Topological Structure and Analysis of Interconnection Networks

Topological Structure and Analysis of Interconnection Networks

Network Theory and Applications Volume 7 Managing Editors: Ding-Zhu Du, University of Minnesota, U.S.A. and Cauligi Raghavendra, University of Southern California, U. S.A.

Topological Structure and Analysis of Interconnection Networks by JunmingXu Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, China Springer-Science+Business Media, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4419-5203-5 ISBN 978-1-4757-3387-7 (ebook) DOI 10.1007/978-1-4757-3387-7 Printed on acid-jreepaper All Rights Reserved 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001. Softcover reprint of the hardcover 1st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Contents Preface 1 Interconnection Networks and Graphs 1.1 Graphs and Interconnection Networks 1.1.1 Graphs.... 1.1.2 Interconnection Networks... 1.1.3 Graph Isomorphism.... 1.2 Basic Concepts and Notations on Graphs 1.2.1 Subgraphs and Operations of Graphs. 1.2.2 Degrees and Edge-Degrees.... 1.2.3 Paths, Cycles and Connected Graphs. 1.2.4 Adjacency Matrices and Other Concepts. 1.3 Trees, Embeddings and Planar Graphs 1.3.1 Trees and k-ary Trees........... 1.3.2 Embedding of Graphs.... 1.3.3 Planar Graphs and Layout of VLSI Circuits. 1.4 Transmission Delay and Diameter.. 1.4.1 Diameter of Graphs... 1.4.2 A verage Distance of Graphs. 1.4.3 Routings in Networks... 1.5 Fault Tolerance and Connectivity. 1.5.1 Menger's Theorem... 1.5.2 Connectivity of Graphs.. 1.5.3 Fault Tolerance of Networks. 1.6 Basic Principles of Network Design. 1.6.1 Introduction.... 1.6.2 Basic Principles of Network Design IX 1 1 2 3 6 8 9 10 12 14 16 16 18 21 22 23 26 28 30 30 31 33 35 35 37

VI CONTENTS 2 Design Methodology of Topological Structure of Interconnection Networks 39 2.1 Line Graphical Method... 40 2.1.1 Line Graph of Undirected Graph 40 2.1.2 Line Graph of Digraph...... 42 2.1.3 Connectivity and Diameter of Line Graphs 44 2.1.4 Eulerian and Hamiltonian Properties. 45 2.1.5 Iterated Line Digraphs....... 46 2.1.6 Edge-Connectivity of Line Graphs 48 2.2 Cayley Method.......... 52 2.2.1 Vertex-Transitive Graphs 52 2.2.2 Edge-Transitive Graphs. 57 2.2.3 Atoms of Graphs..... 59 2.2.4 Connectivity of Transitive Graphs 62 2.2.5 Cayley Graphs........... 65 2.2.6 Transitivity of Cayle Graphs... 67 2.2.7 Atoms and Connectivity of Cayley Graphs. 70 2.2.8 Vertex-Transitive Graphs with Prime Order 74 2.3 Cartesian Product Method... 76 2.3.1 Cartesian Product of Undirected Graphs. 76 2.3.2 Cartesian Product of Digraphs...... 78 2.3.3 Some Remarks on Cartesian Products.. 79 2.3.4 Diameter and Connectivity of Cartesian Products 81 2.3.5 Other Properties of Cartesian Products 84 2.3.6 Cartesian Product of Cayley Graphs 86 2.4 A Basic Problem in Optimal Design... 91 2.4.1 Undirected (d, k)-graph Problems 91 2.4.2 Directed (d, k)-graph Problems. 96 2.4.3 Bipartite (d, k)-graph Problems. 99 2.4.4 Planar (d, k)-graph Problems... 101 2.4.5 Relations between Diameter and Connectivity. 102 3 Well-known Topological Structures of Interconnection Networks 105 3.1 Hypercube Networks........ 105 3.1.1 Two Equivalent Definitions 106 3.1.2 Some Basic Properties. 107 3.1.3 Gray Codes and Cycles.. 110

CONTENTS VB 3.1.4 Lengths of Paths.... 3.1.5 Embedding Problems 3.1.6 Generalized Hypercubes 3.1.7 Some Enhancements on Hypercubes 3.2 De Bruijn Networks.... 3.2.1 Three Equivalent Definitions.... 3.2.2 Eulerian and Hamiltonian Properties. 3.2.3 Uniqueness of Shortest Paths.. 3.2.4 De Bruijn Undirected Graphs.. 3.2.5 Generalized de Bruijn Digraphs. 3.2.6 Comparison with Hypercubes 3.3 Kautz Networks.... 3.3.1 Three Equivalent Definitions 3.3.2 Paths in Kautz Digraphs.. 3.3.3 Kautz Undirected Graphs.. 3.3.4 Generalized Kautz Digraphs. 3.3.5 Connectivity of Generalized Kautz Digraphs 3.4 Double Loop Networks.... 3.4.1 Double Loop Networks.... 3.4.2 L-Tiles in the Plane.... 3.4.3 Diameter of Double Loop Networks. 3.4.4 Optimal Design of Double Loop Networks 3.4.5 Circulant Networks and Basic Properties 3.5 Other Topological Structures of Networks 3.5.1 Mesh Networks and Grid Networks. 3.5.2 3.5.3 3.5.4 3.5.5 3.5.6 3.5.7 Pyramid Networks... Cube-Connected Cycles Butterfly Networks. Benes Networks.... n Networks.... Shuffle-Exchange Networks 112 113 116 118 121 121 124 126 131 131 138 139 139 141 142 142 145 148 148 150 154 160 165 171 171 173 175 178 182 185 186 4 Fault-Tolerant Analysis of Interconnection Networks 187 4.1 Routings in Interconnection Networks.. 187 4.1.1 Forwarding Index of Routing... 188 4.1.2 Edge-Forwarding Index of Routing 196 4.1.3 Delay of Fault-Tolerant Routing 198 4.1.4 Some Upper Bounds........ 202

