Showing papers by "Andrei Z. Broder published in 1989"

Generating random spanning trees

Abstract: The author describes a probabilistic algorithm that, given a connected, undirected graph G with n vertices, produces a spanning tree of G chosen uniformly at random among the spanning trees of G. The expected running time is O(n log n) per generated tree for almost all graphs, and O(n/sup 3/) for the worst graphs. Previously known deterministic algorithms are much more complicated and require O(n/sup 3/) time per generated tree. A Markov chain is called rapidly mixing if it gets close to the limit distribution in time polynomial in the log of the number of states. Starting from the analysis of the above algorithm, it is shown that the Markov chain on the space of all spanning trees of a given graph where the basic step is an edge swap is rapidly mixing. >

Bounds on the cover time

Abstract: Consider a particle that moves on a connected, undirected graphG withn vertices. At each step the particle goes from the current vertex to one of its neighbors, chosen uniformly at random. Tocover time is the first time when the particle has visited all the vertices in the graph starting from a given vertex. In this paper, we present upper and lower bounds that relate the expected cover time for a graph to the eigenvalues of the Markov chain that describes the random walk above. An interesting consequence is that regular expander graphs have expected cover time Θ(n logn).

Trading space for time in undirected s-t connectivity

Abstract: Aleliunas et al. [1] posed the following question: “The reachability problem for undirected graphs can be solved in logspace and O(mn) time [m is the number of edges and n 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.