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# Paper of the Week: The Power of Two Random Choices: A Survey of Techniques and Results David Albert

This is part of our “Paper of the Week” series. For more info, check out our introductory blog post.

This week’s paper is The Power of Two Random Choices: A Survey of Techniques and Results by Michael Mitzenmacher, Andréa W. Richa and Ramesh Sitaraman. It was published in 2001 as part of the book Handbook of Randomized Computing.

This week’s paper was submitted by Hacker School alum Dan Luu, who shared the following:

I like this paper because it’s a really simple idea that’s quite powerful and generally applicable. It’s so simple I can even describe the key intuition in my summary. If you randomly throw `n` balls into `n` bins, the maximum number of balls in a single bin is approximately `O(log n)`, with high probability. But if you take two random choices and place the ball in the bin with fewer balls, the maximum drops to `O(log log n)`. The balls into bins model is a natural fit for scheduling / load balancing, hashing, and a number of other common problems. As a result, two random choices often turns out to be really effective despite its simplicity.

It turns out this is also applicable to a wide variety of problems that don’t obviously map to balls / bins, like circuit routing and random graphs (although applying it to things that aren’t “obviously” balls/bins problems isn’t always simple).

This week’s paper doesn’t have an abstract, so here’s the introduction:

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 log n with high probability. 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 log n / log d + Θ(1) with high probability.

The important implication of this result is that even a small amount of choice can lead to drastically different 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 the past several years, there has been a great deal of research investigating this phenomenon. The picture that has emerged from this research is that the power of two choices is not simply an artifact of the simple balls-and-bins model, but a general and robust phenomenon applicable to a wide variety of situations. Indeed, this two-choice paradigm continues to be applied and refined, and new results appear frequently.