We study the problem of extracting a small subset of representative items from a large data stream. Following the convention in many data mining and machine learning applications such as data summarization, recommender systems, and social network analysis, the problem is formulated as maximizing a monotone submodular function subject to a cardinality constraint -- i.e., the size of the selected subset is restricted to be smaller than or equal to an input integer $k$. In this paper, we consider the problem with additional \emph{fairness constraints}, which takes into account the group membership of data items and limits the number of items selected from each group to a given number. We propose efficient algorithms for this fairness-aware variant of the streaming submodular maximization problem. In particular, we first provide a $(\frac{1}{2}-\varepsilon)$-approximation algorithm that requires $O(\frac{1}{\varepsilon} \cdot \log \frac{k}{\varepsilon})$ passes over the stream for any constant $ \varepsilon>0 $. In addition, we design a single-pass streaming algorithm that has the same $(\frac{1}{2}-\varepsilon)$ approximation ratio when unlimited buffer size and post-processing time is permitted.

翻译：我们研究从大数据流中提取代表项目的一小部分的问题。 在很多数据挖掘和机器学习应用中, 如数据总和、推荐系统和社会网络分析等, 公约之后, 问题被表述为在基本限制条件下最大限度地实现单调子模块功能, 即所选子的大小限制小于或等于输入整数美元。 在本文中, 我们考虑额外\ emph{ 公平性限制} 的问题, 这个问题考虑到数据项目的组组成, 以及将每个组选择的项目数量限制在给定数字之内。 我们为流出子模块最大最大化问题的公平度变体提出有效的算法。 特别是, 我们首先提供$( frac{ 1\\\\\\\\\\\\\\\\\ varepsil) $- approcol 算法, 需要$( worcoc{1\\\\\\\\\\\\\\\\\\\ vareplassl%\ laus) 后, 在任何恒流流 流流流流中, =\ pal sal sal sal =xluslationslation= =xxxxxxxxxxxxl =xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx