M.-S. Shen and K. Shin, "Processor allocation in an n-cube multiprocessor using gray codes", IEEE Trans. Computers C-36 (1987), pp. 1396--1407.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are conflicting, our challenge is to design a strategy that achieves a reduction in overall turn around time. There has been considerable prior research into each of the two topics of i) scheduling of parallel jobs [1,2,6,10,14,19,20,23,24,26] and ii) contiguous node allocation strategies [3,4,5,11,13,27]. There have also been a few studies that have considered both these issues in combination [12,15,18] However, only [15] addresses the impact of contiguous node allocation schemes in conjunction with a job scheduling policy that takes fairness into consideration by use of a FCFS (First Come ....

....from figure 12. Thus, the Selective Buddy allocation strategy outperforms the non contiguous allocation scheme. 6. Related Work Several algorithms have been proposed previously for contiguous node allocation on hypercube and mesh computers. The Buddy strategy has been widely studied in the past [3,5,11]. Several studies [4,13,26] have focused on node allocation strategies for mesh connected systems. A fragmentation free allocation strategy for mesh connected parallel computers that minimizes message passing contention is presented in [17] 16] presents a strategy that minimizes network ....

M. Chen and K.G. Shin, "Processor Allocation in an n-CUBE Multiprocessor Using Gray Codes", IEEE Trans. Computers, vol. 36, no.12, pp.1,396-1,407, Dec.1987.

....sharing or multitasking runtime environment gains more attention recently as parallel systems are used as general purpose computing servers where sequential as well as highly parallel programs execute concurrently. Considerable number of allocation algorithms have been proposed for hypercubes [1] [6] and meshes [7] 11] An allocation algo rithm dynamically partitions the interconnection topology and the corresponding processors and assigns subsets of processors to the requesting jobs. The problem of the processor allo cation assumes importance so as to enable higher system utilization ....

....in Figure 1.b. 3 Isomorphic Allocation 3.1 Hypercube Allocation Algorithms Before we move into the details of the Isomorphic Allocation and its implementation, we briefly introduce data structures used in the allocation algorithms for hypercubes. For a requesting job, the simplest buddy scheme [1] partitions an n dimensional hypercube B into two Bn l S , or two buddies, along one dimension. The partitioning process is repeated along the next dimensions until a subcube of the required size is obtained. Upon a job s completion, the buddy of the released subcube is searched. If two buddies ....

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M.S.Chen and K.G.Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. Cornput., Vol. C-36 pp.1396-1407, Dec. 1987.

....1 If X and Y are isomorfizic edge matrices, then G(X) and G(Y) are iso morphic subgraphs of the hglpercube. 3 Binary Reflected Trees Many spanning trees on the z cube can be described by z x z edge matrices. These include the well known linear paths defined by the binary reflected Gray codes [4, 11], the binomial trees as define in [6] and many others. Fig. i shows an example edge matrix and the spanning tree defined by it. The matrix has non , bits only in the lower triangle. In general, we have the following. Two graphs G and H are isomorphic if there is a one to one correspondence z ....

....edges in dimension 0, each node has a degree no more than two. It is true that not every HamiltonJan path on the cube can be represented as an n x n matrix. Fortunately, the most important ones, that is, those which follow the Gary codes as defined below, do can be so represented. Definition 3 [4] Let L = 10, 1, 1 ) be any permutation of (0, 1, n 1) r be the partial rank of li in L (i.e. ri = I lj: lj li,j i l) and B = b0, b ) be any sequence of 0 and or 1. The Gray code G defined by L and B is recursively defined as follows: r 1 n l, where G; ....

M.-S. Chen and K. G. Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. on Computers, Vol. C-36, No. 12, Dec. 1987, pp. 1396-1407.

....machines have to provide a dynamic processor management facility comparable to state of the art storage management. The dynamic partitioning problem in parallel systems has been subject of a number of papers. Among the regular architectures, the hypercube has been the most attractive so far [1, 3, 6]. Partitioning of mesh connected systems is addressed by Chuang and Tzeng [2] who present a general rectangular partitioning. Li and Cheng [7] are using 2D buddy schemes, but their main interest is on scheduling, i.e. on the problem of how to schedule a given sequence of requests to achieve ....

