article

Almost tight upper bounds for vertical decompositions in four dimensions

Author:

Vladlen KoltunAuthors Info & Claims

Journal of the ACM (JACM), Volume 51, Issue 5

Pages 699 - 730

https://doi.org/10.1145/1017460.1017461

Published: 01 September 2004 Publication History

Abstract

We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is O(n^4+ε), for any ε > 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n^2d−4+ε), for any ε > 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems.

References

[1]

Agarwal, P. K., Aronov, B., and Sharir, M. 1997. Computing envelopes in four dimensions with applications. SIAM J. Comput. 26, 1714--1732.

[2]

Agarwal, P. K., Efrat, A., and Sharir, M. 1999. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput 29, 912--953.

[3]

Agarwal, P. K., and Erickson, J. 1999. Geometric range searching and its relatives. In Advances in Discrete and Computational Geometry, B. Chazelle, J. E. Goodman, and R. Pollack, Eds. Contemporary Mathematics, vol. 223. American Mathematical Society, Providence, RI, 1--56.

[4]

Agarwal, P. K., and Matoušek, J. 1994. On range searching with semialgebraic sets. Disc. Comput. Geom. 11, 393--418.

[5]

Agarwal, P. K., Schwarzkopf, O., and Sharir, M. 1996. The overlay of lower envelopes and its applications. Disc. Comput. Geom. 15, 1--13.

[6]

Agarwal, P. K., and Sharir, M. 1996. Efficient randomized algorithms for some geometric optimization problems. Disc. Comput. Geom. 16, 317--337.

[7]

Agarwal, P. K., and Sharir, M. 2000. Arrangements and their applications. In Handbook of Computational Geometry, J.-R. Sack and J. Urrutia, Eds. Elsevier Science Publishers B.V., North-Holland, Amsterdam, The Netherlands, 49--119.

[8]

Agarwal, P. K., Sharir, M., and Toledo, S. 1994. Applications of parametric searching in geometric optimization. J. Algorithms 17, 292--318.

[9]

Chazelle, B., Edelsbrunner, H., Guibas, L. J., and Sharir, M. 1991. A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. Theoret. Comput. Sci. 84, 77--105. (Also in Proc. International Colloquium on Automata, Languages and Programming (1989), pp. 179--193.)

[10]

Chazelle, B., Edelsbrunner, H., Guibas, L. J., and Sharir, M. 1993. Diameter, width, closest line pair and parametric searching. Disc. Comput. Geom. 10, 183--196.

[11]

Chazelle, B., and Sharir, M. 1990. An algorithm for generalized point location and its applications. J. Symb. Comput. 10, 281--309.

[12]

Clarkson, K. L. 1988. A randomized algorithm for closest-point queries. SIAM J. Comput. 17, 830--847.

[13]

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

[14]

Collins, G. E. 1975. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages. 134--183.

[15]

de Berg, M. 1993. Ray Shooting, Depth Orders and Hidden Surface Removal. Lecture Notes in Computer Science, vol. 703. Springer-Verlag, Berlin, Germany.

[16]

de Berg, M., Guibas, L. J., and Halperin, D. 1996. Vertical decompositions for triangles in 3-space. Disc. Comput. Geom. 15, 35--61.

[17]

de Berg, M., Halperin, D., Overmars, M., Snoeyink, J., and van Kreveld, M. 1994. Efficient ray shooting and hidden surface removal. Algorithmica 12, 30--53.

[18]

Guibas, L. J., Halperin, D., Matoušek, J., and Sharir, M. 1995. On vertical decomposition of arrangements of hyperplanes in four dimensions. Disc. Comput. Geom. 14, 113--122.

[19]

Halperin, D. 1997. Arrangements. In Handbook of Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke, Eds. CRC Press LLC, Boca Raton, FL, Chap. 21, 389--412.

[20]

Halperin, D., and Sharir, M. 1994. New bounds for lower envelopes in three dimensions, with applications to visibility in terrains. Disc. Comput. Geom. 12, 313--326.

[21]

Koltun, V. 2004. Sharp bounds for vertical decompositions of linear arrangements in four dimensions. Disc. Comput. Geom. 31, 435--460.

[22]

Mohaban, S., and Sharir, M. 1997. Ray shooting amidst spheres in three dimensions and related problems. SIAM J. Comput. 26, 654--674.

[23]

Pellegrini, M. 1993. Ray shooting on triangles in 3-space. Algorithmica 9, 471--494.

[24]

Schwarzkopf, O., and Sharir, M. 1997. Vertical decomposition of a single cell in a three-dimensional arrangement of surfaces and its applications. Disc. Comput. Geom. 18, 269--288.

[25]

Sharir, M. 1994. Almost tight upper bounds for lower envelopes in higher dimensions. Disc. Comput. Geom. 12, 327--345.

[26]

Sharir, M. 1997. Algorithmic motion planning. In Handbook of Discrete and Computational Geometry, J. E. Goodman and J. O'Rourke, Eds. CRC Press LLC, Boca Raton, FL, Chap. 40, 733--754.

[27]

Sharir, M., and Agarwal, P. K. 1995. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York.

