The s-t connectivity problem for undirected graphs is to decide whethe
r two designated vertices, s and t, are in the same connected componen
t. This paper presents the first known deterministic algorithms solvin
g undirected s-t connectivity using sublinear space and polynomial tim
e. Our algorithms provide a nearly smooth time-space tradeoff between
depth-first search and Savitch's algorithm. For n vertex, m edge graph
s, the simplest of our algorithms uses space O(s), n(1/2)log(2)n less
than or equal to s less than or equal to nlog(2)n, and time O(((m + n)
n(2)log(2)n)/s). We give a variant of this method that is faster at th
e higher end of the space spectrum. For example, with space Theta(n lo
g n), its time bound is O((m + n) log n), close to the optimal time fo
r the problem. Another generalization uses less space, but more time:
space O(lambda n(1/lambda) log n), for 2 less than or equal to lambda
less than or equal to log(2) n, and time n(O(lambda)). For constant la
mbda the time remains polynomial.