Chen,M.-S.; Shin, K.G.: Processor Allocation in an NCube Multiprocessor Using Gray Codes. IEEE TOC 36, 12 (Dec. 1987), pp. 1396-1407

....system into subcubes. Each user or job is assigned an independent subcube by the operating system. It is known that optimal subcube allocation in a dynamic environment is an NP complete problem [8] Several heuristic algorithms reported in literature are buddy [4] modified buddy [5] gray code [6], free list [7] MSS [8] tree collapsing [9] and PC graph [10] These schemes differ from each other in terms of the subcube recognition ability and or time complexity. Better subcube recognition ability reduces external fragmentation and in turn improves the system utilization. However, ....

....ff n Gammak 1 p which is also free. Detail description of the scheme is given in [5] This scheme has better subcube recognition ability than buddy. The complexities of allocation and deallocation are O(n2 n ) and O(2 k ) respectively. Gray Code The gray code strategy proposed in [6] stores the allocation bits using a binary reflected gray code (BRGC) Here the least integer m is determined such that all the (i mod 2 n ) bits indicate availability of nodes, where i 2 [m2 k Gamma1 ; m 2)2 k Gamma1 Gamma 1] Thereafter, the allocation and deallocation are the same as ....

M. S. Chen and K. G. Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. on Computers, pp. 1396-1407, Dec. 1987.

....with the simulation results. keywords: Hypercube computer, processor allocation, queue delay, topological delay, space sharing 1 Introduction Processor allocation has been an active research area for directly interconnected parallel computers. There proposed allocation algorithms for hypercubes [1][2] 3] 4] and for meshes [5] 6] 7] 8] 9] They partition the interconnection topology and the corresponding processors and assign subsets of processors to requesting jobs. Multiple jobs share the topological space of the system. The multitasking or space sharing runtime environment gain more ....

....known at the decision instant since we do not have a priori information on job execution times. It will be beneficial, however, if we can predict the queue delay. In this paper, we propose a job based method of predicting queue delay in space shared hypercube systems. We assume buddy allocation [1] as a baseline algorithm due to its simplicity, but it can be extended to use with complex other algorithms. The buddy allocator partitions a hypercube interconnection topology into two subcubes of one less dimension, each of which in turn can be also partitioned into two smaller subcubes, and so ....

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M.S.Chen and K.G.Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes", IEEE Trans. Comput., Vol.C-36 pp.1396-1407, Dec.1987.

....recursively as s j = s j Gamma1 x j s j Gamma1 and s 1 =x 1 . Clearly, oe( is a string of length 2 n . For any node v i , let P (v i , denote the walk traversed in Qn as one starts from v i and traverses the edges in the dimensions of oe( in the order they appear in oe( It is known [1] that for any v i and , P (v i , is a Hamiltonian cycle in Qn . Assuming v 0 denote the node represented by the zero binary vector, P (v 0 ; will be referred to as the canonical Hamiltonian cycle (HC) for the permutation and will be denoted, in short, by P ( It is simple to see that the ....

Chen. M. S. and Shin. K. G., "Processor allocation in an N-cube multiprocessor using gray codes", IEEE Trans. Computers. vol. C-36, pp. 1396-1407, 1987.

....it into independent subcubes. Each user or job is assigned an appropriate subcube by the operating system. It is known that optimal allocation in a dynamic environment is an NP complete problem [8] Several heuristic algorithms reported in literature are buddy [4] modified buddy [5] gray code [6], free list [7] MSS [8] tree collapsing [9] and PC graph [10] These schemes differ from each other in terms of the subcube recognition ability and or time complexity. Better subcube recognition ability reduces external fragmentation and in turn improves the system utilization. However, ....

....is found, then the nodes corresponding to that region are allocated to job I k and the 2 k bits are set to 1. When a subcube is deallocated, all the bits in that region are set to 0 to represent the availability of the nodes. The buddy strategy is conceptually simple and statically optimal [6]. This scheme does not have complete subcube recognition ability in a dynamic environment. The time complexities of allocation and deallocation are O(2 n ) and O(2 k ) respectively. The allocation and deallocation complexity can be reduced to O(n) by using an efficient data structure [15] ....

