### 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 θ(n) and the maximum number of balls in a bin is θ(logn/log log n). It is w 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 = ω(logn). Moreover, it is possible w.h.p. to avoid any collisions between n/2 balls if k > log_{2} n. 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 » n one can achieve a constant maximal load, while for km « no substantial improvement can be gained over the case k = 1 (i.e., a random allocation). For any k = ω(logn) and m = ω(log^{2} n), one can achieve a constant load w.h.p. if km = ω(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 < εn there are typically ω(n) collisions. Furthermore, we show that the load is w.h.p. at least log(n/m)/log k+log log((n/m) In particular, for k ≤ polylog(n), if m = n ^{1-δ} the optimal maximal load is θ(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.

Original language | English (US) |
---|---|

Pages (from-to) | 1470-1511 |

Number of pages | 42 |

Journal | Annals of Applied Probability |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2010 |

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### Keywords

- Balanced allocations
- Balls and bins paradigm
- Lower bounds on memory
- Online perfect matching
- Space/performance tradeoffs

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Statistics and Probability

### Cite this

*Annals of Applied Probability*,

*20*(4), 1470-1511. https://doi.org/10.1214/09-AAP656

**Choice-memory tradeoff in allocations.** / Alon, Noga; Gurel-Gurevich, Ori; Lubetzky, Eyal.

Research output: Contribution to journal › Article

*Annals of Applied Probability*, vol. 20, no. 4, pp. 1470-1511. https://doi.org/10.1214/09-AAP656

}

TY - JOUR

T1 - Choice-memory tradeoff in allocations

AU - Alon, Noga

AU - Gurel-Gurevich, Ori

AU - Lubetzky, Eyal

PY - 2010/8

Y1 - 2010/8

N2 - 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 θ(n) and the maximum number of balls in a bin is θ(logn/log log n). It is w 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 = ω(logn). Moreover, it is possible w.h.p. to avoid any collisions between n/2 balls if k > log2 n. 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 » n one can achieve a constant maximal load, while for km « no substantial improvement can be gained over the case k = 1 (i.e., a random allocation). For any k = ω(logn) and m = ω(log2 n), one can achieve a constant load w.h.p. if km = ω(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 < εn there are typically ω(n) collisions. Furthermore, we show that the load is w.h.p. at least log(n/m)/log k+log log((n/m) In particular, for k ≤ polylog(n), if m = n 1-δ the optimal maximal load is θ(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.

AB - 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 θ(n) and the maximum number of balls in a bin is θ(logn/log log n). It is w 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 = ω(logn). Moreover, it is possible w.h.p. to avoid any collisions between n/2 balls if k > log2 n. 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 » n one can achieve a constant maximal load, while for km « no substantial improvement can be gained over the case k = 1 (i.e., a random allocation). For any k = ω(logn) and m = ω(log2 n), one can achieve a constant load w.h.p. if km = ω(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 < εn there are typically ω(n) collisions. Furthermore, we show that the load is w.h.p. at least log(n/m)/log k+log log((n/m) In particular, for k ≤ polylog(n), if m = n 1-δ the optimal maximal load is θ(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.

KW - Balanced allocations

KW - Balls and bins paradigm

KW - Lower bounds on memory

KW - Online perfect matching

KW - Space/performance tradeoffs

UR - http://www.scopus.com/inward/record.url?scp=77955140314&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955140314&partnerID=8YFLogxK

U2 - 10.1214/09-AAP656

DO - 10.1214/09-AAP656

M3 - Article

VL - 20

SP - 1470

EP - 1511

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 4

ER -