## BALANCED ALLOCATIONS: THE HEAVILY LOADED CASE (2006)

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Citations: | 59 - 8 self |

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@MISC{Berenbrink06balancedallocations:,

author = {Petra Berenbrink and Artur Czumaj and Angelika Steger and Berthold Vöcking},

title = { BALANCED ALLOCATIONS: THE HEAVILY LOADED CASE },

year = {2006}

}

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### Abstract

We investigate balls-into-bins processes allocating m balls into n bins based on the multiple-choice paradigm. In the classical single-choice variant each ball is placed into a bin selected uniformly at random. In a multiple-choice process each ball can be placed into one out of d ≥ 2 randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when m ≈ n. In this paper we present the first tight analysis in the heavily loaded case, that is, when m ≫ n rather than m ≈ n. The best previously known results for the multiple-choice processes in the heavily loaded case were obtained using majorization by the single-choice process. This yields an upper bound of the maximum load of bins of m/n + O ( √ m ln n/n) with high probability. We show, however, that the multiple-choice processes are fundamentally different from the single-choice variant in that they have “short memory. ” The great consequence of this property is that the deviation of the multiple-choice processes from the optimal allocation (that is, the allocation in which each bin has either ⌊m/n ⌋ or ⌈m/n ⌉ balls) does not increase with the number of balls as in the case of the single-choice process. In particular, we investigate the allocation obtained by two different multiple-choice allocation schemes,

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Citation Context ...ques used in [1] ("layered induction") and [11] ("witness trees") inherently assume a load of 2m/n already in their base case. Altemative proof techniques using differential equations as suggested in =-=[7; 8; 10; 12]-=- fail for the heavily loaded case, too, because the concentration results obtained by Kurtz's theorem hold only for a limited number of balls. Therefore, the analysis of the heavily loaded case requir... |

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Citation Context ...ced into a bin chosen independently and uniformly at random (i.u.r.). For the case of n bins and m > n in n balls it is well known that there exists a bin receiving m/n + O(V~ lnn/ n) balls (e.g. see =-=[9]-=-). This result holds not only on expectation but with high probability 1. Let the max height above average denote the difference between the number of balls in the fullest bin and the average number o... |

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2 |
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Citation Context ...larly, conditioned on .A4o = Y it holds Yt = .A4t. Our main tool in the analysis of this Markov chain is a variant of the path coupling argument of Bubley and Dyer [4]. This variant was introduced in =-=[5]-=- and is described in the following lemma. LEMMA 1. (Neighboring-Coupling Lemma) Let ~J~ = (Yt)ter~ be a discrete-time Markov chain with a state space fL Let f~* C_ fL Let F be any subset off~* × f~* (... |

1 |
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Citation Context ...nto an arbitrary one among them.) Azar et al. show that the max height above average of Left[d] is only (ln inn)/ in d + O(m/n), w.h.p. • Algorithm Left[d] has been introduced and analyzed by V6cking =-=[11]-=-. This algorithm partitions the set of bins into d groups of equal size. These groups are ordered from left to right. For each ball, we choose the ith location for each ball from the ith group i.u.r. ... |

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