J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: two-level memories. Algorithmica, 12(2--3):110--147, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are also good in practice: We need a high performance implementation and we have to reconsider the model of computation when talking about constant factors. Perhaps the main issue for sorting is that I O and internal work are completely separate issues in the I O model of Vitter and Shriver [29]. In this paper we therefore refine an algorithm from [12] so that I O and computation are overlapped and give an e#cient implementation. Perhaps the most widely used external sorting algorithm is k way merge sort: During run formation, chunks of #(M) elements are read, sorted internally, and ....

J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two level memories. Algorithmica, 12(2/3):110--147, 1994.

....aspects of magnetic disks, and in particular the high latency cost associated with disk head movement. In these models, space is measured in disk blocks, and time is measured in number of block I O operations. A popular model is the Parallel Disk Model (PDM) introduced by Vitter and Shriver [VS94] Arguably the most important, and certainly the most studied area of range search, is geometric range search, where the domain D and the range set R are spaces of geometric objects (points, lines, circles, rectangles, etc. and the incidence relation p is some geometric relation (containment, ....

J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994. 179

....multi level memory hierarchies in general. 1.2 Review We outline a few results on EM algorithms which relate directly to our work. A comprehensive survey can be found in [39] A well studied model of computation for EM algorithms is the Parallel Disk Model (PDM) introduced by Vitter and Shriver [41]. It is used to model the two level memory hierarchy consisting of parallel disks connected to one or more processors which communicate via a shared internal memory or a hypercube like network (see Figure 1) The 1 P1 P2 Pp Processors Processors Router Network . CPU Memory Disk 1 ....

....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 ....

Vitter, J. S., and Shriver, E. A. M. Algorithms for parallel memory, I: Two-level memories. Algorithmica 12, 2--3 (1994), 110-- 147.

....want to model is their extremely long access time relative to that of internal memory. In order to amortize the access time over a large amount of data, typical disks read or write large blocks of contiguous data at once and therefore the standard two level disk model has the following parameters [13, 153, 104]: N = number of objects in the problem instance; T = number of objects in the problem solution; M = number of objects that can fit into internal memory; B = number of objects per disk block; where B M N . An I O operation (or simply I O) is the operation of reading (or writing) a block ....

J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994.

....file servers in production today are on the order of M = 10 6 or 10 7 and B = 10 3 . Large scale problem instances can be in the range N = 10 10 to N = 10 12 . An increasingly popular approach to further increase the throughput of the I O system is to use a number of disks in parallel [63, 65, 133]. Several authors have considered an extension of the above model with a parameter D denoting the number of disks in the system [19, 98, 99, 100, 133] In the parallel disk model [133] one can read or write one block from each of the D disks simultaneously in one I O. The number of disks D range ....

....N = 10 10 to N = 10 12 . An increasingly popular approach to further increase the throughput of the I O system is to use a number of disks in parallel [63, 65, 133] Several authors have considered an extension of the above model with a parameter D denoting the number of disks in the system [19, 98, 99, 100, 133]. In the parallel disk model [133] one can read or write one block from each of the D disks simultaneously in one I O. The number of disks D range up to 10 2 in current disk arrays. Our I O model corresponds to the one shown in Figure 1.2, where we only count the number of blocks of B elements ....

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994.

.... are developed further in a parallel model in which memory consists of a tree of modules, where computation takes place at the leaves [6] I O complexity models start with a single disk and CPU with block transfer [18, 4] and continue through parallel disks with flat memory and hierarchical memory [35, 36]. Our analytical model draws on ideas from several of these papers, though our intent is not to characterize the complexity of problems, but rather to predict performance on many real architectures. 2.3 Related Database Studies Our work builds on the framework proposed by Shekita and Carey [33] ....

Vitter, J. S. and Shriver, E. A. M. Algorithms for Parallel Memory I. Algorithmica, 12(2/3):110--147, Aug/Sep 1994.

.... that the only case in which a derandomization of such procedure is known is for the case of the upper cell in an arrangement of hyperplanes [18] 36 The techniques presented in this paper are likely to lead to an optimal deterministic algorithm in the external memory model of computation [68] in which the internal memory has size M and the I O operations are performed on blocks of size B. There, the aim is an algorithm that uses O( N=B) log M=B (N=B) K=B) I O operations. A randomized solution has been found recently [26] but the question remains open as far as deterministic ....

J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica 12 (1994), 110-147.

....problems. This paper examines the implementation and performance of one such algorithm, Rajasekaran s (l,m) mergesort (LMM) 3] Technical Report PCS TR99 351 2 of 20 The remainder of this paper is organized as follows. In Section 2, we describe Vitter and Shriver s Parallel Disk Model (PDM) [4], under which LMM is asymptotically optimal for large problems. Section 3 presents a theoretical outline of LMM. Section 4 is an in depth discussion of our implementation of the algorithm where we focus on practical matters of implementation. Test results from a DEC workstation are presented in ....

J. S. Vitter, E. A. M. Shriver, Algorithms for parallel memory. I: Two-level memories, Algorithmica 12 (2/3) (1994) 110--147.

....turns out to be a hot topic of research in the external memory context. In this paper, we study the I O complexity of the problem of computing the trapezoidal decomposition of a set of line segments [18] We give a randomized algorithm and analyze its I Ocomplexity in the external memory model [20], in which M is the available internal memory size and B is the size of the block transfer (where 1 B M=2) As a by product, the algorithm also solves the segment intersections problem. Let S be a set of N segments in the plane with a set K(S) of pairwise intersection points. The trapezoidal ....

J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994.

....available disk drives, 4. full utilization of all p available processors, and 5. efficient use of the communication bandwidth available between processors. Most existing EM algorithms are based on what has become known as the Parallel Disk Model (PDM) proposed originally by Vitter and Shriver [87]. For optimality, this model requires that EM algorithms use (asymptotically) the minimum number of disk operations required to solve a problem in the worst case. We will refer to this as the I O optimality criterion. Of course, the running time of an algorithm is a less abstract and more directly ....

....to be accessed. In other words, the blocks could be from consecutive block locations on the disk, or they could be widely separated. In this respect the model was unrealistic, in that no physical disk system allows simultaneous transfer of D blocks under those circumstances. Vitter and Shriver [87, 88] considered a more realistic model, where each of D parallel disks could simultaneously read or write a single block in a single parallel I O operation (see Figure 2.1. This model has become known as the Parallel Disk Model (PDM) Since the previous model of Aggarwal and Vitter was more ....

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994.

....is not fully blocked, the running time can typically be up to a factor of 10 3 (the blocking factor) too high, and if parallel disks are not properly utilized, the running time can be a factor of D too high. More discussion of the traditional EM system model and related issues can be found in [32]. Algorithms designed for internal memory (e.g. RAM and PRAM models) have proven to be generally difficult to adapt to the important EM requirements of blocking and parallel disk I O. However, a blocking requirement also arises in the transmission of data between processors for certain coarse ....

....they required an algorithm s running time due to I O to be asymptotically less than the computation time of the parallel algorithm. By comparison, the traditional cost measure of an EM algorithm has been the number of I O operations performed, and internal computation time is ignored (see [32]) We will call an algorithm I O optimal if the number of I O operations matches the lower bound for the number of I Os required to solve the problem. In multisearch problems [17, 5, 27] a large number of queries are simultaneously processed and satisfied by navigating through a large data ....

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Vitter, J. S., and Shriver, E. A. M. Algorithms for parallel memory, I: Two-level memories. Algorithmica 12, 2--3 (1994), 110--147.

....nd use in higher dimensions, since so far its use is limited because the only tool available for clustering is the separator theorem for planar graphs. The techniques presented in this paper are likely to lead to an optimal deterministic algorithm in the external memory model of computation [72] in which the internal memory has size M and the I O operations are performed on blocks of size B. There, the aim is an algorithm that uses O( N=B) log M=B (N=B) K=B) I O operations. A randomized solution has been found recently [28] but the question remains open for deterministic algorithms. We ....

J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica 12 (1994), 110-147.

....in external memory algorithms was done by Floyd [13] and Hong and Kong [16] who studied matrix operations and fast Fourier transforms. Lower bounds for a number of problems related to sorting were presented by Aggarwal and Vitter [1] The classical I O model was introduced by Vitter and Shriver [28]. The uniprocessor, single disk version of this model represents an EM computer system as a processor, some fixed amount of internal memory, and a disk. It is described by the following parameters: N is the number of elements in the problem instance, M is the number of elements that can fit in ....

....or writing a block of B contiguous data elements to or from the disk. The I O complexity of an algorithm is defined as the total number of I Os that an algorithm performs. It is assumed in this model that the internal computation is free. Several other I O models have been proposed, see e.g. [28, 12, 20]. The theoretical framework of the algorithms in this paper is based on the Parallel Disk I O Model (PDM) proposed by Vitter and Shriver [28] Permutations and sorting have been very widely studied in the context of this model, see [2, 1, 9, 20, 27] Algorithms for problems in computational ....

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994.

.... e.g. 4, 20] While a lot of practical and often heuristic I O efficient algorithms and data structures in adhoc models have been developed in the database community, most theoretical work on I O efficiency in the algorithms community has been done in the Parallel Disk Model of Vitter and Shriver [21]. To investigate the practical viability of the theoretical work, the TPIE project was started at Duke University. The goal of this ongoing project is to provide a portable, extensible, flexible, and easy to use programming environment for efficiently implementing algorithms and data structures ....

....transfer a large block of contiguous data items at a time. Accessing a block involves only one seek and wait, so the amortized cost per unit of data is much smaller than the cost of accessing a single unit. Parallel Disk Model. The Parallel Disk Model (PDM) was introduced by Vitter and Shriver [21] (see also [3] in order to more accurately model a two level main memorydisk system with block transfers. PDM has become the standard theoretical model for designing and analyzing I O efficient algorithms. The model abstracts a computer as a three component system: a processor, a fixed amount of ....

J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994.

....head and waiting for the disk to rotate into position; once done, data in subsequent locations on disk can be accessed very quickly. To amortize (or hide) disk latency, each input output operation (or I O) transfers a large block of contiguous data. We use the standard model of I O complex ity [2, 46] and define the following parameters: K = # of strings to sort; N = total # of characters in the K strings; M = # of characters fitting in internal memory; B = # of characters per disk block (or track) where M N and 1 B M=2. We measure the performance of an algorithm in terms of the ....

....sorting) the final sorted sequence can be obtained with O(K 2 N 2 =B) O(N 2 =B) extra I Os by moving the strings into their final position one at a time. 1. 2 Previous Results in I O efficient Computation Early work on I O algorithms concentrated on sorting and permutation related problems [2, 9, 18, 41, 40, 46]. Work has also been done on matrix algebra and related problems arising in scientific computation [2, 45, 46] More recently, researchers have designed I O algorithms for a number of problems in different areas, such as in computational geometry [6, 10, 28] graph theoretic computation [6, 7, 16] ....

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110-- 147, 1994.

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: two-level memories. Algorithmica, 12(2--3):110--147, 1994.

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: two-level memories. Algorithmica, 12(2--3):110--147, 1994.

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: two-level memories. Algorithmica, 12(2--3):110--147, 1994.

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two-level memories. Algorithmica, 12(2--3):110--147, 1994.

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two level memories. Algorithmica, 12(2/3):110-147, 1994.

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: two-level memories. Algorithmica, 12(2--3):110--147, 1994.

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J.S. Vitter and E.A.M. Shriver. Algorithms for parallel memory.i:Two-level memories. Algorithmica, 12(2-3):110--147, 1994.

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J.S. Vitter and E.A.M. Shriver. Algorithms for parallel memory. i: Two-level memories. Algorithmica, 12(2-3):110--147, 1994.

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J. S. Vitter and E. A. M. Shriver. Algorithms for parallel memory, I: Two level memories. Algorithmica, 12(2/3):110-147, 1994.

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VITTER, J. S., AND SHRIVER, E. A. M. Algorithms for parallel memory, I: Two-level memories. Algorithmica 12, 2--3 (1994), 110--147.

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