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New techniques for computing order statistics in Euclidean space (extended abstract)

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Bernard ChazelleAuthors Info & Claims

SCG '85: Proceedings of the first annual symposium on Computational geometry

Pages 125 - 134

https://doi.org/10.1145/323233.323251

Published: 01 June 1985 Publication History

Abstract

Given a finite point-set S in E², how hard is it to compute the κ^th largest interdistance, or say, the κ^th largest slope or κ^th largest triangular area formed by points of S? We examine the complexity of a general class of problems built from these examples, and present a number of techniques for deriving nontrivial upper bounds. Surprisingly, these bounds often match or come very close to the complexity of the corresponding extremal problems (e.g. computing the largest or smallest interdistance, slope, etc.)

References

[1]

[BS] Bentley, J.L., Shamos, M.I. Divide-and-conquer in multidimensional space, Proc. 8th ACM Annu. Symp. Theory Comput. (1976), pp. 220-230.

Digital Library

[2]

[Bl] Blum, M., Floyd, R.W., Pratt, V.R., Rivest, R.L., Tarjan, R.E. Time bounds for selection, JCSS, 7 (4), pp. 448-461, 1973.

[3]

[Bo] Boyce, J.E., Dobkin, D.P., Drysdale, R.L., Guibas, L.J. Finding extremal polygons, Proc. 14th ACM Annu. Symp. Theory Comput. (1982), pp. 282- 289.

Digital Library

[4]

[CGL] Chazelle, B., Guibas, L.J., Lee, D.T. The power of geometric duality, Proc. 24th IEEE Annu. Symp. Found. Comput. Sci. (1983), pp. 217-225.

Digital Library

[5]

[CW] Chin, F., Wang, CA. Minimum vertex distance between separable convex polygons, Info. Proc. Lett., 18 (1984), pp. 41-45.

[6]

[C] Cole, R. Searching and storing similar lists, Tech. Rep. No. 88, New York Univ., Oct. 1983.

[7]

[DL] Dobkin, D.P., Lipton, R.J. Multidimensional searching problems, SIAM J. on Comput. 5 (2), pp. 181- 186, 1976.

[8]

[EOS] Edelsbrunner, H., O'Rourke, J., Seidel, R. Constructing arrangements of lines and hyperplanes with applications, Proc. 24th IEEE Annu. Symp. Found. Comput. Sci. (1983), pp. 83-91.

Digital Library

[9]

[FJ1] Frederickson, G.N., Johnson, D.B. The complexity of selection and ranking in X + Y and matrices with sorted columns, JCSS, 24 (1982), pp. 197-208.

[10]

[FJ2] Frederickson, G.N., Johnson, D.B. Finding kth paths and p-centers by generating and searching good data structures, J. of Alg., 4 (1983), pp. 61-80.

[11]

[FJ3] Frederickson, G.N., Johnson, D.B. Generalized selection and ranking: sorted matrices, SIAM J. on Comput., 13 (1), pp. 14-30, 1984.

[12]

[GM] Galil, Z., Megiddo, N. A fast selection algorithm and the problem of optimum distribution of effort, J. of ACM, 26, pp. 58-64, 1979.

Digital Library

[13]

[H] Henrici, P. Applied and computational complex analysis , Wiley & sons, New York, 1977.

[14]

[JM] Johnson, D.B., Mizoguchi, T. Selecting the k-th element in X + Y and X₁ + X₂ + .. + X_m, SIAM J. on Comput., 7, pp. 147-153, 1978.

[15]

[MT] McKenna, M., Toussaint G.T. Finding the minimum vertex distance between two disjoint convex polygons in linear time, Tech. Rep. SOCS-83-6, April 1983.

[16]

[Me] Meggido, N., Tamir, A., Zemel, E., Chandrasekaran, R. An O(n log² n) algorithm for the kth longest path in a tree with applications to location problems, SIAM J. on Comput., 10 (2), pp. 328-337, May 1981.

[17]

[Or] O'Rourke, J., Aggarwal, A., Maddila, S., Baldwin, M. An optimal algorithm for finding minimal enclosing triangles, Tech. Rep. JHU/EECS-84/08, Dept. of EE/CS. Johns Hopkins Univ. (1984).

[18]

[PH] Preparata, F.P., Hong, S.J. Convex hulls of finite sets of points in two and three dimensions, Comm. ACM, 20, 2 (1977), pp. 87-93.

Digital Library

[19]

[S] Shamos, M.I. Geometry and statistics: problems at the interface, Algorithms and complexity: new directions and recent results, ed. J.F. Traub, Academic Press, New York, pp. 251-280, 1976.

