Choice-memory tradeoff in allocations

Noga Alon, Ori Gurel-Gurevich, Eyal Lubetzky

Research output: Contribution to journalArticle

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 > 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.

Original languageEnglish (US)
Pages (from-to)1470-1511
Number of pages42
JournalAnnals of Applied Probability
Volume20
Issue number4
DOIs
StatePublished - Aug 2010

Fingerprint

Trade-offs
Ball
Collision
Random Allocation
Upper and Lower Bounds
Paradigm
If and only if

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

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

In: Annals of Applied Probability, Vol. 20, No. 4, 08.2010, p. 1470-1511.

Research output: Contribution to journalArticle

Alon, Noga ; Gurel-Gurevich, Ori ; Lubetzky, Eyal. / Choice-memory tradeoff in allocations. In: Annals of Applied Probability. 2010 ; Vol. 20, No. 4. pp. 1470-1511.
@article{ff051063104f414e81af04ff48e7fc94,
title = "Choice-memory tradeoff in allocations",
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 > 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.",
keywords = "Balanced allocations, Balls and bins paradigm, Lower bounds on memory, Online perfect matching, Space/performance tradeoffs",
author = "Noga Alon and Ori Gurel-Gurevich and Eyal Lubetzky",
year = "2010",
month = "8",
doi = "10.1214/09-AAP656",
language = "English (US)",
volume = "20",
pages = "1470--1511",
journal = "Annals of Applied Probability",
issn = "1050-5164",
publisher = "Institute of Mathematical Statistics",
number = "4",

}

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

AN - SCOPUS:77955140314

VL - 20

SP - 1470

EP - 1511

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 4

ER -