H.S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans of Software Engineering, SE-3(1):95--93, January 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....V is not discontinuity preserving, and optimizing an energy function with such a V lead to oversmoothing at the borders of objects. The ability to find the global minimum e#ciently, while theoretically of great value, does not overcome this drawback. This is clearly the task assignment problem [25, 28, 35]. Figure 1: An example of an expansion move. The labeling on the right is a red expansion move from the labeling on the left. visible in the relatively poor performance of such algorithms on the stereo benchmarks described in [32] Moreover, there almost certainly are no e#cient global energy ....

Harold S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, pages 85--93, 1977.

....with each other instead of performing useful computations. The task assignment problem in distributed systems deals with finding a proper assignment of tasks to processors such that total execution and communication costs are minimized. The problem was first introduced and solved by Stone [1]. Stone reduced the task assignment problem to multiway cut problem by which the optimal assignments can be found in polynomial time for two processor systems. Unfortunately the task assignment problem is known to be NP complete [2] for three and more processors systems in general. Stone extends ....

....the fact that each task i should be assigned to one processor. As it can be realized from the formulation, task assignment problem is very similar to some other well known NP complete problems such as graph partitioning [10] and quadratic assignment [13] In addition to those similarities, Stone [1] and Magirou [5] find a close correspondence between task assignment problem and multi way cut problem. 2.2 Previous Work Numerous studies have been performed to solve the task assignment problem. One of the first is by Stone [1] who used network flow algorithms with a graph theoretical ....

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H. S. Stone, "Multiprocessor scheduling with the aid of network flow algorithms, " IEEE Trans. on Software Engineering, vol. SE-3, january 1977.

....of failure in order to provide the computing resources necessary to fulfill the base mission of the system. 3. PRIOR WORK A number of research efforts have studied the Adapter Allocation step as it applies to the allocation of computing tasks to parallel and multiprocessor systems, notably in [Stone77, Shen85, Bokhari81, Bokhari88, Chu87, Indurkhya86, Efe82, Houstis90]. Woodside93] for instance, developed heuristic extensions to the MULTIFIT bin packing algorithm to create a static task allocator for embedded systems. Beck95] used a similar allocation step as part of the specification of a distributed system. Prakash92] showed the usefulness of linear ....

H. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithms," IEEE Transactions on Software Engineering, SE-3(1): 85-93, Jan 1977.

....for scheduling tasks on a parallel MIMD architecture have usually included a communication cost associated with the sending of data between tasks which are located on different processors. Early work on this problem used graph theoretic techniques such as network flow and or enumeration techniques [9, 18, 21]. Later work concentrated on approximation algorithms [1, 13, 16, 20] Research then evolved to more restricted models which allowed an infinite number of processors in the system. Polynomial algorithms were found for the cases where the precedence constraints form a tree under certain constraints ....

H. S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans. on Software Eng., SE-3(1):85--93, 1977.

....the arcs are undirected. Two other graph models, which are suitable for expressing parallelism, are Petri nets and PERT networks, which will also be discussed in this section. Finally, the suitability of queueing network models is analyzed. 2.2. 1 Undirected Graph Models A static process graph [Ston 77b] G = V; E; f; e) consists of a set of V nodes representing the processes in a parallel or distributed application, E is a set of undirected arcs representing two way communication, f is a function associating computation costs to the nodes and e is a function associating communication costs to ....

H. Stone. "Multiprocessor Scheduling with the Aid of Network Flow Al gorithms ". IEEE Trans. on Software Engineering, Vol. SE-3, pp. 85--93, 1977.

....from a performance point of view are rarely in existence today. Performance evaluation and engineering [Smit 90a] in the context of parallel processing has progressed over the past quarter of a century along different lines. Historically first, works on the so called mapping problem appeared [Ston 77a, Bokh 81] which addressed assignment and scheduling issues in distributed memory MIMD systems. A large body of literature (see e.g. Norm 93] for a classification of the particular problems studied) is available on the subject, however, most of the models have become obsolete due to the ....

H. S. Stone. "Multiprocessor Scheduling with the Aid of Network Flow Algorithms ". IEEE Transactions on Software Engineering, Vol. SE-3, No. 1, pp. 85-- 93, Jan. 1977.

