Lecture 4: Derandomization 4.1 Rp Error Reduction

Consider an RP algorithm with constant error probability that uses r random bits. We will improve the error probability to 2−k with r + O(k) random bits. Compare this with (1) the brute force k-independent trials, which would require O(kr) random bits to achieve the same error probability and (2) the technique due to Karp, Pippenger and Sipser [KPS] (discussed in Lecture 1) which uses r random bits (i.e., no extra random bits) and achieves an error probability of 1/poly(r). We will use random walks on expanders to reduce the error of RP algorithms. Ajtai, Komlos and Szemeredi first used random walks on expanders in the context of small-space derandomization [AKS]. The proof we present in lecture is due to Impagliazzo and Zuckerman [IZ]. The KPS technique, though great in terms of the number of extra random bits being used is limited by the fact that the running time of the improved algorithm is at least poly(1/δ) where δ is the (new reduced) error of the algorithm. Hence, we can reduce the error to at most 1/poly. The technique, discussed today, will further reduce the error to 2−k at the cost of only O(k) extra random bits as opposed O(rk) random bits in the k independent trails, while the algorithm still runs in (randomized) polynomial time. As in KPS, we will use a d-regular expander with V = {0, 1}r, thus |V | = 2r and d is a constant. As in KPS, we will assume that there exists an implicit construction of such expanders in the following sense: given any vertex v and any index i in the range 1 . . . d (where d is the degree of the expander), we can in time polynomial in |v| and |i|, compute the ith-neighbor of v. The expanders constructions we will discuss later in the course will satisfy such strong properties. Recall that to find witnesses, KPS began at a random vertex and completely explored all vertices within a ball of radius O(k). Here, we also start at a random vertex but instead of exploring all vertices in a ball, we will walk randomly for k steps and run the original RP algorithm along all vertices along this random walk. Thus the total randomness uses is at most r + k log d since log d bits are required to choose a random neighbor.