On the Complexity of the St-connectivity Problem

On the complexity of the st-connectivity problem Chung Keung Poon Doctor of Philosophy 1996 Department of Computer Science University of Toronto The directed st-connectivity problem is fundamental to computer science. There are many applications which require algorithms to solve the problem in small space and preferably in small time as well. Furthermore, its space and time-space complexities are related to several long-standing open problems in complexity theory. Depthand breadthrst search are well known algorithms that solve the problem in optimal (i.e., O(n+m)) time while using O(n log n) space where n and m are the number of nodes and edges in the graph respectively. It can also be solved inO(log2 n) space and 2O(log2 n) time by Savitch's algorithm. For space S between (log2 n) and (n log n), the best running time is T = 2O(log2(n logn=S)) mn due to Barnes et al.. Establishing matching lower bounds on the Turing machine model has been one of the major challenges in complexity theory. In this thesis, we introduce the natural structured NNJAG (Node-named JAG) model, which is a generalization of Cook and Racko 's JAG (Jumping Automaton for Graphs) model. For the space complexity, Cook and Racko prove an (log2 n= log log n) lower bound on the JAG model. This is extended to a randomized JAG by Berman and Simon. We further extend this to the probabilistic NNJAG model. For the time-space tradeo , we prove an upper bound of T = 2O(log2(n logn=S)) nO(1) on a JAG via a simulation of the Barnes et al. algorithm. We also prove a lower bound of T = 2 ( log2(n log n=S) log log n ) pnS= log n on an NNJAG, improving the previous bounds of T = (n4=3=S1=3) on NNJAGs by Edmonds and T = (n2=(S log n)) on JAGs by Barnes and Edmonds. Our result almost matches the Barnes et al. algorithm and seems to suggest that st-connectivity cannot be determined in polynomial time and polylogarithmic space simultaneously. For example, i when the space is S = n for any constant 0 < < 1, the NNJAG lower bound implies a super-polynomial running time. This lower bound on time-space tradeo is joint work with Edmonds. ii Acknowledgments I would like to thank my supervisor, Allan Borodin, for his guidance and support, especially during the last year of my PhD study. I would also like to thank the professors, the o ce clerks and the large crowd of postdocs and graduate students in the Department of Computer Science who together provided such an enjoyable environment during my stay. Among them, I am especially grateful to Je Edmonds who constantly helped me along and who is also a co-author of the result presented in Chapter 5. I am also grateful to my PhD committee and external examiner, Larry Ruzzo, for their careful reading of my thesis and many suggestions on improvements. Finally but not the least, I thank my parents and family for their support and I dedicated this (belated) thesis to the memory of my father, Wah Poon. iii