Vlll Contents 4.2 Fault-Tolerant Diameter... 4.2.1 Edge-Addition Problems. 4.2.2 Edge-Deletion Problems. 4.2.3 Vertex-Deletion Problems 4.2.4 Fault-Tolerant Diameters of Some Networks 4.3 Menger-Type Problems in Parallel Systems.... 4.3.1 Disjoint Paths for Bounded Length... 4.3.2 Menger Number and Bounded Connectivity 4.3.3 Edge Disjoint Paths for Bounded Length 4.3.4 Disjoint Paths for Exceeded Length 4.3.5 Rabin Numbers of Networks... 4.4 Wide Diameter of Networks.... 4.4.1 Containers and Basic Properties 4.4.2 Wide Diameter and Basic Results 4.4.3 Wide-Diameter on Regular Graphs 4.4.4 Wide-Diameter on Cartesian Products 4.4.5 Wide-Diameter and Independence Number 4.4.6 Wide-Diameter and Fault-Tolerant Diameter 207 207 215 227 232 235 235 242 246 249 251 255 255 256 259 261 265 268 4.4.7 Wide-Diameters of Some Well-Known Networks. 270 4.4.8 Wide Diameter for Edge Variation... 274 4.5 (1, w)-independence and -Dominating Numbers 275 4.5.1 (1, w)-independence Numbers...... 275 4.5.2 (1, w)-dominating Numbers....... 279 4.5.3 (1, I)-Independence and -Dominating Numbers 282 4.5.4 Some (1, w)-dominating Numbers... 287 4.6 Restricted Fault-Tolerance of Networks.... 288 4.6.1 Restricted Connectivity and Diameter 288 4.6.2 Restricted Edge-Connectivity..... 292 4.6.3 Restricted Edge-Atoms... 295 4.6.4 Restricted Edge-Connectivity of Transitive Graphs 299 4.6.5 Generalized Restricted Edge-Connectivity..... 302 Bibliography 307 List of Symbols 329 Subject Index 337

Preface The advent of very large scale integrated circuit technology has enabled the construction of very complex and large interconnection networks. By most accounts, the next generation of supercomputers will achieve its gains by increasing the number of processing elements, rather than by using faster processors. The most difficult technical problem in constructing a supercomputer will be the design of the interconnection network through which the processors communicate. Selecting an appropriate and adequate topological structure of interconnection networks will become a critical issue, on which many research efforts have been made over the past decade. The book is aimed to attract the readers' attention to such an important research area. Graph theory is a fundamental and powerful mathematical tool for designing and analyzing interconnection networks, since the topological structure of an interconnection network is a graph. This fact has been universally accepted by computer scientists and engineers. This book provides the most basic problems, concepts and well-established results on the topological structure and analysis of interconnection networks in the language of graph theory. The material originates from a vast amount of literature, but the theory presented is developed carefully and skillfully. The treatment is generally self-contained, and most stated results are proved. No exercises are explicitly exhibited, but there are some stated results whose proofs are left to the reader to consolidate his understanding of the material. The book consists of four chapters. The first chapter introduces how to model an interconnection network by a graph and provides a self-contained exposition of the basic graph-theoretic concepts, terminology, notation and the corresponding backgrounds of networks as well as the basic principles of network design. Some basic results on graph theory used in the book are stated. The second chapter presents three major methods for large-scale network design: line graph method, Cayley method and cartesian product

x Preface method. The fundamental properties of the graphs constructed by these methods are presented in details. The (d, k )-graph problem is briefly discussed. As applications of the methods, the third chapter provides four classes of the most well-known network structures: hypercube, de Bruijn, Kautz and double loop networks and their many desirable properties as well. At the end of this chapter, other common network structures such as mesh, grid, pyramid, cube-connected cycle, butterfly, omega, and shuffle-exchange networks are simply mentioned. The fourth chapter is a focal point of the book, from which the reader can easily find some interesting research issues to study further. It presents some basic issues and research results in analysis of fault-tolerant network consisting of six research aspects, involving routing, Menger-type problems in parallel systems, fault-tolerant diameter, wide diameter, (l, w)-dominating number and restricted fault tolerance. Reading the book is not difficult for readers familiar with elementary graph theory. The book will be useful to those readers who intend to start research in design and analysis of interconnection network structures, and students in computer science and applied mathematics, theoretic computer scientists, engineers, designer of interconnection networks, applied mathematicians and other readers who are interested in interconnection networks. The book is developed from the text for an advanced undergraduate and first-year postgraduate course in graph theory and computer science in one semester given at University of Science and Technology of China (USTC). I would like to thank Graduate School and Department of Mathematics of USTC for their support and encouragement. I avail myself of this opportunity to express my heartfelt gratitude to Professor Qiao Li for his continuous help, to Professor Ding-Zhu Du for his encouragement and recommendation of the book as a member of the book series" Network Theory and Applications", to Professor F. K. Hwang for his valuable suggestions, to Processor D. Frank Hsu for his bringing my research interest to the subject when he was inviting USTC in 1992, and to Professor Yuke Wang for his help whenever possible. Finally, I would like to thank my son and postgraduate student, Keli Xu, for his very concrete help, and my wife, Jingxia Qiu, for her support, understanding and love, without which this work would have been impossible. Jun-Ming Xu (xujm@ustc.edu.cn) May 2001