Choice-Memory Tradeoff in Allocations

In the classical balls-and-bins setup, n balls are thrown independently and uniformly into n bins. The most loaded bin then has log n/log log n balls with high probability. A famous result of Aztar, Brooder, Karolin and Opal states that, when given k uniformly and independently selected bins to choose from for the location of each ball, one can achieve a maximal load of log_k log n, simply by putting each ball in the least loaded of the k bins. To implement this simple algorithm, one needs to keep track of the status of the entire array of n bins, which requires about n bits of memory. In this work, we find a tradeoff between the number of choices, k, and the number of memory bits available, m. This tradeoff has a sharp threshold governing the performance: If km≫≫n then one can achieve a constant maximal load.