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

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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