....differently. However, most prominent methods for task assignment in distributed systems are concerned with minimizing the sum of task processing costs on all assigned processors and interprocessor communications (IPC) costs. Examples of such methods typically include graph theoretic solutions [18], 19] and integer programming solutions [20] among others [20] 21] 22] 23] Note that minimizing the aggregated processing of, and or communication cost among, all tasks does not guarantee that individual deadlines will be met. Generally, it is difficult to add real time constraints when a ....

H.S. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithms" IEEE Trans. Software Eng., vol. 3, no. 1, pp. 8593, Jan. 1977.

....nodes in the graph (called terminals) find a cheapest multiway cut, i.e. a subset of the edges whose removal disconnects each terminal from the rest. This is one of several generalizations of the classical undirected s t cut problem, and it has applications in parallel and distributed computing [24], as well as in chip design. Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis [8] initiated the study of Multiway Cut. In the published version of their paper [9] they prove that Multiway Cut is MAX SNPhard even when restricted to instances with three terminals and unit edge costs. ....

H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. In IEEE Transactions on Software Engineering SE-3, pages 85-93, 1977. 10

....cost. The general problem of clustering comes up in various contexts in computer based systems. In a multiprocessor or a distributed computing system, tasks to be processed may have to clustered onto a set of processors so that interprocessor communication cost can be reduced [15] Stone [23] has used a graph theoretical approach to model intermodule communications in a program and adapted the maximum flow algorithm to find an optimal partitioning of a modular program in a two processor system. In distributed DBMS [5] the relations of a database can be fragmented in three ways, ....

H.S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, 3(1):85--93, Jan. 1977.

....as the problem is NP complete. The general problem of clustering comes up in various contexts in computer based systems. In a distributed computing system, tasks to be processed may have to be clustered onto a set of processors so that interprocessor communication cost can be reduced [16] Stone [24] has used a graph theoretical approach to model intermodule communications in a program and adapted the maximum flow algorithm to find an optimal partitioning of a modular program in a two processor system. A heuristic scheme to assign inter dependent tasks to distributed computers nonpreemptively ....

H.S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, 3(1):85--93, Jan. 1977.

....allows a parallel computer to obtain a better flow as well, meaning that the parallel solution is closer to optimal than any solution arrived at sequentially. The real time paradigm can be used to express dynamic variants of static computations. One example is the module allocation problem [27] which relies on advance knowledge about a system of processes. Such a system may be modeled more realistically by allowing its characteristics to change with time in an unpredictable way. A real time computational setting is therefore a more suitable environment for treating This research was ....

....real time network flow problem studied, the use of a parallel approach represents the difference between success and failure of the computation. 6 An Application: Process Scheduling In what follows, the real time maximum flow solutions will be applied to a real time (dynamic) extension of Stone s [27] (static) module allocation problem. In a distributed environment, n processes have to be scheduled to run on m processors. A process is a program entity defined by an executable code and private data. An objective function is defined to evaluate the communication cost and the process execution ....

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H. S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, 3(1):85--93, 1977.

....in turn. 2.1 Scheduling Scheduling is the process of mapping units of work to processors. Research in scheduling parallel computations generally falls into one of two categories scheduling a directed acyclic graph (DAG) 1] 22] 59] 97] or scheduling a static task graph (STG) 9] 10] 54][80]. The DAG based precedence graph often arises from the parallelization of sequential code. In the DAG the nodes represent computations, typically fine grained, and the arcs represent data dependencies. Scheduling an arbitrary precedence graph is NPcomplete for P 2 processors [87] Polynomial time ....

....the communication structure of the application when it is implemented as a collection of processes. In the STG the nodes represent modules or tasks, typically coarse grain, and the arcs represent communications. There are two variants of the STG, the module assignment graph introduced by Stone [80] for non precedence constrained sequential programs and the communication graph for parallel computations. Scheduling an arbitrary STG of the first type is NP complete for P 4 processors [80] Polynomial time algorithms exist for restricted STG s, trees [9] series parallel graphs [86] and linear ....

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H.S. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithms, " IEEE Transactions on Software Engineering, Vol. SE-3, No. 1, January 1977.

....programs. A non directed graph will only represent information exchange and concurrent execution between tasks, while a directed graph will deal also with precedence relations between tasks. There is a very substantial literature about the graph theoretic approach to task assignment. Stone et al. [St77] uses the max flow minimal cut Theorem of Ford and Fulkerson [Fo62] to search for an optimal assignment which will minimize the communication cost of a system with two processors. In [Lo88a] an extension of this approach to a system with n processors is proposed, by recursively using the same ....