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M. S. Chen and K. G. Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. on Computers, pp. 1396-1407, Dec. 1987.

....the same general mechanism but from two different viewpoints consumer s (process) and resource s (processor) 1] However, different selection of processors (processor allocation) may affect system performance, as will be discussed in this thesis. 2 architectures such as hypercubes and meshes [3] [11] Since the path length between processors is different in a distributed memory architecture, the cooperating processes of a job need to be allocated in nearby locations to reduce the interconnection latency. Usually first come first served (FCFS) scheduling is used to assign the incoming ....

....or if the underlying allocation algorithm is unable to detect an available subcube. Most of the previous research on hypercube allocation has been devoted to removing the second type of overflow by increasing the subcube recognition ability. Examples in this category are buddy [2] gray code [3], modified buddy [4] tree collapsing [5] free list [6] MSS Coalescing [7] and PC graph [8] The buddy scheme has low time complexity but it does not have complete subcube recognition ability. On the other hand, schemes like MSS Coalescing and free list have complete subcube recognition ability ....

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M.S.Chen and K.G.Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. Comput., Vol.C-36 pp.1396-1407, Dec.1987.

....or if the underlying allocation algorithm is unable to detect an available subcube. Most of the previous research on hypercube allocation has been devoted to removing the second type of overflow by increasing the subcube recognition ability. Examples in this category are buddy [1] gray code [2], modified buddy [3] free list [4] MSS Coalescing [5] PC graph [6] and tree collapsing [7] The buddy scheme has low time complexity but it does not have complete subcube recognition ability. On the other hand, schemes like MSS Coalescing and free list have complete subcube recognition ability ....

M.S.Chen and K.G.Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. Comput., Vol.C-36 pp.1396-1407, Dec.1987.

....We show that the baseline MIN with the BR matching pattern can be mapped onto the hypercube topology. We can therefore use any multitasking algorithm developed for the hypercube for MIN based multiprocessors also. Examples of allocation schemes are buddy strategy [1] gray code scheme [2], modified buddy strategy [3] tree collapsing [4] free list strategy [5] and coalescing MSS strategy [6] The new matching pattern can be realized with any MIN which is topologically equivalent to the baseline MIN. Next, a noncubic allocation (NCA) algorithm is proposed for assigning incoming ....

....pattern (a) 2 cube task in 4 cube MIN 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1. 0 Conv.matching pattern (Compact) BR matching pattern P0 PA (b) Probability of Acceptance Figure 12: PA for different Matching Patterns of a 2 cube in a 4 cube MIN These assumptions are similar to those used in [2] and [5] except the job arrival pattern and service time distribution. Here, the job arrival rate ( is varied to observe the performance changes. The rate is based on the average service demand (average request size Theta average residence time) and the system power (number of processors) For ....

M.S.Chen and K.G.Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. Comput., Vol.C-36 pp.1396-1407, Dec.1987.

....perform certain tasks, the composite hypercube recognition problem is to find m free nodes forming a composite hypercube CH(m) in Qn , for any given m 2 n . This problem is of critical importance in task allocation and scheduling algorithms and is similar to hypercube 5 recognition problems [Che87]. The exponential number of subcubes in a hypercube makes this problem computationally difficult. Furthermore, the dynamic nature of allocating and deallocating tasks to nodes in a hypercube algorithm leads to fragmentation of the hypercube, i.e. a total of m free nodes may be available but they ....

....of the hypercube, i.e. a total of m free nodes may be available but they may not form a composite hypercube. An allocation strategy minimizing fragmentation depends on an efficient recognition algorithm. We simply note here that the buddy tree strategy for the hypercube recognition problems [Che87] can be extended to efficiently solve the composite hypercube recognition problem as well. For further details, we refer the interested reader to [Boa92] 3 Algorithms for Composite Hypercubes Composite hypercubes are subgraphs of complete hypercubes and it may initially appear that many ....

M. Chen and K. G. Shin. "Processor Allocation in an N-CUBE Multiprocessor using Gray Codes." IEEE Transactions on Computer, C-36(12):1396--1407, 1987.

