Time-space lower bounds for directed s-t connectivity on JAG models

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 $ST = \Omega(n^{2}/\log n)$ and $S^{1/2}T = \Omega(m n^{1/2})$ for Cook and Rackoff's JAG model, 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 time-space lower bound of $S^{1/3}T = \Omega(m^{2/3}n^{2/3})$ on the more powerful node-named JAG model of Poon. These bounds approach the known upper bound of $T = O(m)$ when $S = \Theta(n \log n)$