Com S 633: Randomness in Computation Lecture 7 Scribe: Ankit Agrawal Summary: Com S 633: Randomness in Computation Lecture 7 Scribe: Ankit Agrawal In the last lecture, we looked at random walks on line and used them to devise randomized algorithms for 2-SAT and 3-SAT. For 2-SAT we could design a randomized algorithm taking O n2 steps; for 3-SAT, we were able to reduce the number of steps from O (2n) to O 4 3 n . Today we will extend the concept of random walks to graphs. 1 Random Walks on Graphs Consider a directed graph G = (V, E), |V | = n, |E| = m. Each edge (u, v) of the graph has a weight Muv > 0. Muv denotes the probability to reach v from u in one step. A natural restriction, therefore is that for each vertex, the sum of the weights on outgoing edges is 1, i.e., u vN(u) Muv = 1 (1.1) where N(u) is the set of neighbor vertices of u. A random walk on graph, therefore implies starting at some vertex, and traversing the graph according to the probabilities Muv. We are interested in the long term behavior of traversing like this on the graph, i.e., the probability Collections: Computer Technologies and Information Sciences