## Balanced allocations with heterogenous bins (2007)

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Venue: | In Proc. of Symp. on Parallelism in Algorithms and Architectures (SPAA |

Citations: | 8 - 3 self |

### BibTeX

@INPROCEEDINGS{Wieder07balancedallocations,

author = {Udi Wieder},

title = {Balanced allocations with heterogenous bins},

booktitle = {In Proc. of Symp. on Parallelism in Algorithms and Architectures (SPAA},

year = {2007}

}

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### Abstract

Balls-into-bins processes are a useful and common abstraction for many load-balancing related problems. A well known paradigm for load balancing in distributed or parallel servers is the ”multiple choice paradigm ” where an item (ball) is put in the less loaded out of d uniformly chosen servers (bins). In many applications however the uniformity of the sampling probability is not guaranteed. If the system is heterogenous or dynamic it may be the case that some bins are sampled with a higher probability than others. We investigate the power of the multiple choice paradigm in the setting where bins are not sampled from the uniform distribution. Byers et al [5] showed that a logarithmic imbalance in the sampling probability could be tolerated, as long as the number of balls is linear in the number of bins. We show that if the number of balls is much larger than the number of bins, this ceases to be the case. Given a probability over bins, we prove tight upper and lower bounds for the number of choices needed in the 1-out-of-d scheme in order to maintain a balanced allocations when the number of items is arbitrarily high. 1

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Citation Context ...e a biased sampling distribution. Uneven allocation of the key space In many applications the set of servers is dynamic and may change over time. Such applications naturally include P2P systems (c.f. =-=[16]-=-,[14], [11]), but also include centrally owned data centers which occasionally add servers to the system in order to increase capacity [10]. A typical scheme for using hash functions in dynamic settin... |

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Citation Context ...he key space it “owns”. The partition of the key space between the servers is done dynamically and changes as servers enter and leave the system. This technique is sometimes called consistent hashing =-=[6]-=- and is used in many storage systems. There are various techniques to divide the key space between the servers as evenly as possible. The simplest thing to do is to let each server choose its i.d. uni... |

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Citation Context ...sampling distribution. Uneven allocation of the key space In many applications the set of servers is dynamic and may change over time. Such applications naturally include P2P systems (c.f. [16],[14], =-=[11]-=-), but also include centrally owned data centers which occasionally add servers to the system in order to increase capacity [10]. A typical scheme for using hash functions in dynamic settings such as ... |

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Citation Context ...ote the algorithm where each ball is inserted into the less loaded among d ≥ 2 bins, independently sampled from U where U denotes the uniform distribution over the bins. In a seminal paper Azar et al =-=[2]-=- proved that when m = n balls are inserted by GREEDY(U, d) the log log n heaviest bin has load of + Θ(1) with high probability, the case d = 2 was implicitly proved log d by Karp et al in [8]. Berenbr... |

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Citation Context ...iased sampling distribution. Uneven allocation of the key space In many applications the set of servers is dynamic and may change over time. Such applications naturally include P2P systems (c.f. [16],=-=[14]-=-, [11]), but also include centrally owned data centers which occasionally add servers to the system in order to increase capacity [10]. A typical scheme for using hash functions in dynamic settings su... |

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Citation Context ... as some servers are sampled with probability log n/n and some with probability 1/n log n [11]. A more balanced allocation of the key space is obtained by using subtler algorithms c.f. [11] [14] [12] =-=[7]-=-, none of them however guarantee a perfectly uniform sampling distribution. In centrally owned data bases there are better schemes [9] but even they only guarantee a gap of factor 2 between the sampli... |

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Citation Context ...n the maximum load and the average load is independent of the number of balls thrown. + log log n The multiple choice algorithm has numerous variations and applications in many settings, see e.g. [1],=-=[18]-=-,[17],[10]. See [13] for a survey. A typical application is the following. Assume data items (balls) are to be inserted into servers (bins) in a distributed manner. Upon an arrival of a data item d ha... |

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Citation Context ... is well known that the load of the heaviest bin is at most ln n (1 + o(1)) balls with high probability. If m ≥ n ln n then the load in the heaviest bin is at most m n + �ln ln n m log n n , see e.g. =-=[15]-=-. A substantial decrease in load is achieved by the use of the multiple choice paradigm: Let GREEDY(U, d) denote the algorithm where each ball is inserted into the less loaded among d ≥ 2 bins, indepe... |

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Citation Context ...at when m = n balls are inserted by GREEDY(U, d) the log log n heaviest bin has load of + Θ(1) with high probability, the case d = 2 was implicitly proved log d by Karp et al in [8]. Berenbrink et al =-=[3]-=- generalized this result and proved the following. 1sTheorem 1.1 ([3]). Let γ denote a suitable constant. If m balls are allocated into n bins using GREEDY(U, d) with d ≥ 2 then the number of bins wit... |

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Citation Context ...tween the maximum load and the average load is independent of the number of balls thrown. + log log n The multiple choice algorithm has numerous variations and applications in many settings, see e.g. =-=[1]-=-,[18],[17],[10]. See [13] for a survey. A typical application is the following. Assume data items (balls) are to be inserted into servers (bins) in a distributed manner. Upon an arrival of a data item... |

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Citation Context ...space is obtained by using subtler algorithms c.f. [11] [14] [12] [7], none of them however guarantee a perfectly uniform sampling distribution. In centrally owned data bases there are better schemes =-=[9]-=- but even they only guarantee a gap of factor 2 between the sampling probabilities. It is easy to see that such a sampling imbalance is an inherent property of the approach. There is no way to guarant... |

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Citation Context ...ility as some servers are sampled with probability log n/n and some with probability 1/n log n [11]. A more balanced allocation of the key space is obtained by using subtler algorithms c.f. [11] [14] =-=[12]-=- [7], none of them however guarantee a perfectly uniform sampling distribution. In centrally owned data bases there are better schemes [9] but even they only guarantee a gap of factor 2 between the sa... |

32 | Geometric generalizations of the power of two choices
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Citation Context ...e bins are sampled with a higher probability than others. We investigate the power of the multiple choice paradigm in the setting where bins are not sampled from the uniform distribution. Byers et al =-=[5]-=- showed that a logarithmic imbalance in the sampling probability could be tolerated, as long as the number of balls is linear in the number of bins. We show that if the number of balls is much larger ... |

14 |
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Citation Context ...ar et al [2] proved that when m = n balls are inserted by GREEDY(U, d) the log log n heaviest bin has load of + Θ(1) with high probability, the case d = 2 was implicitly proved log d by Karp et al in =-=[8]-=-. Berenbrink et al [3] generalized this result and proved the following. 1sTheorem 1.1 ([3]). Let γ denote a suitable constant. If m balls are allocated into n bins using GREEDY(U, d) with d ≥ 2 then ... |

4 |
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Citation Context ...nd the average load is independent of the number of balls thrown. + log log n The multiple choice algorithm has numerous variations and applications in many settings, see e.g. [1],[18],[17],[10]. See =-=[13]-=- for a survey. A typical application is the following. Assume data items (balls) are to be inserted into servers (bins) in a distributed manner. Upon an arrival of a data item d hash functions are emp... |

1 |
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(Show Context)
Citation Context ...mum load and the average load is independent of the number of balls thrown. + log log n The multiple choice algorithm has numerous variations and applications in many settings, see e.g. [1],[18],[17],=-=[10]-=-. See [13] for a survey. A typical application is the following. Assume data items (balls) are to be inserted into servers (bins) in a distributed manner. Upon an arrival of a data item d hash functio... |

1 |
Balanced allocaitons: The weighted case
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Citation Context ... maximum load and the average load is independent of the number of balls thrown. + log log n The multiple choice algorithm has numerous variations and applications in many settings, see e.g. [1],[18],=-=[17]-=-,[10]. See [13] for a survey. A typical application is the following. Assume data items (balls) are to be inserted into servers (bins) in a distributed manner. Upon an arrival of a data item d hash fu... |