Article

Free access

Coordinate representation of order types requires exponential storage

Authors:

B. SturmfelsAuthors Info & Claims

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

Pages 405 - 410

https://doi.org/10.1145/73007.73046

Published: 01 February 1989 Publication History

Abstract

We give doubly exponential upper and lower bounds on the size of the smallest grid on which we can embed every planar configuration of n points in general position up to order type. The lower bound is achieved by the construction of a widely dispersed “rigid” configuration which is then modified to one in general position by recent techniques of Sturmfels and White, while the upper bound uses recent results of Grigor'ev and Vorobjou on the solution of simultaneous inequalities. This provides a sharp answer to a question first posed by Chazelle.

References

[1]

L. J. Billera and B. S. Munson, Triangulations of oriented matroids and convez polytopes, SIAM J. Alg. Disc. Meth. 5 (1984), 515-525.

[2]

B. Chazelle, Problem 3-6, Third Computational Geometry Day, Courant Institute, Jan. 16, 1987.

[3]

G. Collins, Qua~tifier elimination for real closed fields by cylindrical algebraic decompos~'tion, in "Second GI Conference on Automata Theory and Formal Languages," Lecture Notes in Computer Science, Springer- Verlag, Berlin, 1!)75, pp. 134-183.

Digital Library

[4]

D. Dobkin and D. Silver, Recipes for geometry and numerical analysis, Part 1: an empirical study, Proc. Fourth Annual ACM Syrup. on Computational Geometry (1988).

Digital Library

[5]

H. Edelsbrunner and E. P. M/icke, Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms, Proc. Fourth Annum ACM Syrup. on Computational Geometry (1988).

Digital Library

[6]

J. E. Goodman and R. Pollack, Multidimensional sorting, SIAM J. Comput. 12 (1983), 484-507.

Digital Library

[7]

J. E. Goodman and It. Pollack, Upper bounds for configurations and polytopes in Ra, Discrete Comput. Geom. I (1986), 219- 227.

Digital Library

[8]

D. H. Greene and F. F. Yao, Finite-resolution computational geometry, in "2?th Annum IEEE Syrup. on the Foundations of Computer Science," 1986, pp. 143-152.

[9]

D. Yu. Grigor'ev and N. N. Vorobjov, Jr., Solving systems of polynomial inequalities in subexponential time, J. Symbolic Comput. 5 (1988), 37-64.

Digital Library

[10]

B. Griinbaum, "Arrangements and Spreads," Amer. Math. Soc., Providence, RI, 1972.

[11]

D. Hilbert, "The Foundations of Geometry,'' translated by E. J. Townsend, 3rd edition, Open Court, La SaUe, IL, 1938.

[12]

C. Hoffmann and J. Hopcroft, Towards implementing robust geometric computations, Fourth Annum ACM Symp. on Computational Geometry (1988).

Digital Library

[13]

V. J. Milenkovic, Verifiable implementations of geometric algorithms using finite precision arithmetic, Proc. 1986 Oxford Workshop on Geometric Reasoning, AI Journal (to appear).

Digital Library

[14]

D. Salesin, J. Stolii, and L. Guibas, Epsilon geometry: building robust algorithms from imprecise computations, Fifth Annual ACM Symp. on Computational Geometry (1989) (to appear).

Digital Library

[15]

B. Sturmfels and N. White, Constructing uniform oriented matroids without the isotopy property, # 432, IMA Preprint Series, University of Minnesota.

[16]

C. Yap, A geometric consistency theorem for a symbolic perturbation scheme, Fourth Annum ACM Syrup. on ComputationM Geometry (1988).

Digital Library

Cited By

Scheucher M(2022)Many Order Types on Integer Grids of Polynomial SizeComputational Geometry10.1016/j.comgeo.2022.101924(101924)Online publication date: Aug-2022
https://doi.org/10.1016/j.comgeo.2022.101924
Chiu MKorman MSuderland MTokuyama T(2022)Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital RaysDiscrete & Computational Geometry10.1007/s00454-021-00349-668:3(902-944)Online publication date: 15-Mar-2022
https://doi.org/10.1007/s00454-021-00349-6
Oropeza MTóth C(2021)Reconstruction of the Crossing Type of a Point Set from the Compatible Exchange Graph of Noncrossing Spanning TreesInformation Processing Letters10.1016/j.ipl.2021.106116(106116)Online publication date: Mar-2021
https://doi.org/10.1016/j.ipl.2021.106116
Show More Cited By