....of owning the machine exclusively, and several runs of the same job result in approximately equal execution times. Examples of contiguous allocation strategies for mesh topologies are 2D Buddy [9] Frame Sliding [5] and First Fit Best Fit [18] For hypercube topologies, examples are Gray Code [4], Partners [1] and Cyclic Buddy [11] However, contiguous processor allocation strategies suffer from fragmentation. Consider an additional job in Fig. 1 that requests four nodes. Six nodes are unallocated at this time, but there is no contiguous block of four nodes available. The nodes (2,3) ....

.... different indexing schemes) as well as the contiguous strategies First Fit [18] Best Fit [18] and Frame Sliding [5] For k ary n cube topologies we use the non contiguous strategies Random [15] MBS [15] Paging [15] and Multipartner [15] as well as the contiguous strategies Buddy [9] Gray Code [4] and Partner [1] Our simulation model uses wormhole switching and minimal dimension ordered routing (XY routing for mesh and Lee routing for k ary n cube topologies) Concerning architecture, we have two uni directional links between adjacent nodes and either a 16 x 32 mesh topology or a 8 ary ....

M. Chen and K. G. Shin. Processor allocation in an nCUBE multiprocessor using Gray codes. IEEE Transactions on Computers, C-36(12):1396--1407, December 1987.

....have localised software interfaces and special purpose hardware. Another generalisation of the model, by Bl a : zewicz et al. 1984,1986] allows tasks to require more than one processor and corresponds closely to the subcube allocation problem for hypercubes studied by Chen and Lai [1988] and Chen and Shin [1987]. Bl a : zewicz et al. 1986] show that the problem is NP complete if tasks can require arbitrary numbers of processors, but give linear time complexity algorithms for an exact solution in the case of each task either requiring one or k processors. Du and Leung [1989] show that the problem is ....

Chen, M.-S. and Shin, K. (1987). Processor allocation in a n-cube multiprocessor using gray codes. IEEE Trans. Comput., C-36(12):1396--1407.

....if the allocation fails only when there is no available subcube of the requested dimension. There are several algorithms that exist that have This represents work done at the University of Missouri Rolla the complete subcube recognition property, such as the multiple graycode (multiple GC) [1], the maximal set of subcubes [2] tree collapsing (TC) 3] and missing combination (MC) 4] Parallel complete subcube recognition algorithms are also proposed [5] The fastest of these algorithm run in approximately O(2 n Delta Gamma n j Delta , which for j = n 2 is worse than O( ....

....total number of processors requested is less than the number of processors in the hypercube, the algorithm will satisfy all requests. Several other allocation schemes have been proposed that, while statically optimal, do not have complete subcube recognition. These include the buddy and graycode [1] strategies. These have time complexities of O(2 n ) but recognize 1= Gamma n k Delta and (n Gamma k 1) Gamma n k Delta of the possible subcubes respectively. Thus, they are not examined in this paper, and neither the fast maximum set of subcubes [2] which is a heuristic ....

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M. Chen and K. G. Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Transactions on Computers, vol. C-36, no. 12, pp. 1396-1407, Dec. 1987.

....computation. The extra complication of the two levels of granularity in the model is not gratuitous: a number of concrete multicomputer mapping problems may be viewed in these terms. The models correspond closely to the subcube allocation problem for hypercubes studied by Chen and Lai [1988] and Chen and Shin [1987]. To illustrate the problem, we return to our running example, using the directed graph of Figure 2. Consider the sink node and its two predecessors. A common requirement in data processing is to sort two lists of items and then merge them into one larger sorted list, possibly eliminating ....

Chen, M.-S. and Shin, K. (1987). Processor allocation in a n-cube multiprocessor using gray codes. IEEE Trans. Comput., C-36(12):1396--1407.

....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 the natural mapping between a hypercube and a buddy system 5 , where dimensions of the cube correspond to levels of the tree [113, 173, 146, 35] (Fig. 6) This scheme is used in the nCUBE system. Implementations can use a bit vector or more sophisticated data structures based on the buddy system tree [331] In the first dimension, or top level, the whole hypercube is viewed as composed of two subcubes: one comprising PEs with a first bit ....

