RL⊆SC

We show that any randomized Logspace algorithm (running in polynomial time, with bounded tw~sided error) can be simulated deterministically in polynomial time and 0(log2 n) space. This puts RL in SC, " Steve's Class ". In particular, we get a polynomial time 0(log2 n) space algorithm for the connectivity problem on undirected graphs. 1 Introduction Given a (directed) graph G and two vertices in it s and t, the connectivity y (or reachability) problem is to decide whether there exists a directed path from s to t in G. The connectivity problem in directed graphs is one of the most basic problems in computer science. Using BFS or DFS, connectivity can be solved in polynomial (actually even linear) time and linear space. On the other hand, Savitch's [Sav70] theorem gives a 0(log2 n)-space algorithm for it, but which requires super-polynomial time. It is a long standing open problem whether there exists an algorithm for connectivity that combines both features: runs in polynomial time and poly-logarithmic space. In fact, there is no known algorithm for the connectivity problem that runs in polynomial time and sublinear space. For connectivity in undirected graphs two more types of algorithms are known. In [AKL*79] a randomized Logspace (and polynomial time) algorithm is given. A zero-error version of this type of Permission to copy without fee all or part of this material ia granted provided that the copiee are not made or distributed for direct commercial advantage, the ACM copyright notiGe and the title of the publication and ita date appear, and notice ia given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. algorithm is given in [BCD*89]. As for deterministic algorithms, Barnes and Ruzzo [BR91] recently presented an algorithm for undirected connectivity that runs in polynomial time and nc space (for any fixed e > o). In this paper we present further progress on this problem, obtaining an exponential improvement (in space). Theorem: There exists a deterministic algorithm for the undirected connectivity problem that runs in polynomial time and 0(log2 n) space. Unfortunately, the running time of the algorithm is a high order polynomial (something like n45). We do not know how to improve the running time significantly , even while increasing the space. The algorithm is obtained by derandomizing the randomized algorithm of [AKL*79]. In fact, the de-randomization result …