Ming-Syan Chen and Kang G. Shin, "Subcube Allocation and Task Migration in Hypercube Multiprocessor," IEEE Trans. on Computers, Vol. C-39, No.9, pp. 1146-1155, Dec, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is a power of by Wagner and West [25] Recently, Bhat and Savage [2] have shown that such codes exist for all . During the years Gray codes and their generalizations have found applications in a variety of areas such as information storage and retrieval [4] processor allocation in the hypercube [5], statistics [7] codes for certain memory devices [8] Manuscript received April 20, 1998; revised May 15, 1999. This work was supported by the Fund for the Promotion of Research at the Technion. The authors are with the Computer Science Department, Technion Israel Institute of Technology, ....

M. Chen and K. G. Shin, "Subcube allocation and task migration in hypercube machines," IEEE Trans. Comput., vol. 39, pp. 1146--1155, 1990.

....by compacting the system to obtain a large subcube, by changing the order of execution, or by reducing the request size so that it fits into the system. Like memory compaction in conventional memory management, the overflow issue can be overcome by relocating the existing jobs. Chen and Shin [25] have proposed a task migration scheme along with the gray code allocation algorithm. Task migration consists of three steps: selecting a target task, locating a destination and finding a deadlock free migration path. By relocating and compacting active jobs within a hypercube at one end, a large ....

M.S.Chen and K.G.Shin, "Subcube Allocation and Task Migration in Hypercube Multiprocessors," IEEE Trans. Comput., Vol.39 pp.1146-1155, Sep.1990.

....compacting the system to obtain a large subcube, by changing the order of execution, or by reducing the request size so that it fits into the system. First, like memory compaction in conventional memory management, the overflow issue can be overcome by relocating the existing jobs. Chen and Shin [10] have proposed a task migration scheme along with the gray code allocation algorithm. Task migration consists of three steps: selecting a target task, locating a destination and finding a deadlock free migration path. By relocating and compacting active jobs within a hypercube at one end, a 1 ....

M.S.Chen and K.G.Shin, "Subcube Allocation and Task Migration in Hypercube Multiprocessors, " IEEE Trans. Comput., Vol.39 pp.1146-1155, Sep.1990.

.... and the DQT will also be implemented in the operating system on the RWC 1, called SCore, under development in our RWC project 1 Introduction So far, many techniques of job scheduling for partitionable parallel machines involve finding an idle partition and mapping a newly entered job onto it [3, 4, 12, 20]. In most cases, however, processor utilization is far from optimal because of fragmentation. Krueger et al. proposed a new job scheduling scheme, called scan , and found that job scheduling order, not mapping, is more important to achieve higher processor utilization [11] All methods, however, ....

M.-S. Chen and K. G. Shin. Subcube Allocation and Task Migration in Hypercube Multiprocessors. IEEE Transactions on Computers, 39(9):1146--1155, 1990.

....algorithms executable in G(k) Task migration strategies between the multiple copies of G(k) also become possible, providing advantages such as load balancing and support of fault tolerance. Note that we will not be discussing task migration or node allocation strategies in this paper (see [14], 15] and [16] for more on this topic) Rather, we present efficient packing techniques that can support these strategies in star graphs when multiple hypercubes are being simulated. A well devised packing technique can be readily extended to support variable dilation embeddings. A ....

....by the partitioning strategies presented throughout this section. S(n) S(3) S(4) S(5) S(6) S(7) S(8) S(9) S(10) Number of packed Q(n Gamma 1) s 1 2 4 12 36 144 576 2,880 Expansion ratio 1.50 1.50 1.88 1.88 2.19 2.19 2.46 2. 46 Number of packed Q(n Gamma 2) s 3 6 12 36 108 432 1,728 8,640 [14] [42] 138] 552] 2,352] 11,760] Expansion ratio 1.00 1.00 1.25 1.25 1.46 1.46 1.64 1.64 [1.07] 1.07] 1.14] 1.14] 1.20] 1.20] Number of packed Q(n Gamma 3) s 6 12 30 90 270 1,080 4,320 21,600 [306] 1,224] 5,328] 26,640] Expansion ratio 1.00 1.00 1.00 1.00 1.17 1.17 1.31 1.31 ....

M.-S. Chen and K. G. Shin, "Subcube Allocation and Task Migration in Hypercube Multiprocessors," IEEE Transactions on Computers, Vol. 39, No. 9, September 1990, pp. 1146-1155.

....of fault tolerance. Note that we will not be discussing task migration or processor allocation strategies in this paper. Rather, we present efficient packing techniques that are required to support these strategies in star graphs when multiple hypercubes are being simulated. We refer the reader to [11] and [12] if the particular issues of task migration and processor allocation in hypercubes is of interest. Let G(k) be a k dimensional graph with hierarchical structure [2] such that a (k 1) dimensional graph G(k 1) can be obtained recursively from c(k) copies of G(k) Several graphs ....

