or when tasks do not all take the same time to finish, when individual tasks have separate deadlines, and so on) This paper only considers the simplest case defined earlier. Upper Bound Results: The best algorithm known for precedence constrained scheduling in general is due to Coffman and Graham[3, 1]. This algorithm runs in polynomial time and gives a schedule whose length is guaranteed to be within a factor 2 Gamma 2=p of the length of the optimal (shortest length) schedule. Earlier, polynomial time algorithms that constructed optimal schedules were developed for two cases: i) Hu[5] gave ....

....is ready if it is itself unlabelled, and it has no unlabelled predecessors. Phase i then simply selects upto p ready vertices and for each such vertex v assigns T (v) i. If several vertices are ready, then some priority scheme is needed to decide which ones should be selected first. Historically[3] the priority scheme was described by putting the vertices into a priority list, and selecting vertices for execution according to their order in the list. 1 Algorithm design in this framework is simply fixing the priority list. In Coffman Graham Hu s algorithms, for example, the priority list ....

E. Coffman and R. Graham. Optimal Scheduling for Two-Processor Systems. Acta Informatica, 1:200--213, 1972.

....to prioritise the ready tasks and guide it towards an optimal solution. This has lead to a profusion of LS based heuristics [5, 11, 16, 20, 25] The MAP solution adopted here is based on the optimal, greedy scheduling algorithm for list scheduling which was proposed by Coffman and Graham [8]. This is an O(n 2 ) algorithm for arbitrary precedence constraints for two processors with unit execution costs. A MAP scheduler has to deal with heterogeneous resources and can no longer just choose the ready instruction with the highest priority, but must also consider whether the correct ....

E. G. Coffman and R. L. Graham. Optimal scheduling for two-processor systems. Acta. Informatica, 1:200--213, 1972.

....e. Such a traversal will visit each PEO exactly twice. However, from Theorem 5.5 from [11] if we print only every second PEO visited in the Hamilton cycle, we obtain every PEO exactly once. As an example, Figure 2 illustrates the graph H Theta e obtained from the graph G which is a simple path [1,2,3,4]. Notice that there are two copies of the graph H with edges between the vertices corresponding to the same PEOs. A Hamilton cycle is also illustrated starting from the PEO 1423. 6 1 3 2 6 5 4 7 Figure 3: A chordal graph with 7 vertices. 4.1 Initialization The algorithm assumes that the ....

....time. The vertices are constrained so that only a simplicial vertex can be run. An optimal schedule is one of minimal length, length dn=2e in light of Theorem 1. This scheduling is analogous to, but simpler than, the scheduling of partially ordered sets on two machines; e.g. Coffman and Graham [4]. 5.1 Clique trees A clique K is maximal if K is not properly contained in another clique, or equivalently, if no vertex in V Gamma K is adjacent to every vertex in K. A clique tree of G is a tree T on the maximal cliques of G such that T has the clique intersection property : for any two ....

E. Coffman and R.L. Graham, Optimal scheduling for two-processor systems, Acta Informatica, 1 (1972) 200--213.

....the precedence constraints are satisfied. The goal of the makespan problem is to find a feasible schedule of the n jobs that minimizes Cmax = max i=1: n fC i g. The problem is NP hard for arbitrary m [4] but efficient solutions exist for tree structured precedence constraints [5] or when m = 2 [6]. A natural extension of the makespan problem is the weighted completion time scheduling problem, where in addition each job i has a positive weight w i that expresses the importance of that job. The objective here is to minimize the total weighted job completion time, i.e. under the same ....

E. Coffman and R. Graham. Optimal scheduling for two-processor systems. Acta Inform., 1:200--213, 1972.

....the precedence constraints are satisfied. The goal of the makespan problem is to find a feasible schedule of the n jobs that minimizes C max = max i=1: n fC i g. The problem is NP hard for arbitrary m [14] but efficient solutions exist for tree structured precedence constraints [9] or when m = 2 [5]. A natural extension of the makespan problem is the weighted completion time scheduling problem, where in addition each job i has a positive weight w i that expresses the importance of that job. The objective here is to minimize the total weighted job completion time, i.e. under the same ....

E. Coffman and R. Graham. Optimal scheduling for two-processor systems. Acta Inform., 1:200--213, 1972. 9

....[31] However, a number of related problems with unit time operations can be solved e#ciently in the absence of register constraints. If all latencies equal one, polynomial algorithms are described by Co#man and Graham that schedule operations on two processors when the dependence graph is a DAG [15] and by Hu for arbitrarily many processors when the dependence graph is a tree [27] If all latencies are equal and the dependence graph is a tree, Bruno, Jones, and So give an e#cient algorithm to schedule operations on multiple processors [10] Finally, if all latencies equal 1 or 2, Bernstein ....

