### 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) |
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Pages (from-to) | 1470-1511 |

Number of pages | 42 |

Journal | Annals of Applied Probability |

Volume | 20 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2010 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

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

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## Cite this

*Annals of Applied Probability*,

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