M.-S. Chen and K. G. Shin, "Subcube Allocation and Task Migration in Hypercube Multiprocessors," IEEE Transactions on Computers, Vol. 39, No. 9, September 1990, pp. 1146-1155.

....two jobs terminate after being executed on disjoint subcubes, we need to be able to recognize whether or not these subcubes can be joined into a larger subcube. If they cannot be joined directly, it might be advantageous to migrate a job running next to one free subcube to the other free subcube [115, 114, 110], thereby creating two adjacent free subcubes that can be joined together. Such migration is also guided by the recognition of the potential to free a large subcube. Therefore the issue of subcube recognition has received much attention in the literature. The first and simplest method is based on ....

....n possible ways to do the partitioning. In the example of Fig. 6, the recognized 2 D subcubes are marked with arrows. These are the horizontal 2 D subcubes of the 4 D hypercube on the left. Other 2 D subcubes are not recognized. Recognition is improved if the PE IDs are regarded as a Gray code [113, 114], rather than as a reflection of the cube s dimensions. In a Gray code, successive elements differ by exactly one bit position. Following the order of the Gray code, we can visit all the PEs in one cycle. As shown at the left of Fig. 7, this can also be interpreted as a tree with 5 In a buddy ....

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M-S. Chen and K. G. Shin, "Subcube allocation and task migration in hypercube multiprocessors ". IEEE Trans. Comput. 39(9), pp. 1146--1155, Sep 1990.

....be united. In the example of Fig. 6, the recognized 2 D subcubes are marked with arrows. These are the horizontal 2 D subcubes of the 4 D hypercube on the left. Other 2 D subcubes are not recognized. Many schemes that offer improved recognition have been proposed, including the use of Gray codes [61, 62], free lists [187, 81] or a lattice structure. A lattice is a partially ordered 3 If the application does not require a subcube, it is still convenient to allocate a subcube, and then possibly to reclaim the leftover nodes [345] 4 In a buddy system, a set of resources is allocated in blocks ....

M-S. Chen and K. G. Shin, "Subcube allocation and task migration in hypercube multiprocessors ". IEEE Trans. Comput. 39(9), pp. 1146--1155, Sep 1990. 76

.... by a reflection [CSW89] Gray codes have found applications in such diverse areas as circuit testing [RC81] signal encoding [Lud81] ordering of documents on shelves [Los92] data compression [Ric86] statistics [DH94] graphics and image processing [ASD90] processor allocation in the hypercube [CS90], hashing [Fal88] computing the permanent [NW78] information storage and retrieval [CCC92] and puzzles, such as the Chinese Rings and Towers of Hanoi [Gar72] In recent variations on combinatorial Gray codes, generation problems have been considered in which the difference between successive ....

M. Chen and K. G. Shin. Subcube allocation and task migration in hypercube machines. IEEE Transactions on Computers, 39(9):1146--1155, 1990.

....to the ability of a scheduler to move an executing job or some of its components to other processors. As such it is an extension of preemption: a task stops running on a certain processor, and it restarts on another processor. Reasons for migration include packing in order to reduce fragmentation [6,12], and the need to withdraw from a workstation when its owner returns [42] Migration is simple on shared memory machines, because threads do not have any state that is local to the processor except for their cache and TLB footprints. The challenge is to ensure that interacting threads map to ....

M-S. Chen and K. G. Shin, "Subcube allocation and task migration in hypercube multiprocessors". IEEE Trans. Comput. 39(9), pp. 1146--1155, Sep 1990.

.... also defined by considering the heterogeneity and time sharing effects [11] Scheduling policies on multiprocessor multicomputer systems are classical topics, and have been studied by many people on different types of systems, such as shared memory systems [8] 10] distributed memory hypercube [3] and mesh [6] Since these policies algorithms are designed for a dedicated homogeneous system, they may not be directly applicable to heterogeneous NOW systems. In order to effectively manage the interaction between parallel jobs and user jobs and well utilize the resources of a heterogeneous ....

M.-S. Chen and K. G. Shin, "Subcube allocation and task migration in hypercube multiprocessor ", IEEE Transactions on Computers, Vol. C-39, No. 9, 1990, pp. 1146-1153. 21

.... that can be obtained from such a system are far outweighed by the development effort [89] Some migration systems have been developed on tightly coupled multi processor systems but such implementations do not have to address the same range of issues as those presented by loosely coupled systems [17, 44]. In such systems, memory and device access is typically shared between all processors. As a consequence, moving a process between processors is relatively straightforward because most process information is equally meaningful on all cpus. 2.3 Motivations for Migration Theoretical and modeling ....

