An Improved Strategy for Exploring a Grid Polygon

  • Agnieszka Kolenderska
  • Adrian Kosowski
  • Michał Małafiejski
  • Paweł Żyliński
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5869)


We study the problem of exploring a simple grid polygon using a mobile robot. The robot starts from a location which is adjacent to the boundary of the polygon, and after exploring all the squares, has to return to its starting location. The robot is equipped with memory, but has no prior knowledge of the explored terrain. The view of the terrain is restricted to the four squares directly adjacent to the robot’s current location. The performance of the exploration strategy is measured in terms of the competitive ratio, with respect to the length of the optimal path for an exploration with complete knowledge of the terrain.

We propose a new exploration strategy which achieves a competitive ratio of 5/4, whereas the previously best approach [Icking, Kamphans, Klein, and Langetepe; Proc. COCOON’05] has a competitive ratio of 4/3. The analysis for our algorithm is tight. Moreover, we show that no exploration strategy is ever better than 20/17-competitive, thus improving the previous lower bound of 7/6.


Exploration problem Path planning Mobile robot Grid polygon Competitive ratio 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arkin, E.M., Fekete, S.P., Islam, K., Meijer, H., Mitchell, J.S.B., Nunez-Rodriguez, Y., Polishchuk, V., Rappaport, D., Xiao, H.: Not being (super)thin or solid is hard: A study of grid hamiltonicity. Computational Geometry: Theory and Applications 42(6-7), 582–605 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arkin, E.M., Fekete, S.P., Mitchell, J.S.B.: Approximation algorithms for lawn mowing and milling. Computational Geometry: Theory and Applications 17(1-2), 25–50 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5), 753–782 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. Theoretical Computer Science 345(2-3), 331–344 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gabriely, Y., Rimon, E.: Competitive on-line coverage of grid environments by a mobile robot. Computational Geometry: Theory and Applications 24(3), 197–224 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gąsieniec, L., Pelc, A., Radzik, T., Zhang, X.: Tree exploration with logarithmic memory. In: Proceedings of the 19th ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 585–594 (2007)Google Scholar
  7. 7.
    Gordon, V.S., Orlovich, Y.L., Werner, F.: Hamiltonian properties of triangular grid graphs. Discrete Mathematics 308(24), 6166–6188 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grigni, M., Koutsoupias, E., Papadimitriou, C.: An approximation scheme for planar graph TSP. In: Proceedings of the Thirty-Sixth Annual IEEE Symposium on the Foundations of Computer Science (FOCS 1995), pp. 387–411 (1995)Google Scholar
  9. 9.
    Herrmann, D., Kamphans, T., Langetepe, E.: Exploring simple triangular and hexagonal grid polygons online. In: Abstracts of the 24th European Workshop on Computational Geometry, pp. 177–180 (2008)Google Scholar
  10. 10.
    Icking, C., Kamphans, T., Klein, R., Langetepe, E.: Exploring an unknown cellular environment. In: Abstracts of the 16th European Workshop on Computational Geometry, pp. 140–143 (2000)Google Scholar
  11. 11.
    Icking, C., Kamphans, T., Klein, R., Langetepe, E.: Exploring simple grid polygons. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 524–533. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamilton paths in grid graphs. SIAM Journal on Computing 11(4), 676–686 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kamphans, T.: Models and Algorithms for Online Exploration and Search. Ph.D. thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn (2005)Google Scholar
  14. 14.
    Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing 28(4), 1298–1309 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Reingold, O.: Undirected ST-Connectivity in Log-Space. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005), pp. 376–385 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Agnieszka Kolenderska
    • 1
  • Adrian Kosowski
    • 1
    • 2
  • Michał Małafiejski
    • 1
  • Paweł Żyliński
    • 3
  1. 1.Dept of Algorithms and System ModelingGdańsk University of TechnologyGdańskPoland
  2. 2.LaBRIUniversité Bordeaux 1 - CNRSTalenceFrance
  3. 3.Institute of Computer ScienceUniversity of GdańskGdańskPoland

Personalised recommendations