文章基本信息

标题：Smallest k-Enclosing Rectangle Revisited
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作者：Timothy M. Chan ; Sariel Har-Peled
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2019
卷号：129
页码：1-15
DOI：10.4230/LIPIcs.SoCG.2019.23
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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. [Haim Kaplan et al., 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 either perimeter or area) that can make the time bound sensitive to k, giving near O(n k) time. We also present a near linear time (1+epsilon)-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.
关键词：Geometric optimization; outliers; approximation algorithms; conditional lower bounds