Randomized Algorithms, Carnegie Mellon: Spring 2011
Lecture #8: Oh, and one more thing
I forgot to mention something about the two choices paradigm: recall from HW #2 that if you throw balls into bins randomly and , the maximum load is about . In fact, you can show that this variance term is about right—with high probability, the highest loaded bin will indeed be about above the average.
On the other hand, if you throw balls into bins using two-choices, then you can show that the highest load is about with high probability. So not only do we gain in the low-load case (when ), we get more control over the variance in the high load case (when ): the additive gap between the average and the maximum loads is now independent of the number of balls! The proofs to show this require new ideas: check out the paper Balanced Allocations: the Heavily Loaded Case by Berenbrink, Czumaj, Steger and Vöcking for more details.
Here is a recent paper of Peres, Talwar and Weider that gives an analysis of the -choice process (where you invoke the two choices paradigm only on fraction of the balls). It also refers to more recent work in the area (weighted balls, weighted bins, etc), in case you’re interested.
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