Lecture 4 : Explicit constructions using zig-zag product and SL=L

We started out with a goal to reduce the probability of error in randomized computation. We saw that using independent runs of the randomized algorithm and then taking majority helps us amplify the success probability. However, this method uses a lot more randomness, often a costly resource, and we would like to amplify the success probability using almost the same amount of randomness as the original algorithm. In some sense, we wanted correlated samples which appeared close to random on a larger space. We tried placing an expander on the space of all possible random strings that the algorithm can use, and then choose a random vertex and selected its neighbours as our strings for subsequent runs. This gave us a way to use the same amount of randomness as in one run of the original algorithm, since the only randomness in thi procedure was in choosing one vertex. Once the vertex was chosen, the set of strings used for sequential runs was completely determined. The expander mixing lemma guaranteed us that the probability of a majority of neighbours being bad was low. Despite the randomness being the same, this method did not amplify the success probability as good as independent runs did. We turned our attention to random walks. Random walks involve correlated random steps and we can thus save the total randomness used. If we take a sufficiently long random walk on a graph, the distribution becomes close to uniform (converges). We proved that random walks on expanders converge quickly, and thus if we take one on an expander, it doesn’t get stuck in the set of (bad) strings which give the wrong answer and eventually heads out into the (good) set, in probability. Given a good expander graph, we showed that the probability of error could be reduced significantly using this method and thus the problem boiled down to the construction of a family of expanders, deterministically. It’s easy to get expanders of high (polynomial) degree by repeated squaring of small graphs, however, that means our random walks require more randomness at each step. We are thus looking for expander graphs of constant degree which can be found deterministically. Zig-zag product is one way to decrease the degree while keeping the expansion close to the original. Thus, we can interleave squaring and zig-zag product to get a constant degree expander, as we shall see.