lecture15 - Notes on Complexity Theory Last updated October...

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Notes on Complexity Theory Last updated: October, 2011 Lecture 15 Jonathan Katz 1 Randomized Space Complexity 1.1 Undirected Connectivity and Random Walks 1.1.1 Markov Chains We now develop some machinery that gives a different, and somewhat more general, perspective on random walks. In addition, we get better bounds for the probability that we hit t . (Note that the previous analysis calculated the probability that we end at vertex t . But it would be sufficient to pass through vertex t at any point along the walk.) The drawback is that here we rely on some fundamental results concerning Markov chains that are presented without proof. We begin with a brief introduction to (finite, time-homogeneous) Markov chains. A sequence of random variables X 0 , . . . over a space Ω of size n is a Markov chain if there exist { p i,j } such that, for all t > 0 and x 0 , . . . , x t - 2 , x i , x j Ω we have: Pr[ X t = x j | X 0 = x 0 , . . . , X t - 2 = x t - 2 , X t - 1 = x i ] = Pr[ X t = x j | X t - 1 = x i ] = p j,i . In other words, X t depends only on X t - 1 (that is, the transition is memoryless ) and is furthermore independent of t . We view X t as the “state” of a system at time t . If we have a probability distribution over the states of the system at time t , represented by a probability vector p t , then the distribution at time t + 1 is given by P · p t (similar to what we have seen in the previous section). Similarly, the probability distribution at time t + is given by P · p t . A finite Markov chain corresponds in the natural way to a random walk on a (possibly directed and/or weighted) graph. Focusing on undirected graphs (which is all we will ultimately be interested in), a random walk on such a graph proceeds as follows: if we are at a vertex v at time t , we move to a random neighbor of v at time t + 1. If the graph has n vertices, such a random walk defines the Markov chain given by: p j,i = k/ deg( i ) there are k edges between j and i 0 otherwise .
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