Balanced allocation: Memory performance tradeoffs

TL;DR

This paper investigates the memory-performance tradeoffs in balanced allocation schemes, showing how limited memory affects the maximum load in bin allocation problems and establishing tight bounds in the communication complexity model.

Contribution

It provides tight bounds on the maximum bin load achievable with limited memory in the balanced allocation problem, extending understanding of memory constraints in load balancing.

Findings

Limited memory increases maximum load proportionally to  ext{log} n/ ext{log} ext{log} n

Heaviest bin contains  ext{delta} ext{log} n/ ext{log} ext{log} n with bits of memory

Bounds are tight in the communication complexity model.

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.