Choice-memory tradeoff in allocations

TL;DR

This paper explores the tradeoff between the number of choices and memory in allocation problems, showing how limited memory impacts the ability to achieve balanced distributions in balls-and-bins models.

Contribution

It establishes a fundamental relationship between choices, memory, and load balancing, extending classical results to memory-constrained settings.

Findings

Achieving constant maximum load requires the product of choices and memory to be proportional to the number of balls.

Limited memory significantly hampers the ability to avoid collisions and balance loads.

Nonadaptive algorithms have tight bounds, showing limitations under memory constraints.

Abstract

In the classical balls-and-bins paradigm, where $n$ balls are placed independently and uniformly in $n$ bins, typically the number of bins with at least two balls in them is $\Theta(n)$ and the maximum number of balls in a bin is $\Theta(\frac{\log n}{\log \log n})$. It is well known that when each round offers $k$ independent uniform options for bins, it is possible to typically achieve a constant maximal load if and only if $k=\Omega(\log n)$. Moreover, it is possible w.h.p. to avoid any collisions between $n/2$ balls if $k>\log_2n$. In this work, we extend this into the setting where only $m$ bits of memory are available. We establish a tradeoff between the number of choices $k$ and the memory $m$, dictated by the quantity $km/n$. Roughly put, we show that for $km\gg n$ one can achieve a constant maximal load, while for $km\ll n$ no substantial improvement can be gained over the case $k=1$ (i.e., a random allocation). For any $k=\Omega(\log n)$ and $m=\Omega(\log^2n)$, one can achieve a constant load w.h.p. if $km=\Omega(n)$, yet the load is unbounded if $km=o(n)$. Similarly, if $km>Cn$ then $n/2$ balls can be allocated without any collisions w.h.p., whereas for $km<\epsilon n$ there are typically $\Omega(n)$ collisions. Furthermore, we show that the load is w.h.p. at least $\frac{\log(n/m)}{\log k+\log\log(n/m)}$. In particular, for $k\leq\operatorname {polylog}(n)$, if $m=n^{1-\delta}$ the optimal maximal load is $\Theta(\frac{\log n}{\log\log n})$ (the same as in the case $k=1$), while $m=2n$ suffices to ensure a constant load. Finally, we analyze nonadaptive allocation algorithms and give tight upper and lower bounds for their performance.