Submodular functions have many real-world applications, such as document summarization, sensor placement, and image segmentation. For all these applications, the key building block is how to compute the maximum value of a submodular function efficiently. We consider both the online and offline versions of the problem: in each iteration, the data set changes incrementally or is not changed, and a user can issue a query to maximize the function on a given subset of the data. The user can be malicious, issuing queries based on previous query results to break the competitive ratio for the online algorithm. Today, the best-known algorithm for online submodular function maximization has a running time of $O(n k d^2)$ where $n$ is the total number of elements, $d$ is the feature dimension and $k$ is the number of elements to be selected. We propose a new method based on a novel search tree data structure. Our algorithm only takes $\widetilde{O}(nk + kd^2 + nd)$ time.

翻译：子模函数在诸多实际应用中具有重要价值，例如文档摘要、传感器布局和图像分割。对于这些应用，核心构建模块是如何高效计算子模函数的最大值。我们考虑该问题的在线与离线两种版本：在每次迭代中，数据集会增量变化或保持不变，用户可发起查询以在给定数据子集上最大化函数。用户可能具有恶意性，基于先前查询结果发起查询，从而破坏在线算法的竞争比。目前，在线子模函数最大化最著名的算法运行时间为$O(n k d^2)$，其中$n$为元素总数，$d$为特征维度，$k$为待选元素数量。我们提出了一种基于新型搜索树数据结构的方法，该算法仅需$\widetilde{O}(nk + kd^2 + nd)$时间。