A fast randomized LOGSPACE algorithm for graph connectivity

Abstract

We study the relationship between undirected graph reachability and graph connectivity, in the context of randomized LOGSPACE algorithms. Aleluinas et al. [2] show that graph reachability (checking whether there is a path connecting vertices S and T) can be decided in logarithmic space and polynomial time, by starting a random walk at S, and checking whether T is hit within some time limit. The random algorithm has one-sided error (with small probability, it fails to determine that S and T are connected). The reachability algorithm may be used in order to decide (with one sided error) whether a graph is connected, by running it n−1 times, each time with a different target vertex T. This increases the running time by a factor of n. In this paper we give an alternative RLOGSPACE algorithm for graph connectivity. Its running time varies between O(n ²) steps and O(n³) steps, depending on the structure of the input graph. This matches the fastest known RLOGSPACE algorithm for reachability, up to a constant factor. Our algorithm has two-sided error.

Supported by a Koret Foundation fellowship.