双色点集中最大矩形识别的变化
Variations of largest rectangle recognition amidst a bichromatic point set
摘要：
多色点集的经典可分性问题是计算几何的一个重要研究领域。本文研究了平面上双色点集P=P_r \cup P_b的不同可分性问题，其中$P_r$和$P_b$分别表示n个红点和m个蓝点的集合，目标是计算所需类型和最大尺寸的单色物体.我们提出了就地算法，用于计算（i）R^2中最大尺寸的任意定向单色矩形，（ii）R^3中最大尺寸的轴平行单色长方体。问题（i）和（ii）的算法的时间复杂度分别为O（m（m+n）（m \sqrt{n} +m \log m+n \log n））和O（m^3 \sqrt{n} +m^2n \log n）。作为一个先决条件，我们提出了一个就地构造的经典数据结构的k-d树，这是最初发明的J。1975年的宾利我们的$k$ -d树的就地变体对于R^k中的一组n个点支持使用O（1）额外工作空间的正交范围报告和计数查询，并且这些查询的时间复杂度与经典复杂度相同，即，时间复杂度为O（n^{1-1/k}+μ），时间复杂度为O（n^{1-1/k}），其中μ是报告查询的输出大小。这个数据结构的构造时间是O（n \log n）。与已有的原地k-d树的构造算法和查询算法相比，该构造算法和查询算法本质上都是非递归的，不需要O（\log n）大小的递归栈，我们相信这一结果是独立的.对于点集P=P_r \cup P_b中任意方向的最大权矩形的计算问题，我们也提出了一个算法。P_r）与负（相应地，这是一个O（m^2（n+m）\log（n+m））时间复杂度为O（n）额外空间的实值权重。
英文摘要：
Classical separability problem involving multi-color point sets is an important area of study in computational geometry. In this paper, we study different separability problems for bichromatic point set P=P_r\cup P_b on a plane, where $P_r$ and $P_b$ represent the set of n red points and m blue points respectively, and the objective is to compute a monochromatic object of the desired type and of maximum size. We propose in-place algorithms for computing (i) an arbitrarily oriented monochromatic rectangle of maximum size in R^2, (ii) an axis-parallel monochromatic cuboid of maximum size in R^3. The time complexities of the algorithms for problems (i) and (ii) are O(m(m+n)(m\sqrt{n}+m\log m+n \log n)) and O(m^3\sqrt{n}+m^2n\log n), respectively. As a prerequisite, we propose an in-place construction of the classic data structure the k-d tree, which was originally invented by J. L. Bentley in 1975. Our in-place variant of the $k$-d tree for a set of n points in R^k supports both orthogonal range reporting and counting query using O(1) extra workspace, and these query time complexities are the same as the classical complexities, i.e., O(n^{1-1/k}+μ) and O(n^{1-1/k}), respectively, where μis the output size of the reporting query. The construction time of this data structure is O(n\log n). Both the construction and query algorithms are non-recursive in nature that do not need O(\log n) size recursion stack compared to the previously known construction algorithm for in-place k-d tree and query in it. We believe that this result is of independent interest. We also propose an algorithm for the problem of computing an arbitrarily oriented rectangle of maximum weight among a point set P=P_r \cup P_b, where each point in P_b (resp. P_r) is associated with a negative (resp. positive) real-valued weight that runs in O(m^2(n+m)\log(n+m)) time using O(n) extra space.
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