Tight Lower Bounds for st-Connectivity on the NNJAG Model

• Authors: Jeff Edmonds;Chung Keung Poon;Dimitris Achlioptas
• Venue: SIAM Journal on Computing
• Year: 1999

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 $\delta 0$, if an NNJAG uses space $S \in O(n^{1-\delta})$, then $T \in 2^{ \Omega(\log^2 (n/S)) }$; otherwise $T \in 2^{ \Omega( \log^2({n\log n \over S}) / \log\log n )} \times (nS / \log n)^{1/2}$. (In a preliminary version of this paper by Edmonds and Poon [Proc. 27th Annual ACM Symposium on Theory of Computing, Las Vegas, NV, 1995, pp. 147--156.], a lower bound of $T \in 2^{ \Omega( \log^2({n\log n \over S}) / \log\log n )} \times (nS/\log n)^{1/2}$ was proved.) Our result greatly improves the previous lower bound of $ST \in \Omega(n^2/\log n)$ on the JAG model by Barnes and Edmonds [ Proc. 34th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, 1993, pp. 228--237] and that of $S^{1/3}T \in \Omega(n^{4/3})$ on the NNJAG model by Edmonds [ Time-Space Lower Bounds for Undirected and Directed ST-Connectivity on JAG Models, Ph. D. thesis, University of Toronto, Toronto, ON, Canada, 1993]. Our lower bound is tight for $S \in O(n^{1-\delta})$, for any $\delta 0$, matching the upper bound of Barnes \etal [ Proc. 7th Annual IEEE Conference on Structure in Complexity Theory, Boston, MA, 1992, pp. 27--33]. As a corollary of this improved lower bound, we obtain the first tight space lower bound of $\Omega( \log^2 n )$ on the NNJAG model. No tight space lower bound was previously known even for the more restricted JAG model.