TY - GEN
T1 - Smallest k-enclosing rectangle revisited
AU - Chan, Timothy M.
AU - Har-Peled, Sariel
N1 - Publisher Copyright:
© Timothy M. Chan and Sariel Har-Peled.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - 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(n5/2)-time algorithm by Kaplan et al. [23]. 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(nk) time. We also present a near linear time (1 + ε)-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.
AB - 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(n5/2)-time algorithm by Kaplan et al. [23]. 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(nk) time. We also present a near linear time (1 + ε)-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.
KW - Approximation algorithms
KW - Conditional lower bounds
KW - Geometric optimization
KW - Outliers
UR - http://www.scopus.com/inward/record.url?scp=85068066014&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85068066014&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2019.23
DO - 10.4230/LIPIcs.SoCG.2019.23
M3 - Conference contribution
AN - SCOPUS:85068066014
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 35th International Symposium on Computational Geometry, SoCG 2019
A2 - Barequet, Gill
A2 - Wang, Yusu
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 35th International Symposium on Computational Geometry, SoCG 2019
Y2 - 18 June 2019 through 21 June 2019
ER -