We show that non-deterministic time NT IME(n) is not contained in deterministic time n # 2-# and polylogarithmic space, for any # > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n # 2-# ) and poly-logarithmic space. A similar result is presented for uniform circuits.

We prove time-space tradeoffs for traversing undi-rected graphs. One of these is a quadratic lower bound on a deterministic model that closely matches the recent probabilistic upper bound of Broder, Karlin, Raghavan, and Upfal. The models used are variants of Cook and Rackoff’s “Jumping Automata for Graphs".

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...

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 (log n) space. The first 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 tradeoff of T \Theta S 1 2 2\Omega i mn 1 2 j to compute directed st-connectivity. The third result proves a time-space tradeoff of T \Theta S 1 3 2\Omega i m 2 3 n 2 3 j on a version of the...

We consider the problem of determining, given a graph G and specified nodes s and t, whether or not there is a path of at most k edges in G from s to t. We show that solving this problem on polynomial-size unbounded fan-in circuits, requires depth \Omega\Gammapth log k), improving on a depth lower bound of\Omega\Gamma/22 k) when k = log O(1) n given in [2, 8]. In addition we show that there is a constant c such that for k log n, any depth d unbounded fan-in circuit for this problem requires size at least n ck ffl d where ffl d = OE \Gamma2d =3 and OE is the golden mean. This latter result improves on an n\Omega\Gamma432 (d+3) k) bound from [2, 8] where log (i) is the i-fold composition of log with itself. The key to our technique is a new form of `switching lemma' which combines some of the features of iteratively shortening terms due to Furst, Saxe, and Sipser [13] and Ajtai [1] with the kinds of switching lemma arguments introduced by Yao [18], Hastad [14], and ...

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...