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=Ω(n ²/log n) and S^1/2T Ω(mn^1/2) for Cook and Rackoff´s JAG model (1980), 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 ^1/3T=Ω(m^2/3n(2/3)) on the more powerful node-named JAG model of Poon (1993). These bounds approach the known upper bound of T=O(m) when S=Θ(n log n)