H. Stone, Multiprocessor scheduling with the aid of network flow algorithms, IEEE Trans. on Software Eng, Vol SE-3, pp. 85-93, 1977.

.... [80] or intelligent fixed priority algorithm [114] Classical methods for task assignment in distributed systems have been developed by minimizing the sum of task processing costs on all assigned processors and interprocessor communications (IPC) costs, including graph theoretic solutions [132, 133] and integer programming solutions [84] among others [82, 84, 117, 124] Since the problem of assigning tasks subject to precedence constraints is generally NP hard [40, 50, 73, 77] some form of enumerative optimization or approximation us151 ing heuristics needed to be developed for this problem ....

H. S. Stone, "Multiprocessor scheduling with the aid of network flow algorithm," IEEE Trans. on Software Engineering, vol. SE-3, no. 1, pp. 85--93, January 1977.

....can be used as performance indices. For the static allocation problem, the objective functions mostly used are [Hou92] Minimization of total computation and communication times in the system; i.e minimization of the total cost incurred by executing the application on a multiprocessor system. [Stone77, Ma82, Lo88, Houstis90]. In the case of homogeneous system, the objective function reduces to the minimization of the total interprocessor communication time. Load Balancing by minimizing the statistical variance of processor utilizations [Bannister83,Tantawi85] The less the variance, the better the load balance. ....

....system, the problem of process assignment is NP complete [Bokhari81, Lo88] Hence, the problem of finding a minimum cost assignment is computationally intractable for all but small systems. Accordingly, several researchers focused on the special cases of the general assignment problem. Stone [Stone77] suggested an efficient optimal algorithm for the problem of assigning processes to two processor (two processor Page 35 problem) by making use of the well known network flow algorithm in two terminal network graphs. He showed how the network model can be extended to systems made up of three or ....

Harold S. Stone, "Multiprocessor scheduling with the aid of network flow algorithm", IEEE Transactions on Software Engineering, 3(1):85-93, Jan. 1979

.... mathematical formulations for matching, collectively called selection theory, that have been proposed to choose the appropriate machine for each subtask of an application program (e.g. 4] 8] 15] 22] A collection of algorithms, called graph based algorithms in this paper (e.g. 2] 15] [19], 21] have been developed to solve matching related problems based on a subtask flow graph that describes the data dependencies among subtasks of an application program. As shown in Fig. 1a, each vertex of the subtask flow graph represents a subtask. Let S[k] denote the kth subtask. Let a data ....

.... same subtask (which implies that the two data items have the same value) In general, most of the graph based algorithms for matching related problems assume that the pattern of data transfers among subtasks is known a priori and can be illustrated using some type of graph (e.g. 2] 14] 15] [19], 21] Thus, no matter which machine is used for executing each subtask of a specific application program, the locations (subtasks) from which each subtask obtains its corresponding input data items are determined by the subtask flow graph and independent of any particular matching scheme ....

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H.S. Stone, "Multiprocessor Scheduling with the Aid of the Network Flow Algorithms," IEEE Trans. Software Eng., vol. 3, no. 1, pp. 85-93, Jan. 1977.

....on two mapping algorithms, each composed of two different phases. The first phase solves the clustering problem by grouping the tasks into a number of clusters equal to the processors available, the second one assigns clusters to processors, Lo [13] continues along the lines traced by Stone [22] in transforming task and host graphs into a single network to which a Min Cut Max Flow algorithm is applied. This paper presents a new mapping framework aimed at compile time determination of satisfactory sub optimal solutions to the process based model of the mapping problem onto massively ....

H. S. Stone. Multiprocessor Scheduling with the aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, 3(1):85--93, January 1977.

....distributes tasks from heavily loaded workstations to idle workstations without user intervention. This user transparent process provides a greater amount of processing power to all users of the system. Early studies showed that automatic load balancing can be achieved in multiprocessing systems [4] (before distributed systems were commonly available) Since then, several different algorithms have been presented for load balancing of multiprocessing and distributed system. The load from one workstation can be transferred to a lightly loaded workstation through two different methods. One ....

H. S. Stone, "Multiprocessor scheduling with the aid of network flow algorithms," IEEE Transactions on Software Engineering, vol. 32, pp. 85-- 93, January 1977.

