Directed s-t connectivity is the problem of detecting whether there is a path from a distinguished vertex s to a distinguished vertex t in a directed graph. We prove time-space lower bounds of ST = \Omega\Gamma n 2 = log n) and S 1=2 T = \Omega\Gamma mn 1=2 ) for Cook and Rackoff's JAG model [8], where n is the number of vertices and m the number of edges in the input graph, and S is the space and T the time used by the JAG. We also prove a timespace lower bound of S 1=3 T = \Omega\Gamma m 2=3 n 2=3 ) on the more powerful node-named JAG model of Poon [13]. These bounds approach the known upper bound of T = O(m) when S = \Theta(n log n).

Abstract. Directed st-connectivity is the problem of deciding whether or not there exists a path from a distinguished node s to a distinguished node t in a directed graph. We prove a time– space lower bound on the probabilistic NNJAG model of Poon [Proc. 34th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, 1993, pp. 218–227]. Let n be the number of nodes in the input graph and S and T be the space and time used by the NNJAG, respectively. We show that, for any δ>0, if an NNJAG uses space S ∈ O(n1−δ), then T ∈ 2Ω(log2 (n/S)) ; otherwise n log n) / log log n) S

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. Depth- and breadth-first 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 in O(log 2 n) space and 2 O(log 2 n) time by Savitch's algorithm. For space S between \Theta(log 2 n) and \Theta(n log n), the best running time is T = 2 O(log 2 (n log n=S)) \Theta mn due to Barnes et al.. Establishing matching lower bounds on the Turing machine model ha...

The following is a second proof of (basically) the same undirected st-connectivity result using recursive flyswatters as given in my thesis and in STOC-93 [Ed93a, Ed-PHD]. The input graph and the reduction techniques in the two proofs are similar. The main difference is that JAG result is reduced to a different game. In this paper, the game consists of a pebble walking on a line. The movements of the pebble are directed by a player and a random input. The conjecture is that the player cannot get the pebble across the line much faster than that done by a random walk. Likely, however, this is hard to prove. What can be proven is that this game becomes equivalent to the game in the original paper, if the player who is directing the pebble always knows where in the line pebble is. Therefore, the lower bound for the original game applies to this new game. Hence, the JAG lower bound proved in this paper is the same as that proven before. Two advantages of this new proof are that it is a litt...