....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, "Processor allocation in an n-cube multiprocessor using Gray codes". IEEE Trans. Comput. C-36(12), pp. 1396--1407, Dec 1987.

....subcubes can be joined into a larger subcube. Therefore the issue of subcube recognition has received much attention in the literature. The simplest method is based on the natural mapping between a hypercube and a buddy system 4 , where dimensions of the cube correspond to levels of the tree [61, 99, 79, 23] (Fig. 6) This scheme is used in the nCUBE system. Implementations can use a bit vector or more sophisticated data structures based on the buddy system tree [203] At each level, only one bit position is considered for the partitioning, so there is only one way in which subcubes can be united. In ....

....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, "Processor allocation in an n-cube multiprocessor using Gray codes". IEEE Trans. Comput. C-36(12), pp. 1396--1407, Dec 1987.

....recognition and fragmentation problems. In case of scheduling a single task requiring a subcube of dimension d i on a hypercube, in which several tasks are already executing, one needs to not only find the required number of free processors but they should also form a subcube of dimension d i (Chen et al. 1987; Al Bassam et al. 1990; Al Dhelaan et al. 1989) The problem of fragmentation of hypercubes arises due to dynamic nature of allocation and deallocation of subcubes. In a fragmented hypercube one may have a sufficient number of nodes but they may not form a subcube (Chen et al. 1988) A third ....

Chen, M. and K. G. Shin, "Processor allocation in an n-cube multiprocessor using Gray Codes," IEEE Transactions on Computers, vol. C-36, no. 12, Dec. 1987, pp. 1396-1407.

....of allocation and deallocation in the above case are O(2 n ) and O(2 k ) respectively. Other Algorithms for Hypercube: Many other allocation algorithms have been proposed for the hypercubes to implement perfect subcube recognition ability. Some examples include the multiple gray code [4], maximal subset of subcubes (MSS) 5] free list [6] tree collapsing [7] and PC graph [8] These algorithms have perfect subcube recognition abilities at the price of high implementation complexities. Our intension is to show that a simple allocation algorithm such as buddy can be modified to ....

....abilities at the price of high implementation complexities. Our intension is to show that a simple allocation algorithm such as buddy can be modified to provide a good performance. The detailed discussion of these algorithms are beyond the scope of this paper and the readers are referred to [4] [8] for the details. 2.1.3 Problems of the Allocation Schemes The allocation algorithms vary in terms of recognition ability and implementation complexity. However, studies [12, 13] show that a better allocation algorithm does not guarantee a significant performance improvement because of the ....

M. S. Chen and K. G. Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. on Computers, vol. C-36, No. 12, Dec. 1987.

....and the time when the allocated jobs start their execution. The job scheduler decides the order in which jobs are considered for processor allocation. Several schemes for processor allocation for non real time parallel tasks have been proposed such as buddy [12] single and multiple gray code [3], free list [6] MSS [2] tree collapsing [4] PC graph[18] and the partitioned based scheme [19] These allocation algorithms vary in terms of their subcube recognition ability and time complexity. The buddy allocation scheme is simple and has low time complexity. In spite of its low subcube ....

....Both of these schemes are based on heuristics and are applicable to hypercube systems. The Buddy RT processor allocation algorithm for real time systems proposed by Babbar and Krueger [1] is an extension of the buddy scheme proposed earlier for memory systems [12] and processor allocations [3]. The basic idea of the scheme is to include an additional restriction time, while doing allocation. The Earliest Available Time (EAT) of a processor is defined as the earliest time at which that processor will become available for allocation. Instead of an array of bits (as used in the ....

M. S. Chen and K. G. Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. on Computers, pp. 1396-1407, Dec. 1987.

....have studied strategies for the location of subcubes in hypercube multicomputers. Some approaches involve enlarging the set of subcubes recognized over that of the buddy system, other approaches involve computation and or memoryintensive algorithms for complete subcube recognition. Chen and Shin [4] introduced a gray coded buddy strategy for subcube location that improves upon the buddy strategy, and proposed a multiple gray coded scheme using Gamma d bd=2c Delta gray codes to effect complete subcube recognition. Al Dhelaan and Bose [2] introduced a modified buddy strategy which ....

