tutorial

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Authors:

Timothy M. Chan,

Patrick LeeAuthors Info & Claims

SOCG'14: Proceedings of the thirtieth annual symposium on Computational geometry

Pages 40 - 49

https://doi.org/10.1145/2582112.2582166

Published: 08 June 2014 Publication History

Abstract

Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the number of comparisons needed. In this paper, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, including:

• an algorithm for the 2D maxima problem that uses n lg h +O(n√lg h) comparisons, where h denotes the output size;

• a randomized algorithm for the 3D maxima problem that uses n lg h + O(n lg2/3 h) expected number of comparisons;

• a randomized algorithm for detecting intersections among a set of orthogonal line segments that uses n lg n + O(n√lg n) expected number of comparisons;

• a data structure for point location among 3D disjoint axis-parallel boxes that can answer queries in (3/2) lg n + O(lg lg n) comparisons;

• a data structure for point location in a 3D box subdivision that can answer queries in (4/3)lg n + O(√lgn) comparisons.

Some of the results can be adapted to solve nonorthogonal problems, such as 2D convex hulls and general line segment intersection.

Our algorithms and data structures use a variety of techniques, including Seidel and Adamy's planar point location method, weighted binary search, and height-optimal BSP trees.

References

[1]

P. Afshani, J. Barbay, and T. M. Chan. Instance-optimal geometric algorithms. In Proc. 50th Annu. IEEE Sympos. Found. Comput. Sci., pages 129--138, 2009.

Digital Library

[2]

S. Arya, T. Malamatos, D. M. Mount, and K. C. Wong. Optimal expected-case planar point location. SIAM J. Comput., 37:584--610, 2007.

Digital Library

[3]

J. L. Bentley and T. A. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Trans. Comput., C-28(9):643--647, 1979.

Digital Library

[4]

T. M. Chan. Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete Comput. Geom., 16:361--368, 1996.

Digital Library

[5]

T. M. Chan. Output-sensitive results on convex hulls, extreme points, and related problems. Discrete Comput. Geom., 16:369--387, 1996.

Digital Library

[6]

T. M. Chan. Klee's measure problem made easy. In Proc. 54th Annu. IEEE Sympos. Found. Comput. Sci., pages 410--419, 2013.

Digital Library

[7]

T. M. Chan. Persistent predecessor search and orthogonal point location on the word RAM. ACM Trans. Algorithms, 9(3):22, 2013.

Digital Library

[8]

T. M. Chan, K. G. Larsen, and M. Pătraşcu. Orthogonal range searching on the RAM, revisited. In Proc. 27th Annu. ACM Sympos. Comput. Geom., pages 1--10, 2011.

Digital Library

[9]

T. M. Chan and M. Pătraşcu. Transdichotomous results in computational geometry, I: Point location in sublogarithmic time. SIAM J. Comput., 39:703--729, 2009.

Digital Library

[10]

T. M. Chan, J. Snoeyink, and C. K. Yap. Primal dividing and dual pruning: Output-sensitive construction of 4-d polytopes and 3-d Voronoi diagrams. Discrete Comput. Geom., 18:433--454, 1997.

[11]

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

Digital Library

[12]

B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir. Algorithms for bichromatic line segment problems and polyhedral terrains. Algorithmica, 11:116--132, 1994.

Digital Library

[13]

B. Chazelle and J. Matoušek. Derandomizing an output-sensitive convex hull algorithm in three dimensions. Comput. Geom. Theory Appl., 5:27--32, 1995.

Digital Library

[14]

K. L. Clarkson. New applications of random sampling in computational geometry. Discrete Comput. Geom., 2:195--222, 1987.

Digital Library

[15]

K. L. Clarkson. More output-sensitive geometric algorithms. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., pages 695--702, 1994.

Digital Library

[16]

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

Digital Library

[17]

W. Cunto and J. I. Munro. Average case selection. J. ACM, 36:270--279, 1989.

Digital Library

[18]

D. Dor and U. Zwick. Selecting the median. SIAM J. Comput., 28:1722--1758, 1999.

Digital Library

[19]

D. Dor and U. Zwick. Median selection requires (2 + &epsis;)n comparisons. SIAM J. Discrete Math., 14:312--325, 2001.

Digital Library

[20]

T. Dubé. Dominance range-query: The one-reporting case. In Proc. 9th Annu. ACM Sympos. Comput. Geom., pages 208--217, 1993.

Digital Library

[21]

A. Dumitrescu, J. S. B. Mitchell, and M. Sharir. Binary space partitions for axis-parallel segments, rectangles, and hyperrectangles. Discrete Comput. Geom., 31:207--227, 2004.

Digital Library

[22]

J. Ford, R. Lester, and S. M. Johnson. A tournament problem. Amer. Math. Monthly, 66:387--389, 1959.

[23]

G. N. Frederickson and D. B. Johnson. The complexity of selection and ranking in X + Y and matrices with sorted rows and columns. J. Comput. Syst. Sci., 24:197--208, 1982.

[24]

M. J. Golin. A provably fast linear-expected-time maxima-finding algorithm. Algorithmica, 11:501--524, 1994.

Digital Library

[25]

G. H. Gonnet and J. I. Munro. Heaps on heaps. SIAM J. Comput., 15:964--971, 1986.

Digital Library

[26]

J. Hershberger, S. Suri, and C. D. Tóth. Binary space partitions of orthogonal subdivisions. SIAM J. Comput., 34:1380--1397, 2005.

