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lecture14 - Notes on Complexity Theory Last updated October...

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Notes on Complexity Theory Last updated: October, 2011 Lecture 14 Jonathan Katz 1 Randomized Space Complexity 1.1 Undirected Connectivity and Random Walks A classic problem in RL is undirected connectivity ( UConn ). Here, we are given an undirected graph and two vertices s, t and are asked to determine whether there is a path from s to t . An RL algorithm for this problem is simply to take a “random walk” (of sufficient length) in the graph, starting from s . If vertex t is ever reached, then output 1; otherwise, output 0. (We remark that this approach does not work for directed graphs.) We analyze this algorithm (and, specifically, the length of the random walk needed) in two ways; each illustrates a method that is independently useful in other contexts. The first method looks at random walks on regular graphs, and proves a stronger result showing that after sufficiently many steps of a random walk the location is close to uniform over the vertices of the graph. The second method is more general, in that it applies to any (non-bipartite) graph; it also gives a tighter bound. 1.1.1 Random Walks on Regular Graphs Fix an undirected graph G on n vertices where we allow self-loops and parallel edges (i.e., integer weights on the edges). We will assume the graph is d -regular and has at least one self-loop at every vertex; any graph can be changed to satisfy these conditions (without changing its connectivity) by adding sufficiently many self-loops. Let G also denote the (scaled) adjacency matrix corresponding to this graph: the ( i, j )th entry is k/d if there are k edges between vertices i and j . Note that G is symmetric ( G i,j = G j,i for all i, j ) and doubly stochastic (all entries are non-negative, and all rows and columns sum to 1). A probability vector
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