On the Competitive Complexity of Navigation Tasks

  • Christian Icking
  • Thomas Kamphans
  • Rolf Klein
  • Elmar Langetepe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2238)

Abstract

A strategy S solving a navigation task T is called competitive with ratio r if the cost of solving any instance t of T does not exceed r times the cost of solving t optimally. The competitive complexity of task T is the smallest possible value r any strategy S can achieve. We discuss this notion, and survey some tasks whose competitive complexities are known. Then we report on new results and ongoing work on the competitive complexity of exploring an unknown cellular environment.

References

  1. 1.
    S. Albers, K. Kursawe, and S. Schuierer. Exploring unknown environments with obstacles. In Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, pages 842–843, 1999.Google Scholar
  2. 2.
    E. M. Arkin, S. P. Fekete, and J. S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universität zu Köln, 1997.Google Scholar
  3. 3.
    S. Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pages 2–11, 1996.Google Scholar
  4. 4.
    R. Baeza-Yates, J. Culberson, and G. Rawlins. Searching in the plane. Inform. Comput., 106:234–252, 1993.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. Bellman. Problem 63-9*. SIAM Review, 5(2), 1963.Google Scholar
  6. 6.
    C. Bröcker and A. López-Ortiz. Position-independent street searching. In Proc. 6th Workshop Algorithms Data Struct., volume 1663 of Lecture Notes Comput. Sci., pages 241–252. Springer-Verlag, 1999.Google Scholar
  7. 7.
    C. Bröcker and S. Schuierer. Searching rectilinear streets completely. In Proc. 6th Workshop Algorithms Data Struct., volume 1663 of Lecture Notes Comput. Sci., pages 98–109. Springer-Verlag, 1999.Google Scholar
  8. 8.
    S. Carlsson and B. J. Nilsson. Computing vision points in polygons. Algorithmica, 24:50–75, 1999.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. Das, P. Heffernan, and G. Narasimhan. LR-visibility in polygons. Comput. Geom. Theory Appl., 7:37–57, 1997.MATHMathSciNetGoogle Scholar
  10. 10.
    P. Dasgupta, P. P. Chakrabarti, and S. C. DeSarkar. A new competitive algorithm for agent searching in unknown streets. In Proc. 16th Conf. Found. Softw. Tech. Theoret. Comput. Sci., volume 1180 of Lecture Notes Comput. Sci., pages 147–155. Springer-Verlag, 1996.Google Scholar
  11. 11.
    A. Datta, C. A. Hipke, and S. Schuierer. Competitive searching in polygons: Beyond generalised streets. In Proc. 6th Annu. Internat. Sympos. Algorithms Comput., volume 1004 of Lecture Notes Comput. Sci., pages 32–41. Springer-Verlag, 1995.Google Scholar
  12. 12.
    A. Datta and C. Icking. Competitive searching in a generalized street. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 175–182, 1994.Google Scholar
  13. 13.
    A. Datta and C. Icking. Competitive searching in a generalized street. Comput. Geom. Theory Appl., 13:109–120, 1999.MATHMathSciNetGoogle Scholar
  14. 14.
    H. Everett. Hamiltonian paths in non-rectangular grid graphs. Report 86-1, Dept. Comput. Sci., Univ. Toronto, Toronto, ON, 1986.Google Scholar
  15. 15.
    A. Fiat and G. Woeginger, editors. On-line Algorithms: The State of the Art, volume 1442 of Lecture Notes Comput. Sci. Springer-Verlag, 1998.Google Scholar
  16. 16.
    S. Gal. Search Games, volume 149 of Mathematics in Science and Engeneering. Academic Press, New York, 1980.Google Scholar
  17. 17.
    S. K. Ghosh and S. Saluja. Optimal on-line algorithms for walking with minimum number of turns in unknown streets. Comput. Geom. Theory Appl., 8(5):241–266, Oct. 1997.MATHMathSciNetGoogle Scholar
  18. 18.
    M. Grigni, E. Koutsoupias, and C. H. Papadimitriou. An approximation scheme for planar graph TSP. In Proc. 36th Annu. IEEE Sympos. Found. Comput. Sci., pages 640–645, 1995.Google Scholar
  19. 19.
    C. Hipke, C. Icking, R. Klein, and E. Langetepe. How to find a point on a line within a fixed distance. Discrete Appl. Math., 93:67–73, 1999.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    F. Hoffmann, C. Icking, R. Klein, and K. Kriegel. The polygon exploration problem. SIAM J. Comput., 2001. to appear.Google Scholar
  21. 21.
    C. Icking. Motion and Visibility in Simple Polygons. PhD thesis, Department of Computer Science, FernUniversität Hagen, 1994.Google Scholar
  22. 22.
    C. Icking, R. Klein, and E. Langetepe. An optimal competitive strategy for walking in streets. In Proc. 16th Sympos. Theoret. Aspects Comput. Sci., volume 1563 of Lecture Notes Comput. Sci., pages 110–120. Springer-Verlag, 1999.Google Scholar
