article

Dynamic planar convex hull operations in near-logarithmic amortized time

Author:

Timothy M. ChanAuthors Info & Claims

Journal of the ACM (JACM), Volume 48, Issue 1

Pages 1 - 12

https://doi.org/10.1145/363647.363652

Published: 01 January 2001 Publication History

Abstract

We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P, such as membership and tangent-finding. Updates take O(log^1+εn) amori tzed time and queries take O (log n time each, where n is the maximum size of P and ε is any fixed positive constant. For some advanced queries such as bridge-finding, both our bounds increase to O(log^3/2n). The only previous fully dynamic solution was by Overmars and van Leeuwen from 1981 and required O(log²n) time per update and O(log n) time per query.

References

[1]

AGARWAL, P. K., DE BERG, M., MATOUSEK, J., AND SCHWARZKOPF, O. 1998. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27, 654-667.]]

Digital Library

[2]

AGARWAL, P. K., AND MATOUSEK, J. 1995. Dynamic half-space range reporting and its applications. Algorithmica 13, 325-345.]]

[3]

AGARWAL, P. K., AND SHARIR, M. 2000. Arrangements and their applications. In Handbook of Computational Geometry, J. Urrutia and J. Sack, eds. North-Holland, Amsterdam, The Netherlands.]]

[4]

ANDRZEJAK, A., AND WELZI., E. 1997. k-sets and j -facets: A tour of discrete geometry. Manuscript.]]

[5]

BASCH, J., GUIBAS, L. J., AND RAMKUMAR, G. 1996. Reporting red-blue intersections between two sets of connected line segments. In Proceedings of the 4th European Symposium on Algorithms. Lecture Notes in Computer Science, vol. 1136. Springer-Verlag, New York, pp. 302-319.]]

Digital Library

[6]

BENTLEY, J., AND SAXE, J. 1980. Decomposable searching problems. I: Static-to-dynamic transformation. J. Algorithms 1, 301-358.]]

[7]

B~HRINGER, K.-F., DONALD,B.R.,AND HALPERIN, D. 1999. On the area bisectors of a polygon. Disc. Comput. Geom. 22, 269-285.]]

[8]

BRODAL, G. S., AND JACOB, R. 2000. Dynamic planar convex hull with optimal query time and O(log n log log n) update time. In Proceedings of the 7th Scandinavian Workshop on Algorithm Theory. Lecture Notes in Computer Science, vol. 1851. Springer-Verlag, Berlin, Germany, pp. 57-70.]]

Digital Library

[9]

CHAN, T. M. 1996. Output-sensitive results on convex hulls, extreme points, and related problems. Disc. Comput. Geom. 16, 369-387.]]

Digital Library

[10]

CHAN, T. M. 1999. Remarks on k-level algorithms in the plane. Manuscript.]]

[11]

CHAN, T. M. 2001. Random sampling, halfspace range reporting, and construction of (\leqk)-levels in three dimensions. SIAM J. Comput. 30, 561-575.]]

Digital Library

[12]

CHAZELLE, B. 1985. On the convex layers of a planar set. IEEE Trans. Inf. Theory IT-31, 509-517.]]

Digital Library

[13]

CHAZELLE, B., AND EDELSBRUNNER, H. 1987. An improved algorithm for constructing kth-order Voronoi diagrams. IEEE Trans. Comput. C-36, 1349-1354.]]

Digital Library

[14]

CHIANG, Y.-J., AND TAMASSIA, R. 1992. Dynamic algorithms in computational geometry. Proc. IEEE 80, 1412-1434.]]

[15]

CORMEN, T. H., LEISERSON, C. E., AND RIVEST, R. L. 1990. Introduction to Algorithms. MIT Press, Cambridge, Mass.]]

Digital Library

[16]

DE BERG, M., VAN KREVELD, M., OVERMARS, M., AND SCHWARZKOPF, O. 1997. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, Germany.]]

[17]

DEVILLERS, O., AND KATZ, M. J. 1999. Optimal line bipartitions of point sets. Int. J. Comput. Geom. Appl. 1, 39-51.]]

[18]

DOBKIN, D., EPPSTEIN, D., AND MITCHELL, D. P. 1996. Computing the discrepancy with applications to supersampling patterns. ACM Trans. Graph. 15, 354-376.]]

Digital Library

[19]

DOBKIN, D. P., AND KIRKPATRICK, D. G. 1983. Fast detection of polyhedral intersection. Theoret. Comput. Sci. 27, 241-253.]]

