Cooeman Jr., E.G., Garey, M.R., and Johnson, D.S., Bin packing with divisible item sizes, Journal of Complexity, 1987, 3, 406-428.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....size. In particular it is the case of hypercube computers and buddy processor allocation schemes where tasks require number of processors which is a power of 2. This case is simpler than the previous ones because matching sizes of tasks to be processed in parallel can be done in polynomial time [7]. For the simplicity reasons we assume in the further presentation that sizes of tasks are powers of 2. The algorithm we propose for this problem builds processor feasible sets in the nonincreasing order of the total value of included tasks. The number of processors occupied by the PFS built ....

Cooeman Jr., E.G., Garey, M.R., and Johnson, D.S., Bin packing with divisible item sizes, Journal of Complexity, 1987, 3, 406-428.

....a uniform distribution over powers of 2, the DHC scheme does not impose any restrictions; the gang sizes are directly mapped onto groups of power of2 processors. Therefore there is no waste, implying a utilization of 100 . This is a special case of packing items with sizes that divide each other [10]. 4.2. Unrestricted Best Fit Algorithms To analyze the waste due to the partitioning into powers of 2, consider an unrestricted best fit algorithm that does not abide by such a restriction. We first examine the optimal off line case, and then a strict on line case, that services gangs on a ....

Coffman, E. G., Jr., Garey, M. R., and Johnson, D. S. Bin packing with divisible item sizes. J. Complex. 3, 4 (Dec. 1987), 406--428.

....of such classes are: I O intensive, Synchronization intensive, and Computation intensive. Each one of these classes is similar to a fuzzy set [37] A fuzzy set associated with a class A is characterized by a membership function f A (x) which associates each task T to a real number in the interval [0,1], with the value of fA (T ) representing the degree of membership of T in A. Thus, the nearer the value of fA (T ) to unity, the higher the degree of membership of T in A. For instance, consider the class of I O intensive tasks, with its respective characteristic function f IO (T ) A value of f ....

....J is the number of tasks (which is equal to number of processors in Gang service algorithms) of job J and M J is the maximum amount of memory required by each task, which will be the same for all tasks. Observe that this problem is different from the two dimensional (geometric) binpacking problem [1], in which rectangles are to be packed into a fixed width strip so as to minimize the weight of packing, since the memory segments required by a job do not need to be contiguous. It also differs from the vector bin packing problem. In the d dimensional version of the vector packing problem, the ....

E.G. Coffman, M.R. Garey, and D.S. Johnson. Bin Packing with Divisible Item 18 sizes. Journal of Complexity, 3:406--428, 1987.

....J is the number of tasks (which is equal to number of processors in gang service algorithms) of job J and M J is the maximum amount of memory required by each task, which will be the same for all tasks. Observe that this problem is different from the twodimensional (geometric) bin packing problem [2], in which rectangles are to be packed into a fixed width strip so as to minimize the weight of packing, since the memory segments required by a job do not need to be contiguous. It also differs from the vector bin packing problem. In the d dimensional version of the vector packing problem, the ....

E.G. Coffman, M.R. Garey, and D.S. Johnson. Bin Packing with Divisible Item sizes. Journal of Complexity, 3:406--428, 1987.

....uniform distribution over powers of two the DHC scheme does not impose any restrictions; the gang sizes are directly mapped onto groups of power of two processors. Therefore there is no waste, implying a utilization of 100 . This is a special case of packing items with sizes that divide each other [10]. 4.2 Unrestricted Best Fit Algorithms To analyze the waste due to the partitioning into powers of two, consider an unrestricted best fit algorithm that does not abide by such a restriction. We first examine the optimal off line case, and then a strict on line case, that services gangs on a ....

E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson, "Bin packing with divisible item sizes". J. Complex. 3(4), pp. 406--428, Dec 1987.

....as needed in order to unite slots and improve run fractions. Specifically, our algorithm re maps all jobs upon every job arrival and termination, using a first fit decreasing allocation to slots [4] this algorithm is optimal if all job sizes divide each other, e.g. if they are powers of two [5]) It is debatable whether this algorithm is realistic, because of the expected overhead, especially on distributed memory machines. It is true that systems that support migration have been implemented successfully [1, 6] but these systems do not attempt to perform migration at such a high rate. ....

