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Algorithms for Embedding Graphs in Books TR 85-028 ALGORITHMS FOR EMBEDDING GRAPHS IN BOOKS by Lenwood Scott Heath A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Computer Science. Chapel Hill 1985 Reader: n © 1985 Lenwood Scott Heath ALL RIGHTS RESERVED m LENWOOD SCOTT HEATH. Algorithms for Embedding Graphs in Books (Under the direction or ARNOLD L. ROSENBERG, Duke University.) ABSTRACT We investigate the problem ol embedding graphs in boob. A book is some number or half­ planes (the page• or the book), which share a common line as boundary (the qine or the book). A book embedding or a graph embeds the vertices on the spine in some order and embeds each edge in some page so that in each page no two edges intersect. The pagenumber ol a graph is the number or pages in a minimum-page embedding or the graph. The pagewidth or a book embed­ ding is the maximum cutwidth or the embedding in any page. A practical application is in the realization or a fault-tolerant array or VLSI processors. Our results are efficient algorithms for embedding certain classes or planar graphs in books or small pagenumber or small pagewidth. The first result is a linear time algorithm that embeds any planar graph in a book or seven pages. This establishes the smallest upper bound known for the pagenumber or the class or planar graphs. The algorithm uses three main ideas. The first is to level the planar graph. The second is to eztend a cycle at one level to the next level by doing micro-surgery. The third is to neat the embedding or successive levels to obtain finite pagenumber. The second result is a linear time algorithm that embeds any trivalent planar graph in a book or two pages. The algorithm edge-augments the graph to make it hamiltonian while keeping it planar. The third result is an 0( n logn) time algorithm for embedding any outerplanar graph with small pagewidth. Our algorithm embeds any ,._valent outerplanar graph in a two-page boolr. with O(dlogn) pagewidth. This result is optimal in pagewidth to within a constant factor. The significance for VLSI design is that any outerplanar graph can be implemented in small area in a fault-tolerant fashion. lv ACKNO~EDGEMENTS I dedicate this dissertation to my wife Sheila, who is my reason Cor perseverins. In the past four years, she has been my source or love, joy, support, inspiration, and meaoios. I express endless gratitude to Arnold Rosenberg for the hours he has devoted to my educ,.. tioo and Cor the belief he expressed in my abilities. Without him, this work would not have been possible. He is also responsible for pointing me in the direction which I now follow. The time and patience or the other members or my committee, Dean Brock, Tom Brylawski, Kye Hedlund, David Plaisted, and Don Stanat, are much appreciated. Don Stanat was my faith­ ful guide through the P"rils or my graduate career. I especially thank Tom Brylawski for giving me the combinatorist's view or things. My mother has always had more confidence in me than I have. Her unfailing love and belief traveled with me alons the road to this accomplishment. I thank the National Science Foundation (grant MCS-83-01213) and the Semiconductor Research Corporation (grant 41258) Cor support or this research. v TABLE OF CONTENTS 1 THE PROBLEM AND ITS MOTIVATIONS................................................................. 1 1.1 The Problem ............................................................................................................... 1 1.2 Motivations ................................................................................................................. 3 1.2.1 Multilayer VLSI Layout ........................................................................................... 3 1.2.2 Design or Fault-Tolerant Processor Arrays ............................................................... 5 1.2.3 Sorting with Parallel Stacks ..................................................................................... 6 1.3 Structure or the Di...,rtation .......... ..... .... ..... .. .. ..... .... ................ ...................... ....... .. ... 7 2 PREVIOUS RESULTS AND TOOLS ............................................................................ 8 2.1 Previous Results .......................................................................................................... 8 2.1.1 Circular Embedding ................................................................................................. 9 2.1.2 One-Page Graphs ..................................................................................................... 11 2.1.3 Two-Pase Graphs ..................................................................................................... 11 2.1.4 Planar Graphs .......................................................................................................... 12 2.2 Tools ........................................................................................................................... 12 3 EMBEDDING PLANAR GRAPHS IN SEVEN PAGES ................................................. 14 3.1 Overview ol the AJsorithm ........................................................ ............................. ..... 14 3.2 Previous Approa.:hes ................................................................................................... 19 3.2.1 The Bemhart-Kainen Conjecture ............................................................................. 19 3.2.2 Birurcators .................. ......................... ..... .... ..... ...... ..... ........................................... 25 vi 3.2.3 Separating Triangles ............. ..................... ........... ......... ........... .. ... .................. .. .. .. ... 26 3.3 Elements or the Seven-Page Algorithm ....................................................................... Z7 3.3.1 Levels ....................................................................................................................... Z7 3.3.2 D-Cycles ...................... ......... .................................... ......... ... ........................ ............ 35 3.3.3 Nesting ...... ..... ............................... ............ ........................ ........................... ....... .. ... 39 3.4 Levels Without Cycles .................................................................................... ............ 44 3.5 The Algorithm .. ....... .. .. .. ....... ....... ............. ............ ......... ...... ... .. .... ... .... .. ... .. .... ............ 46 3.5.1 The Statement ......................................................................................................... 49 3.5.2 Why Seven! ............................................................................................................ 54 3.5.3 Further Analysis .... .. .. .................... ....... ..... ........... ....... .. .. ....... .. .. ..... ......... .. .. ... .... .. ... 55 3.6 Conclusions .. ... .. ....... .. .... .............. ................ .... ....... ....... ......... .. .. .. ... .. .. .. ... .... .. ............ 56 4 EMBEDDING TRIVALENT PLANAR GRAPHS IN TWO PAGES.............................. 58 4.1 Overview or the Algorithm .......................................................................................... 58 4.2 Structure or Biconnected Planar Graphs ..................................................................... 61 4.3 The Main Theorem .. .. .. .. ... .. .. .. ..... ....... ............. .. ..... ....... .. .. .. ... .. .. .. .. .. .. ..... .. .. .. ..... .. .. ... 68 4.4 The Algorithm ........... .. .. ... .... .. ... .. ....... .. ........... ..... ...... ... .. .... ... .. .. .. ..... .. .. ... .... .. ... .. ....... 73 4.5 Oriented Face Traversal .............................................................................................. 77 4.6 Conclusions ..................................................................................................... ............ 82 5 EMBEDDING OUTERPLANAR GRAPHS IN SMALL BOOKS ....................... ............ 84 5.1 Tradeolfs ..................................................................................................................... 84 5.2 Overview or the Algorithm .... .. ..... ....... ...... ............ ...... ......... ... .. .. .. ... .. ....... .. .. .. ... .. .... ... 88 5.2.1 String Construction .................................................................................................. 92 5.2.2 Ladder Construction .. ......... ........... ....... ..... ............. ......... ..... ...... ... .... ..... .... .. ............ 05 5.3 The Algorithm ............................................................................................... :............ 101 5.4 Correctness ................................................................................................................. 105 vll 5.5 Performance ........................ .................. .. ................ .................................... .... ............ 110 5.6 Conclusion .................................................................................................................. Ill 6 CONCLUSIONS ............................................................................................................ 112 REFERENCES .................................................................................................................. 116 GLOSSARY ......................................................................................................................
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