Lecture 14 : Pseudorandom Generators for Logarithmic Space

Given an undirected graph G and vertices s, t ∈ V (G), the undirected connectivity problem is to decide whether there is a path from s to t in G. If G has N vertices and M edges, then the existence of a path between s and t implies the existence of a path of length at most N . If A is the adjacency matrix of G and G has, without loss of generality, self-loops on every vertex, then s and t are connected if and only if (AN )t,s > 0 holds. In other words, the goal of the connectivity problem is to decide if there is a non-zero probability that, starting at vertex s and after traversing N steps in G, we will arrive at vertex t.