  23. 23.
    C. Icking, R. Klein, and E. Langetepe. Searching a goal on m rays within a fixed distance. In Abstracts 15th European Workshop Comput. Geom., pages 137–139. INRIA Sophia-Antipolis, 1999.Google Scholar
  24. 24.
    C. Icking, A. López-Ortiz, S. Schuierer, and I. Semrau. Going home through an unknown street. Technical Report 228, Department of Computer Science, FernUniversität Hagen, Germany, 1998.Google Scholar
  25. 25.
    A. Itai, C. H. Papadimitriou, and J. L. Szwarcfiter. Hamilton paths in grid graphs. SIAM J. Comput., 11:676–686, 1982.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    R. Klein. Walking an unknown street with bounded detour. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 304–313, 1991.Google Scholar
  27. 27.
    R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325–351, 1992.MATHGoogle Scholar
  28. 28.
    J. M. Kleinberg. On-line search in a simple polygon. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 8–15, 1994.Google Scholar
  29. 29.
    E. Kranakis and A. Spatharis. Almost optimal on-line search in unknown streets. In Proc. 9th Canad. Conf. Comput. Geom., pages 93–99, 1997.Google Scholar
  30. 30.
    E. Langetepe. Design and Analysis of Strategies for Autonomous Systems in Motion Planning. PhD thesis, Department of Computer Science, FernUniversität Hagen, 2000.Google Scholar
  31. 31.
    A. López-Ortiz and S. Schuierer. Going home through an unknown street. In Proc. 4th Workshop Algorithms Data Struct., volume 955 of Lecture Notes Comput. Sci., pages 135–146. Springer-Verlag, 1995.Google Scholar
  32. 32.
    A. López-Ortiz and S. Schuierer. Generalized streets revisited. In Proc. 4th Annu. European Sympos. Algorithms, volume 1136 of Lecture Notes Comput. Sci., pages 546–558. Springer-Verlag, 1996.Google Scholar
  33. 33.
    A. López-Ortiz and S. Schuierer. Walking streets faster. In Proc. 5th Scand. Workshop Algorithm Theory, volume 1097 of Lecture Notes Comput. Sci., pages 345–356. Springer-Verlag, 1996.Google Scholar
  34. 34.
    A. López-Ortiz and S. Schuierer. The exact cost of exploring streets with CAB. Technical report, Institut für Informatik, Universität Freiburg, 1998.Google Scholar
  35. 35.
    A. López-Ortiz and S. Schuierer. Lower bounds for searching on generalized streets. University of New Brunswick, Canada, 1998.Google Scholar
  36. 36.
    J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric k-MST problem. In Proc. 7th ACM-SIAM Sympos. Discrete Algorithms, pages 402–408, 1996.Google Scholar
  37. 37.
    J. S. B. Mitchell. Geometric shortest paths and network optimization. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 633–701. Elsevier Science Publishers B.V. North-Holland, Amsterdam, 2000.CrossRefGoogle Scholar
  38. 38.
    S. Ntafos. Watchman routes under limited visibility. Comput. Geom. Theory Appl., 1(3):149–170, 1992.MATHMathSciNetGoogle Scholar
  39. 39.
    S. Schuierer. Lower bounds in on-line geometric searching. In 11th International Symposium on Fundamentals of Computation Theory, volume 1279 of Lecture Notes Comput. Sci., pages 429–440. Springer-Verlag, 1997.Google Scholar
  40. 40.
    S. Schuierer and I. Semrau. An optimal strategy for searching in unknown streets. In Proc. 16th Sympos. Theoret. Aspects Comput. Sci., volume 1563 of Lecture Notes Comput. Sci., pages 121–131. Springer-Verlag, 1999.Google Scholar
  41. 41.
    I. Semrau. Analyse und experimentelle Untersuchung von Strategien zum Finden eines Ziels in Straβenpolygonen. Diploma thesis, FernUniversität Hagen, 1996.Google Scholar
  42. 42.
    D. D. Sleator and R. E. Tarjan. Amortized efficiency of list update and paging rules. Commun. ACM, 28:202–208, 1985.CrossRefMathSciNetGoogle Scholar
  43. 43.
    L. H. Tseng, P. Heffernan, and D. T. Lee. Two-guard walkability of simple polygons. Internat. J. Comput. Geom. Appl., 8(1):85–116, 1998.MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    C. Umans and W. Lenhart. Hamiltonian cycles in solid grid graphs. In Proc. 38th Annu. IEEE Sympos. Found. Comput. Sci., pages 496–507, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Christian Icking
    • 1
  • Thomas Kamphans
    • 2
  • Rolf Klein
    • 2
  • Elmar Langetepe
    • 2
  1. 1.Praktische Informatik VIFernUniversität HagenHagenGermany
  2. 2.Institut für Informatik IUniversität BonnBonnGermany

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