arXiv 2102.084762021-02-16cs.DS

数据流模型中的最大覆盖：参数化和广义化

Maximum Coverage in the Data Stream Model: Parameterized and Generalized

Andrew McGregor, David Tench, Hoa T. Vu

备注：: Conference version to appear at ICDT 2021

摘要：

我们提出了算法的最大覆盖和最大唯一覆盖问题的数据流模型。这两个问题的输入是大小为$n$和值为$k\in [m]$的论域的$m$子集。在Max-Cover中，问题是找到最多$k$集合的集合，使得至少一个集合覆盖的元素数量最大化。在Max-Unique-Cover中，问题是找到最多$k$集合的集合，使得恰好一个集合所覆盖的元素的数量最大化。我们的目标是设计单遍算法，使用的空间是次线性的输入大小。我们的主要算法结果是：如果集合的大小最多为$d$，存在使用$\tilde{O}(d^{d+1} k^d)$空间的单遍算法，可以精确地解决这两个问题。对于常数$d$，这是最佳的多对数因子。如果每个元素出现在最多的$r$集，我们提出了一个单程算法使用$\tilde{O}(k^2 r/ε^3)$空间，返回一个$1+ε$近似的情况下，最大覆盖。我们还提出了一个单遍算法，使用稍微多一点的内存，即，在$\tilde{O}(k^3 r/ε^{4})$空间中，该$1+ε$近似于Max-Unique-Cover。与上述结果相反，当$d$和$r$是任意的时，对于任一问题的任何恒定通道$1+ε$近似算法都需要$Ω(ε^{-2}m)$空间，但存在单通道$O(ε^{-2}mk)$空间算法。事实上，当$d$和$r$不受限制时，任何对最大覆盖和最大唯一覆盖分别具有比$e/(e-1)$和$e^{1-1/k}$更好的近似的恒定通算法都需要$Ω(m/k^2)$空间。途中，我们还获得了一个算法的参数化版本的流集覆盖问题。

英文摘要：

We present algorithms for the Max-Cover and Max-Unique-Cover problems in the data stream model. The input to both problems are $m$ subsets of a universe of size $n$ and a value $k\in [m]$. In Max-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by at least one set is maximized. In Max-Unique-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by exactly one set is maximized. Our goal is to design single-pass algorithms that use space that is sublinear in the input size. Our main algorithmic results are: If the sets have size at most $d$, there exist single-pass algorithms using $\tilde{O}(d^{d+1} k^d)$ space that solve both problems exactly. This is optimal up to polylogarithmic factors for constant $d$. If each element appears in at most $r$ sets, we present single pass algorithms using $\tilde{O}(k^2 r/ε^3)$ space that return a $1+ε$ approximation in the case of Max-Cover. We also present a single-pass algorithm using slightly more memory, i.e., $\tilde{O}(k^3 r/ε^{4})$ space, that $1+ε$ approximates Max-Unique-Cover. In contrast to the above results, when $d$ and $r$ are arbitrary, any constant pass $1+ε$ approximation algorithm for either problem requires $Ω(ε^{-2}m)$ space but a single pass $O(ε^{-2}mk)$ space algorithm exists. In fact any constant-pass algorithm with an approximation better than $e/(e-1)$ and $e^{1-1/k}$ for Max-Cover and Max-Unique-Cover respectively requires $Ω(m/k^2)$ space when $d$ and $r$ are unrestricted. En route, we also obtain an algorithm for a parameterized version of the streaming Set-Cover problem.

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。

摘要：

英文摘要：

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