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All the needles in a haystack: Can exhaustive search overcome combinatorial chaos?

  • Jürg Nievergelt
  • Ralph Gasser
  • Fabian Mäser
  • Christoph Wirth
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1000)

Abstract

For half a century since computers came into existence, the goal of finding elegant and efficient algorithms to solve “simple” (welldefined and well-structured) problems has dominated algorithm design. Over the same time period, both processing and storage capacity of computers have increased roughly by a factor of 106. The next few decades may well give us a similar rate of growth in raw computing power, due to various factors such as continuing miniaturization, parallel and distributed computing. If a quantitative change of orders of magnitude leads to qualitative changes, where will the latter take place? Many problems exhibit no detectable regular structure to be exploited, they appear “chaotic”, and do not yield to efficient algorithms. Exhaustive search of large state spaces appears to be the only viable approach. We survey techniques for exhaustive search, typical combinatorial problems that have been solved, and present one case study in detail.

Keywords

State Space Exhaustive Search Travel Salesman Problem Game Tree Forward Search 
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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References

  1. [Allen 89]
    J. D. Allen, A Note on the Computer Solution of Connect-Four, in: Heuristic Programming in Artificial Intelligence 1: the First Computer Olympiad (eds. D.N.L. Levy and D.F. Beal), Ellis Horwood, Chichester, England. 1989, pp. 134–135.Google Scholar
  2. [Allis 94]
    L.V. Allis, Searching for Solutions in Games and Artificial Intelligence, Doctoral dissertation, University of Limburg, Maastricht, (1994).Google Scholar
  3. [Appell 77]
    K. Appell and W. Haken, The Solution of the Four-Color-Map Problem, Scientific American (Oct. 1977) 108–121.Google Scholar
  4. [Avis 92]
    D. Avis and K. