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180
Parallel merge sort
 SIAM Journal of Computing
, 1988
"... Abstract. We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and O(logn) time. The constant in the runnin ..."
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Cited by 298 (2 self)
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Abstract. We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and O(logn) time. The constant in the running time is still moderate, though not as small. 1.
Algorithms for Parallel Memory I: TwoLevel Memories
, 1992
"... We provide the first optimal algorithms in terms of the number of input/outputs (I/Os) required between internal memory and multiple secondary storage devices for the problems of sorting, FFT, matrix transposition, standard matrix multiplication, and related problems. Our twolevel memory model is n ..."
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Cited by 249 (32 self)
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We provide the first optimal algorithms in terms of the number of input/outputs (I/Os) required between internal memory and multiple secondary storage devices for the problems of sorting, FFT, matrix transposition, standard matrix multiplication, and related problems. Our twolevel memory model is new and gives a realistic treatment of parallel block transfer, in which during a single I/O each of the P secondary storage devices can simultaneously transfer a contiguous block of B records. The model pertains to a largescale uniprocessor system or parallel multiprocessor system with P disks. In addition, the sorting, FFT, permutation network, and standard matrix multiplication algorithms are typically optimal in terms of the amount of internal processing time. The difficulty in developing optimal algorithms is to cope with the partitioning of memory into P separate physical devices. Our algorithms' performance can be significantly better than those obtained by the wellknown but nonopti...
A Comparison of Sorting Algorithms for the Connection Machine CM2
"... We have implemented three parallel sorting algorithms on the Connection Machine Supercomputer model CM2: Batcher's bitonic sort, a parallel radix sort, and a sample sort similar to Reif and Valiant's flashsort. We have also evaluated the implementation of many other sorting algorithms pro ..."
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Cited by 183 (7 self)
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We have implemented three parallel sorting algorithms on the Connection Machine Supercomputer model CM2: Batcher's bitonic sort, a parallel radix sort, and a sample sort similar to Reif and Valiant's flashsort. We have also evaluated the implementation of many other sorting algorithms proposed in the literature. Our computational experiments show that the sample sort algorithm, which is a theoretically efficient "randomized" algorithm, is the fastest of the three algorithms on large data sets. On a 64Kprocessor CM2, our sample sort implementation can sort 32 10 6 64bit keys in 5.1 seconds, which is over 10 times faster than the CM2 library sort. Our implementation of radix sort, although not as fast on large data sets, is deterministic, much simpler to code, stable, faster with small keys, and faster on small data sets (few elements per processor). Our implementation of bitonic sort, which is pipelined to use all the hypercube wires simultaneously, is the least efficient of the three on large data sets, but is the most efficient on small data sets, and is considerably more space efficient. This paper analyzes the three algorithms in detail and discusses many practical issues that led us to the particular implementations.
Scans as primitive parallel operations
 IEEE Transactions on Computers
, 1989
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The Power of Reconfiguration
, 1998
"... This paper concerns the computational aspects of the reconfigurable network model. The computational power of the model is investigated under several network topologies and assuming several variants of the model. In particular, it is shown that there are reconfigurable machines based on simple netwo ..."
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Cited by 90 (7 self)
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This paper concerns the computational aspects of the reconfigurable network model. The computational power of the model is investigated under several network topologies and assuming several variants of the model. In particular, it is shown that there are reconfigurable machines based on simple network topologies, that are capable of solving large classes of problems in constant time. These classes depend on the kinds of switches assumed for the network nodes. Reconfigurable networks are also compared with various other models of parallel computation, like PRAM's and Branching Programs. Part of this work is to be presented at the 18th International Colloquium on Automata, Languages, and Programming (ICALP), July 1991, Madrid. y Department of Computer Science, The Hebrew University, Jerusalem 91904, Israel. Email: yosi@humus.huji.ac.il, Supported by Eshcol Fellowship. z Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehovot 76100, Israel. Email: p...
Randomized routing and sorting on fixedconnection networks
 JOURNAL OF ALGORITHMS
, 1994
"... This paper presents a general paradigm for the design of packet routing algorithms for fixedconnection networks. Its basis is a randomized online algorithm for scheduling any set of N packets whose paths have congestion c on any boundeddegree leveled network with depth L in O(c + L + log N) steps ..."
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Cited by 89 (13 self)
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This paper presents a general paradigm for the design of packet routing algorithms for fixedconnection networks. Its basis is a randomized online algorithm for scheduling any set of N packets whose paths have congestion c on any boundeddegree leveled network with depth L in O(c + L + log N) steps, using constantsize queues. In this paradigm, the design of a routing algorithm is broken into three parts: (1) showing that the underlying network can emulate a leveled network, (2) designing a path selection strategy for the leveled network, and (3) applying the scheduling algorithm. This strategy yields randomized algorithms for routing and sorting in time proportional to the diameter for meshes, butterflies, shuffleexchange graphs, multidimensional arrays, and hypercubes. It also leads to the construction of an areauniversal network: an Nnode network with area Θ(N) that can simulate any other network of area O(N) with slowdown O(log N).
Special Purpose Parallel Computing
 Lectures on Parallel Computation
, 1993
"... A vast amount of work has been done in recent years on the design, analysis, implementation and verification of special purpose parallel computing systems. This paper presents a survey of various aspects of this work. A long, but by no means complete, bibliography is given. 1. Introduction Turing ..."
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Cited by 77 (5 self)
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A vast amount of work has been done in recent years on the design, analysis, implementation and verification of special purpose parallel computing systems. This paper presents a survey of various aspects of this work. A long, but by no means complete, bibliography is given. 1. Introduction Turing [365] demonstrated that, in principle, a single general purpose sequential machine could be designed which would be capable of efficiently performing any computation which could be performed by a special purpose sequential machine. The importance of this universality result for subsequent practical developments in computing cannot be overstated. It showed that, for a given computational problem, the additional efficiency advantages which could be gained by designing a special purpose sequential machine for that problem would not be great. Around 1944, von Neumann produced a proposal [66, 389] for a general purpose storedprogram sequential computer which captured the fundamental principles of...
CommunicationEfficient Parallel Sorting
, 1996
"... We study the problem of sorting n numbers on a pprocessor bulksynchronous parallel (BSP) computer, which is a parallel multicomputer that allows for general processortoprocessor communication rounds provided each processor sends and receives at most h items in any round. We provide parallel sort ..."
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Cited by 71 (2 self)
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We study the problem of sorting n numbers on a pprocessor bulksynchronous parallel (BSP) computer, which is a parallel multicomputer that allows for general processortoprocessor communication rounds provided each processor sends and receives at most h items in any round. We provide parallel sorting methods that use internal computation time that is O( n log n p ) and a number of communication rounds that is O( log n log(h+1) ) for h = \Theta(n=p). The internal computation bound is optimal for any comparisonbased sorting algorithm. Moreover, the number of communication rounds is bounded by a constant for the (practical) situations when p n 1\Gamma1=c for a constant c 1. In fact, we show that our bound on the number of communication rounds is asymptotically optimal for the full range of values for p, for we show that just computing the "or" of n bits distributed evenly to the first O(n=h) of an arbitrary number of processors in a BSP computer requires\Omega\Gammaqui n= log(h...
Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers
 Journal of Computer and System Sciences
, 1996
"... This paper presents a deterministic sorting algorithm, called Sharesort, that sorts n records on an nprocessor hypercube, shuffleexchange, or cubeconnected cycles in O(log n (log log n) 2 ) time in the worst case. The algorithm requires only a constant amount of storage at each processor. Th ..."
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Cited by 68 (10 self)
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This paper presents a deterministic sorting algorithm, called Sharesort, that sorts n records on an nprocessor hypercube, shuffleexchange, or cubeconnected cycles in O(log n (log log n) 2 ) time in the worst case. The algorithm requires only a constant amount of storage at each processor. The fastest previous deterministic algorithm for this problem was Batcher's bitonic sort, which runs in O(log 2 n) time. Supported by an NSERC postdoctoral fellowship, and DARPA contracts N0001487K825 and N00014 89J1988. 1 Introduction Given n records distributed uniformly over the n processors of some fixed interconnection network, the sorting problem is to route the record with the ith largest associated key to processor i, 0 i ! n. One of the earliest parallel sorting algorithms is Batcher's bitonic sort [3], which runs in O(log 2 n) time on the hypercube [10], shuffleexchange [17], and cubeconnected cycles [14]. More recently, Leighton [9] exhibited a boundeddegree,...
Fast Parallel Sorting Under LogP: Experience with the CM5
 IEEE Transactions on Parallel and Distributed Systems
, 1996
"... In this paper, the LogP model is used to analyze four parallel sorting algorithms (bitonic, column, radix, and sample sort). LogP characterizes the performance of modern parallel machines with a small set of parameters: the communication latency (L), overhead (o), bandwidth (g), and the number of pr ..."
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Cited by 54 (10 self)
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In this paper, the LogP model is used to analyze four parallel sorting algorithms (bitonic, column, radix, and sample sort). LogP characterizes the performance of modern parallel machines with a small set of parameters: the communication latency (L), overhead (o), bandwidth (g), and the number of processors (P). We develop implementations of these algorithms in SplitC, a parallel extension to C, and compare the performance predicted by LogP to actual performance on a CM5 of 32 to 512 processors for a range of problem sizes and input sets. The sensitivity of the algorithms is evaluated by varying the distribution of key values and the rank ordering of the input. The LogP model is shown to be a valuable guide in the development of parallel algorithms and a good predictor of implementation performance. The model encourages the use of data layouts which minimize communication and balanced communication schedules which avoid contention. Using an empirical model of local processor performance, LogP predictions closely match observed execution times on uniformly distributed keys across a broad range of problem and machine sizes for all four algorithms. Communication performance is oblivious to the distribution of the keys values, whereas the local sort performance is not. The communication phases in radix and sample sort are sensitive to the ordering of keys, because certain layouts result in contention. 1