....in the sense that it fails to grant a request only if there is an insufficient number of available nodes to satisfy the request. In dynamic allocation and deallocation, however, it can perform quite poorly. Studies of the behavior of subcube allocation in a dynamic situation have been made in [4, 5, 10] for the purpose of evaluating various policies governing the selection of which subcube to allocate when there are several available subcubes. Variations of the single buddy system have been suggested which allocate the union of several single buddy systems. For example, an orthogonal double ....

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M.S. Chen and K.G. Shin, "Processor allocation in an n-cube multiprocessor using Gray codes", IEEE Trans. on Computers 36 (1987), pp. 1396-1407.

....[3] in which 2D meshes are used as clusters in hypercubes. The AFL strategy achieves a more effective utilization of processors in hypercubes. Details of this method are given in s Chapters 3 and 4. The most well known subcube allocation methods are the buddy system [23] the gray code method [24], and the free list method [25] Other variations include the modified buddy strategy [26] the associative memory method [27] and the MPP method [28] The buddy system was first proposed in 1965 as a memory storage allocation scheme. For a hypercube Q n , it uses 2 n bits to keep track of the ....

....Definition The problem of subcube allocation has been studied extensively to maximize processor utilization and minimize system fragmentation in hypercubes. Several strategies have been proposed and implemented for subcube allocation, including the buddy strategy [23] the gray code (GC) strategy [24], the modified buddy strategy [26] the MPP method [28] and the free list strategy [25] Of these approaches, only the MPP method and the free list strategy have been shown to perform optimally, since they provide perfect subcube 26 recognition. Additionally, the free list strategy operates ....

[Article contains additional citation context not shown here]

M.-S. Chen and K. G. Shin, "Processor allocation in an n-cube multiprocessor using gray codes," IEEE Transactions on Computers, vol. C-36, pp. 1396--1407, December 1987.

....in the sense that it fails to grant a request only if there is an insufficient number of available nodes to satisfy the request. In dynamic allocation and de allocation, however, it is no longer optimal. Studies of the behavior of subcube allocation in this dynamic situation have been made in [ChSh, DuHa] for the purpose of evaluating various policies governing the selection of which subcube to allocate when there are several available subcubes. One of the problems in using a serial algorithm to implement any allocation system is in maintaining the availability information when faults occur. This ....

....is a j such that D(j) s ff and g Gamma1 d Gammas ff (a (1) a (d Gammas ff ) and g Gamma1 d Gammas ff (a (1) a (j Gamma1) 1 Gamma a (j) a (j 1) a (d Gammas ff ) are consecutive mod 2 d Gammas ff . The double Gray coded buddy system, first suggested in [ChSh], allocates q subcubes by using the Gray coded buddy system corresponding to the identity permutation, plus the Gray coded buddy system corresponding to the permutation which reverses the order of the bits. This allocation scheme could be implemented in Theta(d) time by running Algorithm 3.2 ....

M.-S. Chen and K. Shin, "Processor allocation in an n-cube multiprocessor using gray codes", IEEE Trans. Computers C-36 (1987), 1396-1407.

....computation. The extra complication of the two levels of granularity in the model is not gratuitous: a number of concrete multicomputer mapping problems may be viewed in these terms. The models correspond closely to the subcube allocation problem for hypercubes studied by Chen and Lai [1988] and Chen and Shin [1987]. To illustrate the problem, we return to our running example, using the directed graph of Figure 2. Consider the sink node and its two predecessors. A common requirement in data processing is to sort two lists of items and then merge them into one larger sorted list, possibly eliminating ....

Chen, M.-S. and Shin, K. (1987). Processor allocation in a n-cube multiprocessor using gray codes. IEEE Trans. Comput., C-36(12):1396--1407.

....schemes fall into two major types, bottom up or bit mapped techniques and topdown or list based methods. 6, 7] While the former are usually less complex, they tend to suffer from poorer recognition ability. The Buddy Strategy in particular has been used for bottom up allocation for hypercubes [8] and can be readily adapted for other interconnection networks, such as the substar. It has been proven that the buddy strategy is statically optimal under a LIFO release scheme [9] A general algorithm for a Buddy system is as shown below: function Buddy (r,n) searches for an r sub graph in an n ....

