Advertisement

A Lower Bound for Randomized Searching on m Rays

  • Sven Schuierer
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2598)

Abstract

We consider the problem of on-line searching on m rays. A point robot is assumed to stand at the origin of m concurrent rays one of which contains a goal g that the point robot has to find. Neither the ray containing g nor the distance to g are known to the robot. The only way the robot can detect g is by reaching its location. We use the competitive ratio as a measure of the performance of a search strategy, that is, the worst case ratio of the total distance D R traveled by the robot to find g to the distance D from the origin to g. We present a new proof of a tight lower bound of the competitive ratio for randomized strategies to search on m rays. Our proof allows us to obtain a lower bound on the optimal competitive ratio for a fixed m even if the distance of the goal to the origin is bounded from above. Finally, we show that the optimal competitive ratio converges to 1+2(e α - 1)/α2 m∼1+2·1.544m, for large m where á minimizes the function (e x-1)/x 2.

Keywords

Step Length Competitive Ratio Simple Polygon Goal Position Hiding Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Baeza-Yates, J. Culberson, and G. Rawlins. Searching in the plane. Information and Computation, 106:234–252, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Margrit Betke, Ronald L. Rivest, and Mona Singh. Piecemeal learning of an unknown environment. In Sixth ACM Conference on Computational Learning Theory (COLT 93), pages 277–286, July 1993.Google Scholar
  3. 3.
    K-F. Chan and T. W. Lam. An on-line algorithm for navigating in an unknown environment. International Journal of Computational Geometry & Applications, 3:227–244, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. Datta, Ch. Hipke, and S. Schuierer. Competitive searching in polygons-beyond generalized streets. In Proc. Sixth Annual International Symposium on Algorithms and Computation, pages 32–41. LNCS 1004, 1995.Google Scholar
  5. 5.
    A. Datta and Ch. Icking. Competitive searching in a generalized street. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 175–182, 1994.Google Scholar
  6. 6.
    A. Datta and Ch. Icking. Competitive searching in a generalized street. Comput. Geom. Theory Appl, 13:109–120, 1999.zbMATHMathSciNetGoogle Scholar
  7. 7.
    S. Gal. Search Games. Academic Press, 1980.Google Scholar
  8. 8.
    Christian Icking and Rolf Klein. Searching for the kernel of a polygon: A competitive strategy. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 258–266, 1995.Google Scholar
  9. 9.
    M. Y. Kao, Y. Ma, M. Sipser, and Y. Yin. Optimal constructions of hybrid algorithms. J. of Algorithms, 29:142–164, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Y. Kao, J. H. Reif, and S. R. Tate. Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem. Information and Computation, 131(1):63–80, 1997.CrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325–351, 1992.zbMATHGoogle Scholar
  12. 12.
    J. M. Kleinberg. On-line search in a simple polygon. In Proc. of 5th ACM-SIAM Symp. on Discrete Algorithms, pages 8–15, 1994.Google Scholar
  13. 13.
    A. López-Ortiz and S. Schuierer. Position-independent near optimal searching and on-line recognition in star polygons. In Proc. 4th Workshop on Algorithms and Data Structures, pages 284–296. LNCS, 1997.Google Scholar
  14. 14.
    A. López-Ortiz und S. Schuierer. Lower bounds for streets and generalized streets. Intl. Journal of Computational Geometry & Applications, 11(4):401–422, 2001.zbMATHCrossRefGoogle Scholar
  15. 15.
    C. H. Papadimitriou and M. Yannakakis. Shortest paths without a map. In Proc. 16th Internat. Colloq. Automata Lang. Program., volume 372 of Lecture Notes in Computer Science, pages 610–620. Springer-Verlag, 1989.Google Scholar
  16. 16.
    S. Schuierer. Efficient robot self-localization in simple polygons. In H. Christensen, H. Bunke, and H. Noltemeier, editors, Sensor Based Intelligent Robots, volume 1724, pages 220–239. LNAI, 1999.Google Scholar
  17. 17.
    D. D. Sleator and R. E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28:202–208, 1985.CrossRefMathSciNetGoogle Scholar
  18. 18.
    A. Yao. Probabilistic computations: Towards a unified measure of complexity. In Proc. 18th IEEE Symp. on Foundations of Comp. Sci., pages 222–227, 1977.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Sven Schuierer
    • 1
  1. 1.Novartis Pharma AGBaselSwitzerland

Personalised recommendations