Com S 633: Randomness in Computation Lecture 6 Scribe: Hongyu Sun Summary: Com S 633: Randomness in Computation Lecture 6 Scribe: Hongyu Sun From the last lecture we know, in the random walk on a line, the expected time to reach n starting at 0 is n 2 . In today's lecture we use random walks on line to devise algorithms for 2SAT and 3SAT. A Randomized algorithm for 2-SAT Let  be a 2CNF formula. Consider the following algorithm. 1. Input (y 1 ;    ; y n ) 2. Pick an arbitrary assignment x if (x) = 1 output x and Accept else pick a clause that x does not satisfy (this clause has 2 literals and x sets both to be false) Randomly pick one of the literals and update x by making that literal true 3. Repeat Step 2 for m times If the input formula is not satis able, the the algorithm does not accept. From now, we assume that the input formula is satis able. Let a = a 1    a n be a satisfying assignment. Let x t denote the assignment at the beginning of t th iteration of the loop. Let X t be a random variable that represents the number of bits at which x t and a match. Observe that the algorithm accepts when X t reaches n or when x becomes another satisfying assignment of . Observe that If x t and a matches at j places, then x t+1 matches with a at either j + 1 or j 1 places. Thus Collections: Computer Technologies and Information Sciences