[20]

[T] Toussaint, G.T. An optimal algorithm for computing the minimum vertex distance between two crossing convex polygons, Proc. 21st Allerton Conf. on Comm. Control and Comput. (1983), pp. 457-458.

[21]

[Y] Yao, A.C. On constructing minimum spanning tree in k-dimensional space and related problems, SIAM J. on Comput. 11 (4), pp. 721-736, 1982.

Cited By

Haase CKiefer S(2016)The complexity of the Kth largest subset problem and related problemsInformation Processing Letters10.1016/j.ipl.2015.09.015116:2(111-115)Online publication date: 1-Feb-2016
https://dl.acm.org/doi/10.1016/j.ipl.2015.09.015
Glozman AKedem KShpitalnik G(2005)On some geometric selection and optimization problems via sorted matricesAlgorithms and Data Structures10.1007/3-540-60220-8_48(26-37)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-60220-8_48
Salowe J(2005)Selecting the Kth largest-area convex polygonAlgorithms and Data Structures10.1007/3-540-51542-9_22(243-250)Online publication date: 26-May-2005
https://doi.org/10.1007/3-540-51542-9_22
Show More Cited By

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New techniques for computing order statistics in Euclidean space (extended abstract)

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cover image ACM Conferences

SCG '85: Proceedings of the first annual symposium on Computational geometry

June 1985

322 pages

ISBN:0897911636

DOI:10.1145/323233

Chairman:
Joseph O'Rourke
Johns Hopkins Univ., Baltimore, MD

Copyright © 1985 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 01 June 1985

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SCG85: The 1st Annual ACM SIGACT/SIGGRAPH Symposium on Computational Geometry 1985

June 5 - 7, 1985

Maryland, Baltimore, USA

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Cited By

Haase CKiefer S(2016)The complexity of the Kth largest subset problem and related problemsInformation Processing Letters10.1016/j.ipl.2015.09.015116:2(111-115)Online publication date: 1-Feb-2016
https://dl.acm.org/doi/10.1016/j.ipl.2015.09.015
Glozman AKedem KShpitalnik G(2005)On some geometric selection and optimization problems via sorted matricesAlgorithms and Data Structures10.1007/3-540-60220-8_48(26-37)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-60220-8_48
Salowe J(2005)Selecting the Kth largest-area convex polygonAlgorithms and Data Structures10.1007/3-540-51542-9_22(243-250)Online publication date: 26-May-2005
https://doi.org/10.1007/3-540-51542-9_22
Cole RSalowe JSteiger WSzemerédi E(2005)Optimal slope selectionAutomata, Languages and Programming10.1007/3-540-19488-6_112(133-146)Online publication date: 31-May-2005
https://doi.org/10.1007/3-540-19488-6_112
Langerman SSteiger W(2003)The Complexity of Hyperplane Depth in the PlaneDiscrete & Computational Geometry10.1007/s00454-003-0011-x30:2(299-309)Online publication date: 1-Aug-2003
https://dl.acm.org/doi/10.1007/s00454-003-0011-x
Katoh NIwano KAvis D(1992)Finding k farthest pairs and k closest/farthest bichromatic pairs for points in the planeProceedings of the eighth annual symposium on Computational geometry10.1145/142675.142740(320-329)Online publication date: 1-Jul-1992
https://dl.acm.org/doi/10.1145/142675.142740
Dickerson MDrysdale RDrysdale R(1991)Enumerating k distances for n points in the planeProceedings of the seventh annual symposium on Computational geometry10.1145/109648.109674(234-238)Online publication date: 1-Jun-1991
https://dl.acm.org/doi/10.1145/109648.109674
Agarwal PAronov BSharir MSuri SSeidel R(1990)Selecting distances in the planeProceedings of the sixth annual symposium on Computational geometry10.1145/98524.98597(321-331)Online publication date: 1-May-1990
https://dl.acm.org/doi/10.1145/98524.98597
Paterson MYao FMehlhorn K(1989)Binary partitions with applications to hidden surface removal and solid modellingProceedings of the fifth annual symposium on Computational geometry10.1145/73833.73836(23-32)Online publication date: 5-Jun-1989
https://dl.acm.org/doi/10.1145/73833.73836
Chazelle B(1987)Some techniques for geometric searching with implicit set representationsActa Informatica10.1007/BF0026329524:5(565-582)Online publication date: 1-Sep-1987
https://dl.acm.org/doi/10.1007/BF00263295

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