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T-Theory and Analysis of Online Algorithms

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T-Theory and Analysis of Online Algorithms UNLV Retrospective Theses & Dissertations 1-1-2007 T-theory and analysis of online algorithms James Oravec University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/rtds Repository Citation Oravec, James, "T-theory and analysis of online algorithms" (2007). UNLV Retrospective Theses & Dissertations. 2182. http://dx.doi.org/10.25669/no3p-11tt This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Retrospective Theses & Dissertations by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. T-THEORY AND ANALYSIS OF ONLINE ALGORITHMS by James Oravec Bachelor of Arts in Computer Science University of Nevada, Las Vegas 2W05 Master of Science in Computer Science University of Nevada, Las Vegas 2007 A thesis submitted in partial fulfillment of the requirements for the Master of Science in Computer Science Department of Computer Science Howard R. Hughes College of Engineering Graduate College University of Nevada, Las Vegas August 2007 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1448414 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI UMI Microform 1448414 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thesis Approval The Graduate College University of Nevada, Las Vegas JULY 2nd ■ 20 07 The Thesis prepared by JAMES A. ORAVEC Entitled T-THEORY AND ANALYSIS OF ONLINE ALGORITHMS is approved in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN COMPUTER SCIENCE Examination Committee Chair Dean of the Graduate College Z--i Examination CommitteeCoiamination Member Exami/ation Catnn Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT T-Theory and Analysis of Online Algorithms by James Oravec Lawrence L. Larmore, Examination Committee Chair Professor of Computer Science University of Nevada, Las Vegas Several advancements in Online Algorithms can be credited to T-theory, a field of discrete mathematics. T-theory has aided in the development of several online algorithms for the t-server problem, although the standard notation of T-theory was not used at the time of their creation. A summary of the ^-server problem, and some important concepts of T-theory, are given. A number of known k-server results are restated using the established terminology of T-theory. Included is a 3-competitiveness proof, using T-theory, for the HARMONIC algorithm for two servers, which was presented in a paper by Larmore and Oravec [71]. Previously, the Knowledge State Method was documented in Kurlinski’s thesis [70]. Additional research and analysis was done by Larmore and Oravec. Summaries of that work, as well as prior work by Larmore and Bein are given. Research supported by NSF grant CCR-0312093. Ill Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ABSTRACT .................................................................................................................... iii ACKNOWLEDGMENTS .............................................................................................. v CHAPTER 1 INTRODUCTION ............................................................................ 1 Thesis Introduction ................................................................................................. 1 CHAPTER 2 ONLINE ALGORITHMS.................................................................. 10 Introduction .............................................................................................................. 10 Deterministic versus Randomized ....................................................................... 10 The Work Function Algorithm ............................................................................. 11 CHAPTER 3 PARTICULAR PROPERTIES OF T-THEORY ........................... 12 Introduction ................................................................ 12 Basic Definitions........................................................................................................ 12 Exam ples .................................................................................................................... 13 P ro p e rtie s ................................................................................................................. 14 CHAPTER 4 THE AT-SERVER PROBLEM AND T-THEORY ..................... 16 Introduction to the fc-Server P roblem ................................................................. 16 Elementary T-Theory Concepts .......................................................................... 20 The Virtual Server Construction .......................................................................... 26 Tree A lgorithm s........................................................................................................ 27 Balance Algorithms .................................................................................... 31 Server Algorithms in the Tight Span .................................................................... 33 Definition and Analysis of HANDICAP .......................................................... 37 Harmonic Algorithms .............................................................................................. 44 Summary and Future Applications of T-theory ................................................... 50 BIBLIOGRAPHY ........................................................................................................... 53 APPENDIX A MATHEMATICA CALCULATIONS . 61 APPENDIX B FIGURES AND TABLES................................................................. 64 VITA ..................................................................................................................................... 105 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS The DT^]X document class was made by Steve Lumos. I would also like to give thanks to Dean Bartkiw and Marek Chrobak for reviewing the final manuscript coau­ thored by Larmore and Oravec. This thesis is dedicated to those who make a difference in the life of others. I have been blessed in having a great group of friends and mentors. To my family and friends, for their love, encouragement and support. To my ad­ visor, Lawrence L. Larmore, for giving me a genuine academic experience. To Angel Muleshkov, for explaining mathematics to me in depth. To Ronald Corbett, for en­ couraging me to think outside the box. To Chad Lexis, a childhood friend who has encouraged me to achieve more in life. To all of my computer science friends, for making my time at UNLV enjoyable. To Colonel Robert Luberacki and the men and women of the United States Military Services, for all they have done for this great country. To CJ Dowell, who left us before his time. He is a reminder to me and others that life is short, so enjoy your life and those who are important to you, while you can. Lastly, I would like to thank all of the members of my committee: A joy Datta, Lawrence Larmore, John Minor, and Angel Muleshkov. V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION Thesis Introduction This thesis surveys the research performed by Larmore and Oravec between February 2006 and April 2007. Chapter 4 is the heart of the thesis. This chapter is based on the topics covered in the paper by Larmore and Oravec titled “T-Theory Applications to Online Algorithms for the Server Problem” [71], which is currently available on arXiv, a prepublishing service provided by Cornell University. The remainder of the thesis covers research topics that Larmore and Oravec have worked on. Those topics may lead to future published papers. In order for the reader to better understand the research in this paper, we provide additional background information. Lastly, Eppstein [50] suggested a possible generalization of a previous results by Bein, Chrobak, and Larmore [15]. The suggested generalization would extend pre­ vious results for Manhattan planes to Manhattan or bifolds, a class of metric spaces introduced by Eppstein.
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