In this paper, we study the following “balls in buckets” problem. Suppose there is a sequence $B_1,B_2,\dots ,B_n$ of buckets having integer sizes $s_1,s_2,\dots ,s_n$, respectively. For a given target fraction α, $0<\alpha <1$, our goal is to sequentially place balls in buckets until at least $\lceil \alpha n\rceil$ buckets are full, so as to minimize the number of balls used, which we shall denote by $OP{T}_{\alpha }(I)$ for a given instance I.

If we knew the size of each bucket, we could obtain an optimal assignment, simply by filling the buckets in order of increasing size until the desired number had been filled. Here we consider the case where, although we know n and α, we do not know the specific bucket sizes $s_i$, and when we place a ball in bucket $B_j$, we only learn whether or not the bucket $B_j$ is now full.

We study what can be done under four variants of incomplete information:

1.

We know nothing at all about the bucket sizes;

2.

we know the maximum bucket size;

3.

we know the sizes $s_1\le s_2\le \cdots \le s_m$ that occur in the instance; and

4.

we know the profile of the sizes: the size list as above, and, for each size, $s_i$, the number $k_i$ of buckets that have that size,

The game above showcases the rich variety of interesting combinatorial and algorithmic questions that this setup gives rise to, and in addition has applications in an area of cryptography known as secure multi-party computation, where taking over (“corrupting”) a party by an adversary has a cost, and where a hidden diversity—corresponding to lack of information on the amount of computational resources the adversary should invest to corrupt a participant—translates into robustness and efficiency benefits.

在本文中，我们研究以下“桶中的球”问题。假设有一个序列${乙}_{\mathrm{1个}}，{乙}_2，\dots ，{乙}_{ñ}$ 整数大小的桶数 $s_{\mathrm{1个}}，s_2，\dots ，s_{ñ}$， 分别。对于给定的目标分数α，$0<\alpha <\mathrm{1个}$，我们的目标是依次将球放入桶中，直到至少 $\lceil \alpha ñ\rceil$ 桶已满，以尽量减少使用的球数，我们将用 $ØP{Ť}_{\alpha }（\mathrm{一世}）$对于给定的实例我。

如果我们知道每个存储桶的大小，则只需按存储桶的大小递增顺序填充，直到填充了所需的数量，就可以获得最佳分配。在这里，我们考虑以下情况：尽管我们知道n和α，但我们不知道具体的铲斗尺寸$s_{\mathrm{一世}}$，以及当我们在桶中放一个球时 ${乙}_{Ĵ}$，我们只会了解水桶是否 ${乙}_{Ĵ}$ 现在已经满了。

我们研究在不完整信息的四种变体下可以做什么：

1。

我们对铲斗尺寸一无所知。

2。

我们知道最大的铲斗尺寸；

3。

我们知道大小 $s_{\mathrm{1个}}\le s_2\le \cdots \le s_{米}$发生在实例中；和

4。

我们知道尺寸的概况：上面的尺寸列表，对于每种尺寸，$s_{\mathrm{一世}}$， 号码 ${ķ}_{\mathrm{一世}}$ 这么大的水桶

上面的游戏展示了此设置引起的各种有趣的组合和算法问题，此外，在加密领域（称为安全多方计算）中也有应用，在该领域中，交易方接管（“破坏”）一方对手要付出代价，而隐藏的多样性（对应于缺乏对手应该投资的计算资源来破坏参与者的信息）会转化为健壮性和效率优势。