skip to main content
10.1145/1185448.1185463acmotherconferencesArticle/Chapter ViewAbstractPublication Pagesacm-seConference Proceedingsconference-collections
Article

Chessboard domination on programmable graphics hardware

Published: 10 March 2006 Publication History

Abstract

In this paper we present an algorithm to compute the minimum dominating number of a chessboard graph given any chess piece. We use the CPU to compute possible minimally dominating sets, which we then send to programmable graphics hardware to determine the set's domination. We find that the GPU accelerated algorithm performs better than a comparable CPU based algorithm for board sizes greater than 9. To our knowledge, this paper presents the first algorithm to determine the minimum domination number of a chessboard graph using the GPU.

References

[1]
J. Bolz, I. Farmer, E. Grinspun, and P. Schröder. Sparse matrix solvers on the gpu: conjugate gradients and multigrid. ACM Trans. Graph., 22(3):917--924, 2003.
[2]
A. P. Buger and C. M. Mynhardt. An upper bound for the minimum number of queens covering the n x n chessboard. Discrete Appl. Math., 121:51--60, 2002.
[3]
I. Bugk, T. Foley, D. Horn, J. Sugerman, K. Fatahalian, M. Houston, and P. Hanrahan. Brook for gpus: stream computing on graphics hardware. ACM Trans. Graph., 23(3):777--786, 2004.
[4]
A. P. Burger, E. J. Cockayne, and C. M. Mynhardt. Domination numbers for the queen's graph. Bull. Inst. Combin. Appl., 10:73--82, 1994.
[5]
A. P. Burger, E. J. Cockayne, and C. M. Mynhardt. Domination and irredundance in the queen's graph. Discrete Math., 163:47--66, 1997.
[6]
A. P. Burger and C. M. Mynhardt. Properties of dominating sets of the queens graph Q4k+3.Utilitas Math., 57:237-253, 2000.
[7]
A. P. Burger and C. M. Mynhardt. Symmetry and domination in queens graphs. Bull. Inst. Combin. Appl., 29:11--14, 2000.
[8]
A. P. Burger and C. M. Mynhardt. An improved upper bound for queens domination numbers. Discrete Math., 266(1--3):119--131, 2003.
[9]
E. J. Cockayne. Chessboard domination problems. Discrete Math., 86:13--20, 1990.
[10]
C. F. A. de Jaenisch. Traité des Applications de l'Analyse Mathématique au Jeu des Échecs. St. Pétersbourg, 1862.
[11]
P. B. Gibbons and J. A. Webb. Some new results for the queens domination problem. Australas. J. Combin., 15:145--160, 1997.
[12]
M. J. Harris, W. V. Baxter, T. Scheuermann, and A. Lastra. Simulation of cloud dynamics on graphics hardware. In Proceedings of the ACM SIGGRAPH/EUROGRAPHICS conference on Graphics hardware, pages 92--101. Eurographics Association, 2003.
[13]
T. W. Haynes, S. T. Hedetniemi, and P. J. Slater. Funamentals of Domination in Graphs. Marcel-Dekker, New York, 1998.
[14]
S. T. Hedetniemi and R. C. Laskar. Introduction. Discrete Math., 86:3--9, 1990.
[15]
M. D. Kearse and P. B. Gibbons. Computational methods and new results for chessboard problems. Australas. J. Combin., 23:253--284, 2001.
[16]
B. Khailany, W. J. Dally, S. Rixner, U. J. Kapasi, J. D. Owens, and B. Towles. Exploring the vlsi scalability of stream processors. In Proceedings of the Ninth Symposium on High Performance Computer Architecture, February 2003.
[17]
J. Krüger and R. Westermann. Linear algebra operators for gpu implementation of numerical algorithms. ACM Trans. Graph., 22(3):908--916, 2003.
[18]
P. R. J. Östergård and W. D. Weakley. Values of domination numbers of the queen's graph. The Electronic Journal of Combinatorics, 8(1), 2001.
[19]
T. J. Purcell, I. Buck, W. R. Mark, and P. Hanrahan. Ray tracing on programmable graphics hardware. ACM Trans. Graph., 21(3):703--712, 2002.
[20]
Semiconductor Industry Association. International technology roadmap for semiconductors. http://public.itrs.net/, December 2002.
[21]
R. A. Wagner and R. M. Geist. The crippled queen placement problem. The Science of Computer Programming, 4:221--248, 1984.
[22]
W. D. Weakley. Domination in the queen's graph. In Y. Alavi and A. J. Schwenk, editors, Graph Theory, Combinatorics, and Algorithms, volume 2, pages 1223--1232, New York, 1995. Wiley-Interscience.
[23]
W. D. Weakley. A lower bound for domination numbers of the queen's graph. The Electronic Journal of Combinatorics, 8(1), 2001.
[24]
W. D. Weakley. Upper bounds for domination numbers of the queen's graph. Discrete Math., 242:229--243, 2002.

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences
ACMSE '06: Proceedings of the 44th annual ACM Southeast Conference
March 2006
823 pages
ISBN:1595933158
DOI:10.1145/1185448
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: 10 March 2006

Permissions

Request permissions for this article.

Check for updates

Qualifiers

  • Article

Conference

ACM SE06
ACM SE06: ACM Southeast Regional Conference
March 10 - 12, 2006
Florida, Melbourne

Acceptance Rates

ACMSE '06 Paper Acceptance Rate 100 of 244 submissions, 41%;
Overall Acceptance Rate 502 of 1,023 submissions, 49%

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • 0
    Total Citations
  • 247
    Total Downloads
  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 20 Feb 2025

Other Metrics

Citations

View Options

Login options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Figures

Tables

Media

Share

Share

Share this Publication link

Share on social media