## The (1 + β)-Choice Process and Weighted Balls-into-Bins

Citations: | 3 - 0 self |

### BibTeX

@MISC{Peres_the(1,

author = {Yuval Peres and Kunal Talwar and Udi Wieder},

title = {The (1 + β)-Choice Process and Weighted Balls-into-Bins},

year = {}

}

### OpenURL

### Abstract

Suppose m balls are sequentially thrown into n bins where each ball goes into a random bin. It is well-known that the gap between the load of the most loaded bin m log n and the average is Θ( n), for large m. If each ball goes to the lesser loaded of two random bins, this gap dramatically reduces to Θ(log log n) independent of m. Consider now the following “(1 + β)-choice ” process for some parameter β ∈ (0, 1): each ball goes to a random bin with probability (1−β) and the lesser loaded of two random bins with probability β. How does the gap for such a process behave? Suppose that the weight of each ball was drawn from a geometric distribution. How is the gap (now defined in terms of weight) affected? In this work, we develop general techniques for analyzing such balls-into-bins processes. Specifically, we show that for the (1 + β)-choice process above, the gap is Θ(log n/β), irrespective of m. Moreover the gap stays at Θ(log n/β) in the weighted case for a large class of weight distributions. No non-trivial explicit bounds were previously known in the weighted case, even for the 2-choice paradigm. 1

### Citations

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(Show Context)
Citation Context ...6]) that if m = n the maxlog n imum load is Θ( log log n ) with high probability, and if m >> n log n the maximum ) load is with high proba+ Θ . In a seminal paper Azar et bility m n (√ m log n n al. =-=[1]-=- show that when d ≥ 2 and m = n the maximum log log n load is log d + O(1) w.h.p. (the case d = 2 was implicitly shown by Karp et al. in [3]). There are many variants and applications of this result, ... |

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Citation Context ...expected lookup cost is only (1 + β). The question of course is how much imbalance is introduced due to this modified procedure. It is easy to verify that in this + 1−β n . Mitzenmacher in his thesis =-=[4]-=- suggested studying the (1 + β)-process in the context of queueing theory. He observes that this process may also serve to model the case where all balls perform the best of 2 strategy, but some fract... |

98 | The power of two random choices: A survey of the techniques and results
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Citation Context ...= n the maximum log log n load is log d + O(1) w.h.p. (the case d = 2 was implicitly shown by Karp et al. in [3]). There are many variants and applications of this result, see for instance the survey =-=[5]-=- and the references therein. Berenbrink et al. [2] generalize the bound for arbitrarily large m showing the load to be m n + O(1) with probability 1 − 1/poly(n). Thus, surprisingly, the additive gap b... |

76 | balls into bins” - a simple and tight analysis
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(Show Context)
Citation Context ...on of setting d = 1, also known as the single choice scheme, is well understood and relatively easy to analyze, as the location of each ball is an independent random variable. It is well known ( e.g. =-=[6]-=-) that if m = n the maxlog n imum load is Θ( log log n ) with high probability, and if m >> n log n the maximum ) load is with high proba+ Θ . In a seminal paper Azar et bility m n (√ m log n n al. [1... |

57 | Balanced allocations: The heavily loaded case
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(Show Context)
Citation Context ....p. (the case d = 2 was implicitly shown by Karp et al. in [3]). There are many variants and applications of this result, see for instance the survey [5] and the references therein. Berenbrink et al. =-=[2]-=- generalize the bound for arbitrarily large m showing the load to be m n + O(1) with probability 1 − 1/poly(n). Thus, surprisingly, the additive gap between the heaviest bin and the average is indepen... |

14 |
Friedhelm Meyer auf der Heide. Efficient PRAM simulation on a distributed memory machine. Algorithmica
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(Show Context)
Citation Context ... Θ . In a seminal paper Azar et bility m n (√ m log n n al. [1] show that when d ≥ 2 and m = n the maximum log log n load is log d + O(1) w.h.p. (the case d = 2 was implicitly shown by Karp et al. in =-=[3]-=-). There are many variants and applications of this result, see for instance the survey [5] and the references therein. Berenbrink et al. [2] generalize the bound for arbitrarily large m showing the l... |

8 | Ballanced allocations with heterogeneous bins
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(Show Context)
Citation Context ...istributions overvectors x and y, we say that x is majorized by y, written as x ≼ y if for every Schur-convex function γ it holds that E[γ(x)] ≤ E[γ(y)]. The following theorem is implicit in [1] and =-=[8]-=-, it establishes a way of comparing different processes in the case of uniform weights. Theorem 2.3. Let p and q be probability vectors associated with processes x(·) and y(·) respectively, and assume... |

6 | Balanced allocations: the weighted case
- TALWAR, WIEDER
(Show Context)
Citation Context ...server is the total size of files assigned to it. We model this case by letting each ball receive a weight by independently case pi = β(2i−1) n2 sampling from a distribution D. Talwar and Wieder show =-=[7]-=- that in the two choice scheme, if the weight distribution has a finite variance and is ’smooth’ in some mild sense, then the gap is independent of the number of balls thrown. However, no non-trivial ... |