[20]

EDELSBRUNNER, H. 1987. Algorithms in Combinatorial Geometry. Springer-Verlag, Berlin, Germany.]]

Digital Library

[21]

EDELSBRUNNER, H., AND WELZI., E. 1986. Constructing belts in two-dimensional arrangements with applications. SIAM J. Comput., 15, 271-284.]]

[22]

FRIEDMAN, J., HERSHBERGER, J., AND SNOEYINK, J. 1996. Efficiently planning compliant motion in the plane. SIAM J. Comput. 25, 562-599.]]

Digital Library

[23]

GHALI, S., AND STEWART, A. J. 1996. Maintenance of the set of segments visible from a moving viewpoint in two dimensions. In Proceedings of the 12th Annual ACM Symposium on Computational Geometry (Philadelphia, Pa., May 24-26). ACM, New York, pp. V3-V4.]]

Digital Library

[24]

HAR-PELED, S. 1999. Taking a walk in a planar arrangement. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 100-110.]]

Digital Library

[25]

HERSHBERGER, J., AND SURI, S. 1991. Finding tailored partitions. J. Algorithms 12, 431-463.]]

Digital Library

[26]

HERSHBERGER, J., AND SURI, S. 1992. Applications of a semi-dynamic convex hull algorithm. BIT 32, 249-267.]]

Digital Library

[27]

HERSHBERGER, J., AND SURI, S. 1996. Off-line maintenance of planar configurations. J. Algorithms 21, 453-475.]]

Digital Library

[28]

JANARDAN, R. 1993. On maintaining the width and diameter of a planar point-set online. Int. J. Comput. Geom. Appl. 3, 331-344.]]

[29]

KAPOOR, S. 1998. Dynamic maintenance of 2-d convex hulls and order decomposable problems. Manuscript.]]

[30]

MATOUSEK, J. 1995. On geometric optimization with few violated constraints. Disc. Comput. Geom. 14, 365-384.]]

Digital Library

[31]

MEGIDDO, N. 1983. Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30, 852-865.]]

Digital Library

[32]

MEHLHORN, K. 1984. Data Structures and Algorithms 3: Multi-Dimensional Searching and Computational Geometry, Springer-Verlag, Heidelberg, Germany.]]

Digital Library

[33]

MULMULEY, K. 1991. Randomized multidimensional search trees: Lazy balancing and dynamic shuffling. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, Calif., pp. 180-196.]]

Digital Library

[34]

MULMULEY, K. 1994. Computational Geometry: An Introduction Through Randomized Algorithms. Prentice-Hall, Englewood Cliffs, N.J.]]

[35]

OVERMARS, M. H. 1983. The Design of Dynamic Data Structures. Lecture Notes in Computer Science, Vol. 156. Springer-Verlag, New York.]]

Digital Library

[36]

OVERMARS, M. H., AND VAN LEEUWEN, J. 1981. Maintenance of configurations in the plane. J. Comput. Sys. Sci. 23, 166-204.]]

[37]

PREPARATA, F. P., AND SHAMOS, M. I. 1985. Computational Geometry: An Introduction. Springer- Verlag, New York.]]

Digital Library

[38]

RAMOS, E. A. 1999. On range reporting, ray shooting, and k-level construction. In Proceedings of the 15th Annual ACM Symposium on Computational Geometry (Miami Beach, Fla., June 13-16). ACM, New York, pp. 390-399.]]

Digital Library

[39]

ROTE, G., SCHWARZ, C., AND SNOEYINK, J. 1993. Maintaining the approximate width of a set of points in the plane. In Proceedings of the 5th Canadian Conference on Computational Geometry, pp. 258-263.]]

[40]

SCHWARZKOPF, O. 1991. Dynamic maintenance of geometric structures made easy. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computing Science. IEEE, New York, pp. 197-206.]]