E. Coffman and R. Graham, Optimal scheduling for two-processor systems, Acta Inform., 1 (1972), pp. 200--213.

....# 3 machines for which c # (I) 1; and a simple greedy algorithm can be used to establish that c # (I) # 2 for every instance with m # 3 machines. This dichotomy between m = 2 and m # 3 forms an interesting parallel to the fact that the 2 processor case is known to be polynomially solvable [4, 6], while the complexity of the m processor case for fixed m # 3 is an open question. As a final note, one can view traveling salesman problems within this perspective on scheduling, since tours yield vectors of arrival times at each city. In this context, the approximation of the minimum latency ....

E.G. Co#man, R.L. Graham, "Optimal scheduling for two-processor systems," Acta Informatica 1(1972), pp. 200-213.

.... optimum with growing number of tasks even if more than two processors exist, as explained in [14] Figure 7: Examples of task graphs structured as an in tree (left) or an out tree (right) Coffman and Graham develop an algorithm to assign intree constrained tasks onto two identical processors [6]. It is optimal but suspends and migrates running tasks (preemptive scheduling) This method is extended in [5] for two heterogeneous processors and arbitrary precedence constraint graphs. It is proofed to be still optimal. The best case behavior of some preemptive and nonpreemptive scheduling ....

E. Coffman, R. Graham, Optimal Scheduling for Two-Processor Systems, Acta Informatica 1, 1972.

....For example, let us consider the uniform precedence graph G depicted in figure 7. Figure 8 shows the graphs G 1 and G 2 . Consider the classical (non cyclic) scheduling problem of jT 1 j tasks with the same processing time and subject to a precedence graph G 1 on 2 machines. Coffman and Graham [4] have shown that this problem can be polynomially solved by a list algorithm. Figure 9 illustrates an optimal schedule of the tasks from T 1 for the precedence constraints G 1 . The minimal makespan 11 f 1 2 3 4 5 6 7 8 9 10 11 2 1 1 2 5 4 3 2 1 6 10 11 9 7 8 G 1 2 G ....

E.G Coffman, JR., R.L.Graham, Optimal scheduling for two processors systems. Acta Informatica, 13, p 200-213, 1972.

.... Lawler [5] he developed a polynomial algorithm with a relative performance of 1 (m Gamma 2) w opt (w opt denotes the makespan of an optimal schedule) It is well known that if there are no communication delays, the Coffman Graham list algorithm provides an optimal solution on two processors [2]. Moreover, Lam and Sethi [4] proved that its performance ratio is bounded by 2 Gamma 2 m on m processors. Our purpose was here to study the behavior of this algorithm for the UET UCT problem. Using a particular decomposition of the CG schedule inspired by the one of Lam and Sethi [4] and by ....

E.G.Coffman,JR. and R.L.Graham, Optimal scheduling for two processor systems, Acta Informatica 1, 200-213 (1972).

....arbitrary precedence constraints, individual processing times p i 1 and zero or unit latencies; the work in [5] proves this fact for unit latencies only. 2. Our algorithm specializes to most of the previously known cases such as the classical two processor scheduling due to Coffman and Graham [4], again via the unification in [15] 4. Computing the Modified Deadlines The modified deadlines computation algorithm we describe below repeatedly invokes a backward scheduling subroutine backschedule, which is a generalization of the backward scheduling process described in Palem and Simons ....

.... processor problems, since precedence constrained multiprocessor scheduling is NP complete, even when restricted to two processors and processing times of 1 and 2 [21] A similar problem with unit latencies, 1 j prec(l ij = 1) p j 2 IN j Cmax , has been addressed in [5] using the Coffman Graham [4] lexicographical rank function for listscheduling priorities. See also the recent work of Brucker and Knust [2] for other related results. The optimality proof makes use of a certain rigid structure of list schedules generated by this rank function, and is quite involved. In contrast, our ....

E. Coffman and R. Graham. Optimal scheduling for twoprocessor systems. Acta Informatica, 1:200--213, 1972.

....We address only the static scheduling problem. Hereafter we refer to the static 2 scheduling problem as simply scheduling. The scheduling problem is NP complete for most of its variants except for a few highly simplified cases (these cases will be elaborated in Chapter 2) 37] 40] [41], 51] 52] 63] 74] 78] 90] 100] 150] 151] 152] 158] 162] 185] and therefore, many heuristics with polynomial time complexity have been suggested [8] 9] 31] 40] 51] 52] 66] 102] 134] 147] 154] 165] 174] However, most of these heuristics are based on ....

