3 Relations Between Randomized Complexity Classes and Other Complexity Classes

Proof Fix a satisfying assignment a. Define the distance of r to a, denoted dist(r, a), as the number of coordinates where ri 6= ai (this is called Hamming distance). Let r = r, r, r, · · · be the assignments generated by the algorithm, and let di = dist(r , a). Note that as we only change one bit of the assignment, di+1 = di + ∆i where ∆i ∈ {−1, 1}. We claim that Pr[∆i = −1] ≥ 1/2. Once this is established, we have a random walk of d1, d2, . . . on the range {0, 1, 2, . . . , n}, which decreases in each step with probability at least 1/2. It can be shown that such a random walk will reach 0 after O(n) steps whp. To finish, we need to prove that Pr[bi = −1] ≥ 1/2. To see that, assume the value of r on the current two bits is (α1, α2) and the value of a is (β1, β2), where α1, α2, β1, β2 ∈ {0, 1} and (α1, α2) 6= (β1, β2) since r does not satisfy φ. Then, it can be verified that • If α1 = β1, α2 6= β2 then Pr[bi = −1] = 1/2. • If α1 6= β1, α2 = β2 then Pr[bi = −1] = 1/2. • If α1 6= β1, α2 6= β2 then Pr[bi = −1] = 1.