Coupling any number of balls in the infinite-bin model

On the Average Maximal Number of Balls in a Bin Resulting from Throwing r Balls into n Bins T times

We use the holonomic ansatz to estimate the asymptotic behavior, in $T$, of the average maximal number of balls in a bin that is obtained when one throws uniformly at random (without replacement) $r$ balls into $n$ bins, $T$ times. Our approach works, in principle, for any fixed $n$ and $r$. We were able to do the cases $(n,r)$ = $(2,1),(3,1),(4,1), (4,2)$, but things get too complicated for larger values of $n$ and $r$. We are pledging a \$150 donation to the OEIS for an explicit expression, (in terms of $n$, $r$, and $π$) for the constant $C_{n,r}$ such that that average equals $\frac{n}{r}\…

On the Combinatorics of Placing Balls into Ordered Bins

In this paper, we use techniques of enumerative combinatorics to study the following problem: we count the number of ways to split $n$ balls into nonempty, ordered bins so that the most crowded bin has exactly $k$ balls. We find closed forms for three of the different cases that can arise: $k > \frac{n}{2}$, $k = \frac{n}{2}$, and when there exists $j < k$ such that $n = 2k + j$. As an immediate result of our proofs, we find a closed form for the number of positive integer solutions to $x_1 + x_2 + \dots + x_{\ell} = n$ with the attained maximum of $\{x_1, x_2, \dots, x_{\ell}\}$ being equal t…

Even More Infinite Ball Packings from Lorentzian Root Systems

Boyd (1974) proposed a class of infinite ball packings that are generated by inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms of root systems in Lorentz space. In particular, he showed that the space-like weight vectors correspond to a ball packing if and only if the associated Coxeter graph is of "level $2$." In Maxwell's work, the simple roots form a basis of the representations space of the Coxeter group. In several recent studies, the more general based root system is considered, where the simple roots are only required to be positively independent. In this paper,…

Tight Bounds for Repeated Balls-into-Bins

We study the repeated balls-into-bins process introduced by Becchetti, Clementi, Natale, Pasquale and Posta (2019). This process starts with $m$ balls arbitrarily distributed across $n$ bins. At each round $t=1,2,\ldots$, one ball is selected from each non-empty bin, and then placed it into a bin chosen independently and uniformly at random. We prove the following results: $\quad \bullet$ For any $n \leq m \leq \mathrm{poly}(n)$, we prove a lower bound of $Ω(m/n \cdot \log n)$ on the maximum load. For the special case $m=n$, this matches the upper bound of $O(\log n)$, as shown in [BCNPP19]. I…

Unbalanced Allocations

We consider the unbalanced allocation of $m$ balls into $n$ bins by a randomized algorithm using the "power of two choices". For each ball, we select a set of bins at random, then place the ball in the fullest bin within the set. Applications of this generic algorithm range from cost minimization to condensed matter physics. In this paper, we analyze the distribution of the bin loads produced by this algorithm, considering, for example, largest and smallest loads, loads of subsets of the bins, and the likelihood of bins having equal loads.

On perfect colorings of infinite multipath graphs

A coloring of vertices of a given graph is called perfect if the color structure of each ball of radius $1$ in the graph depends only on the color of the ball center. Let $n$ be a positive integer. We consider a lexicographic product of the infinite path graph and a graph $G$ that can be either the complete or empty graph on $n$ vertices. We give a complete description of perfect colorings with an arbitrary number of colors of such graph products.

均衡分配：缓存与装箱、孪生与减薄

Balanced Allocations: Caching and Packing, Twinning and Thinning

本文提出一个通用分析框架，统一处理缓存、装箱、孪生和减薄四种略微偏好欠载箱的均衡分配过程，证明在任意m≥n的重负载情况下最大负载与平均负载的差距均为O(log n)。

面向均衡分配的均值偏置过程

Mean-Biased Processes for Balanced Allocations

提出均值偏置的均衡分配过程，通过概率或权重偏置欠载箱子，证明其最大与最小负载差距以高概率为对数级，并利用势函数与均值分位数稳定化现象进行分析。