We consider a variation of balls-into-bins which randomly allocates m balls into n bins. Following Godfrey's model (SODA, 2008), we assume that each ball t, $1⩽t⩽m$, comes with a hypergraph ${\mathcal{H}}^{(t)}=\{B_1,B_2,\dots ,B_{s_t}\}$, and each edge $B\in {\mathcal{H}}^{(t)}$ contains at least a logarithmic number of bins. Given $d⩾2$, our d-choice algorithm chooses an edge $B\in {\mathcal{H}}^{(t)}$, uniformly at random, and then chooses a set D of d random bins from the selected edge B. The ball is allocated to a least-loaded bin from D. We prove that if the hypergraphs ${\mathcal{H}}^{(1)},\dots ,{\mathcal{H}}^{(m)}$ satisfy a balancedness condition and have low pair visibility, then after allocating $m=\mathrm{\Theta }(n)$ balls, the maximum load of any bin is at most ${\log }_d\log n+\mathcal{O}(1)$, with high probability. Moreover, we establish a lower bound for the maximum load attained by the balanced allocation for a sequence of hypergraphs in terms of pair visibility.

我们考虑球入箱的一种变体，它将m 个球随机分配到n 个箱中。根据戈弗雷模型（SODA，2008），我们假设每个球t，$1⩽t⩽米$，带有超图${\mathcal{H}}^{（t）}=\{{乙}_1,{乙}_2,\dots \dots ,{乙}_{s_t}\}$，以及每条边$乙\epsilon {\mathcal{H}}^{（t）}$至少包含对数个 bin。给定$d⩾2$，我们的d选择算法选择一条边$乙\epsilon {\mathcal{H}}^{（t）}$，均匀随机，然后从所选边B中选择一组D，其中d个随机 bin 。球被分配到D中负载最少的容器中。我们证明如果超图${\mathcal{H}}^{（1）},\dots \dots ,{\mathcal{H}}^{（米）}$满足平衡性条件并且对可见性较低，然后分配后$米=\mathrm{\theta }（n）$球，任何箱子的最大负载最多为${\mathrm{日志}}_d\mathrm{日志}n+\mathcal{氧}（1）$，概率很高。此外，我们根据对可见性建立了一系列超图的平衡分配所获得的最大负载的下限。