S. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37, 38--57, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... partitioning problems have been studied extensively in application areas including databases (e.g. histograms, grid files, index compression) parallel computing (e.g. load balancing) computer graphics (e.g. spatial data structures) and video compression (e.g. block matching) see [17, 22, 2, 4, 9] for some discussion. Following are a few examples that explicitly study partitionings. In databases, several authors [1, 11] have used partitionings to construct histograms in two or more dimensions. In this case, the metric used is often either MAX SUM for V Optimal histograms [30] or ....

S. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37, 38--57, 1988.

....be assigned to processors numbered at most j and jobs with higher index than i must be assigned to processors numbered at least j. Our goal is to minimize the time before the last processor nishes. To motivate why this particular problem is of interest consider the following example from Bokhari [2]: In communication systems it is often the case that a continuous stream of data packages have to be received and processed in real time. The processing can among other things include demodulation, error correction and possibly decryption of each incoming data package before the contents of the ....

....able to utilize the processors more eOEciently we get the following problem: Given n consecutively ordered tasks, each taking f(i) time, and p processors. Partition the tasks into p consecutive intervals such that the maximum time needed to execute the tasks in each interval is minimized. Bokhari [2] also described how a solution to this problem can be used in parallel processing as compared to pipelined. Mehrmann [9] shows how this particular partitioning problem arises when solving a block tridiagonal system on a parallel computer. We now give the formal denition of the problem: Let the ....

[Article contains additional citation context not shown here]

S. H. Bokhari, Partitioning problems in parallel, pipelined, and distributed computing, IEEE Trans. Comput., 37 (1988), pp. 4857.

.... of one architecture into another one of the same topology, but smaller size have previously been studied in [5, 12, 25] Furthermore, a number of solutions for mapping task graphs onto a parallel machine can be viewed as embeddings into architectures of smaller size and different topology [9, 11, 19, 26]. A complete binary tree is a binary tree in which all leaves are on the same level and every interior node has two children. Since n m, one processor of H is assigned a number of processors of T . Let the load of a processor of H be the number of processors of T assigned to it. Minimizing the ....

S. Bokhari. Partitioning Problems in Parallel, Pipelined, and Distributed Computing. IEEE Transaction on Computers, pages 48-57, Vol. 37, 1988.

....the processors execution costs. See Nicol [1989] for the limitations of the results in this paper. Shen and Tsai [1985] consider a function which assigns communication costs to both the sending and receiving processor in a way that it is difficult to justify in the case of a multicomputer 9 . Bokhari [1988] describes polynomial time algorithms for solving the above mini max problem in the case of chains of tasks and chains of processors with the constraint that communicating tasks must be mapped to adjacent processors, and also in various other constrained task formulations for host satellite ....

Bokhari, S. (1988). Partitioning problems in parallel, pipelined and distributed computing. IEEE Trans. Comput., C-37(1):48--57.

....assigned to each processor. See Nicol [1989] for the limitations of the results in this paper. A variation on this is the work of Efe [1982] who does not explicitly propose this model but whose mapping algorithm iteratively improves this quantity from an initial heuristic assignment of processes. Bokhari [1988] describes polynomial time algorithms for solving the mini max problem in the case of chains of processes and chains of processors with the constraint that communicating processes must be mapped to adjacent processors, and also in various other constrained process formulations for host satellite ....

Bokhari, S. (1988). Partitioning problems in parallel, pipelined and distributed computing. IEEE Trans. Comput., C-37(1):48--57.

....Partitioning problems have been studied extensively in various application areas including databases, parallel computing (e.g. load balancing) computational geometry (e.g. clustering) video compression (e.g. block matching) etc. Some related papers from a variety of application areas include [17, 23, 25, 24, 1, 3, 10]. Here we review a selection of related work most relevant to us. Hardness Results. Hardness results exist only for a simple metric function, namely, MAX SUM ID [18] proved it to be NP hard for arbitrary partitions, and [11, 6] proved it to be NP hard for p Theta p partition. Our NP hardness ....

S. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37, 38--57, 1988.

....It is commonly accepted that a number of applications can bene t from exploiting both kinds of parallelism. For instance, applications in signal and image processing are usually composed of a set of potentially data parallel tasks interacting to compute a stream of homogeneous data sets [2, 3, 20, 21]. In these applications data parallel modules usually do not scale beyond a given threshold due to the limited size of the data involved. In this case, using di erent sets of processors to compute independently di erent elements of the input stream can considerably increase the system throughput ....

