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 2­SAT and 3­SAT. For 2­SAT we could design a randomized algorithm taking O # n 2 # steps; for 3­SAT, we were able to reduce the number of steps from O (2 n ) 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 M uv > 0. M uv 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 # v#N(u) M uv = 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 M uv . 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