LIPIcs.SoCG.2019.23.pdf
- Filesize: 0.67 MB
- 15 pages
Document
Smallest k-Enclosing Rectangle Revisited
Authors
-
Part of:
Volume:
35th International Symposium on Computational Geometry (SoCG 2019)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2019-06-11
Cite AsGet BibTex
Timothy M. Chan and Sariel Har-Peled. Smallest k-Enclosing Rectangle Revisited. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.23
https://doi.org/10.4230/LIPIcs.SoCG.2019.23
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.
Subject Classification
ACM Subject Classification
- Theory of computation → Computational geometry
Keywords
- Geometric optimization
- outliers
- approximation algorithms
- conditional lower bounds
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
- Alok Aggarwal, Hiroshi Imai, Naoki Katoh, and Subhash Suri. Finding k points with minimum diameter and related problems. J. Algorithms, 12(1):38-56, 1991. URL: http://dx.doi.org/10.1016/0196-6774(91)90022-Q.
- Rossen Atanassov, Prosenjit Bose, Mathieu Couture, Anil Maheshwari, Pat Morin, Michel Paquette, Michiel H. M. Smid, and Stefanie Wuhrer. Algorithms for optimal outlier removal. J. Discrete Algorithms, 7(2):239-248, 2009. URL: http://dx.doi.org/10.1016/j.jda.2008.12.002.
-
Arturs Backurs, Nishanth Dikkala, and Christos Tzamos. Tight hardness results for maximum weight rectangles. In Proc. 43rd Int. Colloq. Automata Lang. Prog. (ICALP), pages 81:1-81:13, 2016.
- Luis Barba, Stephane Durocher, Robert Fraser, Ferran Hurtado, Saeed Mehrabi, Debajyoti Mondal, Jason Morrison, Matthew Skala, and Mohammad Abdul Wahid. On k-enclosing objects in a coloured point set. In Proc. 25th Canad. Conf. Comput. Geom.\CNFCCCG, 2013. URL: http://cccg.ca/proceedings/2013/papers/paper_35.pdf.
- Jérémy Barbay, Timothy M. Chan, Gonzalo Navarro, and Pablo Pérez-Lantero. Maximum-weight planar boxes in O(n²) time (and better). Inform. Process. Lett., 114(8):437-445, 2014. URL: http://dx.doi.org/10.1016/j.ipl.2014.03.007.
- Marshall W. Bern. Approximate closest-point queries in high dimensions. Inf. Process. Lett., 45(2):95-99, 1993. URL: http://dx.doi.org/10.1016/0020-0190(93)90222-U.
- David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Pătraşcu, and Perouz Taslakian. Necklaces, convolutions, and X+Y. Algorithmica, 69(2):294-314, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9734-3.
- Timothy M. Chan. Approximate nearest neighbor queries revisited. Discrete & Computational Geometry, 20(3):359-373, 1998. URL: http://dx.doi.org/10.1007/PL00009390.
- Timothy M. Chan. Geometric applications of a randomized optimization technique. Discrete Comput. Geom., 22(4):547-567, 1999. URL: http://dx.doi.org/10.1007/PL00009478.
- Timothy M. Chan and Sariel Har-Peled. Smallest k-enclosing rectangle revisited. CoRR, abs/1903.06785, 2019. URL: http://arxiv.org/abs/1903.06785.
- Timothy M. Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In Proc. 47th Annu. ACM Sympos. Theory Comput. (STOC), pages 31-40, 2015. URL: http://dx.doi.org/10.1145/2746539.2746568.
- Timothy M. Chan and Mihai Pătraşcu. Counting inversions, offline orthogonal range counting, and related problems. In Proc. 21st ACM-SIAM Sympos. Discrete Algs. (SODA), pages 161-173, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.15.
- Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing Razborov-Smolensky. In Proc. 27th ACM-SIAM Sympos. Discrete Algs. (SODA), pages 1246-1255, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
- Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk. On problems equivalent to (min, +)-convolution. In Proc. 44th Int. Colloq. Automata Lang. Prog. (ICALP), volume 80 of LIPIcs, pages 22:1-22:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.22.
- Sandip Das, Partha P. Goswami, and Subhas C. Nandy. Smallest k-point enclosing rectangle and square of arbitrary orientation. Inform. Process. Lett., 94(6):259-266, 2005. URL: http://dx.doi.org/10.1016/j.ipl.2005.02.013.
- Amitava Datta, Hans-Peter Lenhof, Christian Schwarz, and Michiel H. M. Smid. Static and dynamic algorithms for k-point clustering problems. J. Algorithms, 19(3):474-503, 1995. URL: http://dx.doi.org/10.1006/jagm.1995.1048.
- Mark de Berg, Sergio Cabello, Otfried Cheong, David Eppstein, and Christian Knauer. Covering many points with a small-area box. CoRR, abs/1612.02149, 2016. URL: http://arxiv.org/abs/1612.02149.
- David Eppstein and Jeff Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete Comput. Geom., 11:321-350, 1994. URL: http://jeffe.cs.illinois.edu/pubs/small.html.
- Harold N. Gabow and Robert Endre Tarjan. A linear-time algorithm for a special case of disjoint set union. J. Comput. Sys. Sci., 30(2):209-221, 1985. URL: http://dx.doi.org/10.1016/0022-0000(85)90014-5.
- Sariel Har-Peled and Soham Mazumdar. Fast algorithms for computing the smallest k-enclosing disc. Algorithmica, 41(3):147-157, 2005. URL: https://sarielhp.org/p/03/min_disk/.
- Sariel Har-Peled and Micha Sharir. Relative (p,ε)-approximations in geometry. Discrete Comput. Geom., 45(3):462-496, 2011. URL: http://dx.doi.org/10.1007/s00454-010-9248-1.
- Allan Grønlund Jørgensen and Kasper Green Larsen. Range selection and median: Tight cell probe lower bounds and adaptive data structures. In Proc. 22nd ACM-SIAM Sympos. Discrete Algs. (SODA), pages 805-813, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.63.
- Haim Kaplan, Sasanka Roy, and Micha Sharir. Finding axis-parallel rectangles of fixed perimeter or area containing the largest number of points. In Proc. 26th Annu. Euro. Sympos. Alg. (ESA), volume 87 of LIPIcs, pages 52:1-52:13, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2017.52.
-
Jiří Matoušek. Reporting points in halfspaces. Comput. Geom. Theory Appl., 2:169-186, 1992.
- Mihai Pătraşcu. Towards polynomial lower bounds for dynamic problems. In Proc. 42nd Annu. ACM Sympos. Theory Comput. (STOC), pages 603-610, 2010. URL: http://dx.doi.org/10.1145/1806689.1806772.
- Michael Segal and Klara Kedem. Enclosing k points in the smallest axis parallel rectangle. Inform. Process. Lett., 65(2):95-99, 1998. URL: http://dx.doi.org/10.1016/S0020-0190(97)00212-3.
- Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. J. Assoc. Comput. Mach., 22(2):215-225, 1975. URL: http://dx.doi.org/10.1145/321879.321884.
- Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Proc. 46th Annu. ACM Sympos. Theory Comput. (STOC), pages 664-673, 2014. URL: http://dx.doi.org/10.1145/2591796.2591811.