Digital Library

Cited By

Baumann AKaplan HKlost KKnorr KMulzer WRoditty LSeiferth P(2024)Dynamic Connectivity in Disk GraphsDiscrete & Computational Geometry10.1007/s00454-023-00621-x71:1(214-277)Online publication date: 3-Jan-2024
https://doi.org/10.1007/s00454-023-00621-x
Wang H(2023)Dynamic Convex Hulls Under Window-Sliding UpdatesAlgorithms and Data Structures10.1007/978-3-031-38906-1_46(689-703)Online publication date: 31-Jul-2023
https://dl.acm.org/doi/10.1007/978-3-031-38906-1_46
Dumitrescu A(2022)Peeling SequencesMathematics10.3390/math1022428710:22(4287)Online publication date: 16-Nov-2022
https://doi.org/10.3390/math10224287
Show More Cited By

Index Terms

Dynamic planar convex hull operations in near-logarithmic amortized time

Recommendations

A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries

We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log³n) expected amortized time, deletions take O(log⁶n) expected amortized time, and ...
Read More
Three problems about dynamic convex hulls
SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry

We present three results related to dynamic convex hulls: A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update ...
Read More
Dynamic Planar Convex Hull Operations in Near-Logarithmic Amortized Time
FOCS '99: Proceedings of the 40th Annual Symposium on Foundations of Computer Science

We give a data structure that allows arbitrary insertions and deletions on a planar point set P and supports basic queries on the convex hull of P, such as membership and tangent-finding. Updates take O(log{1+eps}n) amortized time and queries take O(log ...
Read More

Comments

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 48, Issue 1

Jan. 2001

147 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/363647

Editor:
F. Thomson Leighton
Massachusetts Institute of Technology, Cambridge

Issue’s Table of Contents

Copyright © 2001 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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 2001

Published in JACM Volume 48, Issue 1

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
1,883
Total Downloads

Downloads (Last 12 months)41
Downloads (Last 6 weeks)4

Other Metrics

View Author Metrics

Citations

Cited By

Baumann AKaplan HKlost KKnorr KMulzer WRoditty LSeiferth P(2024)Dynamic Connectivity in Disk GraphsDiscrete & Computational Geometry10.1007/s00454-023-00621-x71:1(214-277)Online publication date: 3-Jan-2024
https://doi.org/10.1007/s00454-023-00621-x
Wang H(2023)Dynamic Convex Hulls Under Window-Sliding UpdatesAlgorithms and Data Structures10.1007/978-3-031-38906-1_46(689-703)Online publication date: 31-Jul-2023
https://dl.acm.org/doi/10.1007/978-3-031-38906-1_46
Dumitrescu A(2022)Peeling SequencesMathematics10.3390/math1022428710:22(4287)Online publication date: 16-Nov-2022
https://doi.org/10.3390/math10224287
Agarwal PCohen RHalperin DMulzer W(2022)Maintaining the Union of Unit Discs under Insertions with Near-Optimal OverheadACM Transactions on Algorithms10.1145/352761418:3(1-27)Online publication date: 11-Oct-2022
https://dl.acm.org/doi/10.1145/3527614
Chan T(2020)Dynamic Geometric Data Structures via Shallow CuttingsDiscrete & Computational Geometry10.1007/s00454-020-00229-564:4(1235-1252)Online publication date: 1-Dec-2020
https://dl.acm.org/doi/10.1007/s00454-020-00229-5
Chan THershberger JPratt S(2019)Two Approaches to Building Time-Windowed Geometric Data StructuresAlgorithmica10.1007/s00453-019-00588-381:9(3519-3533)Online publication date: 1-Sep-2019
https://dl.acm.org/doi/10.1007/s00453-019-00588-3
Gyongyosi LImre SNguyen H(2018)A Survey on Quantum Channel CapacitiesIEEE Communications Surveys & Tutorials10.1109/COMST.2017.278674820:2(1149-1205)Online publication date: Oct-2019
https://doi.org/10.1109/COMST.2017.2786748
Sunita Garg D(2018)Dynamizing Dijkstra: A solution to dynamic shortest path problem through retroactive priority queueJournal of King Saud University - Computer and Information Sciences10.1016/j.jksuci.2018.03.003Online publication date: Mar-2018
https://doi.org/10.1016/j.jksuci.2018.03.003
Basu RBhattacharya BTalukdar T(2012)The Projection Median of a Set of Points in źdDiscrete & Computational Geometry10.5555/3115956.311602547:2(329-346)Online publication date: 1-Mar-2012
https://dl.acm.org/doi/10.5555/3115956.3116025
CHAN T(2012)THREE PROBLEMS ABOUT DYNAMIC CONVEX HULLSInternational Journal of Computational Geometry & Applications10.1142/S021819591260009622:04(341-364)Online publication date: Aug-2012
https://doi.org/10.1142/S0218195912600096
Show More Cited By

View Options

Get Access

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.

Media

Figures

Other

Tables

View Issue’s Table of Contents