## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 22, Number 4 (2012), 1642-1649.

### Balanced allocation: Memory performance tradeoffs

Itai Benjamini and Yury Makarychev

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 22, Number 4 (2012), 1642-1649.

**Dates**

First available in Project Euclid: 10 August 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1344614206

**Digital Object Identifier**

doi:10.1214/11-AAP804

**Mathematical Reviews number (MathSciNet)**

MR2985172

**Zentralblatt MATH identifier**

1246.68163

**Subjects**

Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]

Secondary: 60C05: Combinatorial probability

**Keywords**

Balls–and–bins process load balancing memory performance tradeoffs

#### Citation

Benjamini, Itai; Makarychev, Yury. Balanced allocation: Memory performance tradeoffs. Ann. Appl. Probab. 22 (2012), no. 4, 1642--1649. doi:10.1214/11-AAP804. https://projecteuclid.org/euclid.aoap/1344614206