M. S. Chen and K. G. Shin, "Processor allocation in an n-cube multiprocessor using gray codes," IEEE Transactions on Computers, vol. C-36, pp. 1396--1407, December 1987.

....it into independent subcubes. Each user or job is assigned an appropriate subcube by the operating system. It is known that optimal allocation in a dynamic environment is an NP complete problem [8] Several heuristic algorithms reported in literature are buddy [4] modified buddy [5] gray code [6], free list [7] MSS [8] tree collapsing [9] and PC graph [10] These schemes differ from each other in terms of the subcube recognition ability and or time complexity. Comparison of all the hypercube allocation policies shows that the performance improvement due to better subcube recognition ....

....k 1) ff n Gammak 1 p which is also free. Detail description of the scheme is given in [5] This scheme has better subcube recognition ability than buddy. The complexities of allocation and deallocation are O(n2 n ) and O(2 k ) respectively. Gray Code The gray code strategy proposed in [6] stores the allocation bits using a binary reflected gray code (BRGC) Here the least integer m is determined such that all the (i mod 2 n ) bits indicate availability of nodes, where i 2 [m2 k Gamma1 ; m 2)2 k Gamma1 Gamma1] Thereafter, the allocation and deallocation are the same as in ....

M. S. Chen and K. G. Shin, "Processor Allocation in an N-Cube Multiprocessor Using Gray Codes," IEEE Trans. on Computers, pp. 1396-1407, Dec. 1987.

....mappings, are found to possess some desirable properties. For some special case, optimal parallel mappings are also proved to be optimal among all mappings. 1 Introduction Subcube allocation the problem of finding a subcube in a large target hypercube has been studied extensively [1, 2] under the assumption that incoming subcube requests are independent. The commonlyused objective of subcube allocation is to minimize hypercube fragmentation. In certain applications, it may be necessary to cluster task modules into small groups, and each group is assigned to a subcube so as to ....

M. S. Chen and K. G. Shin, "Processor allocation in an n-cube multiprocessor using gray codes", IEEE Trans. on Computers, vol. C-36, pp. 1396-- 1407, Dec. 1987.

....our model, the extension is straightforward as long as some restrictions are observed. Finally, complexities and performance evaluation of the algorithms are presented. 1 Introduction Dynamic processor allocation for multicomputer systems has been an active research area in the past several years [3, 4, 6, 7, 8]. In most research works, tasks are assumed to arrive dynamically and their execution time cannot be determined beforehand. A task will run to end once allocated and dispatched, i.e. no preemption is allowed. The processors are assumed to be interconnected by direct networks. Among the ....

....to arrive dynamically and their execution time cannot be determined beforehand. A task will run to end once allocated and dispatched, i.e. no preemption is allowed. The processors are assumed to be interconnected by direct networks. Among the topologies, hypercube and 2D mesh are studied the most [3, 6, 7, 8]. Representative allocation algorithms include the frame sliding strategy for 2D meshes and the free list strategy for hypercubes [7, 8] The key to an efficient allocation algorithm lies in the modeling of the multicomputer system. For example, in the free list strategy, processors in a hypercube ....

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M.S. Chen and K.G. Shin, "Processor allocation in an N-cube multiprocessor using gray code," IEEE Trans. on Computers, Vol.C-36,pp.1396-1407,1987

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M.-S. Shen and K. Shin, "Processor allocation in an n-cube multiprocessor using gray codes", IEEE Trans. Computers C-36 (1987), pp. 1396--1407.

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M. S. Chen and K. G. Shin, "Processor Allocation in an n-Cube Multiprocessor using Gray Codes," IEEE Transactions on Computers, vol. C-36, pp. 1396--1407, December 1987.

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Ming-Syan Chen and Kang G. Shin, "Processor Allocation in an N-cube Multiprocessor Using Gray Godes," IEEE Trans. on Computers, Vol. C-36, No.12, pp. 1396-1407, Dec, 1987.

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