M.S. Chen and K.G. Shin. Subcube Allocation and Task Migration in Hypercube Multiprocessors. IEEE Transactions on Computers, 39:1146--1155, 1990.

....a job finishes, its subcube is combined with free neighboring subcubes, if any, so that a larger idle subcube becomes available. The goal of their work is to describe efficient schemes for finding sub cubes to satisfy arriving jobs and minimizing the fragmentation of available subcubes. Later work [9] describes a global parallel task migration strategy to reduce the available subcube fragmentation. Other studies consider batch scheduling algorithms where the number of jobs, the parallelism of the jobs, and jobs execution time needs are known in advance. Instead of minimizing overall response ....

M.-S. Chen and K.G. Shin. Subcube allocation and task migration in hypercube multiprocessors. IEEE Transactions on Computers, C-39(9):1146--1153, September 1990.

....of managing threads, as exposed by the maximum load of any PE. Related work. There has been a significant amount of prior work in processor allocation; all such work view the computational load as a sequence of tasks, each requiring certain computational resources. A number of prior studies in [12, 9, 10, 11, 13, 14, 18] allocate processors considering topology constraint of each task. In [12, 9, 10, 11] they consider the problem of subcube recognition for hypercube machines, but they do not formally analyze their algorithms; further [12, 10] give task reallocating strategies, but there is no formal measure on ....

....significant amount of prior work in processor allocation; all such work view the computational load as a sequence of tasks, each requiring certain computational resources. A number of prior studies in [12, 9, 10, 11, 13, 14, 18] allocate processors considering topology constraint of each task. In [12, 9, 10, 11], they consider the problem of subcube recognition for hypercube machines, but they do not formally analyze their algorithms; further [12, 10] give task reallocating strategies, but there is no formal measure on the tradeoff between the reallocation frequency and resulting fragmentation. The ....

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M. Chen and K. Shin (1990): Subcube Allocation and Task Migration in Hypercube Multiprocessors. IEEE Trans. Comp. C-39, 1146-1155.

.... F 1I NTRODUCTION UBCUBES in the hypercube are assigned to execute jobs according to a processor allocation scheme, with each job occupying one subcube of an appropriate size. Example processor allocation algorithms can be found in [1], 2] 3] 4] 5] When a job completes its execution, the occupied subcube is released. After repeated subcube assignments and releases, it is likely for a hypercube to become fragmented so that an incoming job cannot be allocated even if a sufficient number of free nodes are present, because ....

....over the communication channels and the sizes of programs data to be transmitted are often large. To reduce the compaction duration, the third cost component has to be lowered effectively. It is common that processes in a subcube (which belong to one task) are moved in parallel, as addressed in [1], 8] Hypercubes of the second generation support either circuit switching (such as Intel s iPSC 2 and iPSC 860) or wormhole routing (such as NCUBE s n Cube 2) for efficient message transmission. Task migration in a hypercube with circuit switching or wormhole routing requires that the ....

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M.S. Chen and K.G. Shin, "Subcube Allocation and Task Migration in Hypercube Multiprocessors," IEEE Trans. Computers, vol. 39, no. 9, pp. 1,146-1,155, Sept. 1990.

....directed edges from v i to v j , where w ij denotes the weight on the edge, and represents the length of the message from v i to v j . We will first discuss a simple case where all subcubes are of the same dimension, using the subcube communication model defined for uniform size subcubes as in [6, 7]. Given (v i ; v j ; w ij ) w ij 0, suppose OE i = anan Gamma1 : a 1 is the subcube address which v i is mapped to, and OE j = b n b n Gamma1 : b 1 is the subcube address v j is mapped to. We define an instance of subcube communication as each node in OE i sends a message of length w ij ....

....proposed in [6] where a 1 to 1 mapping function is found between source and destination nodes, and a message between each sourcedestination pair is routed through a shortest path. Also, all messages in an instance of subcube communication are routed through edge disjoint paths. From [7], we get the sum of lengths of these paths as T (OE i ; OE j ) M (OE i ; OE j )2 d , where M (OE i ; OE j ) is defined as M (OE i ; OE j ) P n i=1 m(a i ; b i ) where m(a i ; b i ) 8 : 1 if a i = b i 6= 0 if a i = b i 1=2 otherwise. Therefore, we define the bandwidth of such ....

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M. S. Chen and K. G. Shin, "Subcube allocation and task migration in hypercube multiprocessor", IEEE Trans. on Computers, vol. C-39, pp. 1146-- 1155, Sep. 1990.

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Ming-Syan Chen and Kang G. Shin, "Subcube Allocation and Task Migration in Hypercube Multiprocessor," IEEE Trans. on Computers, Vol. C-39, No.9, pp. 1146-1155, Dec, 1990.

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