Exploring Simple Grid Polygons

  • Christian Icking
  • Tom Kamphans
  • Rolf Klein
  • Elmar Langetepe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3595)


We investigate the online exploration problem of a short-sighted mobile robot moving in an unknown cellular room without obstacles. The robot has a very limited sensor; it can determine only which of the four cells adjacent to its current position are free and which are blocked, i.e., unaccessible for the robot. Therefore, the robot must enter a cell in order to explore it. The robot has to visit each cell and to return to the start. Our interest is in a short exploration tour, i.e., in keeping the number of multiple cell visits small. For abitrary environments without holes we provide a strategy producing tours of length \(S \leq C + \frac{1}{2} E -- 3\), where C denotes the number of cells – the area – , and E denotes the number of boundary edges – the perimeter – of the given environment. Further, we show that our strategy is competitive with a factor of \(\frac43\), and give a lower bound of \(\frac76\) for our problem. This leaves a gap of only \(\frac16\) between the lower and the upper bound.


Robot navigation exploration covering online algorithms competitive analysis lower bounds grid polygons 


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  1. 1.
    Arkin, E.M., Fekete, S.P., Mitchell, J.S.B.: Approximation algorithms for lawn mowing and milling. Technical report, Mathematisches Institut, Universität zu Köln (1997)Google Scholar
  2. 2.
    Arora, S.: Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In: Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pp. 2–11 (1996)Google Scholar
  3. 3.
    Deng, X., Kameda, T., Papadimitriou, C.: How to learn an unknown environment I: The rectilinear case. J. ACM 45(2), 215–245 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Everett, H.: Hamiltonian paths in non-rectangular grid graphs. Report 86-1, Dept. Comput. Sci., Univ. Toronto, Toronto, ON (1986)Google Scholar
  5. 5.
    Fiat, A. (ed.): Dagstuhl Seminar 1996. LNCS, vol. 1442. Springer, Heidelberg (1998)MATHGoogle Scholar
  6. 6.
    Gabriely, Y., Rimon, E.: Competitive on-line coverage of grid environments by a mobile robot. Comput. Geom. Theory Appl. 24, 197–224 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Grigni, M., Koutsoupias, E., Papadimitriou, C.H.: An approximation scheme for planar graph TSP. In: Proc. 36th Annu. IEEE Sympos. Found. Comput. Sci., pp. 640–645 (1995)Google Scholar
  8. 8.
    Handel, U., Icking, C., Kamphans, T., Langetepe, E., Meiswinkel, W.: Gridrobot – an environment for simulating exploration strategies in unknown cellular areas. Java Applet (2000), http://www.geometrylab.de/Gridrobot/
  9. 9.
    Hoffmann, F., Icking, C., Klein, R., Kriegel, K.: The polygon exploration problem. SIAM J. Comput. 31, 577–600 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Icking, C., Kamphans, T., Klein, R., Langetepe, E.: Exploring an unknown cellular environment. In: Abstracts 16th European Workshop Comput. Geom., Ben-Gurion University of the Negev, pp. 140–143 (2000)Google Scholar
  11. 11.
    Icking, C., Kamphans, T., Klein, R., Langetepe, E.: On the competitive complexity of navigation tasks. In: Hager, G.D., Christensen, H.I., Bunke, H., Klein, R. (eds.) Dagstuhl Seminar 2000. LNCS, vol. 2238, pp. 245–258. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Icking, C., Klein, R., Langetepe, E.: Searching for the kernel of a polygon: A competitive strategy using self-approaching curves. Technical Report 211, Department of Computer Science, FernUniversität Hagen, Germany (1997)Google Scholar
  13. 13.
    Icking, C., Klein, R., Langetepe, E., Schuierer, S., Semrau, I.: An optimal competitive strategy for walking in streets. SIAM J. Comput. 33, 462–486 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Itai, C.H.: Papadimitriou, and J. L. Szwarcfiter. Hamilton paths in grid graphs. SIAM J. Comput. 11, 676–686 (1982)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kamphans, T.: Models and Algorithms for Online Exploration and Search. PhD thesis, University of Bonn (to appear)Google Scholar
  16. 16.
    Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP. SIAM J. Comput. 28, 1298–1309 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier Science Publishers B.V, North-Holland (2000)CrossRefGoogle Scholar
  18. 18.
    Ntafos, S.: Watchman routes under limited visibility. Comput. Geom. Theory Appl. 1(3), 149–170 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Umans, C., Lenhart, W.: Hamiltonian cycles in solid grid graphs. In: Proc. 38th Annu. IEEE Sympos. Found. Comput. Sci., pp. 496–507 (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Christian Icking
    • 1
  • Tom Kamphans
    • 2
  • Rolf Klein
    • 2
  • Elmar Langetepe
    • 2
  1. 1.Praktische Informatik VIUniversity of HagenHagenGermany
  2. 2.Computer Science IUniversity of BonnBonnGermany

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