S. H. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37(1):48-57, January 1988.

....the chains on chains problem can be classified as probe and dynamic programming (DP) approaches. The probe approach relies on repeated investigations for the existence of a partition with a bottleneck value no greater than a given value. The probe approach goes back to Iqbal s [11] and Bokhari s [2] works describing approximate and optimal algorithms running in log and 3 times, respectively. Here, denotes the sum of the weights in the workload array , i.e. 1 . Iqbal and Bokhari [12] and Nicol and O Hallaron [18] later proposed an log algorithm, and finally Nicol [19] proposed an log 2 ....

Bokhari, S. H. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Trans. Computers, 37, 1 (1988), 48--57.

....the partitioning of the computations inside these modules. There is a large volume of literature on mapping and scheduling parallel programs, and some excellent references are [1, 16, 26] Solutions to the problem of partitioning a chain of tasks among a set of processors have been presented in [2, 12, 13]. Our research extends this body of research to include data parallel tasks that can execute on a variable number of processors. Choudhary et al. 6] address the problem of optimal processor assignment to data parallel tasks and this paper addresses the same fundamental problem and borrows the ....

BOKHARI, S. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers 37 (Jan 1988), 48--57.

....execution. The lift to front min cut algorithm, in our current implementation, can produce only two machine, client server applications. The problem of partitioning applications across three or more machines is provably NP hard [13] Numerous heuristic algorithms exist for multi way graph cutting [7, 10, 12, 33, 38]. To more accurately evaluate the rest of the system, we restrict ourselves to an exact, two way algorithm for clientserver computing. After analysis, the application s ICC graph and component classification data (to be described later) are written into the configuration record in the application ....

Bokhari, Shahid. Partitioning Problems in Parallel, Pipelined, and Distributed Computing. IEEE Transactions on Computers, 37(1):48-57, 1988.

....on three specific array partitioning problems. Here we briefly describe the application context for each; further details of modeling will be discussed in the journal version. One dimensional case under F . This problem was abstracted for load balancing in pipelined, parallel environments in [B88] and studied in [OM95, AF91, HL92, MS95, M93, CN91, HNC92, N91] etc. Two dimensional case under F . This problem arises in balanced data distribution as implemented in the Superb environment [ZBG86] and HPF2 [HPF] High Performance Fortran) See [M93, CM 95] for more applications to particlein ....

....our results for each of the three problems of our interest. 1D p partition under F . This problem has been extensively researched. We summarize the previous work and our results in the table below, providing all citations where identical bounds were obtained independently. Reference Bound Bokhari [B88] O(n 3 p) Anily Federgruen [AF91] O(n 2 p) Hansen Liu [HL92] O(n 2 p) Manne Sorevik [MS95] O(np log p) Choi Narahari [CN91] O(np) Olstad Manne [OM95] O(np) Nicol [N91] O(n p 2 log 2 n) Charikar, Chekuri Motwani [CCM96] O(n p 2 log 2 n) Han, Narahari Choi ....

S. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37, 38--57, 1988.

....of size n into p intervals such that the maximum number of nonzero elements in any interval is minimized. This gives our 1D p partition problem under F . Further details on the application of the 1D p partition problem under F for load balancing in pipelined, parallel environments can be found in [B88, OM95, AF91, HL92, MS95, M93, MH95]. Two dimensional case under F . For data stored in two dimensional arrays, several high performance computing languages allow the user to specify a partitioning and distribution of data onto a logical set of processors. An example of such a scheme is what is known as the generalized block ....

....ut We focus on these three problems in this paper. 1.3 Results We state our results for each of the three problems of our interest. 1D p partition under F . This problem has been extensively researched. We summarize the previous work and our results in the table below. Reference Bound Bokhari [B88] O(n 3 p) Anily Federgren [AF91] O(n 2 p) Hansen Liu [HL92] O(n 2 p) Manne Sorevik [MS95] O(np log p) Olstad Manne [OM95] O(np) This paper O(n p 2 log 2 n) This paper O(n log n) It is not difficult to design an O(n 2 p) time algorithm for this problem using dynamic ....

S. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37, 38--57, 1988.

....or undirected, uniform or non uniform, there are basically four types of static mappings based on the topological structures of the task and system graphs. These are (1) mapping of specialized tasks onto specialized systems (e.g. mapping of a chain structured task onto chain linked processors) [8, 24, 4], 2) mapping of specialized tasks onto arbitrary systems (e.g. mapping of trees onto any architectures) 3, 13] 3) mapping of arbitrary tasks onto specialized systems (e.g. mapping of any tasks onto a hypercube or a completely connected network) 19, 22, 11, 25, 14, 26] and (4) mapping of ....

S. H. Bokhari. Partitioning problem in parallel, pipelined, and distributed computing. IEEE Trans. on Computers, 37(1):48--57, January 1988.

....Unlike many other objective functions in the mapping literature, this one explicitly considers parallelism in both computation and communication. Fast algorithms for finding optimal mappings are known when the LPs are arranged in a linear order, and the mapping satisfies the contiguity constraint [3, 11, 27, 6]. This means that the workload assigned to a processor must be a contiguous subchain of LPs in the linear order. If we are to use these techniques we must rationally order the LPs, attempting to force the highest rates of communication to be between co resident or nearby LPs. Since the mapping ....

S. H. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Trans. on Computers, 37(1):48--57, January 1988.

....n intervals and w into m intervals so as to minimize the maximal sum of the elements in each interval. In Section 5 we will show a practical example of such a method. For an example of how solutions to the partitioning problem can be used to speed up computations in pipelined environments see [4]. The outline of this paper is as follows: In Section 2 we describe the main partitioning problem and give an overview of recent work. In Section 3 we develop a new efficient algorithm for solving this problem. A number of variants of the partitioning problem are described and solved in Section ....

....To simplify the notation we write f(i; j) instead of f(oe i ; oe i 1 ; oe j ) We also denote the interval foe i ; oe i 1 ; oe j g by [i; j] The cost of a partition is the cost of the most expensive interval. The first reference to the MinMax problem that we are aware of is by Bokhari [4] who presented an O(n 3 p) algorithm using a bottleneck path algorithm. Anily and Federgruen [2] and Hansen and Lih [9] independently presented the same dynamic programming algorithm with time complexity O(n 2 p) Manne and S revik [10] then presented an O(p(n Gamma p) log p) algorithm based ....

S. H. Bokhari, Partitioning problems in parallel, pipelined, and distributed computing, IEEE Trans. Comput., 37 (1988), pp. 48--57.

....of 100. Our techniques may not be applicable when communication costs that depend on the particular sets of processors assigned to a task (e.g. contention) contribute significantly to overall performance. A large literature exists on the topic of mapping workload to processors, see, for instance [1, 3, 4, 6, 15, 17, 18, 23, 24, 26, 27, 31, 33]. A new problem has recently emerged, that of scheduling of tasks on multitasked parallel architectures where each task can be assigned a set of processors. Some formulations consider scheduling policies with the goal of achieving good average response time and good throughput, given an arrival ....

.... T 1 (p) because of the monotonicity of each u (t i ) Since F (p) is the minimum cost among all assignments in T (p) we have F 2 (p) F 1 (p) This result can be viewed as a generalization of Bokhari s graph based argument for monotonicity of the minimal sum cost, given a bottleneck cost [4]. Suppose for a given pipeline computation we are able to solve for F (p) given any . The set of all possible throughput values is f1=f i (x) j i = 1; n; x = 1; pg; O(pn log(pn) time is needed to generate and sort them. Given response time constraint fl, and tentative ....

S. H. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Trans. on Computers, 37(1):48--57, January 1988.

....NP hard [6] Thus if scheduling problems are to be solved optimally in polynomial time, they must contain enough restrictions to make them tractable. In this paper we will consider one such problem. To motivate why this particular problem is of interest consider the following example from Bokhari[3]: In communication systems it is often the case that a continuous stream of data packages have to be received and processed in real time. The processing can among other things include demodulation, error correction and possibly decryption of each incoming data package before the contents of the ....

....be able to utilize the processors more efficiently we get the following problem: Given n consecutively ordered tasks, each taking f(i) time, and p processors. Partition the tasks into p consecutive intervals such that the maximum time needed to execute the tasks in each interval is minimized. In [3] it is also described how a solution to this problem can be used in parallel processing as compared to pipelined. The outline of this paper is as follows: In Section 2 we describe the main partitioning problem and give an overview of recent work. In Section 3 we develop a new efficient algorithm ....

[Article contains additional citation context not shown here]

S. H. Bokhari, Partitioning problems in parallel, pipelined, and distributed computing, IEEE Trans. Comput., 37 (1988), pp. 48--57.

.... Algorithms for a Chain like Task System Chan, Chi Lok ,Gilbert, Young y The Chinese University of Hong Kong March 21, 1995 Abstract The optimal allocation of a chain like task system on the chain like network computers was first presented by Bokhari with time complexity O(m 3 n)[1], where m and n denote the number of modules and the number of processors respectively. Sheu and Chiang improved it and gave an O(minfm;ngm 2 ) algorithm[2] Hsu had further developed a two phase approach with the worst case time complexity of O(m (m 0 Gamma n) 2 n) 3] where m 0 denotes ....

