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Book

An introduction to parallel algorithms

01 Oct 1992-
TL;DR: This book provides an introduction to the design and analysis of parallel algorithms, with the emphasis on the application of the PRAM model of parallel computation, with all its variants, to algorithm analysis.
Abstract: Written by an authority in the field, this book provides an introduction to the design and analysis of parallel algorithms. The emphasis is on the application of the PRAM (parallel random access machine) model of parallel computation, with all its variants, to algorithm analysis. Special attention is given to the selection of relevant data structures and to algorithm design principles that have proved to be useful. Features *Uses PRAM (parallel random access machine) as the model for parallel computation. *Covers all essential classes of parallel algorithms. *Rich exercise sets. *Written by a highly respected author within the field. 0201548569B04062001

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Citations
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Journal ArticleDOI
TL;DR: Deterministic parallel methods for constructing the upper envelope of segments on the weakest shared-memory model, the EREW PRAM, are presented and it is shown that it can be found optimally in 0(logn) time using 0(n) processors.
Abstract: Given a collection of segments in the plane, if we regard the segments as opaque barriers, their upper envelope consists of the portions of the segments visible from point (0, +/spl infin/). In this paper, we present deterministic parallel methods for constructing the upper envelope of segments on the weakest shared-memory model, the EREW PRAM. We show that we can find the upper envelope of n line segments optimally in 0(logn) time using 0(n) processors. Furthermore, if the segments are nonintersecting and their endpoints are sorted in x-coordinate, then we can reduce the number of processors to 0(n/ logn). Our method implies that we can find the upper envelope sequentially in 0(n log log n) time, which improves previous results. We also show that we can find the upper envelope of n k-intersecting segments (any pair of the segments intersects at most k times) with a slightly larger time and processor bound.

5 citations

Proceedings ArticleDOI
15 Apr 2002
TL;DR: This paper presents a methodology for designing efficient algorithms for SMP-clusters on top of the ?
Abstract: Clusters of symmetric multiprocessor (SMP) nodes are one of the most important parallel architectures now and in the future. The architecture consists of shared-memory nodes with multiple processors and a fast interconnection network between the nodes. New programming models try to exploit this architecture by using threads in the nodes and using message-passing-libraries for internode communication. In order to develop efficient algorithms it is necessary to consider the hybrid nature of the architecture and of the programming models. In this paper, we present a methodology for designing efficient algorithms for SMP-clusters on top of the ?NUMA-model. The ?NUMA-model is a computational model that extends the bulk-synchronous parallel (BSP) model with the characteristics of SMP-clusters. The ?NUMA-methodology is a top-down method, which suggests to develop an optimal overall algorithm by developing optimal algorithms for each level in the machine hierarchy. We use the problem of dense matrix-vector-multiplication for presentation. The theoretical results of our analysis are verified practically. We show results of experiments, which were made on a Linux-cluster of dual Pentium-III nodes.

5 citations

Book ChapterDOI
TL;DR: This work presents fast algorithms for merging and sorting of data on a multiprocessor system connected through a Partitioned Optical Passive Stars (POPS) network and their sorting algorithm is more efficient compared to a simulated hypercube sorting algorithm on the POPS.
Abstract: We present fast algorithms for merging and sorting of data on a multiprocessor system connected through a Partitioned Optical Passive Stars (POPS) network. In a POPS(d,g) network there are n = dg processors and they are divided into g groups of d processors each. There is an optical passive star (OPS) coupler between every pair of groups. Each OPS Coupler can receive an optical signal from any one of its source nodes and broadcast the signal to all the destination nodes. The time needed to perform this receive and broadcast is referred to as a slot and the complexity of an algorithm using the POPS network is measured in terms of number of slots it uses. Our sorting algorithm is more efficient compared to a simulated hypercube sorting algorithm on the POPS.

5 citations

Book ChapterDOI
TL;DR: In this article, the potentials of automatic parallelization and vectorization of the sequential C++ applications from the SPEC CPU2006 suite are discussed, and the effects of paralllelization are evaluated by profiling and executing on two representative parallel machines.
Abstract: Being multiprocessors (both on-chip and/or off-chip), modern computer systems can automatically exploit the benefits of parallel programs, but their resources remain underutilized in executing still-Prevailing sequential applications. An obvious solution is in the parallelization of such applications. The first part overviews he broad issues in parallelization. Various parallelization approaches and contemporary software and hardware tools for extracting parallelism from sequential applications are studied. It also attempts to identify typical code patterns amenable for parallelization. The second part represents a case study where the SPEC CPU2006 suite-is considered as a representative collection of typical sequential applications. Following that, it discusses the possibilities and potentials of automatic parallelization and vectorization of the sequential C++ applications from the CPU2006 suite. Since these potentials are generally limited, it explores the issues in manual parallelization of these applications. After previously identified patterns are applied by source-to-source code modifications, the effects of paralllelization are evaluated by profiling and executing on two representative parallel machines. Finally, the presented results are carefully discussed.

5 citations

Proceedings ArticleDOI
10 Aug 1998
TL;DR: Efficient parallel methods for finding the upper envelope of k-intersecting segments for any integer k/spl ges/0, in the weakest shared memory model, the EREW PRAM are given.
Abstract: Given a collection of segments in the plane that intersect pairwise at most k times, regarding the segments as opaque barriers, their upper envelope consists of the portions of the segments visible from point (0,+/spl infin/). We give efficient parallel methods for finding the upper envelope of k-intersecting segments for any integer k/spl ges/0, in the weakest shared memory model, the EREW PRAM. We show that the upper envelope of n k-intersecting segments can be found in 0(log/sup 1+/spl epsiv//n) time using 0(/spl lambda//sub k+1/(n)/log/sup /spl epsiv//n) processors for any /spl epsiv/>0, where /spl lambda//sub k+2/(n)/sup 1/ is the size of the upper envelope. In particular, for line segments we show the following optimal algorithms: the upper envelope of n line segments can be found in O(log n) time using O(n) processors, and if the line segments are nonintersecting and sorted, the envelope can be found in O(log n) time using O(n/log n) processors. We also show that our methods imply a fast sequential result: the upper envelope of n sorted line segments can be found in O(n log log n) time sequentially, which improves the known lowest upper bound O(n log n).

5 citations


Cites background or methods from "An introduction to parallel algorit..."

  • ...In phase (2), E[li; li+1] (1 i m) can be found by computing the intersection of Ej and li for any i and j (one should make k copies of lines l1, l2, : : :, lm+1 to avoid concurrent reading), and then for every j compute the intersections of Ej with l1, l2, : : :, lm+1 by merging the endpoints of the segments of Uj with l1, l2, : : :, lm+1 by their x-coordinates, in O(logn) time using O(n= logn) processors [15]....

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  • ...The prefix maxima computation of Y (P ) can be done in O(logn) time usingO(n= logn) processors[15]....

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  • ...We first sort S in increasing slope and sort P (S) in increasing x-coordinate in O(logn) time using O(n) processors in the EREW PRAM [15]....

