Seminar on Sublinear Time Algorithms

Proof Idea For any two specific inputs there exists an algorithm that is adapted to these inputs. For example, if we take the two inputs x = (1, 2, ..., n) as a “yes” example, and y = (1, 1, 2, 2, ..., n, n, n + 1, n + 2, ..., n(1 − )) as a “no” example, then an algorithm that checks the second coordinate can distinguish between them. Therefore, we look on distributions over inputs. We need two distributions, one over “yes” instances and the other over “no” instances. Then, we require that the algorithm accepts every “yes” instance and rejects every “no” instance with high probability. We assume that the algorithm is random, meaning it looks on random coordinates of the input. Given √ n samples we get ( √ n 2 ) ≈ n correlated pairs. It should find one out of n pairs that are “interesting” (witnesses for collisions). The probability that a random pair is one of these witnesses is only n n = n . By looking at less than n pairs we might not catch any of them. We will assume that the algorithm makes less than Θ( √ n ) queries and show that statistically, it is likelt to get the same answers to its queries on inputs from D and D, so it cannot distinguish between them with high probability. Hence, we need different distributions D,D (the ones above are not “hard enough”).