13 : The Probabilistic Method IV CS 598 shp - Randomized Algorithms

Once I sat on the steps by a gate of David's Tower, I placed my two heavy baskets at my side. A group of tourists was standing around their guide and I became their target marker. " You see that man with the baskets? Just right of his head there's an arch from the Roman period. Just right of his head. " " But he's moving, he's moving! " I said to myself: redemption will come only if their guide tells them, " You see that arch from the Roman period? It's not important: but next to it, left and down a bit, there sits a man who's bought fruit and vegetables for his family. " — Yehuda Amichai, Tourists 1 The Method of Conditional Probabilities In previous lecture, we encountered the following problem: Problem 1.1 (Set Balancing) Given a binary matrix A of size n × n, find a vector τ ∈ {−1, +1} n , such that Aτ ∞ is minimized. Using random assignment and the Chernoff inequality, we showed that there exists τ , such that Aτ ∞ ≤ 4 √ n ln n. Can we derandomize this algorithm? Namely, can we come up with an efficient deterministic algorithm that has low discrepancy? To derandomize our algorithm, construct a computation tree of depth n, where in the ith level we expose the ith coordinate of τ. This tree T has depth n. The root represents all possible random choices, while a node at depth i, represents all computations when the first i bits are fixed. For a node v ∈ T , let P (v) be the probability that a random computation starting from v succeeds. Let v l and v r be the two children of v. Clearly, P (v) = (P (v l) + P (v r))/2. In particular, max(P (v l), P (v r)) ≥ P (v). Thus, if we could could compute P (·) quickly (and deterministically), then we could derandomize the algorithm.