2026-07-23T21:51:32Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:10427 2024-03-06T10:45:57Z ddc:004 open_access

Smallest k-Enclosing Rectangle Revisited Chan, Timothy M. Har-Peled, Sariel Geometric optimization outliers approximation algorithms conditional lower bounds 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Timothy M. Chan and Sariel Har-Peled 2019 Is Part Of LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2019.23 urn:nbn:de:0030-drops-104276 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.23 eng https://creativecommons.org/licenses/by/3.0/legalcode