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References
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Book
01 Sep 1991
TL;DR: This chapter discusses sorting on a Linear Array with a Systolic and Semisystolic Model of Computation, which automates the very labor-intensive and therefore time-heavy and expensive process of manually sorting arrays.
Abstract: Preface Acknowledgments Notation 1 Arrays and Trees 1.1 Elementary Sorting and Counting 1.1.1 Sorting on a Linear Array Assessing the Performance of the Algorithm Sorting N Numbers with Fewer Than N Processors 1.1.2 Sorting in the Bit Model 1.1.3 Lower Bounds 1.1.4 A Counterexample-Counting 1.1.5 Properties of the Fixed-Connection Network Model 1.2 Integer Arithmetic 1.2.1 Carry-Lookahead Addition 1.2.2 Prefix Computations-Segmented Prefix Computations 1.2.3 Carry-Save Addition 1.2.4 Multiplication and Convolution 1.2.5 Division and Newton Iteration 1.3 Matrix Algorithms 1.3.1 Elementary Matrix Products 1.3.2 Algorithms for Triangular Matrices 1.3.3 Algorithms for Tridiagonal Matrices -Odd-Even Reduction -Parallel Prefix Algorithms 1.3.4 Gaussian Elimination 1.3.5 Iterative Methods -Jacobi Relaxation -Gauss-Seidel Relaxation Finite Difference Methods -Multigrid Methods 1.4 Retiming and Systolic Conversion 1.4.1 A Motivating Example-Palindrome Recognition 1.4.2 The Systolic and Semisystolic Model of Computation 1.4.3 Retiming Semisystolic Networks 1.4.4 Conversion of a Semisystolic Network into a Systolic Network 1.4.5 The Special Case of Broadcasting 1.4.6 Retiming the Host 1.4.7 Design by Systolic Conversion-A Summary 1.5 Graph Algorithms 1.5.1 Transitive Closure 1.5.2 Connected Components 1.5.3 Shortest Paths 1.5.4 Breadth-First Spanning Trees 1.5.5 Minimum Weight Spanning Trees 1.6 Sorting Revisited 1.6.1 Odd-Even Transposition Sort on a Linear Array 1.6.2 A Simple Root-N(log N + 1)-Step Sorting Algorithm 1.6.3 A (3 Root- N + o(Root-N))-Step Sorting Algorithm 1.6.4 A Matching Lower Bound 1.7 Packet Routing 1.7.1 Greedy Algorithms 1.7.2 Average-Case Analysis of Greedy Algorithms -Routing N Packets to Random Destinations -Analysis of Dynamic Routing Problems 1.7.3 Randomized Routing Algorithms 1.7.4 Deterministic Algorithms with Small Queues 1.7.5 An Off-line Algorithm 1.7.6 Other Routing Models and Algorithms 1.8 Image Analysis and Computational Geometry 1.8.1 Component-Labelling Algorithms -Levialdi's Algorithm -An O (Root-N)-Step Recursive Algorithm 1.8.2 Computing Hough Transforms 1.8.3 Nearest-Neighbor Algorithms 1.8.4 Finding Convex Hulls 1.9 Higher-Dimensional Arrays 1.9.1 Definitions and Properties 1.9.2 Matrix Multiplication 1.9.3 Sorting 1.9.4 Packet Routing 1.9.5 Simulating High-Dimensional Arrays on Low-Dimensional Arrays 1.10 problems 1.11 Bibliographic Notes 2 Meshes of Trees 2.1 The Two-Dimensional Mesh of Trees 2.1.1 Definition and Properties 2.1.2 Recursive Decomposition 2.1.3 Derivation from KN,N 2.1.4 Variations 2.1.5 Comparison With the Pyramid and Multigrid 2.2 Elementary O(log N)-Step Algorithms 2.2.1 Routing 2.2.2 Sorting 2.2.3 Matrix-Vector Multiplication 2.2.4 Jacobi Relaxation 2.2.5 Pivoting 2.2.6 Convolution 2.2.7 Convex Hull 2.3 Integer Arithmetic 2.3.1 Multiplication 2.3.2 Division and Chinese Remaindering 2.3.3 Related Problems -Iterated Products -Rooting Finding 2.4 Matrix Algorithms 2.4.1 The Three-Dimensional Mesh of Trees 2.4.2 Matrix Multiplication 2.4.3 Inverting Lower Triangular Matrices 2.4.4 Inverting Arbitrary Matrices -Csanky's Algorithm -Inversion by Newton Iteration 2.4.5 Related Problems 2.5 Graph Algorithms 2.5.1 Minimum-Weight Spanning Trees 2.5.2 Connected Components 2.5.3 Transitive Closure 2.5.4 Shortest Paths 2.5.5 Matching Problems 2.6 Fast Evaluation of Straight-Line Code 2.6.1 Addition and Multiplication Over a Semiring 2.6.2 Extension to Codes with Subtraction and Division 2.6.3 Applications 2.7 Higher-Dimensional meshes of Trees 2.7.1 Definitions and Properties 2.7.2 The Shuffle-Tree Graph 2.8 Problems 2.9 Bibliographic Notes 3 Hypercubes and Related Networks 3.1 The Hypercube 3.1.1 Definitions and Properties 3.1.2 Containment of Arrays -Higher-Dimensional Arrays -Non-Power-of-2 Arrays 3.1.3 Containment of Complete Binary Trees 3.1.4 Embeddings of Arbitrary Binary Trees -Embeddings with Dilation 1 and Load O(M over N + log N) -Embeddings with Dilation O(1) and Load O (M over N + 1) -A Review of One-Error-Correcting Codes -Embedding Plog N into Hlog N 3.1.5 Containment of Meshes of Trees 3.1.6 Other Containment Results 3.2 The Butterfly, Cube-Connected-Cycles , and Benes Network 3.2.1 Definitions and Properties 3.2.2 Simulation of Arbitrary Networks 3.2.3 Simulation of Normal Hypercube Algorithms 3.2.4 Some Containment and Simulation Results 3.3 The Shuffle-Exchange and de Bruijn Graphs 3.3.1 Definitions and Properties 3.3.2 The Diaconis Card Tricks 3.3.3 Simulation of Normal Hypercube Algorithms 3.3.4 Similarities with the Butterfly 3.3.5 Some Containment and Simulation Results 3.4 Packet-Routing Algorithms 3.4.1 Definitions and Routing Models 3.4.2 Greedy Routing Algorithms and Worst-Case Problems 3.4.3 Packing, Spreading, and Monotone Routing Problems -Reducing a Many-to-Many Routing Problem to a Many-to-One Routing Problem -Reducing a Routing Problem to a Sorting Problem 3.4.4 The Average-Case Behavior of the Greedy Algorithm -Bounds on Congestion -Bounds on Running Time -Analyzing Non-Predictive Contention-Resolution Protocols 3.4.5 Converting Worst-Case Routing Problems into Average-Case Routing Problems -Hashing -Randomized Routing 3.4.6 Bounding Queue Sizes -Routing on Arbitrary Levelled Networks 3.4.7 Routing with Combining 3.4.8 The Information Dispersal Approach to Routing -Using Information Dispersal to Attain Fault-Tolerance -Finite Fields and Coding Theory 3.4.9 Circuit-Switching Algorithms 3.5 Sorting 3.5.1 Odd-Even Merge Sort -Constructing a Sorting Circuit with Depth log N(log N +1)/2 3.5.2 Sorting Small Sets 3.5.3 A Deterministic O(log N log log N)-Step Sorting Algorithm 3.5.4 Randomized O(log N)-Step Sorting Algorithms -A Circuit with Depth 7.45 log N that Usually Sorts 3.6 Simulating a Parallel Random Access Machine 3.6.1 PRAM Models and Shared Memories 3.6.2 Randomized Simulations Based on Hashing 3.6.3 Deterministic Simulations using Replicated Data 3.6.4 Using Information Dispersal to Improve Performance 3.7 The Fast Fourier Transform 3.7.1 The Algorithm 3.7.2 Implementation on the Butterfly and Shuffle-Exchange Graph 3.7.3 Application to Convolution and Polynomial Arithmetic 3.7.4 Application to Integer Multiplication 3.8 Other Hypercubic Networks 3.8.1 Butterflylike Networks -The Omega Network -The Flip Network -The Baseline and Reverse Baseline Networks -Banyan and Delta Networks -k-ary Butterflies 3.8.2 De Bruijn-Type Networks -The k-ary de Bruijn Graph -The Generalized Shuffle-Exchange Graph 3.9 Problems 3.10 Bibliographic Notes Bibliography Index Lemmas, Theorems, and Corollaries Author Index Subject Index

2,895 citations


"An introduction to parallel algorit..." refers background in this paper

  • ...Multiprocessorbased computers have been around for decades and various types of computer architectures [2] have been implemented in hardware throughout the years with different types of advantages/performance gains depending on the application....

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  • ...Every location in the array represents a node of the tree: T [1] is the root, with children at T [2] and T [3]....

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  • ...The text by [2] is a good start as it contains a comprehensive description of algorithms and different architecture topologies for the network model (tree, hypercube, mesh, and butterfly)....

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Book
01 Jan 1984
TL;DR: The authors have divided the use of computers into the following four levels of sophistication: data processing, information processing, knowledge processing, and intelligence processing.
Abstract: The book is intended as a text to support two semesters of courses in computer architecture at the college senior and graduate levels. There are excellent problems for students at the end of each chapter. The authors have divided the use of computers into the following four levels of sophistication: data processing, information processing, knowledge processing, and intelligence processing.

1,410 citations


"An introduction to parallel algorit..." refers background in this paper

  • ...Parallel architectures have been described in several books (see, for example, [18, 29])....

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Journal ArticleDOI
TL;DR: The success of data parallel algorithms—even on problems that at first glance seem inherently serial—suggests that this style of programming has much wider applicability than was previously thought.
Abstract: Parallel computers with tens of thousands of processors are typically programmed in a data parallel style, as opposed to the control parallel style used in multiprocessing. The success of data parallel algorithms—even on problems that at first glance seem inherently serial—suggests that this style of programming has much wider applicability than was previously thought.

1,000 citations


"An introduction to parallel algorit..." refers background in this paper

  • ...Recent work on the mapping of PRAM algorithms on bounded-degree networks is described in [3,13,14, 20, 25], Our presentation on the communication complexity of the matrix-multiplication problem in the sharedmemory model is taken from [1], Data-parallel algorithms are described in [15]....

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Proceedings ArticleDOI
01 May 1978
TL;DR: A model of computation based on random access machines operating in parallel and sharing a common memory is presented and can accept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines.
Abstract: A model of computation based on random access machines operating in parallel and sharing a common memory is presented. The computational power of this model is related to that of traditional models. In particular, deterministic parallel RAM's can accept in polynomial time exactly the sets accepted by polynomial tape bounded Turing machines; nondeterministic RAM's can accept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines. Similar results hold for other classes. The effect of limiting the size of the common memory is also considered.

951 citations


"An introduction to parallel algorit..." refers background in this paper

  • ...Rigorous descriptions of shared-memory models were introduced later in [11,12]....

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Journal ArticleDOI
TL;DR: It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log 2 + 10(n - 1) using processors which can independently perform arithmetic operations in unit time.
Abstract: It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log2n + 10(n - 1)/p using p ≥ 1 processors which can independently perform arithmetic operations in unit time. This bound is within a constant factor of the best possible. A sharper result is given for expressions without the division operation, and the question of numerical stability is discussed.

864 citations


"An introduction to parallel algorit..." refers methods in this paper

  • ...The WT scheduling principle is derived from a theorem in [7], In the literature, this principle is commonly referred to as Brent's theorem or Brent's scheduling principle....

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