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Aleliunas <italic>et al.</italic> [1] posed the following question: “The reachability problem for undirected graphs can be solved in logspace and <italic>O</italic>(<italic>mn</italic>) time [<italic>m</italic> is the number of edges and <italic>n</italic> is the number of vertices] by a probabilistic algorithm that simulates a random walk, or in linear time and space by a conventional deterministic graph traversal algorithm. Is there a spectrum of time-space trade-offs between these extremes?” We answer this question in the affirmative for linear-sized graphs by presenting an algorithm which is faster than the random walk by a factor essentially proportional to the size of its workspace. For denser graphs, the algorithm is faster than the random walk but the speed-up factor is smaller.

Trading space for time in undirected s-t connectivity