Index Terms

Coordinate representation of order types requires exponential storage
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
    2. Graph theory

Recommendations

Convex Hulls of Random Order Types
We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3):
- The number of extreme points in an n-point order type, ...
Powering requires threshold depth 3

We study the circuit complexity of the powering function, defined as POW"m(Z)=Z^m for an n-bit integer input Z and an integer exponent m= =2. Specifically, we prove a 2^@W^(^n^/^l^o^g^n^) lower bound on the size of any depth-2 majority circuit that ...
Symmetric Exponential Time Requires Near-Maximum Circuit Size
STOC 2024: Proceedings of the 56th Annual ACM Symposium on Theory of Computing

We show that there is a language in S₂E/₁ (symmetric exponential time with one bit of advice) with circuit complexity at least 2ⁿ/n. In particular, the above also implies the same near-maximum circuit lower bounds for the classes Σ₂E, (Σ₂E∩Π₂E)/₁, and ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC '89: Proceedings of the twenty-first annual ACM symposium on Theory of computing

February 1989

600 pages

ISBN:0897913078

DOI:10.1145/73007

Editor:
D. S. Johnson

Copyright © 1989 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]

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 February 1989

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

STOC89

Sponsor:

SIGACT

STOC89: 21st Annual ACM Symposium on the Theory of Computing

May 14 - 17, 1989

Washington, Seattle, USA

Acceptance Rates

STOC '89 Paper Acceptance Rate 56 of 196 submissions, 29%;

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

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

29
Total Citations
View Citations
543
Total Downloads

Downloads (Last 12 months)51
Downloads (Last 6 weeks)11

Reflects downloads up to 04 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Scheucher M(2022)Many Order Types on Integer Grids of Polynomial SizeComputational Geometry10.1016/j.comgeo.2022.101924(101924)Online publication date: Aug-2022
https://doi.org/10.1016/j.comgeo.2022.101924
Chiu MKorman MSuderland MTokuyama T(2022)Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital RaysDiscrete & Computational Geometry10.1007/s00454-021-00349-668:3(902-944)Online publication date: 15-Mar-2022
https://doi.org/10.1007/s00454-021-00349-6
Oropeza MTóth C(2021)Reconstruction of the Crossing Type of a Point Set from the Compatible Exchange Graph of Noncrossing Spanning TreesInformation Processing Letters10.1016/j.ipl.2021.106116(106116)Online publication date: Mar-2021
https://doi.org/10.1016/j.ipl.2021.106116
Erickson Jvan der Hoog IMiltzow T(2020)Smoothing the gap between NP and ER2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00099(1022-1033)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00099
Caraballo LDíaz-Báñez JFabila-Monroy RHidalgo-Toscano CLeaños JMontejano A(2020)On the number of order types in integer grids of small sizeComputational Geometry10.1016/j.comgeo.2020.101730(101730)Online publication date: Nov-2020
https://doi.org/10.1016/j.comgeo.2020.101730
Çağırıcı OHliněný PPokrývka FSankaran A(2020)Clique-Width of Point ConfigurationsGraph-Theoretic Concepts in Computer Science10.1007/978-3-030-60440-0_5(54-66)Online publication date: 9-Oct-2020
https://doi.org/10.1007/978-3-030-60440-0_5
Han JKohayakawa YSales MStagni HChan T(2019)Extremal and probabilistic results for order typesProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310462(426-435)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310462
Kratsch SPhilip GRay S(2016)Point Line CoverACM Transactions on Algorithms10.1145/283291212:3(1-16)Online publication date: 25-Apr-2016
https://dl.acm.org/doi/10.1145/2832912
Cardinal J(2015)Computational Geometry Column 62ACM SIGACT News10.1145/2852040.285205346:4(69-78)Online publication date: 1-Dec-2015
https://dl.acm.org/doi/10.1145/2852040.2852053
Kratsch SPhilip GRay SChekuri C(2014)Point line coverProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634190(1596-1606)Online publication date: 5-Jan-2014
https://dl.acm.org/doi/10.5555/2634074.2634190
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

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 Publication

Media

Figures

Other

Tables

View Table of Contents