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DOI: 10.4230/LIPIcs.SoCG.2019.23
URN: urn:nbn:de:0030-drops-104276
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10427/
Chan, Timothy M. ; Har-Peled, Sariel
Smallest k-Enclosing Rectangle Revisited
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Abstract
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.
BibTeX - Entry
@InProceedings{chan_et_al:LIPIcs:2019:10427,
author = {Timothy M. Chan and Sariel Har-Peled},
title = {{Smallest k-Enclosing Rectangle Revisited}},
booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
pages = {23:1--23:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-104-7},
ISSN = {1868-8969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10427},
URN = {urn:nbn:de:0030-drops-104276},
doi = {10.4230/LIPIcs.SoCG.2019.23},
annote = {Keywords: Geometric optimization, outliers, approximation algorithms, conditional lower bounds}
}
| Keywords: | Geometric optimization, outliers, approximation algorithms, conditional lower bounds | |
| Seminar: | 35th International Symposium on Computational Geometry (SoCG 2019) | |
| Issue date: | 2019 | |
| Date of publication: | 11.06.2019 |