....three special cases for which there exists optimal polynomial time algorithms. These cases are: 1) scheduling tree structured task graphs with uniform computation costs on arbitrary number of processors [90] 2) scheduling arbitrary task graphs with uniform computation costs on two processors [41] and (3) scheduling an interval ordered task graph [57] with uniform node weights to an arbitrary number of processors [151] However, even in Parallel Program Scheduling Job Scheduling (independent tasks) Scheduling and Mapping (multiple interacting tasks) Dynamic Scheduling Static ....

[Article contains additional citation context not shown here]

E.G. Coffman and R.L. Graham, "Optimal Scheduling for Two-Processor Systems," Acta Informatica, vol. 1, 1972, pp. 200-213.

.... = 1) r i Lmax P p i = 1; intree(l ij = l) Lmax P p i = 1; outtree(l ij = l) r i Cmax P p i = 1; tree(l ij = l) Cmax Hu [18] P p i = p; tree Cmax Garey and Johnson [9] P2 p i = 1; prec Lmax Bernstein and Gertner [1] 1 p i = 1; prec(l ij # 0, 1 ) Cmax Co#man and Graham [5] 1 p i = 1; prec(l ij = 1) Cmax P2 p i = 1; prec Cmax Fig. 5. Specializing our algorithm to known results. Our algorithm can solve all problems listed. 1) In the first group, we have a range of scheduling problems introduced by Bruno, Jones and So wherein all the latencies are equal, and ....

....arbitrary precedence constraints, individual processing times p i # 1 and zero or unit latencies; the work in [6] proves this fact for unit latencies only. 2) Our algorithm specializes to most of the previously known cases such as the classical two processor scheduling due to Co#man and Graham [5], again via the unification in [28] 3.2 NP Hardness The generic TCPS problem is NP complete and subsumes many simpler problems which are also known to be NP complete. Figure 6 summarizes these results. Hennessy and Gross [16] are the first to show that precedence constrained scheduling with ....

[Article contains additional citation context not shown here]

Coffman, E. G., and Graham, R. L. Optimal scheduling for two-processor systems. Acta Informatica 1 (1972), 200--213.

.... simplified cases [29] 36] 38] 59] To tackle the 2 problem, many heuristic algorithms, which are based on simplifying assumptions about the structure of the parallel programs as well as the underlying parallel processing systems, have been reported in the literature [1] 11] 16] [19], 33] 39] 64] 66] 67] 69] For more realistic cases, a scheduling algorithm needs to address a number of issues. It should exploit the parallelism by identifying the task graph structure, and take into consideration task granularity, load balancing, arbitrary computation and ....

....benefited from the approaches employed in the area of operations research since allocation of parallel programs to parallel processors is analogous to allocation of a set of jobs to a set of machines. However, such approaches assume a very simple model of the parallel computer 9 [16] [19]. Over the years, with the rapid advancements in computer architectures, the scheduling problem has evolved through a number of generations. Every scheduling algorithm reported in the early literature works under different circumstances and assumptions. However, there are three fundamental ....

[Article contains additional citation context not shown here]

E.G. Coffman and R.L. Graham, "Optimal Scheduling for Two-Processor Systems," Acta Informatica, vol. 1, pp. 200-213, 1972.

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E.G. Coffman Jr. and R.L. Graham, "Optimal Scheduling for two Processor Systems", Acta Informatica, 1, 1972, pp. 200-213.

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E. G. Coffman, Jr and R. L. Graham, "Optimal Scheduling for Two-Processor Systems", Acta Informatica 1, (1972), pp. 200--213.

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Coffman, E., and Graham, R. Optimal scheduling for two processor systems. Acta Informatica 1 (1972), 200--213.

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Coffman, E., and Graham, R. Optimal scheduling for two processor systems. Acta Informatica 1 (1972), 200--213.

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Coffman, E. G., Jr. and Graham, R. L. (1972). Optimal Scheduling for Two-Processor System, Acta Informatica, 1, 200-213.

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E.G. Coffman Jr., R.L. Graham, "Optimal Scheduling for two Processor Systems", Acta Informatica, 1, 1972.

No context found.

E. Coffman and R. Graham. Optimal scheduling for two-processor systems. Acta Informatica, 1:200--213, 1972.

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