TY - JOUR
T1 - Smallest k-Enclosing Rectangle Revisited
AU - Chan, Timothy M.
AU - Har-Peled, Sariel
N1 - Funding Information:
Work on this paper by T.M.C. was partially supported by an NSF AF award CCF-1814026. Work by S.H. was partially supported by an NSF AF awards CCF-1907400 and CCF-1421231.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2021/9
Y1 - 2021/9
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. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52. Leibniz-Zent Inform, Wadern, 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 both perimeter and 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. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52. Leibniz-Zent Inform, Wadern, 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 both perimeter and 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
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U2 - 10.1007/s00454-020-00239-3
DO - 10.1007/s00454-020-00239-3
M3 - Article
AN - SCOPUS:85090146199
SN - 0179-5376
VL - 66
SP - 769
EP - 791
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
IS - 2
ER -