The Power of One-Retry in Reducing Discrepancy

Consider an on-line system in which a player is sequentially given independent and uniformly chosen points from [0, 1]. After looking at each point the player is allowed to either keep it or replace it with a fresh uniformly chosen point. This choice may depend on the location of all previously kept points. However, only one retry is allowed at every step. The goal of the player is to reduce the discrepancy of the empirical distribution of points, that is – to reduce the maximal deviation of the number of points in an interval from its expectation. Our main result is a retry strategy which guarantees discrepancy of O(log N/N) with high probability (compare with Θ(1/ √ N) without the power of one-retry). This answers a question of Benjamini. We also show that even if retry is allowed only with probability η at every step for arbitrary 0 < η, a similar result holds. Finally, we show that the best maximum load obtainable by a one-retry strategy in the classical N balls thrown into N bins setting is Θ( √