MapReduce (MR) frameworks for maximizing monotone, submodular functions subject to a cardinality constraint (SMCC) have currently only been shown to work with linear-adaptive (non-parallelizable) algorithms, that require large number of distributions in order to utilize the available processors, thus resulting in severe restrictions on the cardinality constraint in addition to limited scalability. Low-adaptive algorithms do not currently satisfy the requirements of these distributed MR frameworks, thereby limiting their performance. We study the SMCC problem in a distributed setting and propose the first MR algorithms with sublinear adaptive complexity. Our algorithms, R-DASH, T-DASH and G-DASH provide $0.316-\varepsilon$, $3/8 -\varepsilon$, and $1 - 1/e -\varepsilon$ approximation ratios, respectively, with nearly optimal adaptive complexity and nearly linear time complexity. Additionally, we provide a framework to increase, under some mild assumptions, the maximum permissible cardinality constraint from $O( n / \ell^2)$ of prior MR algorithms to $O( n / \ell )$, where $n$ is the data size and $\ell$ is the number of machines; under a stronger condition on the objective function, we increase the maximum constraint value to $n$. Finally, we provide empirical evidence to demonstrate that our sublinear-adaptive, distributed algorithms provide orders of magnitude faster runtime compared to current state-of-the-art distributed algorithms.

翻译：最大单一调值、受基本调值制约的亚模量功能(SMCC)框架目前只显示与线性调整(不可分)算法合作,这些算法需要大量分配,以便利用现有的处理器,因此除了可缩放性有限外,还严重限制了基本限制。低调算法目前无法满足这些分布式MR框架的要求,从而限制了它们的业绩。我们在分布式设置中研究员工和管理当局协调会的问题,并提出第一个具有亚线性适应性复杂性的MR算法。我们的算法、R-DASH、T-DASH和G-DASH分别提供0.316-Narepsilon美元、3/8-\varepslon美元和1-1-e-varepslon$近似比,几乎是最佳的适应性复杂度和近线性时间。此外,我们提供了一个框架,可以根据一些温和假设,从美元(n/ell2)的可允许性基本调值算法算算算算算算出第一个MUR值的MR-ralal-altialal-alalalalalal-al-al-xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx