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Finding Optimal Solutions to Atomix

  • Falk Hüffner
  • Stefan Edelkamp
  • Henning Fernau
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2174)

Abstract

We present solutions of benchmark instances to the solitaire computer game Atomix found with different heuristic search methods. The problem is PSPACE-complete. An implementation of the heuristic algorithm A* is presented that needs no priority queue, thereby having very low memory overhead. The limited memory algorithm IDA* is handicapped by the fact that, due to move transpositions, duplicates appear very frequently in the problem space; several schemes of using memory to mitigate this weakness are explored, among those, “partial” schemes which trade memory savings for a small probability of not finding an optimal solution. Even though the underlying search graph is directed, backward search is shown to be viable, since the branching factor can be proven to be the same as for forward search.

Keywords

Hash Table Priority Queue State Table Goal Position Runtime Overhead 
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.

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Reference

  1. 1.
    J. C. Culberson. Sokoban is PSPACE-complete. In E. Lodi, L. Pagli, and N. Santoro, editors, Proc. FUN-98, pp. 65–76. Carleton Scientific, Waterloo, 1998.Google Scholar
  2. 2.
    J. C. Culberson and J. Schaeffer. Pattern databases. Computational Intelligence, 14(3):318–334, 1998.CrossRefMathSciNetGoogle Scholar
  3. 3.
    E. D. Demaine, M. L. Demaine, and J. O’Rourke. PushPush and Push-1 are NPhard in 2D. In Proc. 12th Canadian Conf. Computational Geometry, pp. 211–219, Fredericton, 2000.Google Scholar
  4. 4.
    J. Eckerle and S. Schuierer. Efficient memory-limited graph search. In Proc. KI-95, vol. 981 of LNCS/LNAI, pp. 101–112. Springer, Berlin, 1995.Google Scholar
  5. 5.
    S. Edelkamp and R. E. Korf. The branching factor of regular search spaces. In Proc. AAAI-98/IAAI-98, pp. 299–304. AAAI Press, Menlo Park, 1998.Google Scholar
  6. 6.
    S. Edelkamp, A. L. Lafuente, and S. Leue. Protocol verification with heuristic search. In AAAI-Spring Symposium on Model-based Validation of Intelligence, pp. 75–83, 2001.Google Scholar
  7. 7.
    S. Edelkamp and U. Meyer. Theory and Practice of Time-Space Trade-Offs in Memory Limited Search. This volume.Google Scholar
  8. 8.
    S. Edelkamp and S. Schrödl. Localizing A. In Proc. AAAI-00/IAAI-00, pp. 885–890. AAAI Press, Menlo Park, 2000.Google Scholar
  9. 9.
    O. Hansson, A. Mayer, and M. Yung. Criticizing solutions to relaxed models yields powerful admissible heuristics. Information Sciences, 63(3):207–227, 1992.CrossRefGoogle Scholar
  10. 10.
    P. E. Hart, N. J. Nilsson and B. Raphael. A formal basis for heuristic determination of minimum path cost. IEEE Transactions on on Systems Science and Cybernetics, 4:100–107, 1968.CrossRefGoogle Scholar
  11. 11.
    G. J. Holzmann. On limits and possibilities of automated protocol analysis. In Proc. International Conf. Protocol Specification, Testing, and Verification, pp. 339–346, Zürich, 1987. North-Holland, Amsterdam.Google Scholar
  12. 12.
    M. Holzer and S. Schwoon. Assembling Molecules in Atomix is Hard. Technical Report 0101, Institut für Informatik, Technische Universität München, May 2001.Google Scholar
  13. 13.
    A. Junghanns and J. Schaeffer. Sokoban: Enhancing general single-agent search methods using domain knowledge. Artificial Intelligence, 129(1-2):219–251, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    R. E. Korf. Depth-first iterative-deepening: An optimal admissible tree search. Artificial Intelligence, 27(1):97–109, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. E. Korf and L. A. Taylor. Finding optimal solutions to the Twenty-Four Puzzle. In Proc. AAAI-96/IAAI-96, pp. 1202–1207. AAAI Press, Menlo Park, 1996.Google Scholar
  16. 16.
    H. W. Kuhn. The Hungarian Method for the Assignment Problem. In Naval Res. Logist. Quart, pp. 83–98, 1955.Google Scholar
  17. 17.
    D. Ratner and M. K. Warmuth. The (n2-1)-puzzle and related relocation problems. Journal of Symbolic Computation, 10(2):111–137, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    A. Reinefeld and T. A. Marsland. Enhanced iterative-deepening search. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(7):701–710, 1994.CrossRefGoogle Scholar
  19. 19.
    U. Stern and D. L. Dill. Combining state space caching and hash compaction. In Methoden des Entwurfs und der Verifikation digitaler Systeme, 4. GI/ITG/GME Workshop, Berichte aus der Informatik, pp. 81–90, 1996. Shaker, Aachen.Google Scholar
  20. 20.
    P. Wolper and D. Leroy. Reliable hashing without collision detection. In Proc. CAV-93, pp. 59–70. Springer, Berlin, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Falk Hüffner
    • 1
  • Stefan Edelkamp
    • 2
  • Henning Fernau
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Wilhelm-Schickard Institut für InformatikUniversität TübingenTübingenFed. Rep. of Germany
  2. 2.Universität FreiburgInstitut für InformatikFreiburgFed. Rep. of Germany

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