Article

Sublinear geometric algorithms

Authors:

Bernard Chazelle,

Avner MagenAuthors Info & Claims

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

Pages 531 - 540

https://doi.org/10.1145/780542.780620

Published: 09 June 2003 Publication History

Abstract

We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in two-dimensional Delaunay triangulations and Voronoi diagrams, and ray shooting in convex polyhedra, all of which run in time O(√n), where n is the size of the input. We also provide sublinear solutions for the approximate evaluation of the volume of a convex polytope and the length of the shortest path between two points on the boundary.

References

[1]

Agarwal, P.K., Har-Peled, S., Karia, M. Computing approximate shortest paths on convex polytopes, Algorithmica 33 (2) 1999, 227--242.]]

Digital Library

[2]

Agarwal, P.K., Har-Peled, S., Sharir, M., Varadarajan, K. Approximating shortest paths on a convex polytope in three dimensions, J. ACM 44 (1997), 567--584.]]

Digital Library

[3]

Agarwal, P.K., Erickson, J. Geometric range searching and its relatives, in "Advances in Discrete and Computational Geometry," eds. Chazelle, B., Goodman, J.E., Pollack, R., Contemporary Mathematics 223, Amer. Math. Soc., 1999, pp. 1--56.]]

[4]

Barequet, G., Har-Peled, S. Efficiently approximating the minimum-volume bounding box of a point set in three dimensions, J. Algorithms 38 (2001), 91--109.]]

Digital Library

[5]

Chazelle, B. The Discrepancy Method: Randomness and Complexity, Cambridge University Press, 2000; paperback version 2001.]]

Digital Library

[6]

Chazelle, B., Dobkin, D.P. Intersection of convex objects in two and three dimensions, J. ACM 34 (1987), 1--27.]]

Digital Library

[7]

Chen, J., Han, Y. Shortest paths on a polyhedron, Proc. 6th SOCG (1990), 360--369.]]

Digital Library

[8]

Clarkson, K.L., Shor, P.W. Applications of random sampling in computational geometry, II, Disc. Comput. Geom. 4 (1989), 387--421.]]

Digital Library

[9]

Czumaj, A., Ergun, F., Fortnow, L., Magen, A., Newman, I., Rubinfeld, R., Sohler, C. Sublinear-time approximation of Euclidean minimum spanning tree, Proc. 14th SODA (2003), to appear.]]

Digital Library

[10]

Czumaj, A., Sohler, C. Property testing with geometric queries, Proc. 9th ESA (2001), 266--277.]]

Digital Library

[11]

Czumaj, A., Sohler, C., Ziegler, M. Property testing in computational geometry, Proc. 8th ESA (2000), 155--166.]]

Digital Library

[12]

Devroye, L., Mücke, E.P., Zhu, B. A note on point location in Delaunay triangulations of random points, Algorithmica 22 (1998), 477--482.]]

[13]

Dobkin, D.P., Kirkpatrick, D.G. Determining the separation of preprocessed polyhedra -- a unified approach, Proc. 17th ICALP (1990), 400--413.]]

Digital Library

[14]

Dudley, R.M. Metric entropy of some classes of sets with differentiable boundaries, J. Approx. Theory 10 (1974), 227--236.]]

[15]

Eppstein, D. Dynamic Euclidean minimum spanning trees and extrema of binary functions, Disc. Comput. Geom. 13 (1995), 111--122.]]

Digital Library

[16]

Ergun, F., Kannan, S., Kumar, S. Ravi, Rubinfeld, R., Viswanathan, M. Spot-checkers, Proc. STOC (1998), 259--268.]]

Digital Library

[17]

Grötschel, M., Lovász, L., Schrijver, A. Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.]]

[18]

Har-Peled, S. Approximate shortest-path and geodesic diameter on convex polytopes in three dimensions, Disc. Comput. Geom. 21 (1999), 217--231.]]

[19]

Kapoor, S., Efficient computation of geodesic shortest paths, Proc. 31rd STOC (1999), 770--779.]]

Digital Library

[20]

Matoušek, J. Geometric range searching, ACM Comput. Surv. 26 (1994), 421--461.]]

Digital Library

[21]

Mehlhorn, K., Naher, S., Schilz, T., Schirra, S., Seel, M., Seidel, R., Uhrig, C. Checking geometric programs or verification of geometric structures, Comput. Geom.: Theory and Appl. 12 (1999), 85--103.]]

Digital Library

[22]

Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H. The discrete geodesic problem, SIAM J. Comput. 16 (1987), 647--668.]]

Digital Library

[23]

Mitchell, J.S.B. An algorithmic approach to some problems in terrain nevigation, Autonomous Mobile Robots: Perception, Mapping and Nevigation, IEEEE computer society press, Los Alamitos, CA (1991) 408--427.]]

[24]

Mücke, E.P., Saias, I., Zhu, B. Fast randomized point location without preprocessing in two and three-dimensional Delaunay triangulations, Proc. 12th SOCG (1996), 274--283.]]

Digital Library

[25]

Mulmuley, K. Output sensitive and dynamic constructions of higher order Voronoi diagrams and levels in arrangements, JCSS 47 (1993), 437--458.]]

Digital Library

[26]

Pogorelov, A.V. Extrinsic geometry of convex surfaces, Volume 35 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1973.]]

[27]

Ron, D. Property testing, a tutorial, to appear in "Handbook on Randomization."]]

[28]

Seidel, R. Small-dimensional linear programming and convex hulls made easy, Disc. Comput. Geom. 6 (1991), 423--434.]]

Digital Library

[29]

Sharir, M., Schorr, A. On shortest paths in polyhedral spaces, SIAM J. Comput. 15 (1986), 193--215.]]

Digital Library

Cited By

Chazelle BSeshadhri C(2011)Online geometric reconstructionJournal of the ACM10.1145/1989727.198972858:4(1-32)Online publication date: 20-Jul-2011
https://dl.acm.org/doi/10.1145/1989727.1989728
Chandran NMoriarty ROstrovsky RPandey OSafari MSahai A(2008)Improved algorithms for optimal embeddingsACM Transactions on Algorithms10.1145/1383369.13833764:4(1-14)Online publication date: 22-Aug-2008
https://dl.acm.org/doi/10.1145/1383369.1383376
Cheng SVigneron A(2007)Motorcycle Graphs and Straight SkeletonsAlgorithmica10.5555/3118786.311925347:2(159-182)Online publication date: 1-Feb-2007
https://dl.acm.org/doi/10.5555/3118786.3119253
Show More Cited By

Index Terms

Sublinear geometric algorithms
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

Sublinear Geometric Algorithms

We initiate an investigation of sublinear algorithms for geometric problems in two and three dimensions. We give optimal algorithms for intersection detection of convex polygons and polyhedra, point location in two-dimensional triangulations and Voronoi ...
Sublinear geometric algorithms and geometric lower bounds
Efficient/practical algorithms for geometric structures: convex hulls, Delaunay triangulations and Voronoi diagrams

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Published In

cover image ACM Conferences

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

June 2003

740 pages

ISBN:1581136749

DOI:10.1145/780542

Conference Chair:
Lawrence L. Larmore
University of Nevada Las Vegas, Las Vegas, NV
,
Program Chair:
Michel X. Goemans
Massachusetts Institute of Technology, Cambridge, MA

Copyright © 2003 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: 09 June 2003

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STOC03: The 35th Annual ACM Symposium on Theory of Computing

June 9 - 11, 2003

CA, San Diego, USA

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STOC '03 Paper Acceptance Rate 80 of 270 submissions, 30%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Chazelle BSeshadhri C(2011)Online geometric reconstructionJournal of the ACM10.1145/1989727.198972858:4(1-32)Online publication date: 20-Jul-2011
https://dl.acm.org/doi/10.1145/1989727.1989728
Chandran NMoriarty ROstrovsky RPandey OSafari MSahai A(2008)Improved algorithms for optimal embeddingsACM Transactions on Algorithms10.1145/1383369.13833764:4(1-14)Online publication date: 22-Aug-2008
https://dl.acm.org/doi/10.1145/1383369.1383376
Cheng SVigneron A(2007)Motorcycle Graphs and Straight SkeletonsAlgorithmica10.5555/3118786.311925347:2(159-182)Online publication date: 1-Feb-2007
https://dl.acm.org/doi/10.5555/3118786.3119253
Bagchi AChaudhary AEppstein DGoodrich M(2007)Deterministic sampling and range counting in geometric data streamsACM Transactions on Algorithms10.1145/1240233.12402393:2(16-es)Online publication date: 1-May-2007
https://dl.acm.org/doi/10.1145/1240233.1240239
Ahn HBrass PCheong ONa HShin CVigneron A(2006)Inscribing an axially symmetric polygon and other approximation algorithms for planar convex setsComputational Geometry: Theory and Applications10.5555/1139255.164652733:3(152-164)Online publication date: 1-Feb-2006
https://dl.acm.org/doi/10.5555/1139255.1646527
Chazelle BSeshadhri CAmenta NCheong O(2006)Online geometric reconstructionProceedings of the twenty-second annual symposium on Computational geometry10.1145/1137856.1137912(386-394)Online publication date: 5-Jun-2006
https://dl.acm.org/doi/10.1145/1137856.1137912
Zhu B(2006)Voronoi Diagram and Delaunay TriangulationProceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2006.38(2-3)Online publication date: 2-Jul-2006
https://dl.acm.org/doi/10.1109/ISVD.2006.38
Ambainis AIwama KKawachi ARaymond RYamashita S(2006)Improved algorithms for quantum identification of boolean oraclesProceedings of the 10th Scandinavian conference on Algorithm Theory10.1007/11785293_27(280-291)Online publication date: 6-Jul-2006
https://dl.acm.org/doi/10.1007/11785293_27
Chan TChen EMitchell JRote G(2005)Multi-pass geometric algorithmsProceedings of the twenty-first annual symposium on Computational geometry10.1145/1064092.1064121(180-189)Online publication date: 6-Jun-2005
https://dl.acm.org/doi/10.1145/1064092.1064121
Aleksandrov LMaheshwari ASack J(2005)Determining approximate shortest paths on weighted polyhedral surfacesJournal of the ACM10.1145/1044731.104473352:1(25-53)Online publication date: 1-Jan-2005
https://dl.acm.org/doi/10.1145/1044731.1044733
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