Reconfigurations in Graphs and Grids

  • Gruia Calinescu
  • Adrian Dumitrescu
  • János Pach
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3887)


Let G be a connected graph, and let V and V ′ two n-element subsets of its vertex set V(G). Imagine that we place a chip at each element of V and we want to move them into the positions of V ′ (V and V ′ may have common elements). A move is defined as shifting a chip from v 1 to v 2 (v 1,v 2V(G)) on a path formed by edges of G so that no intermediate vertices are occupied. We give upper and lower bounds on the number of moves that are necessary, and analyze the computational complexity of this problem under various assumptions: labeled versus unlabeled chips, arbitrary graphs versus the case when the graph is the rectangular (infinite) planar grid, etc. We provide hardness and inapproximability results for several variants of the problem. We also give a linear-time algorithm which performs an optimal (minimum) number of moves for the unlabeled version in a tree, and a constant-ratio approximation algorithm for the unlabeled version in a graph. The graph algorithm uses the tree algorithm as a subroutine.


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  1. 1.
    Abellanas, M., Bereg, S., Hurtado, F., Olaverri, A.G., Rappaport, D., Tejel, J.: Moving coins. Computational Geometry: Theory and Applications (to appear)Google Scholar
  2. 2.
    Alimonti, P., Kann, V.: Hardness of approximating problems on cubic graphs. In: Bongiovanni, G., Bovet, D.P., Di Battista, G. (eds.) CIAC 1997. LNCS, vol. 1203, pp. 288–298. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  3. 3.
    Archer, A.: A modern treatment of the 15 puzzle. American Mathematical Monthly 106, 793–799 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Arora, S.: Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. J. of the ACM 45(5), 1–30 (1998)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Auletta, V., Monti, A., Parente, M., Persiano, P.: A linear-time algorithm for the feasibility of pebble motion in trees. Algorithmica 23, 223–245 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bar-Yehuda, R.: One for the Price of Two: A Unified approach for approximating covering problems. Algorithmica 27, 131–144 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bereg, S., Dumitrescu, A., Pach, J.: Sliding disks in the plane. In: Akiyama, J., Kano, M., Tan, X. (eds.) Japan Conference on Discrete and Computational Geometry 2004. LNCS, Springer, Heidelberg (2004) (to appear)Google Scholar
  8. 8.
    Bereg, S., Dumitrescu, A.: The lifting model for reconfiguration, Discrete & Computational Geometry (accepted); A preliminary version in Proceedings of the 21st Annual Symposium on Computational Geometry (SOCG 2005), Pisa, Italy, pp. 55–62 (June 2005)Google Scholar
  9. 9.
    Dumitrescu, A., Pach, J.: Pushing squares around, Graphs and Combinatorics (to appear); A preliminary version in Proceedings of the 20-th Annual Symposium on Computational Geometry (SOCG 2004), NY, June 2004, pp. 116–123 (2004)Google Scholar
  10. 10.
    Dumitrescu, A., Suzuki, I., Yamashita, M.: Motion planning for metamorphic systems: feasibility, decidability and distributed reconfiguration. IEEE Transactions on Robotics and Automation 20(3), 409–418 (2004)CrossRefGoogle Scholar
  11. 11.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)MATHGoogle Scholar
  12. 12.
    Goemans, M.X., Williamson, D.P.: The primal-dual method for approximation algorithms and its application to network design problems. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-Hard Problems. PWS Publishing Co. (1995)Google Scholar
  13. 13.
    Johnson, W.W.: Notes on the 15 puzzle. I. American Journal of Mathematics 2, 397–399 (1879)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kornhauser, D., Miller, G., Spirakis, P.: Coordinating pebble motion on graphs, the diameter of permutation groups, and applications. In: Proceedings of the 25th Symposium on Foundations of Computer Science (FOCS 1984), pp. 241–250 (1984)Google Scholar
  15. 15.
    Papadimitriou, C., Raghavan, P., Sudan, M., Tamaki, H.: Motion planning on a graph. In: Proceedings of the 35-th Symposium on Foundations of Computer Science (FOCS 1994), pp. 511–520 (1994)Google Scholar
  16. 16.
    Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43, 425–440 (1991)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ratner, D., Warmuth, M.: Finding a shortest solution for the (N × N)- extension of the 15-puzzle is intractable. Journal of Symbolic Computation 10, 111–137 (1990)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Story, W.E.: Notes on the 15 puzzle. II. American Journal of Mathematics 2, 399–404 (1879)CrossRefGoogle Scholar
  19. 19.
    Wilson, R.M.: Graph puzzles, homotopy, and the alternating group. Journal of Combinatorial Theory, Series B 16, 86–96 (1974)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Gruia Calinescu
    • 1
  • Adrian Dumitrescu
    • 2
  • János Pach
    • 3
  1. 1.Department of Computer ScienceIllinois Institute of TechnologyChicagoUSA
  2. 2.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA
  3. 3.Courant Institute of Mathematical SciencesNew YorkUSA

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