E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson, "Bin packing with divisible item sizes". J. Complex. 3(4), pp. 406--428, Dec 1987.

....job size of J i , and IP indicates the number of idle processors. Let us suppose that s jobs has been dispatched by the LJF algorithm and they are still in execution, that is, processors are executing J 1 : J s . Then, the number of processors that are currently idle is derived by formula (2) [15]. IP = c 1 1 min(P 1 ; P s ) 128 (2) c 1 is an integer and c 1 0 Because the LJF dispatches a larger job prior to a smaller job, J s is the smallest job among jobs in execution. Thus, IP = c 1 1 P s : 3) Here, there is a relation represented by (4) between job size of the job at the top ....

E. G. Coffman, M. R. Garey, and D. S. Johnson. Bin Packing with Divisible Item Sizes. Journal of Complexity, 3:406--428, 1987. 10

.... simple first fit algorithm achieves better packing if the packed items are sorted in decreasing size [127, 125] If the item sizes divide each other and also divide the bin capacity which is the case for jobs that require subcubes from a hypercube, for example a perfect packing is achieved [126]. In terms of scheduling, this means that scheduling the larger jobs first may be expected to cause less fragmentation, and therefore higher resource utilization, than FCFS. However, the fragmentation can still be quite large [363] Despite the intuitive appeal of some of these scheduling ....

E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson, "Bin packing with divisible item sizes". J. Complex. 3(4), pp. 406--428, Dec 1987.

.... a simple first fit algorithm achieves better packing if the packed items are sorted in decreasing size [69, 67] If the item sizes divide each other and also divide the bin capacity which is the case for jobs that require subcubes from a hypercube, for example a perfect packing is achieved [68]. In terms of scheduling, this means that scheduling the larger jobs first may be expected to cause less fragmentation, and therefore higher resource utilization, than FCFS. Despite the intuitive appeal of some of these scheduling policies, studies indicate that they do not necessarily perform ....

E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson, "Bin packing with divisible item sizes". J. Complex. 3(4), pp. 406--428, Dec 1987.

.... most of the weight is at low values, and strong discrete components appear at powers of two (two examples are given in Figure 2) This result is significant because jobs using power of two partitions are easier to pack, and small jobs may be used to fill in holes between larger jobs [6, 13]. Thus including or omitting these features has a significant impact on the performance of the modeled system. 4 Weights In the previous section we espoused the value of characterizing workloads by using distributions rather than just moments. However, care must be taken in collecting the data ....

E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson, "Bin packing with divisible item sizes". J. Complex. 3(4), pp. 406-- 428, Dec 1987.

....of ST ST GammaZ(2V Gamma1) of the optimal off line choices for every sequence of requests. We also show that this is the best possible for any on line algorithm, even when randomness is used. 1. 3 Previous work The power of using divisible sizes in a related context has been demonstrated in [CGJ87], which considers variations on the bin packing problem using the same assumption. The ad placement problem can in fact be described as a bin packing problem where, in addition to the standard notion of the size of an item s i , every item also has a weight w i , and a copy of item i must be ....

E. Coffman, M. Garey, and D. Johnson. Bin packing with divisible item sizes. Journal of Complexity, 3:406 -- 428, 1987.

....of ST ST GammaZ(2V Gamma1) of the optimal off line choices for every sequence of requests. We also show that this is the best possible for any on line algorithm, even when randomness is used. 1. 3 Previous work The power of using divisible sizes in a related context has been demonstrated in [CGJ87], which considers variations on the bin packing problem using the same assumption. The ad placement problem can in fact be described as a bin packing problem where, in addition to the standard notion of the size of an item s i , every item also has a weight w i , and a copy of item i must be ....

E. Coffman, M. Garey, and D. Johnson. Bin packing with divisible item sizes. Journal of Complexity, 3:406 -- 428, 1987.

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