....feasible length K schedule, layered graph, merged module, optimal schedule, schedule length, un mergeable modules. 1 Introduction The problem we investigate in this paper is the allocation of a chain like task system on the chain like network computers which was first presented by Bokhari [1]. A chain like task system consists of m modules scheduled on n processors. Each module is associated with an execution time and each module communicates with its neighboring modules with a communication cost (Fig. 1) E 1 E E 3 2 C 0,1 C 1,2 C 2,3 C 3,4 Fig. 1: A chain like task system E mail: ....

[Article contains additional citation context not shown here]

Shahid H. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37:48--57, January 1988.

....to each processor. See Nicol [1989] for the limitations of the results in this paper. A variation on this is the work of Efe [1982] who does not explicitly propose this model but whose mapping algorithm iteratively improves this quantity from an initial heuristic assignment of processes. Bokhari [1988] describes polynomial time algorithms for solving the mini max problem in the case of chains of processes and chains of processors with the constraint that communicating processes must be mapped to adjacent processors, and also in various other constrained process formulations for host satellite ....

Bokhari, S. (1988). Partitioning problems in parallel, pipelined and distributed computing.

....[6] even with a number of simplifying restrictions. Stone [16] solves the heterogeneous network scheduling problem for the case m = 2 machines using a network flow algorithm. Bokhari [2] solves the problem for arbitrary m and n when the task graph is a tree. Related results are derived in [3, 13, 17]. A (nonpreemptive) mapping of tasks to machines is a function Pi : V H. If Pi maps all n tasks to the same machine, the mapping is sequential. If the codomain of Pi contains more than one machine, the mapping may permit concurrent execution of tasks on distinct machines. Except when ....

Bokhari, S.H. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Trans. Comput. 37, 1 (Jan. 1988), 48--57.

....then it is called task allocation [10] Whether the graphs are directed or undirected, uniform or nonuniform, there are basically four types of static mappings based on the topological structures of the task and system graphs. These are (1) mapping of specialized tasks onto specialized systems [8, 17, 4], 2) mapping of specialized tasks onto arbitrary systems [3, 13] 3) mapping of arbitrary tasks onto specialized systems [11, 18, 19] and (4) mapping of arbitrary tasks onto arbitrary systems [2, 14, 1, 15, 9, 16] One of the earliest mapping algorithms, which can map an arbitrary task onto an ....

S. H. Bokhari. Partitioning problem in parallel, pipelined, and distributed computing. IEEE Trans. on Computers, 37(1):48--57, January 1988.

....be assigned to processors numbered at most j and jobs with higher index than i must be assigned to processors numbered at least j. Our goal is to minimize the time before the last processor nishes. To motivate why this particular problem is of interest consider the following example from Bokhari [2]: In communication systems it is often the case that a continuous stream of data packages have to be received and processed in real time. The processing can among other things include demodulation, error correction and possibly decryption of each incoming data package before the contents of the ....

....able to utilize the processors more eOEciently we get the following problem: Given n consecutively ordered tasks, each taking f(i) time, and p processors. Partition the tasks into p consecutive intervals such that the maximum time needed to execute the tasks in each interval is minimized. Bokhari [2] also described how a solution to this problem can be used in parallel processing as compared to pipelined. Mehrmann [9] shows how this particular partitioning problem arises when solving a block tridiagonal system on a parallel computer. We now give the formal de nition of the problem: Let the ....

[Article contains additional citation context not shown here]

S. H. Bokhari, Partitioning problems in parallel, pipelined, and distributed computing, IEEE Trans. Comput., 37 (1988), pp. 4857.

.... four types of static mappings based on the topological structures of the task and system graphs: 1) mapping of arbitrary tasks onto specialized systems [17, 30, 1, 20, 24] 2) mapping of specialized tasks onto arbitrary systems [7, 19] 3) mapping of specialized tasks onto specialized systems [11, 36, 8], and (4) mapping of arbitrary tasks onto arbitrary systems [6, 14, 26, 4, 5, 28, 30, 15, 29, 32, 10] In this paper, we concentrate on static mapping of arbitrary non uniform directed task graphs onto arbitrary non uniform system graphs. Mapping of arbitrary directed task graphs onto arbitrary ....

S. H. Bokhari. Partitioning problem in parallel, pipelined, and distributed computing. IEEE Trans. on Computers, 37(1):48--57, January 1988.

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S. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Transactions on Computers, 37, 38--57, 1988.

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S. H. Bokhari. Partitioning problems in parallel, pipelined, and distributed computing. IEEE Trans. on Computers, 37(1):48--57, January 1988.

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