快速排序，最大桶和有限独立性的Min-Wise哈希
Quicksort, Largest Bucket, and Min-Wise Hashing with Limited Independence
摘要：
随机算法和数据结构通常是在假设可以访问完美的随机性源的情况下进行分析的。用来衡量哈希函数或随机数生成器的“随机性”的最基本度量是其独立性：如果每个变量都是均匀的，并且每个大小$k$子集都是独立的，则随机变量序列被称为$k$独立的。在本文中，我们考虑三个经典的算法在有限的独立性。我们提供了新的界限随机快速排序，最小明智的哈希和最大桶大小有限的独立性。我们的结果可以总结如下。- 随机快速排序当使用与$5$无关的散列函数计算枢轴元素时，Karloff和Raghavan，J.ACM'93展示了$O ( n \log n)$对于特殊版本的快速排序的预期最坏情况运行时间。我们改进了这一点，表明相同的运行时间是实现只有$4$独立。- 最小值哈希对于集合$A$，考虑特定元素被映射到最小散列值的概率。已知$5$独立性意味着最优概率$O (1 /n)$。Broder等人，STOC'98表明$2$独立性意味着它是$O(1 / \sqrt{|A|})$。我们展示了一个匹配的下界，以及新的紧界$3$和$4$独立的哈希函数。- 最大的桶我们考虑使用独立于$k$的哈希函数将$n$球分配到$n$桶的情况，并分析最大桶大小。Alon et.等人，STOC'97表明存在$2$独立的散列函数，其暗示大小为$Ω( n^{1/2})$的桶。我们推广的界限，提供一个$k$独立的家庭的功能，意味着大小$Ω( n^{1/k})$。
英文摘要：
Randomized algorithms and data structures are often analyzed under the assumption of access to a perfect source of randomness. The most fundamental metric used to measure how "random" a hash function or a random number generator is, is its independence: a sequence of random variables is said to be $k$-independent if every variable is uniform and every size $k$ subset is independent. In this paper we consider three classic algorithms under limited independence. We provide new bounds for randomized quicksort, min-wise hashing and largest bucket size under limited independence. Our results can be summarized as follows. -Randomized quicksort. When pivot elements are computed using a $5$-independent hash function, Karloff and Raghavan, J.ACM'93 showed $O ( n \log n)$ expected worst-case running time for a special version of quicksort. We improve upon this, showing that the same running time is achieved with only $4$-independence. -Min-wise hashing. For a set $A$, consider the probability of a particular element being mapped to the smallest hash value. It is known that $5$-independence implies the optimal probability $O (1 /n)$. Broder et al., STOC'98 showed that $2$-independence implies it is $O(1 / \sqrt{|A|})$. We show a matching lower bound as well as new tight bounds for $3$- and $4$-independent hash functions. -Largest bucket. We consider the case where $n$ balls are distributed to $n$ buckets using a $k$-independent hash function and analyze the largest bucket size. Alon et. al, STOC'97 showed that there exists a $2$-independent hash function implying a bucket of size $Ω( n^{1/2})$. We generalize the bound, providing a $k$-independent family of functions that imply size $Ω( n^{1/k})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
