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Long-term balanced allocation via thinning

Ohad N. Feldheim, Ori Gurel-Gurevich, Jiange Li

TL;DR

It is shown that when m and n are sufficiently large, a typical maximum load of (log n) can be achieved with high probability, asymptotically the same as the optimal maximum load that could be achieved at time m.

Abstract

We study the long-term behavior of the two-thinning variant of the classical balls-and-bins model. In this model, an overseer is provided with uniform random allocation of $m$ balls into $n$ bins in an on-line fashion. For each ball, the overseer could reject its allocation and place the ball into a new bin drawn independently at random. The purpose of the overseer is to reduce the maximum load of the bins, which is defined as the difference between the maximum number of balls in a single bin and $m/n$, i.e., the average number of balls among all bins. We provide tight estimates for three quantities: the lowest maximum load that could be achieved at time $m$, the lowest maximum load that could be achieved uniformly over the entire time interval $[m]:=\{1, 2, \cdots, m\}$, and the lowest \emph{typical} maximum load that could be achieved over the interval $[m]$, where the typicality means that the maximum load holds for $1-o(1)$ portion of the times in $[m]$. We show that when $m$ and $n$ are sufficiently large, a typical maximum load of $(\log n)^{1/2+o(1)}$ can be achieved with high probability, asymptotically the same as the optimal maximum load that could be achieved at time $m$. However, for any strategy, the maximal load among all times in the interval $[m]$ is $Ω\big(\frac{\log n}{\log\log n}\big)$ with high probability. A strategy achieving this bound is provided. An explanation for this gap is provided by our optimal strategies as follows. To control the typical load, we restrain the maximum load for some time, during which we accumulate more and more bins with relatively high load. After a while, we have to employ for a short time a different strategy to reduce the number of relatively heavily loaded bins, at the expanse of temporarily inducing high load in a few bins.

Long-term balanced allocation via thinning

TL;DR

It is shown that when m and n are sufficiently large, a typical maximum load of (log n) can be achieved with high probability, asymptotically the same as the optimal maximum load that could be achieved at time m.

Abstract

We study the long-term behavior of the two-thinning variant of the classical balls-and-bins model. In this model, an overseer is provided with uniform random allocation of balls into bins in an on-line fashion. For each ball, the overseer could reject its allocation and place the ball into a new bin drawn independently at random. The purpose of the overseer is to reduce the maximum load of the bins, which is defined as the difference between the maximum number of balls in a single bin and , i.e., the average number of balls among all bins. We provide tight estimates for three quantities: the lowest maximum load that could be achieved at time , the lowest maximum load that could be achieved uniformly over the entire time interval , and the lowest \emph{typical} maximum load that could be achieved over the interval , where the typicality means that the maximum load holds for portion of the times in . We show that when and are sufficiently large, a typical maximum load of can be achieved with high probability, asymptotically the same as the optimal maximum load that could be achieved at time . However, for any strategy, the maximal load among all times in the interval is with high probability. A strategy achieving this bound is provided. An explanation for this gap is provided by our optimal strategies as follows. To control the typical load, we restrain the maximum load for some time, during which we accumulate more and more bins with relatively high load. After a while, we have to employ for a short time a different strategy to reduce the number of relatively heavily loaded bins, at the expanse of temporarily inducing high load in a few bins.

Paper Structure

This paper contains 30 sections, 36 theorems, 320 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Discussion
  4. Upper bound strategies and lower bound techniques
  5. Outline
  6. Strategies for taming the maximum loads
  7. Two-thinning strategy
  8. A realizability criterion
  9. The basic strategies
  10. Combinations of the basic strategies
  11. Optimal strategies
  12. Preliminaries
  13. Poisson approximation
  14. Poisson tail estimate
  15. Concentration bounds for the drift strategy
  16. ...and 15 more sections

Key Result

Theorem 1

For all $m,n\in \mathbb{N}$, there exists an explicit two-thinning strategy $f:=f_{m,n}$ such that, with high probability, Moreover, in the first two cases the maximum loads are optimal up to some multiplicative constants, while in the third case we have a lower bound of $\Omega(\sqrt{\log n})$ for all two-thinning strategies.

Figures (1)

  • Figure 1: Above: an algorithmic description of the $Q$-multi-scale strategy. Below: the first three scales of this strategy. The first scale is the $L$-relative threshold strategy. The second scale consists of $N_1$ iterations, the $j$-th of which incorporates the strategy of the first scale followed by the multi-stage $(\log^{\alpha'_1}n,Q+Q^{1, j}+\ell_1, \ell_1)$-threshold strategy. The third scale consists of $N_2$ iterations, each of which consists of the second scale strategy with its $Q$ set to be $Q+Q^{2, j}$, followed by the multi-stage $(\log^{\alpha'_2}n, Q+Q^{2, j}+\ell_2, \ell_2)$-threshold strategy.

Theorems & Definitions (68)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1: MU05, Theorem 5.10
  • Lemma 3.2: FGG18, Lemma 2.2
  • ...and 58 more