....during its lifetime. This approach runs counter to more traditional approaches to programming distributed memory machines that focus on mapping the topology of problems (rings, trees, etc. to the topology of machines (hypercube, shuffle exchange etc. to exploit nearest neighbor communication[3, 23]. Exploiting this kind of locality is important for architectures with multi level memory hierarchies like the Intel iPSC 1, but it is not important to our two level hierarchy. While our system can be extended to include this kind of topological information, we believe that the performance ....

H. Stone. Multiprocessor scheduling with the aid of network flow algorithms. ACM Transactions on Software Engineering, 1977.

....Ercal and Ramanujam [22] report on two mapping algorithms, each composed of two different phases. The first phase solves the cardinality variation problem by grouping the tasks into clusters, the second one assigns clusters to processors; Lo [16] continues along the lines traced by Stone [25] in transforming task and host graphs into a single network to which a Min Cut Max Flow algorithm is applied. 3 Preliminaries 3.1 Target parallel architectures The greedy mapping algorithm discussed in this paper is designed for multicomputers composed of large ensembles of processing nodes ....

H. S. Stone. Multiprocessor Scheduling with the aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, 3(1):85--93, January 1977.

....applications, the objective of the allocation problem may be the minimum completion time, processor load balancing, or total cost of execution and communication, etc. For the assignment problem where the objective is to minimize the total cost of execution and interprocessor communication, Stone [11] and Towsley [12] presented O(n 3 M) algorithms for treestructure and series parallel graphs, respectively, of M tasks and n processors. For general task graphs, the assignment problem has been proven [9] to be NP complete. Many papers [8] 9] 10] presented branch and bound methods which yielded ....

....the available processors in the systems. Such replication may lead to a lower total cost than the optimal assignment problem does. An example illustrating this point is presented in Section 3. In the assignment problem, polynomial time algorithms exist for special cases, such as treestructure [11] and series parallel [12] task graphs. This paper represents one of the first few attempts at finding special cases for the replication problem. The class of applications we consider in this paper is computation intensive applications in which the execution cost of a task is greater than its ....

H.S. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithm," IEEE Trans. Soft. Eng. Vol 3, pp 85-93, Jan. 1977.

....by properly mapping the tasks to the processors [27] 119] 127] This requires balancing the computation load uniformly among the processors while simultaneously keeping communication costs as low as possible. The research in this area was pioneered by Stone and Bohkari [24] 26] 27] 28] [177]: Stone [177] applied network flow algorithms to solve the assignment problem while Bokhari described the mapping problem as being equivalent to graph isomorphism, quadratic assignment and sparse matrix bandwidth reduction problems [25] The task precedence graph model, in which the nodes ....

....mapping the tasks to the processors [27] 119] 127] This requires balancing the computation load uniformly among the processors while simultaneously keeping communication costs as low as possible. The research in this area was pioneered by Stone and Bohkari [24] 26] 27] 28] 177] Stone [177] applied network flow algorithms to solve the assignment problem while Bokhari described the mapping problem as being equivalent to graph isomorphism, quadratic assignment and sparse matrix bandwidth reduction problems [25] The task precedence graph model, in which the nodes represent the tasks ....

H.S. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithms," IEEE Transactions on Software Engineering, vol. SE-3, no. 1, Jan. 1977, pp. 85-93.

....processors while simultaneously keeping communication costs as low as possible. The mapping problem is analogous to graph to graph mapping since both the problem and machine models can be represented as graphs. The research in this area was pioneered by Stone and Bohkari [12] 13] 14] Stone [63] applied network flow algorithms to solve the assignment problem. Bokhari described the mapping problem as being equivalent to graph isomorphism, quadratic assignment and sparse matrix bandwidth reduction problems [11] However, this approach does not consider the precedence constraints among the ....

H.S. Stone, "Multiprocessor Scheduling with the Aid of the Network Flow Algorithm," IEEE Trans. on Software Engineering, vol. SE-3, no. 1, pp. 85-93, Jan. 1977.

....5 Sec 6 we show how using these results, scans, scatters, gathers, etc. can be automatically deduced from a given PAS. Finally, we discuss related work and give our conclusions. 2 Specification of LACS In parallel processing, communication activity is typically described as a directed graph [5, 20], where nodes represent source processes, and arcs represent communication from a (single) source node to a (single) destination node. LACS uses two generalizations of this. First, we use multi graphs in order to permit multiple destinations (broadcasts) and multiple sources (reductions) Second, ....

