Lecture 11 : Applications of Expanders Instructors :

In this lecture we discuss a central application of expanders in the context of derandomization, namely randomness-efficient confidence boosting, i.e., reducing the error probability of a randomized algorithm without increasing the need for random bits by too much. We present two approaches – a deterministic one and a randomized one. Deterministic confidence boosting requires no additional randomness but takes more time to achieve a certain level of confidence. Randomized confidence boosting takes less time but requires a small amount of additional randomness. Recall the setting of confidence boosting from Lecture 5. Given a randomized algorithm A(x, ρ) with error at most 12 − η, we want to construct a randomized algorithm A ′(x, ρ′) for the same problem but with error at most δ such that (i) the number of random bits r′ . = |ρ′| that A′ needs is not much larger than the r . = |ρ| random bits A needs, and (ii) the running time of A′ is not much larger than the running time of A. The new algorithm A′(x, ρ′) runs A(x, ρi) for t random bit sequences ρi, i ∈ [t], and combines the results in an appropriate way. If the underlying problem has a unique solution, the combining function is simply the plurality vote. In Lecture 5 we analyzed confidence boosting when the ρi are chosen (a) independently uniformly and (b) pairwise uniformly. In this lecture we do the same for two ways of obtaining the ρi as vertices of an explicit expander G of size N = 2r, equating the vertices of G with the bit strings in {0, 1}r.