A. Aggarwal and G. Plaxton. Optimal parallel sorting in multi-level storage. In Proc. ACM--SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the data blocks residing in memory at a given time are such that the I O required to get them there can be charged to the amount of internal memory work that can be accomplished using that set of memory resident data blocks. Several interesting parallel disk sorting algorithms [VS94, NV95, AP94] performing an optimal number Theta of I O operations have been proposed, but they are somewhat complicated and difficult to implement in practice. As a consequence, an attractive alternative to implement sorting algorithms for parallel disks is to use the technique of disk striping (or ....

....BGV97] for more intuition regarding the difficulty merging with parallel independent disks. Nodine and Vitter [NV95] overcame this difficulty by performing external merging by first approximately merging the runs followed by additional passes to refine the merge. Aggarwal and Plaxton s Sharesort [AP94] technique does repeated merging and has accompanying overheads. Each of these approaches involves extra overheads and are not ideal for practical implementation. 3 SRM Algorithm The SRM algorithm of Barve et al. BGV97] overcomes the difficulties involved in parallel disk merging by using a ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. Proc. Fifth Annual ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

....implemented using only striped writes. This restriction is important in practical parallel disk systems, where striped writes may be required for maintaining redundancy information to make recovery possible in the event of a disk failure [CLG, Sch] Subsequently to our work, Aggarwal and Plaxton [AgP] developed another optimal deterministic external sorting algorithm. Their algorithm is based on the hypercube sorting method of Cypher and Plaxton [CyP] which has higher constant factors. Barve, Grove, and Vitter [BGV] developed a practical sorting method based on merge sort, which does probably ....

Alok Aggarwal & C. Greg Plaxton, "Optimal Parallel Sorting in Multi-Level Storage," Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, Washington, D.C. (January 1994).

....provided matching upper and lower I O bounds for several problems. These bounds apply to the PDM model. The lower bound for sorting states that the worst case number of I O s required for sorting is Theta( N BD log M B N B ) 1 [2, 39] Several EM algorithms exist for sorting, including [1, 2, 3, 29, 30, 41, 42, 31]. Surprisingly, it turns out that performing a permutation requires Theta(minf N D ; N BD logM B N B g) I Os [2, 39] while the same can be performed in linear time in the RAM model. Similarly, the worst case number of I Os required to transpose a p Theta q matrix from row major order ....

Aggarwal, A., and Plaxton, G. Optimal parallel sorting in multi-level storage. Proc. ACM-SIAM Symp. on Discrete Algorithms (1994), 659--668.

.... implementing algorithms designed for the model can lead to significant runtime improvements in practice [40, 41, 126, 129] Finally, it should be mentioned that several authors have considered extended theoretical models that try to model the hierarchical nature of the memory of real machines [1, 2, 3, 4, 7, 77, 116, 131, 132, 134], but such models quickly become theoretically very complicated due to the large number of parameters. Therefore only very basic problems like sorting have been considered in these models. 3 1.2 Outline of the Thesis The rest of this thesis consists of two parts. The first part contains a ....

A. Aggarwal and G. Plaxton. Optimal parallel sorting in multi-level storage. Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

....# blocks each block having # records. A more realistic model was envisioned in [20] Several asymptotically optimal algorithms have been given for sorting on this model. Nodine and Vitter s optimal algorithm [13] involves solving certain matching problems. Aggarwal and Plaxton s optimal algorithm [4] is based on the Sharesort algorithm of Cypher and Plaxton. Vitter and Shriver gave an optimal randomized algorithm for disk sorting [20] All these results are highly nontrivial and theoretically interesting. However, the underlying constants in their time bounds are high. In practice the simple ....

A. Aggarwal and C. G. Plaxton, Optimal Parallel Sorting in Multi-Level Storage, Proc. Fifth Annual ACM Symposium on Discrete Algorithms, 1994, pp. 659-668.

....requires. Although this cost model does not account for the variation in disk access times caused by head movement and rotational latency, programmers often have no control over these factors. The number of disk accesses, however, can be minimized by carefully designed algorithms such as those in [AP94, Arg95, CGG 95, Cor92, Cor93, CSW94, GTVV93, NV93, VS94] and this paper. There are two restrictions implied by the Vitter Shriver model. In order for the memory to accomodate the records transferred in a parallel I O operation to all D disks, we require that BD M . Also, we assume that M N ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 659--668, January 1994.

....will participate in the merge process is input dependent and unknown. Vitter and Shriver [VS94] give further intuition regarding the difficulty of mergesorting on parallel disks. In Greed Sort [NV90] the trick of approximate merging is used to circumvent this difficulty. Aggarwal and Plaxton [AP94] use the Sharesort technique that does repeated merging with accompanying overhead. Recently, Pai et al. [PSV94] considered the average case performance of a simple merging scheme for R = D sorted runs, one run on each disk. They use an approximate model of average case inputs and require that ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. Proc. Fifth Annual ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

....The first is a variation on sample sort and the other is a variation on the approach of sorting by regular sampling. On the other hand, from the perspective of the individual SMP, there are fewer choices for sorting on hierarchical shared memory machines. These include the algorithms described in [3, 11, 12, 2, 6, 13]. Unfortunately, none of these algorithms by itself is sufficient to achieve efficient performance on an SMP cluster. Efficient algorithms for distributed memory machines tend to confine interprocessor communication to a minimum number of regular balanced exchanges. By contrast, algorithms for ....

A. Aggarwal and G. Plaxton. Optimal Parallel Sorting in Multi-Level Storage. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 659--668, 1994.

....matching upper and lower I O bounds for several problems, and these bounds apply to the PDM model. The lower bound for sorting states that the worst case number of I O s required for sorting is Theta( N BD log M B N B ) 1 [3, 47] Several EM algorithms exist for sorting, including [2, 3, 4, 36, 37, 49, 50, 38]. Surprisingly, it turns out that performing a permutation requires Theta(minf N D ; N BD log M B N B g) I Os [3, 47] while the same can be performed in linear time in the RAM model. Similarly, the worst case number of I Os required to transpose a p Theta q matrix from row major ....

Aggarwal, A., and Plaxton, G. Optimal parallel sorting in multi-level storage. Proc. ACM-SIAM Symp. on Discrete Algorithms (1994), 659--668.

.... implementing algorithms designed for the model can lead to significant runtime improvements in practice [40, 41, 126, 129] Finally, it should be mentioned that several authors have considered extended theoretical models that try to model the hierarchical nature of the memory of real machines [1, 2, 3, 4, 7, 77, 116, 131, 132, 134], but such models quickly become theoretically very complicated due to the large number of parameters. Therefore only very basic problems like sorting have been considered in these models. 3 1.2 Outline of the Thesis The rest of this thesis consists of two parts. The first part contains a ....

A. Aggarwal and G. Plaxton. Optimal parallel sorting in multi-level storage. Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

....will participate in the merge process is input dependent and unknown. Vitter and Shriver [VS94] give further intuition regarding the di#culty of mergesorting on parallel disks. In Greed Sort [NV90] the trick of approximate merging is used to circumvent this di#culty. Aggarwal and Plaxton [AP94] use the Sharesort technique that does repeated merging with accompanying overhead. Recently, Pai et al. [PSV94] considered the average case performance of a simple merging scheme for R = D sorted runs, one run on each disk. They use an approximate model of average case inputs and require that the ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. Proc. Fifth Annual ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

....explicitly work with out of core data on parallel disks. Since the introduction of the Parallel Disk Model (PDM) by Vitter and Shriver in 1990 [VS94] there have been significant technical advances on how to carefully plan parallel disk accesses for common data parallel operations and algorithms [AP94, Arg95, AVV95, BGV97, CGG 95, Cor93, Cor97, CN96, CWN97, CSW94, GTVV93, NV93, NV95, Wis96, WGWR93, VS94] The performance improvements gained by using these methods can be tremendous, and their impacts increase with the problem size. They require a degree of coordination among the processors ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 659--668, January 1994.

....and the other is a variation on the approach of sorting by regular sampling. On the other hand, from the perspective of the individual SMP, there are fewer choices for sorting on hierarchical shared memory machines. These include distribution sort [4] greed sort [13] balance sort [14] sharesort [3], simple randomized merge sort [8] radix sort, and the sorting algorithm of Varman et al. 16, 15] Unfortunately, none of these algorithms by themselves is sufficient to achieve efficient performance on an SMP cluster. Efficient algorithms for distributed memory machines tend to confine ....

....sophisticated versions of merge sort, designed to make optimal use of multiple independent disks. However, straightforward merging is inherently sequential, so without modifications these algorithms would not be expected to perform efficiently with multiple processors. Finally, there is sharesort [3], which is a hybrid of the first two approaches. Briefly, the input of n elements is evenly divided into n fl subsets, where fl is some constant between 0 and 1, and the resulting subsets are recursively sorted. A set of precisely evenly spaced splitters are then computed to divide the sorted ....

A. Aggarwal and G. Plaxton. Optimal Parallel Sorting in Multi-Level Storage. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 659--668, 1994.