[28]

Tagansky, B. 1996. A new technique for analyzing substructures in arrangements of piecewise linear surfaces. Disc. Comput. Geom. 16, 455--479.

Cited By

Agarwal PEzra E(2024)Line Intersection Searching Amid Unit Balls in 3-SpaceAlgorithmica10.1007/s00453-024-01284-7Online publication date: 6-Dec-2024
https://doi.org/10.1007/s00453-024-01284-7
Aronov BEzra ESharir M(2022)Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related ProblemsDiscrete & Computational Geometry10.1007/s00454-022-00437-168:4(997-1048)Online publication date: 1-Dec-2022
https://dl.acm.org/doi/10.1007/s00454-022-00437-1
Do T(2022)Nondegenerate Spheres in Four DimensionsDiscrete & Computational Geometry10.1007/s00454-021-00366-568:2(406-424)Online publication date: 1-Sep-2022
https://dl.acm.org/doi/10.1007/s00454-021-00366-5
Show More Cited By

Index Terms

Almost tight upper bounds for vertical decompositions in four dimensions

Recommendations

Almost tight upper bounds for lower envelopes in higher dimensions
SFCS '93: Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science

We show that the combinatorial complexity of the lower envelope of n surfaces or surface patches in d-space (d/spl ges/3), all algebraic of constant maximum degree, and bounded by algebraic surfaces of constant maximum degree, is O(n/sup d-1+/spl epsi//)...
Almost Tight Upper Bounds for Vertical Decompositions in Four Dimensions
FOCS '01: Proceedings of the 42nd IEEE symposium on Foundations of Computer Science

We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is 0(n^{4 + \varepsilon } ), for any \varepsilon > 0. This improves the best previously known upper ...
Complexity Bounds for Vertical Decompositions of Linear Arrangements in Four Dimensions
WADS '01: Proceedings of the 7th International Workshop on Algorithms and Data Structures

We prove tight and near-tight combinatorial complexity bounds for vertical decompositions of arrangements of linear surfaces in four dimensions. In particular, we prove a tight upper bound of Θ(n⁴) for the vertical decomposition of an arrangement of n ...

Comments

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 51, Issue 5

September 2004

151 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1017460

Issue’s Table of Contents

Copyright © 2004 ACM.

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 September 2004

Published in JACM Volume 51, Issue 5

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

41
Total Citations
View Citations
1,256
Total Downloads

Downloads (Last 12 months)10
Downloads (Last 6 weeks)0

Reflects downloads up to 08 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Agarwal PEzra E(2024)Line Intersection Searching Amid Unit Balls in 3-SpaceAlgorithmica10.1007/s00453-024-01284-7Online publication date: 6-Dec-2024
https://doi.org/10.1007/s00453-024-01284-7
Aronov BEzra ESharir M(2022)Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related ProblemsDiscrete & Computational Geometry10.1007/s00454-022-00437-168:4(997-1048)Online publication date: 1-Dec-2022
https://dl.acm.org/doi/10.1007/s00454-022-00437-1
Do T(2022)Nondegenerate Spheres in Four DimensionsDiscrete & Computational Geometry10.1007/s00454-021-00366-568:2(406-424)Online publication date: 1-Sep-2022
https://dl.acm.org/doi/10.1007/s00454-021-00366-5
Azar YGanesh AGe RPanigrahi D(2021)Online Service with DelayACM Transactions on Algorithms10.1145/345992517:3(1-31)Online publication date: 15-Jul-2021
https://dl.acm.org/doi/10.1145/3459925
Aronov BDe Berg MGudmundsson JHorton M(2021)On β-Plurality Points in Spatial Voting GamesACM Transactions on Algorithms10.1145/345909717:3(1-21)Online publication date: 15-Jul-2021
https://dl.acm.org/doi/10.1145/3459097
Viola CŽivný S(2021)The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite DomainsACM Transactions on Algorithms10.1145/345804117:3(1-23)Online publication date: 15-Jul-2021
https://dl.acm.org/doi/10.1145/3458041
Focke JGoldberg LŽivný S(2021)The Complexity of Approximately Counting Retractions to Square-free GraphsACM Transactions on Algorithms10.1145/345804017:3(1-51)Online publication date: 15-Jul-2021
https://dl.acm.org/doi/10.1145/3458040
Agarwal PBoris Aronov Esther Ezra Zahl J(2021)Efficient Algorithm for Generalized Polynomial Partitioning and Its ApplicationsSIAM Journal on Computing10.1137/19M126855050:2(760-787)Online publication date: 15-Apr-2021
https://doi.org/10.1137/19M1268550
Berg MBuchin KJansen BWoeginger G(2020)Fine-grained Complexity Analysis of Two Classic TSP VariantsACM Transactions on Algorithms10.1145/341484517:1(1-29)Online publication date: 31-Dec-2020
https://dl.acm.org/doi/10.1145/3414845
Samanta DChakraborty T(2020)VectorEntryACM Transactions on Accessible Computing10.1145/340653713:3(1-29)Online publication date: 3-Aug-2020
https://dl.acm.org/doi/10.1145/3406537
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Issue’s Table of Contents