The Power of Two Random Choices: A Survey of Techniques and Results

To motivate this survey, we begin with a simple problem that demonstrates a powerful fundamental idea. Suppose that n balls are thrown into n bins, with each ball choosing a bin independently and uniformly at random. Then the maximum load, or the largest number of balls in any bin, is approximately log n= log logn with high probability.1 Now suppose instead that the balls are placed sequentially, and each ball is placed in the least loaded of d 2 bins chosen independently and uniformly at random. Azar, Broder, Karlin, and Upfal showed that in this case, the maximum load is log logn= log d + (1) with high probability [ABKU99]. The important implication of this result is that even a small amount of choice can lead to drastically di erent results in load balancing. Indeed, having just two random choices (i.e., d = 2) yields a large reduction in the maximum load over having one choice, while each additional choice beyond two decreases the maximum load by just a constant factor. Over