....will participate in the merge process is input dependent and unknown. Vitter and Shriver [VS94] give further intuition regarding the difficulty of mergesorting on parallel disks. In Greed Sort [NV90] the trick of approximate merging is used to circumvent this difficulty. Aggarwal and Plaxton [AP94] use the Sharesort technique that does repeated merging with accompanying overhead. Recently, Pai et al. [PSV94] considered the average case performance of a simple merging scheme for R = D sorted runs, one run on each disk. They use an approximate model of average case inputs and require that the ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. Proc. Fifth Annual ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

....items are too far apart. A final application of Columnsort [94] in conjunction with partial striping suffices to restore total order. An optimal deterministic merge sort, with somewhat higher constant factors than those of the distribution sort algorithms, was developed by Aggarwal and Plaxton [9], based upon the Sharesort hypercube sorting algorithm [52] To guarantee even distribution during the merging, it employs two high level merging schemes in which the scheduling is almost oblivious. The most practical method for sorting is the simple randomized merge sort (SRM) algorithm of Barve ....

A. Aggarwal and C. G. Plaxton. Optimal parallel sorting in multi-level storage. Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, 659--668, 1994.

....VS94] In particular, Vitter and Shriver s Parallel Disk Model (PDM) VS94] provides a reasonable model for the design of I O efficient algorithms and analysis at a feasible level of abstraction. A number of I O efficient algorithms have been designed on PDM to solve problems such as sorting [AP94, Arg95, NV93, VS94] general permuting [VS94] BMMC permutations [Cor92, Cor93, CSW94, Wis95] mesh and torus permutations [Cor92, Wis95] matrix matrix multiplication [VS94] matrix transpose [Cor92, Cor93, CSW94, VS94] FFT [VS94] LU decomposition [WGWR93] graph algorithms [CGG 95] and ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 659--668, Arlington, VA, January 1994.

....to significant runtime improvements in practice [32, 33, 89, 92] We will discuss some of these experiments in later sections. Finally, it should be mentioned that several authors have considered extended theoretical models that try to model the hierarchical nature of the memory of real machines [2, 3, 4, 5, 8, 60, 81, 95, 98, 96], but such models quickly become theoretically very complicated due to the large number of parameters. Therefore only very basic problems like sorting have been considered in these models. 1.2 Outline In this note we survey the basic paradigms for designing efficient external memory algorithms ....

.... Vitter [6] and the notion of parallel disks was introduced by Vitter and Shriver [97] The latter papers also deal with fundamental problems such as permutation, sorting and matrix transposition, and a number of authors have considered the difficult problem of sorting optimally on parallel disks [5, 21, 73, 71]. The problem of implementing various classes of permutations has been addressed in [38, 39, 41] More recently researchers have moved on to more specialized problems in the computational geometry [12, 14, 19, 32, 53, 99] graph theoretical [13, 14, 32, 34, 52, 66] and string processing areas [15, ....

[Article contains additional citation context not shown here]

A. Aggarwal and G. Plaxton. Optimal parallel sorting in multi-level storage. Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

....performance near that obtained by solving the same problem on a machine with a much larger RAM, a great deal of money can be saved. Up to this point, a great many I O efficient algorithms have been developed. The problems that have been considered include sorting and permutation related problems [1, 2, 14, 15, 22], computational geometry [3, 4, 11, 23] and graph problems [7] Until recently, there had been virtually no work directed at implementing these algorithms. Some work has now begun to appear [6, 19] but to the best of our knowledge no comprehensive package designed to support I O efficient ....

....are distributed over the D disks. Although the the denominator in (2) is larger than the denominator in (1) by an additive term of lg D, the leading constant factor in (2) is larger than that of (1) by a multiplicative factor of k. A number of independent disk distribution sort algorithms exist [1, 15, 16, 22], with values of k ranging from approximately 3 to 20. Before implementing an external sort on parallel disks, it is useful to examine the circumstances 3 For a recent example, see [3] under which the I O complexity (2) for using the disks independently is less than the I O complexity (1) with ....

A. Aggarwal and G. Plaxton. Optimal parallel sorting in multi-level storage. In Proc. 4th Annual ACM-SIAM Symp. on Discrete Algorithms, Arlington, VA, 1994.

....until the file is sorted, preferably with R as close to M=B as possible. Vitter and Shriver [VS94] give some intuition regarding the difficulty of mergesorting on parallel disks. In Greed Sort [NV90] the trick of approximate merging is used to circumvent this difficulty. Aggarwal and Plaxton [AP94] use the Sharesort technique that does repeated merging with accompanying overhead. Recently, Pai et al. [PSV94] considered the average performance of a simple merging scheme for R = D sorted runs, one run on each disk. Apart from their questionable block random depletion model and their ....

Alok Aggarwal and C. Greg Plaxton. Optimal parallel sorting in multi-level storage. Proc. Fifth Annual ACM-SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

No context found.

A. Aggarwal and G. Plaxton. Optimal parallel sorting in multi-level storage. In Proc. ACM--SIAM Symp. on Discrete Algorithms, pages 659--668, 1994.

No context found.

A. Aggarwal and C. G. Plaxton, Optimal Parallel Sorting in Multi-Level Storage, Proc. Fifth Annual ACM Symposium on Discrete Algorithms, New York, 1994, pp. 659-668.

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