 
Summary: Balanced Allocations
Yossi Azar \Lambda Andrei Z. Broder y Anna R. Karlin z Eli Upfal x
Abstract
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)) ln n= ln ln n 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 ln n= ln 2+O(1)
balls  exponentially less than before. Furthermore, we show that
a similar gap exists in the infinite 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 online load balancing.
1 Introduction
Suppose that we sequentially place n balls into n boxes by putting each ball
into a randomly chosen box. Properties of this random allocation process
have been extensively studied in the probability and statistics literature.
