Streaming Geometric Optimization Using Graphics Hardware

  • Pankaj K. Agarwal
  • Shankar Krishnan
  • Nabil H. Mustafa
  • Suresh Venkatasubramanian
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2832)

Abstract

In this paper we propose algorithms for solving a variety of geometric optimization problems on a stream of points in ℝ2 or ℝ3. These problems include various extent measures (e.g. diameter, width, smallest enclosing disk), collision detection (penetration depth and distance between polytopes), and shape fitting (minimum width annulus, circle/line fitting). The main contribution of this paper is a unified approach to solving all of the above problems efficiently using modern graphics hardware. All the above problems can be approximated using a constant number of passes over the data stream. Our algorithms are easily implemented, and our empirical study demonstrates that the running times of our programs are comparable to the best implementations for the above problems. Another significant property of our results is that although the best known implementations for the above problems are quite different from each other, our algorithms all draw upon the same set of tools, making their implementation significantly easier.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P., Guibas, L., Har-Peled, S., Rabinovitch, A., Sharir, M.: Penetration depth of two convex polytopes in 3d. Nordic J. Comput. 7(3), 227–240 (2000)MathSciNetMATHGoogle Scholar
  2. 2.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.: Approximating extent measures of points (2002) (submitted for publication)Google Scholar
  3. 3.
    Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30, 412–458 (1998)CrossRefGoogle Scholar
  4. 4.
    Chan, T.M.: Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. In: Proc. 16th Annu. Sympos. on Comp. Geom., pp. 300–309 (2000)Google Scholar
  5. 5.
    Duncan, C., Goodrich, M., Ramos, E.: Efficient approximation and optimization algorithms for computational metrology. In: ACM-SIAM Symp. Discrete Algo., pp. 121–130 (1997)Google Scholar
  6. 6.
    Feigenbaum, J., Kannan, S., Zhang, J.: Computing diameter in the streaming and sliding-window models. DIMACS Working Group on Streaming Data Analysis II (2003)Google Scholar
  7. 7.
    Fournier, A., Fussell, D.: On the power of the frame buffer. ACM Transactions on Graphics, 103–128 (1988)Google Scholar
  8. 8.
    Guha, S., Krishnan, S., Munagala, K., Venkatasubramanian, S.: The power of a two-sided depth test and its application to CSG rendering and depth extraction. Tech. rep., AT&T (2002)Google Scholar
  9. 9.
  10. 10.
    Har-Peled, S.: A practical approach for computing the diameter of a point-set. In: Proc. 17th Annu. Symp. on Comp. Geom., pp. 177–186 (2001)Google Scholar
  11. 11.
    Hersberger, J., Suri, S.: Convex hulls and related problems in data streams. In: SIGMOD-DIMACS MPDS Workshop (2003)Google Scholar
  12. 12.
    Hoff III, K.E., Keyser, J., Lin, M., Manocha, D., Culver, T.: Fast computation of generalized Voronoi diagrams using graphics hardware. In: Computer Graphics. Annual Conference Series, vol 33, pp. 277–286 (1999)Google Scholar
  13. 13.
    Indyk, P.: Stream-based geometric algorithms. In: SIGMOD-DIMACS MPDS Workshop (2003)Google Scholar
  14. 14.
    Kim, Y.J., Lin, M.C., Manocha, D.: Fast penetration depth estimation between polyhedral models using hierarchical refinement. In: 6th Intl. Workshop on Algo. Founda. of Robotics (2002)Google Scholar
  15. 15.
    Korn, F., Muthukrishnan, S., Srivastava, D.: Reverse nearest neighbour aggregates over data streams. In: Proc. 28th Conf. VLDB (2002)Google Scholar
  16. 16.
    Krishnan, S., Mustafa, N., Venkatasubramanian, S.: Hardware-assisted computation of depth contours. In: Proc. 13th ACM-SIAM Symp. on Discrete Algorithms, pp. 558–567 (2002)Google Scholar
  17. 17.
    Majhi, J., Janardan, R., Smid, M., Schwerdt, J.: Multi-criteria geometric optimization problems in layered manufacturing. In: Proc. 14th Annu. Symp. on Comp. Geom, pp. 19–28 (1998)Google Scholar
  18. 18.
    Malandain, G., Boissonnat, J.-D.: Computing the diameter of a point set. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, p. 197. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Mustafa, N., Koutsofios, E., Krishnan, S., Venkatasubramanian, S.: Hardware assisted view dependent map simplification. In: Proc. 17th Annu. Symp. on Comp. Geom, pp. 50–59 (2001)Google Scholar
  20. 20.
    Muthukrishnan, S.: Data streams: Algorithms and applications. Tech. rep., Rutgers University (2003)Google Scholar
  21. 21.
    Woo, M., Neider, J., Davis, T., Shreiner, D.: OpenGL(R) Programming Guide: The Official Guide to Learning OpenGL. Version 1.2, 3 edn. Addison-Wesley, Reading (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Shankar Krishnan
    • 2
  • Nabil H. Mustafa
    • 1
  • Suresh Venkatasubramanian
    • 2
  1. 1.Dept. of Computer ScienceDuke UniversityDurhamU.S.A.
  2. 2.AT&T Labs – ResearchFlorham ParkUSA

Personalised recommendations