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 J and ℸ) can be decided in logarithmic space and polynomial time, by starting a random walk at J, and checking whether ℸ is hit within some time limit. The randomized algorithm has one-sided error (with small probability, it fails to determine that J and ℸ 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 ℸ. This increases the running time by a factor of n. In this paper we give an alternative randomized LOGSPACE algorithm for graph connectivity. Its running time varies between O( n 2) steps and O( n 3) 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.
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