F. THOMSON LEIGHTON INTRODUCTION TO PARALLEL ALGORITHMS AND ARCHITECTURES: ARRAYS TREES HYPERCUBES

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1 F. THOMSON LEIGHTON INTRODUCTION TO PARALLEL ALGORITHMS AND ARCHITECTURES: ARRAYS TREES HYPERCUBES MORGAN KAUFMANN PUBLISHERS SAN MATEO, CALIFORNIA

2 Contents Preface Organization of the Material Teaching from the Text Exercises and Bibliographic Notes Errors Preview of Volume II Acknowledgments Notation ix x xi xii xiii xiv xv xix 1 ARRAYS AND TREES Elementary Sorting and Counting Sorting on a Linear Array 5 Assessing the Performance of the Algorithm... 7 Sorting N Numbers with Fewer Than N Processors Sorting in the Bit Model Lower Bounds A Counterexample Counting Properties of the Fixed-Connection Network Model Integer Arithmetic Carry-Lookahead Addition Prefix Computations 37 Segmented Prefix Computations Carry-Save Addition 44

3 1.2.4 Multiplication and Convolution Division and Newton Iteration * Matrix Algorithms Elementary Matrix Products Algorithms for Triangular Matrices Algorithms for Tridiagonal Matrices * 72 Odd-Even Reduction 72 Parallel Prefix Algorithms Gaussian Elimination Iterative Methods * 92 Jacobi Relaxation 93 Gauss-Seidel Relaxation 95 Finite Difference Methods 97 Multigrid Methods Retiming and Systolic Conversion A Motivating Example Palindrome Recognition The Systolic and Semisystolic Models of Computation Retiming Semisystolic Networks Conversion of a Semisystolic Network into a Systolic Network The Special Case of Broadcasting Retiming the Host Design by Systolic Conversion A Summary Graph Algorithms Transitive Closure Connected Components Shortest Paths Breadth-First Spanning Trees Minimum-Weight Spanning Trees Sorting Revisited Odd-Even Transposition Sort on a Linear Array A Simple y/n(log N + 1)-Step Sorting Algorithm A (3v^V + o(v / ]V r ))-Step Sorting Algorithm * A Matching Lower Bound 151

4 iii 1.7 Packet Routing Greedy Algorithms Average-Case Analysis of Greedy Algorithms Routing N Packets to Random Destinations Analysis of Dynamic Routing Problems Randomized Routing Algorithms Deterministic Algorithms with Small Queues An Off-line Algorithm Other Routing Models and Algorithms Image Analysis and Computational Geometry Component-Labelling Algorithms 201 Levialdi's Algorithm * 202 An 0(-//V)-Step Recursive Algorithm Computing Hough Transforms Nearest-Neighbor Algorithms Finding Convex Hulls * Higher-Dimensional Arrays Definitions and Properties Matrix Multiplication Sorting Packet Routing Simulating High-Dimensional Arrays on Low-Dimensional Arrays Problems Bibliographic Notes MESHES OF TREES The Two-Dimensional Mesh of Trees Definition and Properties Recursive Decomposition Derivation from K N<N Variations Comparison With the Pyramid and Multigrid Elementary О (log AT)-Step Algorithms Routing Sorting 289 ч

5 2.2.3 Matrix-Vector Multiplication Jacobi Relaxation Pivoting Convolution Convex Hull* Integer Arithmetic Multiplication Division and Chinese Remaindering Related Problems 306 Iterated Products 306 Root Finding Matrix Algorithms The Three-Dimensional Mesh of Trees Matrix Multiplication Inverting Lower Triangular Matrices Inverting Arbitrary Matrices * 316 Csanky's Algorithm 316 Inversion by Newton Iteration Related Problems Graph Algorithms Minimum-Weight Spanning Trees * Connected Components Transitive Closure Shortest Paths Matching Problems * Fast Evaluation of Straight-Line Code Addition and Multiplication Over a Semiring Extension to Codes with Subtraction and Division Applications Higher-Dimensional Meshes of Trees Definitions and Properties The Shuffle-Tree Graph Problems Bibliographic Notes 386

6 3 HYPERCUBES AND RELATED NETWORKS The Hypercube Definitions and Properties Containment of Arrays 396 Higher-Dimensional Arrays 399 Non-Power-of-2 Arrays Containment of Complete Binary Trees Embeddings of Arbitrary Binary Trees * 410 Embeddings with Dilation 1 and Load 0(f + logiv) 412 Embeddings with Dilation O(l) and LoadO(f+ 1) 416 A Review of One-Error-Correcting Codes * Embedding P\ 0e N into H iogn Containment of Meshes of Trees Other Containment Results The Butterfly, Cube-Connected-Cycles, and Benes Network Definitions and Properties Simulation of Arbitrary Networks * Simulation of Normal Hypercube Algorithms * Some Containment and Simulation Results The Shuffle-Exchange and de Bruijn Graphs Definitions and Properties The Diaconis Card Tricks Simulation of Normal Hypercube Algorithms Similarities with the Butterfly * * Some Containment and Simulation Results Packet-Routing Algorithms Definitions and Routing Models Greedy Routing Algorithms and Worst-Case Problems 515 A General Lower Bound for Oblivious Routing * Packing, Spreading, and Monotone Routing Problems 524 i

7 Reducing a Many-to-Many Routing Problem to a Many-to-One Routing Problem 536 Reducing a Routing Problem to a Sorting Problem The Average-Case Behavior of the Greedy Algorithm 539 Bounds on Congestion 542 Bounds on Running Time 547 Analyzing Non-Predictive Contention-Resolution Protocols Converting Worst-Case Routing Problems into Average-Case Routing Problems 561 Hashing 562 Randomized Routing Bounding Queue Sizes 571 Routing on Arbitrary Levelled Networks Routing with Combining The Information Dispersal Approach to Routing Using Information Dispersal to Attain Fault-Tolerance 604 Finite Fields and Coding Theory Circuit-Switching Algorithms Sorting Odd-Even Merge Sort 622 Constructing a Sorting Circuit with Depth logiv(logiv + l)/ Sorting Small Sets * A Deterministic 0(logiVloglog./V)-Step Sorting Algorithm Randomized 0(logiV)-Step Sorting Algorithms *. 657 A Circuit with Depth 7.45 log N that Usually Sorts Simulating a Parallel Random Access Machine PRAM Models and Shared Memories Randomized Simulations Based on Hashing Deterministic Simulations Using Replicated Data Using Information Dispersal to Improve Performance 709

8 3.7 The Fast Fourier Transform The Algorithm Implementation on the Butterfly and Shuffle-Exchange Graph Application to Convolution and Polynomial Arithmetic Application to Integer Multiplication Other Hypercubic Networks Butterflylike Networks 730 The Omega Network 730 The Flip Network 732 The Baseline and Reverse Baseline Networks Banyan and Delta Networks 736 k-ary Butterflies De Bruijn-Type Networks 739 The fc-ary de Bruijn Graph 741 The Generalized Shuffle-Exchange Graph Problems Bibliographic Notes 777 BIBLIOGRAPHY 785 INDEX 803 Lemmas, Theorems, and Corollaries 804 Author Index 807 Subject Index 811 4