Savitch showed in 1970 that nondeterministic logspace (NL) is contained in deterministic O(log^2(n)) space but his algorithm requires quasipolynomial time. The question whether we can have a deterministic algorithm for every problem in NL that requires polylogarithmic space and simultaneously runs in polynomial time was left open.

In this paper we give a partial solution to this problem and show that for every language in NL there exists an unambiguous nondeterministic algorithm that requires O(log^2(n)) space and simultaneously runs in polynomial time.