Given a set of n points in the plane, and a parameter $$k$$ , we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing $$k$$ points. We present the first near quadratic time algorithm for this problem, improving over the previous near- $$O(n^{5/2})$$ -time algorithm by Kaplan et al. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52. Leibniz-Zent Inform, Wadern, 2017). We provide an almost matching conditional lower bound, under the assumption that $$(\min ,+)$$ -convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for both perimeter and area) that can make the time bound sensitive to $$k$$ , giving near $$O(nk)$$ time. We also present a near linear time $$(1+\varepsilon )$$ -approximation algorithm to the minimum area of the optimal rectangle containing $$k$$ points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.

中文翻译：

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重新审视最小的 k 封闭矩形

给定平面中的一组 n 个点和一个参数 $$k$$ ，我们考虑计算包围 $$k$$ 点的最小（周长或面积）轴对齐矩形的问题。我们针对这个问题提出了第一个近二次时间算法，改进了之前 Kaplan 等人的近 $O(n^{5/2})$$ -time 算法。（第 25 届欧洲算法研讨会。Leibniz Int Proc Inform，第 87 卷，#52。Leibniz-Zent Inform，Wadern，2017 年）。我们提供了一个几乎匹配的条件下界，假设 $$(\min ,+)$$ -convolution 无法在真正的次二次时间内解决。此外，我们提出了一个新的减少（对于周长和面积），它可以使时间限制对 $$k$$ 敏感，给出接近 $$O(nk)$$ 的时间。我们还提出了接近线性时间 $$(1+\varepsilon )$$ - 近似算法到包含 $$k$$ 点的最佳矩形的最小面积。此外，我们研究了相关问题，包括问题的 3 边、任意定向、加权和子集和版本。