多项选择和（1 + Beta）过程的多维均衡分配
Multidimensional Balanced Allocation for Multiple Choice & (1 + Beta) Processes
摘要：
将球分配到箱子中是负载均衡问题的一个很好的研究抽象。文献中有许多关于m个球被扔到n个箱子中的顺序（一维）分配情况的结果。本文研究了非加权球和加权球以及多维和标量模型的对称多重选择过程，并给出了多维球和多维箱的（1+beta）选择过程的间隙界的结果。我们表明，对称d选择过程和m=O（n），上界的差距是O（lnln（n））w. h. p.这个上限的差距是在D=f的因素的下限。这是第一个这样的紧结果.对于一般的情况，m>>n，期望间隙的上界为O（lnln（n））.对于变量f和非均匀分布的填充维数，我们得到期望间隙的上界为O（log（n））.进一步，对于多个圆形平行球和多个平行箱，我们证明了当m=O（n）时，间隙也有O（loglog（n））的界，当m>>n时，同样的界也适用于期望间隙.我们的分析在序贯标量情形下也有很强的意义。对于加权球和加权箱以及一般情况m>>n，我们证明了期望间隙的上界是O（log（n）），这改进了n^c的最佳先验界。此外，我们表明，对于（1 + beta）选择过程和m=O（n）上界（假设f个填充维度在D个总维度上均匀分布）在间隙上是O（log（n）/beta），其在下限的D=f因子内。对于非均匀分布的固定f和二项分布的随机f，预期差距仍为O（log（n）/beta）与投掷的球的总数无关。这是第一个这样的紧密结果（1 +beta）范式与多维球和箱。
英文摘要：
Allocation of balls into bins is a well studied abstraction for load balancing problems.The literature hosts numerous results for sequential(single dimensional) allocation case when m balls are thrown into n bins. In this paper we study the symmetric multiple choice process for both unweighted and weighted balls as well as for both multidimensional and scalar models.Additionally,we present the results on bounds on gap for (1+beta) choice process with multidimensional balls and bins. We show that for the symmetric d choice process and with m=O(n), the upper bound on the gap is O(lnln(n)) w.h.p.This upper bound on the gap is within D=f factor of the lower bound. This is the first such tight result.For the general case of m>>n the expected gap is bounded by O(lnln(n)).For variable f and non-uniform distribution of the populated dimensions,we obtain the upper bound on the expected gap as O(log(n)). Further,for the multiple round parallel balls and bins,we show that the gap is also bounded by O(loglog(n)) for m=O(n).The same bound holds for the expected gap when m>>n. Our analysis also has strong implications in the sequential scalar case.For the weighted balls and bins and general case m>>n,we show that the upper bound on the expected gap is O(log(n)) which improves upon the best prior bound of n^c.Moreover,we show that for the (1 + beta) choice process and m=O(n) the upper bound(assuming uniform distribution of f populated dimensions over D total dimensions) on the gap is O(log(n)/beta),which is within D=f factor of the lower bound.For fixed f with non-uniform distribution and for random f with Binomial distribution the expected gap remains O(log(n)/beta) independent of the total number of balls thrown. This is the first such tight result for (1 +beta) paradigm with multidimensional balls and bins.
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