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August 2012 Balanced allocation: Memory performance tradeoffs
Itai Benjamini, Yury Makarychev
Ann. Appl. Probab. 22(4): 1642-1649 (August 2012). DOI: 10.1214/11-AAP804


Suppose we sequentially put $n$ balls into $n$ bins. If we put each ball into a random bin then the heaviest bin will contain ${\sim}\log n/\log\log n$ balls with high probability. However, Azar, Broder, Karlin and Upfal [SIAM J. Comput. 29 (1999) 180–200] showed that if each time we choose two bins at random and put the ball in the least loaded bin among the two, then the heaviest bin will contain only ${\sim}\log\log n$ balls with high probability. How much memory do we need to implement this scheme? We need roughly $\log\log\log n$ bits per bin, and $n\log\log\log n$ bits in total.

Let us assume now that we have limited amount of memory. For each ball, we are given two random bins and we have to put the ball into one of them. Our goal is to minimize the load of the heaviest bin. We prove that if we have $n^{1-\delta}$ bits then the heaviest bin will contain at least $\Omega(\delta\log n/\log\log n)$ balls with high probability. The bound is tight in the communication complexity model.


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Itai Benjamini. Yury Makarychev. "Balanced allocation: Memory performance tradeoffs." Ann. Appl. Probab. 22 (4) 1642 - 1649, August 2012.


Published: August 2012
First available in Project Euclid: 10 August 2012

zbMATH: 1246.68163
MathSciNet: MR2985172
Digital Object Identifier: 10.1214/11-AAP804

Primary: 68Q87
Secondary: 60C05

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.22 • No. 4 • August 2012
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