lecture15

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

This preview shows pages 1–2. Sign up to view the full content.

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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern