Aggarwal, A., J.S. Vitter, `The Input/Output Complexity of Sorting and Related Problems,' Communications of the ACM, 31(9), pp. 1116--1127, 1988.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....stored on the hard disc. The effect of this cannot be neglected, and thus special algorithms are necessary for solving problems that are so large that their data do not fit into the main memory: external memory algorithms. There are only a few researchers who have intensively considered this field [1, 5, 4, 13]. Parallel Computation and the BSP Model. The situation in parallel computation is different: the lack of a unifying model is viewed as one of the main reasons why progress in this domain has been so much slower than in sequential computation. Here, numerous models exist, and a lot of energy is ....

....of steps for solving the problem on a sequential computer with an infinite memory. It is said to be asymptotically x optimal, if T ext (x o(1) Delta T seq . Here the hidden limit is to be taken for M 1. Note that T seq corresponds to the work that an algorithm has to do. In earlier papers [1, 4], the number of paging operations was considered as a unique quality measure for external memory algorithms. In many cases this may be adequate, but generally, this is only half the story. There are external memory problems, for which asymptotical one optimality is achievable. Not considering the ....

Aggarwal, A., J.S. Vitter, `The Input/Output Complexity of Sorting and Related Problems,' Communications of the ACM, 31(9), pp. 1116--1127, 1988.

....consideration that data must often reside in secondary storage rather than main memory; in a parallel setting this may involve the parallel access of multiple disks. Therefore it is necessary to design parallel algorithms which consider the possible data movement between main and secondary memory [AV88] and more generally which consider multiple levels of memory including register and cache. In the P HMM model, each processor has a memory hierarchy organized into discrete levels, much like the memory organization in the HMM, and all of P separate memories are connected together at the base ....

A. Aggarwal and J. Vitter, "The input/output complexity of sorting and related problems," Communications of the ACM, vol. 31, pp. 1116--1127, Sept. 1988.

....These are useful in computational geometry, as described in [GTV] where sort(n) is the I O complexity of sorting n items. In cases where the block size is not extremely small, any problem that can require an arbitrary permutation to solve takes at least Omega Gamma sort(n) I O operations [AgV]. Thus, TPIE can optimally solve any problem that can be solved by an appropriately structured PRAM algorithm, yet requires arbitrary permutations. Furthermore, it has recently been shown that many interesting problems that don t require arbitrary permutations still require enough permutation that ....

A. Aggarwal and J. S. Vitter, "The Input/Output Complexity of Sorting and Related Problems," Comm. ACM 31 (1988), 1116--1127.

....consideration that data must often reside in secondary storage rather than main memory; in a parallel setting this may involve the parallel access of multiple disks. Therefore it is necessary to design parallel algorithms which consider the possible data movement between main and secondary memory [AV88] and more generally which consider multiple levels of memory including register and cache. In the P HMM model, each processor has a memory hierarchy organized into discrete levels, much like the memory organization in the HMM, and all of P separate memories are connected together at the base ....

A. Aggarwal and J. Vitter, "The input/output complexity of sorting and related problems," Comm. ACM, vol. 31, pp. 1116--1127, Sept. 1988.

....that of the fastest memory) to transfer a block between levels l Gamma 1 and l, it follows that a memory hierarchy will behave as single flat memory if t l b l p s l Gamma1 for 2 l L. Prior Research Models for memory hierarchies have been studied by a number of authors. Aggarwal and Vitter [3] examined a two level memory in which P blocks of contiguous items can be transferred in each step. They obtained tight bounds for sorting related problems, including sorting, FFT, permutation networks, and matrix transposition. They did not handle the I O limited case or multiple levels. ....

....1. The ith output vertex of F (d) is the rth output vertex of F (j) t;s for the unique integers r; s 0 such that i = r2 d Gammaj s and s 2 d Gammaj . 18 A bound on the S span for the FFT is implicit in the work of Hong and Kung [11] and explicit in the work of Aggarwal and Vitter [3]. We state the bound and for completeness give a variant of the latter s proof. Lemma 4.4 The S span of the FFT graph F (d) on n = 2 d inputs satisfies ae(S; G) 2S log S when S n. Proof ae(S; G) is the maximum number of vertices of G that can be pebbled from some initial configuration of S ....

A. Aggarwal and J. S. Vitter, "The Input/Output Complexity of Sorting and Related Problems," Communications of the ACM 31 (September 1988), 1116-- 1127.

.... of the leaf nodes [BDH, Fri] Savage and Vitter examined the effects of block size on a number of problems, including FFT and matrix multiplication [SaV] Aggarwal and Vitter introduced a parallel disk model and gave algorithms for matrix multiplication, permuting, FFT, and sorting in their model [AgV]. Vitter and Shriver gave a more realistic parallel disk model and gave optimal algorithms for the same suite of problems as considered by Aggarwal and Vitter [ViSa] Ullman and Yannakakis investigated the input output complexity of the transitive closure problem [UlY] Carlson showed how to use ....

....a record of data once the disk read write head is positioned. An increasingly popular way to get further speedup is to use many disk drives working in parallel [GHK, GiS, Jil, Mag, PGK, Uni] Initial work in the use of parallel block transfer for sorting was done by Aggarwal and Vitter [AgV]. In their model, they considered the parameters N = # records in the file M = # records that can fit in internal memory B = # records per block D = # blocks transferred per I=O where M N , and 1 DB M=2. In each I O, D blocks of B records can be transferred simultaneously, as illustrated ....

[Article contains additional citation context not shown here]

Alok Aggarwal and Jeffrey Scott Vitter, "The Input/Output Complexity of Sorting and Related Problems," Communications of the ACM (September 1988), 1116--1127.

...., the lower order terms are bounded to O(N Delta log(N Delta s=M 2 ) s) with O(log s Delta N) comparisons. The algorithm is based on the refined deterministic sampling technique from [9] which in turn is similar to the sampling technique in [4] Methods of this type have also been applied in [1, 7]. The variant with N o(N) reads and writes is most interesting, because this variant allows for several refinements. It can be made adaptive: as soon as it can be ruled out that an element is a candidate for the element to select, this element is not written away. On average case inputs, this ....

....separately. The time required for performing a comparison is denoted by t comp , that required for a load by t load and that for a store by t store . Our goal is to minimize the number of load and store operations, while keeping the number of comparisons as small as possible. 2. 2 Sorting In [1], it has been shown that sorting N numbers on a machine with M main memory requires Omega (log N= log M) Delta N=B paging operations. Sorting can be performed as follows: first all subsets of size M are sorted; then an (M=B Gamma 1) way merge is performed as often as necessary. In this way ....

Aggarwal, A., J.S. Vitter, `The Input/Output Complexity of Sorting and Related Problems,' Communications of the ACM, 31(9), pp. 1116--1127, 1988.

....hard disc. The effect of this cannot be neglected, and thus special algorithms are necessary for solving problems that are so large, that their data do not fit into the main memory: external memory algorithms. There is only a limited number of researchers who have intensively considered this field [1, 4, 3, 10]. Parallel Computation and the BSP Model. The situation in parallel computation is different: the lack of a unifying model may be viewed as one of the main reasons why progress in this domain has been so much slower than in sequential computation. Here numerous models are around, and a lot of ....

..... Here T seq gives the minimum number of steps for solving the problem on a sequential computer with an infinite memory. It is said to be asymptotically x optimal, if T ext (x o(1) Delta T seq for M 1. Note that T seq corresponds to the work that an algorithm has to do. In earlier papers [1, 3], the number of paging operations was considered as unique quality measure for external memory algorithms. In many cases this may be adequate, but generally, this is only half the story. There are external memory problems, for which asymptotical oneoptimality is achievable. Not considering the ....

Aggarwal, A., J.S. Vitter, `The Input/Output Complexity of Sorting and Related Problems,' Communications of the ACM, 31(9), pp. 1116--1127, 1988.

.... fit such huge problem sizes. Now, suppose that one has hardware designed for a p sized FFT problem, and one wants to use it to solve an n sized FFT problem, where n p. Clever techniques for optimally using the p sized FFT hardware to solve an n sized FFT were designed by Aggarwal and Vitter [1]. However, these methods introduce serious practical complications of their own, such as necessitating multiple uses of the dedicated FFT chip, and the elaborate combining of the answers returned by these multiple uses of the FFT chip. This would involve impractical constant factors. ....

A. Aggarwal and J.S. Vitter, "The Input/Output Complexity of Sorting and Related Problems," Communications of the ACM, Vol. 31, 1988, pp. 1116--1127.

....Finally, note that only three passes through the external data set are required to perform this algorithm the second step can be performed on a block of data after the first step, before it is returned to memory. This bounded number of passes is in accordance with the I O complexity results in [2]. Main Memory Performance Results using the Four Step FFT Depending on implementation, the four step FFT algorithm may actually require a slightly larger number of floating point arithmetic operations than conventional FFT algorithms. In spite of this slight handicap, it is remarkably efficient ....

Aggarwal, A., and Vitter, J. S., "The Input/Output Complexity of Sorting and Related Problems", Communications of the ACM, vol. 31 (1988), p. 1116 - 1127.

....is at most P times its running time. Following the approach in [AAC] we can imagine superimposing onto the P HMM type hierarchical memory a sequence of two level memories. For each M in the range P M N , we superimpose on the P HMM an internal memory of size M and one infinite sized disk. By [AgV], the I O complexity of sorting N records with one disk, no blocking, and an internal memory of size M is TM (N) Omega N log N log M Gamma M : 1) The Gamma M term permits M records to reside initially in the internal memory. In each individual hierarchy, every transfer done by ....

A. Aggarwal and J. S. Vitter, "The Input/Output Complexity of Sorting and Related Problems," Communications of the ACM (September 1988), 1116--1127.

.... Previous work using lazy batched updates on the B tree yielded algorithms with O( log 2 ) I Os [34] Our new method uses an off line top down implementation of the sweep, which is based upon a novel application of the subdivision technique used in the distribution sort algorithms of [3,27,37]. The central idea is that we divide the input into O( strips, each containing an equal number of input objects. We then scan down these strips simultaneously, looking for components of the solution involving interactions between objects among different strips. Once we have done this, we are left ....

....a persistent tree by the distribution sweep method. We slightly modify our application of distribution sweeping for this construction, however, in that we follow the recursive calls on the sequences of suboperations by a non recursive merge step. We begin by applying using the techniques of [3,37] to divide the set X of elements mentioned in oe into s groups of size roughly N=s each, where s = d p e. This, of course, divides oe into s subsequences, one for each group. We then recursively construct a persistent data structure for each subsequence. Each such recursive call returns a list ....

[Article contains additional citation context not shown here]

A. Aggarwal and J. S. Vitter, "The Input/Output Complexity of Sorting and Related Problems," Comm. ACM 31 (1988), 1116--1127.

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