Balanced allocations

Suppose that we sequentially place n balls into n boxes by putting each ball into a randomly chosen box. It is well known that when we are done, the fullest box has with high probability (1 + o(1)) lnn = ln lnn balls in it. Suppose instead, that for each ball we choose two boxes at random and place the ball into the one which is less full at the time of placement. We show that with high probability, the fullest box contains only ln lnn = ln 2+O(1) balls { exponentially less than before. Furthermore, we show that a similar gap exists in the innite process, where at each step one ball, chosen uniformly at random, is deleted, and one ball is added in the manner above. We discuss consequences of this and related theorems for dynamic resource allocation, hashing, and on-line load balancing.