In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}. One of the extensions, introduced in \cite{Armaselu-FWCG}, is called \textit{MBSR with outliers} or MBSR-O, and is a more general version of the MBSR problem in which the optimal rectangle is allowed to contain up to $k$ outliers, where $k$ is given as part of the input. For MBSR-O, we improve the previous known running time bounds of $O(k^7 m \log m + n)$ to $O(k^3 m + m \log m + n)$. The other extension is called \textit{MBSR among circles} or MBSR-C and asks for the largest axis-aligned rectangle separating red points from blue unit circles. For MBSR-C, we provide an algorithm that runs in $O(m^2 + n)$ time.

中文翻译：

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最大双色分离矩形问题的扩展

在本文中，我们研究了\cite{Armaselu-CCCG, Armaselu-arXiv} 中引入的最大双色分离矩形（MBSR）问题的两个扩展。\cite{Armaselu-FWCG} 中引入的扩展之一称为 \textit{MBSR with outliers} 或 MBSR-O，是 MBSR 问题的更一般版本，其中允许最优矩形包含最多$k$ 离群值，其中 $k$ 作为输入的一部分给出。对于 MBSR-O，我们将先前已知的运行时间界限 $O(k^7 m \log m + n)$ 改进为 $O(k^3 m + m \log m + n)$。另一个扩展名为 \textit{MBSR between circles} 或 MBSR-C，它要求最大的轴对齐矩形将红色点与蓝色单位圆分开。对于 MBSR-C，我们提供了一个在 $O(m^2 + n)$ 时间内运行的算法。