Time-space Lower Bounds for Undirected and Directed St -connectivity on Jag Models

Time-space lower bounds for undirected and directed st-connectivity on JAG models Je A. Edmonds Doctor of Philosophy 1993 Department of Computer Science University of Toronto Directed and undirected st-connectivity are important problems in computing. There are algorithms for the undirected case that use O (n) time and algorithms that use O (logn) space. The rst result of this thesis proves that, in a very natural structured model, the JAG (Jumping Automata for Graphs), these upper bounds are not simultaneously achievable. This uses new entropy techniques to prove tight bounds on a game involving a helper and a player that models a computation having precomputed information about the input stored in its bounded space. The second result proves that a JAG requires a time-space tradeo of T S 1 2 2 mn 12 to compute directed st-connectivity. The third result proves a time-space tradeo of T S 1 3 2 m 23n 2 3 on a version of the JAG model that is considerably more powerful and general than the usual JAG. i Acknowledgments There are many people I wish to thank for helping me along the path from s to t. There were those who held my hand while I walked the treacherous edges from idea to idea and bug to bug. I am also indebted to those who helped me make sure that my ideas changed state from thought to written form in a way that insured they were accepted rather than rejected. As I jumped from deadline to deadline, there were always people who insured that I cleared the hurdles and who gave me the support I needed to prepare for the next leap. Faith Fich, my advisor, provided the perfect balance between directing my path and leaving me to take a random walk through the undirected graph of research. Courses and research groups were important in providing the necessary supplies for my trip. They helped me learn about new papers, hear about the research of my peers, and gave me opportunities to share my own work. Some of my favorite courses were combinatorial methods by Faith and Mauricio Karchmer, logic and non-standard numbers by Russell Impagliazzo and Steve Cook, graph theory by Derek Corneil, distributed systems by Vassos Hadzilacos, and complexity by Al Borodin. The research groups that most in uenced me were graph theory led by Derek Corneil, NC2 led by Mauricio Karchmer, ordinals and graph minors led by Arvind Gupta, the probabilistic method led by Hisao Tamaki, and complexity theory led by Al Borodin. The breadth requirements were important as well. Some of them were actually interesting and taking them with Gara Pruesse made them all the more enjoyable. Especially, the beer afterwards. When I was traversing some of those edges during my mathematical odyssey, there were many people who provided valuable support. Russell Impagliazzo, Arvind Gupta, Toni Pitassi, Hisao Tamaki, Tino Tamon, and Chung Keung Poon were kind enough to listen to my ideas and help me along. Some of my best work was also done jointly with these same people. Hisao, Tino, C.K. and Russell were particularly helpful in providing me with information from the mathematical literature that I wasn't as well versed in as I should have been. Al Borodin, Charlie Racko and Steve Cook helped me by hitting my bugs with a y swatter. And of course, a great deal of thanks goes to Faith Fich for treating me as a peer and for always being excited to discuss research. I will always be grateful to those who helped me convert my research to understandable English. Faith Fich was my most devoted critic and English teacher and spent much of her time beating my writing into shape and combing out the typos. Since criticism is often hard to hear, I sometimes relied on Jeannine St. Jacques to help put it into perspective. Many proof readers were also crucial in revising my work. In addition to Faith Fich, I would like to thank Toni Pitassi, Naomi Nishi, Jennifer Edmonds, Miriam Zachariah, Paul Beame, and some unknown referees who were very hard working. I was also saved many times by Steven Bellantoni, Eric Schenk, and Ken Lalonde when I was lost with Unix and Latex. There were many people who provided the emotional support I needed to continue my journey. ii Faith Fich, Russell Impagliazzo, and Arvind Gupta, who had already become successful researchers, often reassured me when I despaired about my own success. Faith was always there for me with motherly love. Through out the years, Al too gave many touching words of support that were greatly appreciated. I thank Derek Corneil, Arvind Gupta, Faith Fich, Steven Rudich, and Pat Dymond for arranging for me to give talks outside of U. of T. which provided me with some of the exposure and experience I needed. Martha Hendricks lightened things up and provided distractions when I was working too hard. Rick Wallace, Shirley Russ, Roland Mak, and Toni Pitassi were always by my side to help me with ongoing emotional issues. PhD students are responsible for more than their own research. As I learned to teach undergraduates, I found important role models in Jim McInnes, Peter Gibbons, and Faith Fich. Kathy Yen, Teresa Miao, and Vicky Shum led me by the hand when I had administrative obligations. I would be remiss if I did not mention a few others who helped me on my way. My father, Jack Edmonds, deserves some of the credit for my introduction to mathematics. He continually kept me interested in puzzles to ponder. My mother, Patricia Oertel, provided support and encouragement as I made my way through public school and university. Baby Joshua was very obliging in keeping me awake at night so that I could do research. He also provided a lots of joy and love. My wife Miriam was a tremendous support in every way possible and one last thanks goes to Faith Fich for being such a wonderful advisor. iii