文章基本信息

标题：Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points
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作者：Haim Kaplan ; Sasanka Roy ; Micha Sharir 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：87
页码：52:1-52:13
DOI：10.4230/LIPIcs.ESA.2017.52
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n^{5/2} log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk^{3/2} log k) time, and (ii) an approximation algorithm that finds, in O(n+(n/(k epsilon^5))*log^{5/2}(n/k)log((1/epsilon) log(n/k))) time, a rectangle of the given perimeter or diagonal that contains at least (1-epsilon)k points of P, where k is the optimum value. We then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art.
关键词：Computational geometry; geometric optimization; rectangles; perimeter; area