....problem of merely specifying the communication in an abstract, architecture independent manner. The background of this was in our language LaRCS [14] which was developed in the context of mapping. We noted that most of the previous work in mapping used a graph theoretic model of the computation [5, 20], and did not explicitly describe temporal information. LaRCS was developed to address this limitation, and includes the notion of 26 Sanjay Rajopadhye collecting edges into groups that are simultaneously active, and of giving an order for this activity. LACS extends this by using multi graphs ....

H. S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

....parallel program completion time by properly mapping the tasks to the processors. This requires balancing the computation load uniformly among the processors while simultaneously keeping communication costs as low as possible. The research in this area was pioneered by Stone and Bohkari [22] [158]: Stone [158] applied network flow algorithms to solve the assignment problem while Bokhari described the mapping problem as being equivalent to graph isomorphism, quadratic assignment and sparse matrix bandwidth reduction problems [23] The task precedence graph model or simply the DAG, in which ....

....program completion time by properly mapping the tasks to the processors. This requires balancing the computation load uniformly among the processors while simultaneously keeping communication costs as low as possible. The research in this area was pioneered by Stone and Bohkari [22] 158] Stone [158] applied network flow algorithms to solve the assignment problem while Bokhari described the mapping problem as being equivalent to graph isomorphism, quadratic assignment and sparse matrix bandwidth reduction problems [23] The task precedence graph model or simply the DAG, in which the nodes ....

H.S. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithms," IEEE Transactions on Software Engineering, vol. SE-3, no. 1, Jan. 1977, pp. 85-93.

....The task allocation problem can be optimally solved in special cases such as two processor distributed systems, or linear array of any number of processors. If the target architecture contains two processors, then the task allocation problem is stated as a maximum flow minimum cut problem [11] which can be polynomially solved using, for example, the Ford Fulkerson algorithm. There also exists a heuristic presented [7] which addresses the general m processor problem using the 2 way min cut algorithm m times. Most of the theoretical work on mappings considers structured graphs like ....

H. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Trans. Soft. Engin., vol. SE-3, no. 1, pp. 85-93, Jul. 1977.

....and an extensible portion that administrators can use to implement new scheduling policies or to extend the basic descriptions of requirements or abilities. To determine the basis for the fixed portion of the description vector, we reviewed 18 algorithms from the existing scheduling literature [2 5, 7, 13, 17 20, 22 24, 26 28, 30, 31]. Table 3 depicts the resulting data set. We found that only two characteristics processor speed and inter processor communication time estimates were used by more than four algorithms. Therefore, we included processor speed estimates in the description vector and provide a mechanism to ....

H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

.... on new tasks loadave an estimate of the load average for the entire system Procclass information on the different classes of processors in the system To determine the basis for the fixed portion of the description vector, we reviewed 18 algorithms from the existing scheduling literature [2 5, 7, 13, 17 20, 22 24, 26 28, 30, 31]. Table 1 depicts the resulting data set. We found that only two characteristics processor speed and inter processor communication time estimates were used by more than four algorithms. Therefore, we included processor speed estimates in the description vector and provide a mechanism to ....

H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

....time of the application be optimal. Load balancing can be performed statically or dynamically. For the class of parallel applications for which the structure is known and stable, static load balancing 1 is more suitable; before execution, the mapping of load units is definitively decided (see [18], 3] Indeed, when the structure of the application is known in advance, its components (load units) are precisely determined and their amount of CPU time and communication time are determined too. Unfortunately many applications do not satisfy these conditions : their components have ....

H. S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, Jan. 1977.

....both throughput (processor utilization) as well as application performance are optimized. The problem is NP hard in all but the most restrictive of cases and so research has focused on the development of heuristic algorithms to find suboptimal assignments. Stone in his paper on task partitioning [Sto77] takes execution and communication costs into account and uses a Max Flow Min Cut algorithm to find an optimal assignment. However he fails to assign any explicit weight to concurrency. This is remedied in V.M. Lo s model described in [Lo88] which uses interference costs to account for loss of ....

H.S. Stone. Multiprocessor Scheduling with the aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, January 1977.

....optimal algorithms exist only for special cases, like Stone s algorithm for assigning tasks to a system of two or three processors. This algorithm requires that the complete knowledge of the behaviour of a task (e.g. run costs and interprocessor communication costs) is available, see Stone[13]. Global task scheduling heuristics have been proposed in a number of articles. These heuristics differ with regard to the characteristics of the underlying network, the information required to make a scheduling decision, considered constraints, used performance criteria, and the design choices as ....