Digital Library

[27]

R. A. Jarvis. On the identification of the convex hull of a finite set of points in the plane. Inform. Process. Lett., 2:18--21, 1973.

[28]

K. Kaligosi, K. Mehlhorn, J. I. Munro, and P. Sanders. Towards optimal multiple selection. In Proc. 32nd Int. Colloq. Automata, Languages and Programming, volume 3580 of Lecture Notes Comput. Sci., pages 103--114. Springer-Verlag, 2005.

Digital Library

[29]

D. G. Kirkpatrick and R. Seidel. Output-size sensitive algorithms for finding maximal vectors. In Proc. 1st Annu. ACM Sympos. Comput. Geom., pages 89--96, 1985.

Digital Library

[30]

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM J. Comput., 15:287--299, 1986.

Digital Library

[31]

D. E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, Reading, MA, 1973.

[32]

H. T. Kung, F. Luccio, and F. P. Preparata. On finding the maxima of a set of vectors. J. ACM, 22:469--476, 1975.

Digital Library

[33]

Z. Li and B. A. Reed. Heap building bounds. In Proc. 9th Workshop Algorithms Data Struct., volume 3608, pages 14--23. Springer-Verlag, 2005.

Digital Library

[34]

Giuseppe Liotta, Franco P. Preparata, and Roberto Tamassia. Robust proximity queries: An illustration of degree-driven algorithm design. SIAM J. Comput., 28(3):864--889, 1998.

Digital Library

[35]

K. Mehlhorn. Data Structures and Algorithm 1: Sorting and Searching, volume 1 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin/Heidelberg, Germany, 1981.

Digital Library

[36]

T. Ottmann, S. Schuierer, and S. Soundaralakshmi. Enumerating extreme points in higher dimensions. In Proc. 12th Sympos. Theoret. Aspects Comput. Sci., volume 900 of Lecture Notes Comput. Sci., pages 562--570. Springer-Verlag, 1995.

[37]

M. H. Overmars and C.-K. Yap. New upper bounds in Klee's measure problem. SIAM J. Comput., 20:1034--1045, 1991.

Digital Library

[38]

M. S. Paterson and F. F. Yao. Optimal binary space partitions for orthogonal objects. J. Algorithms, 13:99--113, 1992.

Digital Library

[39]

F. P. Preparata. A new approach to planar point location. SIAM J. Comput., 10(3):473--482, 1981.

[40]

F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985.

Digital Library

[41]

R. Seidel and U. Adamy. On the exact worst case complexity of planar point location. J. Algorithms, 37:189--217, 2000.

Digital Library

[42]

C. D. Tóth. Binary space partitions: Recent developments. In Combinatorial and Computational Geometry, volume 52 of MSRI Publications, pages 525--552. Cambridge University Press, 2005.

Cited By

Rahul SIndyk P(2015)Improved bounds for orthogonal point enclosure query and point location in orthogonal subdivisions in R3Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722144(200-211)Online publication date: 4-Jan-2015
https://dl.acm.org/doi/10.5555/2722129.2722144
Groz BMilo TMilo TCalvanese D(2015)Skyline Queries with Noisy ComparisonsProceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/2745754.2745775(185-198)Online publication date: 20-May-2015
https://dl.acm.org/doi/10.1145/2745754.2745775

Index Terms

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Recommendations

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the ...
Instance-Optimal Geometric Algorithms
FOCS '09: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science

We prove the existence of an algorithm $A$ for computing 2-d or 3-dconvex hulls that is optimal for {\em every point set\/} in the following sense: %for every set $S$ of $n$ points and for every algorithm $A'$ in a certain class $\A$, the running time ...
Instance-Optimal Geometric Algorithms

We prove the existence of an algorithm A for computing 2D or 3D convex hulls that is optimal for every point set in the following sense: for every sequence σ of n points and for every algorithm A′ in a certain class A, the running time of A on input σ ...

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences

SOCG'14: Proceedings of the thirtieth annual symposium on Computational geometry

June 2014

588 pages

ISBN:9781450325943

DOI:10.1145/2582112

Program Chairs:
Siu-Wing Cheng
HKUST
,
Olivier Devillers
INRIA

Copyright © 2014 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 the author(s) 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].

In-Cooperation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 June 2014

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Tutorial
Research
Refereed limited

Conference

SOCG'14

SOCG'14: Annual Symposium on Computational Geometry

June 8 - 11, 2014

Kyoto, Japan

Acceptance Rates

SOCG'14 Paper Acceptance Rate 60 of 175 submissions, 34%;

Overall Acceptance Rate 625 of 1,685 submissions, 37%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
147
Total Downloads

Downloads (Last 12 months)5
Downloads (Last 6 weeks)0

Reflects downloads up to 31 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Rahul SIndyk P(2015)Improved bounds for orthogonal point enclosure query and point location in orthogonal subdivisions in R3Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms10.5555/2722129.2722144(200-211)Online publication date: 4-Jan-2015
https://dl.acm.org/doi/10.5555/2722129.2722144
Groz BMilo TMilo TCalvanese D(2015)Skyline Queries with Noisy ComparisonsProceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/2745754.2745775(185-198)Online publication date: 20-May-2015
https://dl.acm.org/doi/10.1145/2745754.2745775

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 Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents