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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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Book ChapterDOI
20 Aug 2001
TL;DR: This paper considers two P-complete geometric problems in the plane and proposes an algorithm for computing the envelope layers of S in O( nα(n) log3 n/p) time using pprocessors, where α( n) is the functional inverse of Ackermann's function which grows extremely slowly.
Abstract: P-complete problems seem to have no parallel algorithm which runs in polylogarithmic time using a polynomial number of processors. A P-complete problem is in class EP (Efficient and Polynomially fast) if and only if there exists a cost optimal algorithm to solve it in T(n) = O(t(n))Ɛ) (Ɛ>1) using P(n) processors such that T(n) × P(n) = O(t(n)), where t(n) is the time complexity of the fastest sequential algorithm which solves the problem. The goal of our research is to find EPparallel algorithms for P-complete problems. In this paper we consider two P-complete geometric problems in the plane. First we consider the convex layers problem of a set S of n points. Let k be the number of the convex layers of S. When 1 ≤ k ≤ nƐ/2 (0 < Ɛ < 1) we can find the convex layers of S in O( nlogn/p) time using p processors, where 1 ≤ p ≤ n1-Ɛ/2. Next, we consider the envelope layers problem of a set S of n line segments. Let k be the number of the envelope layers of S. When 1 ≤ k ≤ nƐ/2 (0 < Ɛ < 1), we propose an algorithm for computing the envelope layers of S in O( nα(n) log3 n/p) time using pprocessors, where 1 ≤p ≤ n1-Ɛ/2, and α(n) is the functional inverse of Ackermann's function which grows extremely slowly. The computational model we use in this paper is the EREW-PRAM. Our first algorithm, for the convex layers problem, belongs to EP, and the second one, for the envelope layers problem, belongs to the class EP if a small factor of log n is ignored.

3 citations


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

  • ...Operation (I) can be done in O(log n) using n processors [ 7 ], operation (II) and (III) can be done in O(log n) time sequentially [9]....

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01 Jan 1996
TL;DR: Computer Science) COMPUTATIONAL MODELS and PROGRAM SYNTHESIS for PARALLEL OUT-OF-CORE COMPUTATION by Zhiyong Li Department of Computer Science Duke University
Abstract: Computer Science) COMPUTATIONAL MODELS AND PROGRAM SYNTHESIS FOR PARALLEL OUT-OF-CORE COMPUTATION by Zhiyong Li Department of Computer Science Duke University

3 citations


Additional excerpts

  • ...in order to simplify algorithm design [43, 70]....

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Book ChapterDOI
01 Jan 2015
TL;DR: “Theoretical Science,” “Experimental science” and “Computational Science” has become the three major types of science to accelerate technological development and social progression (Quinn, 2002).
Abstract: Parallel computing is an important research area with a long development history in computer science. A parallel computer is a collection of processing elements that cooperate and communicate to solve large problems rapidly. In parallel computing, a problem decomposed in multiple parts can be solved concurrently by using multiple compute resources which run on multiple processors; an overall control/coordination mechanism is applied. The multiple instructions of decomposed computational problem parts should be executed in less response time at any moment of time. The computer resources might be a single computer with many processors, or many computers connected by a network, or a combination of both, whereas in serial computing a program run on a single computer with a single Central Processing Unit (CPU) and instructions are executed one by one; at any moment of time only one instruction executes (Wilkinson & Allen, 2009). Parallel computing is parallel with respect to time and space. Figure 1 depicts the general process of solving a problem using parallel computing. Therefore, we can observe that parallel computing consists the following phases: parallel computers (hardware platforms), parallel algorithm (theoretical basis), parallel programming (software supports), and parallel applications (large problems). Now “Theoretical Science,” “Experimental Science” and “Computational Science” has become the three major types of science to accelerate technological development and social progression (Quinn, 2002).

3 citations

Journal ArticleDOI
TL;DR: A simple cost-optimal encoding algorithm for ordered trees is proposed and it is shown that the encoding can be used to obtain an optimal Breadth-First traversal of ordered trees.
Abstract: Encoding the shape of a tree is a basic step in a number of algorithms in integrated circuit design, automated theorem proving, and game playing. We propose a simple cost-optimal encoding algorithm for ordered trees and show that our encoding can be used to obtain an optimal Breadth-First traversal of ordered trees. Specifically, with an n-node ordered tree as input our algorithms run in O(logn) time using O(n/logn) processors in the EREW-PRAM model of computation. We then show that the Breadth-First algorithm can be used to produce new encodings of binary and ordered trees.

3 citations

Book ChapterDOI
30 Jul 2009
TL;DR: A parallel algorithm to find an on-line node ranking for general tree with using O (n 3 / log2 n ) processors on CREW PRAM model is presented.
Abstract: A node ranking of a graph G = (V , E ) is a proper node coloring C : V ****** such that any path in G with end nodes x , y fulfilling C (x ) = C (y ) contains an internal node z with C (z ) > C (x ). In the on-line version of the node ranking problem, the nodes v 1 , v 2 ,..., v n are coming one by one in an arbitrary order; and only the edges of the induced subgraph G [{v 1 , v 2 ,..., v i }] are known when the color for the node v i be chosen. And the assigned color can not be changed later. In this paper, we present a parallel algorithm to find an on-line node ranking for general tree. Our parallel algorithm needs O (n log2 n ) time with using O (n 3 / log2 n ) processors on CREW PRAM model.

3 citations


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

  • ...All of these parallel techniques can be found in [2] or [13]....

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  • ...In [2] or [13], we know the time complexity of Prefix Computation and Interval Broadcasting both are O(log k) with using O(k/ log k) processors, where k is the size of input array....

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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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