Harold S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, pages 85--93, January 1977.

....with processors, also called task placement, and the choice of which task to execute among those available on a processor. Our work concentrates on facilities for the former. Many researchers have studied the problem of task placement in distributed systems, including Sarkar [37] Lo [28] Stone [42], and Blake [4] however, their algorithms have assumed that all processors are similar, that perfect information describing the system and tasks is instantaneously available, or that they have total control of all processors in the system. We have found no evidence in the literature that our more ....

....The flexibility of our mechanism allows us to tailor the vector to other applications. To determine what information should be passed in the vector, we surveyed the existing research and noted the classes of information used for scheduling algorithms (including Sarkar [37] Lo [27, 28] Stone [42], Drexl [12] and Hochbaum and Shmoys [22] Surprisingly, very few of the current scheduling algorithms use any information beyond processor speed. Of the surveyed papers, 50 used the processor speed as input to their algorithms, while only 7 considered the communications structure of the ....

Harold S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

.... parallel and distributed systems has been studied for some time [2, 6] It has been proved that no optimal policy can be obtained within polynominal computation time [15] Various heuristics (algorithms) have been proposed with the aim of producing an efficient scheduling policy for parallel tasks [3, 10, 13, 14]. A more detailed survey is found in [1] The scheduling algorithm determines the attributes of tasks and processors to be considered within the task and processor models, respectively. The most common task attributes addressed are task computation time and task communication time, which are ....

Harold S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, Volume SE-3, Number 1, pages 85--93, January 1977.

....system or the parallel program. The two most commonly used performance measures are parallel execution time of the parallel program (or schedule length) and total cost of communication delay and load balance which has been adopted by a number of commonly used scheduling algorithms such as [2, 7, 11] and [5, 8, 10] respectively. The scheduling algorithm employed in a parallel system also determines attributes of tasks and processors to be considered in the task and processor model. Generally speaking, with the objective of minimizing total cost of communication delay and load balance, the ....

Harold S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, Volume SE-3, Number 1, pages 85--93, January 1977.

....of static mapping, the information regarding processors in the system, as well as processes of an application is assumed to be known a priori. Since this problem is NP hard [Cof74, LK78, GJ79] optimal solutions can only be formulated for some special cases with strict assumptions [Hu61, CG72, Sto77, Bok81] Previous research primarily focussed on finding heuristic solutions under different situations [NT93] Nevertheless, these heuristic solutions apply only to particular communication or hardware characteristics [Sah84, MA86] such as the interconnection of processors. In the case of ....

.... the number of possible task to grain and grain to workstation mappings in heterogeneous clusters is large and the optimal mapping problem is NP complete [Cof74, LK78, 82 The Comedians on Heterogeneous Clusters GJ79] The optimal solution can only be worked out in some special cases [Hu61, CG72, Sto77, Bok81] In a shared network environment, dynamic workload and workstation heterogeneity should also be taken into consideration when a mapping is selected. It is therefore more crucial to find a quick and efficient way of searching for suitable mappings at runtime. To enable fast workstation ....

H.S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans. on Software Engineering, 3(1):85--93, 1977. (Cited on pages 11 and 82).

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H.S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans of Software Engineering, SE-3(1):95--93, January 1977.

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H. S. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithms", IEEE Trans. on Software Engineering, Vol. SE-3, Jan. 1977, pp. 85-93.

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H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

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H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

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H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

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H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

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H. S. Stone. Multiprocessor Scheduling with the Aid of Network Flow Algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977.

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H.S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans of Software Engineering, SE-3(1):95--93, January 1977.

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H. S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, vol. SE-3, pp. 85--93, 1977.

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H. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, vol. SE-3, 1, pages 85--93, January 1977.

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H. S. Stone. Multiprocessor scheduling with the aid of network flow algorithms. IEEE Transactions on Software Engineering, SE-3(1):85--93, January 1977. 21

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H. S. Stone, "Multiprocessor Scheduling with the Aid of Network Flow Algorithms," IEEE Trans. on Software Engineering, vol. 3, no. 1, pp. 85--93, Jan. 1977.

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H. S. Stone, "Multiprocessor Scheduling with the aid of Network Flow Algorithms," IEEE Transactions on Software Engineering, January, 1977.

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Stone, H. (1977a). Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans. Software Engrg., SE-3:85--93.

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Stone, H. (1977a). Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans. Software Engrg., SE-3:85--93.

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