## Balanced Allocations: The Weighted Case (2008)

Citations: | 7 - 2 self |

### BibTeX

@MISC{Talwar08balancedallocations:,

author = {Kunal Talwar and Udi Wieder},

title = {Balanced Allocations: The Weighted Case},

year = {2008}

}

### OpenURL

### Abstract

We investigate balls-and-bins processes where m weighted balls are placed into n bins using the “power of two choices ” paradigm, whereby a ball is inserted into the less loaded of two randomly chosen bins. The case where each of the m balls has unit weight had been studied extensively. In a seminal paper Azar et al. [2] showed that when m = n the most loaded bin has Θ(log log n) balls with high probability. Surprisingly, the gap in load between the heaviest bin and the average bin does not increase with m and was shown by Berenbrink et al. [4] to be Θ(log log n) with high probability for arbitrarily large m. We generalize this result to the weighted case where balls have weights drawn from an arbitrary weight distribution. We show that as long as the weight distribution has finite second moment and satisfies a mild technical condition, the gap between the weight of the heaviest bin and the weight of the average bin is independent of the number balls thrown. This is especially striking when considering heavy tailed distributions such as Power-Law and Log-Normal distributions. In these cases, as more balls are thrown, heavier and heavier weights are encountered. Nevertheless with high probability, the imbalance in the load distribution does not increase. Furthermore, if the fourth moment of the weight distribution is finite, the expected value of the gap is shown to be independent of the number of balls. 1 1

### Citations

282 | A brief history of generative models for power law and lognormal distributions
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Citation Context ... paradigm, whereby a ball is inserted into the less loaded of two randomly chosen bins. The case where each of the m balls has unit weight had been studied extensively. In a seminal paper Azar et al. =-=[2]-=- showed that when m = n the most loaded bin has Θ(log log n) balls with high probability. Surprisingly, the gap in load between the heaviest bin and the average bin does not increase with m and was sh... |

163 |
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- McDiarmid
- 1998
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- 1997
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- 2003
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84 | How Asymmetry Helps Load Balancing
- Vöcking
- 1999
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59 | Balanced Allocations: The Heavily Loaded Case
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Citation Context ...he most loaded bin has Θ(log log n) balls with high probability. Surprisingly, the gap in load between the heaviest bin and the average bin does not increase with m and was shown by Berenbrink et al. =-=[4]-=- to be Θ(log log n) with high probability for arbitrarily large m. We generalize this result to the weighted case where balls have weights drawn from an arbitrary weight distribution. We show that as ... |

57 | L.: Parallel Randomized Load Balancing
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- 1995
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43 |
Load balancing and density dependent jump Markov processes
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- 1996
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Citation Context ...t is therefore essential to analyze the weighted case separately. 1.1 Related Work As mentioned previously, the unweighted case had been the object of extensive study in many contexts (c.f. [14],[1], =-=[11]-=- [12]) and to a large extent is well understood. Vöcking [13] proved the surprising result that a similar process with an asymmetric tie breaking rule called GOLEFT obtains a better bound in the m = n... |

34 | Geometric generalizations of the power of two choices
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- 2004
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Citation Context ...or instance a distributed storage system in which there are n servers and whenever a data item is to be inserted into the system it is assigned to the less loaded among two random servers 1 (c.f. [7] =-=[8]-=-). It is known that many types of data items such as files in a PC file system and multimedia files have sizes distributed by a heavy tailed distribution [10]. While splitting the files into fixed-siz... |

9 | Ballanced allocations with heterogeneous bins
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- 2007
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Citation Context ... tailed. It is therefore essential to analyze the weighted case separately. 1.1 Related Work As mentioned previously, the unweighted case had been the object of extensive study in many contexts (c.f. =-=[14]-=-,[1], [11] [12]) and to a large extent is well understood. Vöcking [13] proved the surprising result that a similar process with an asymmetric tie breaking rule called GOLEFT obtains a better bound in... |

4 |
Handbook of Randomized Computing, chapter The power of two random choices: A survey of the techniques and results
- Mitzenmacher, Richa, et al.
- 2000
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Citation Context ...t the additive gap between the maximum load and the average load does not depend on the number of balls thrown! The two choice paradigm had been investigated in a variety of models and scenarios, see =-=[12]-=- for a survey. Our goal in this paper is to prove similar bounds for the case where balls are weighted. In our model there is a weight distribution W. In each round i a weight wi is sampled from W. A ... |

2 |
Friedhelm Meyer auf der Heide, and Klaus Schröder. Allocating weighted jobs in parallel
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- 1999
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Citation Context .... The weighted case had been investigated in a somewhat different model, where the balls arrive in parallel and are allowed to communicate with one another prior to making the allocation decision [1],=-=[3]-=-. In this model, the weights are arbitrary, but the additive gap may be large. 1.2 Our Contributions Our main result shows that under mild assumptions on the weight distribution, the differences from ... |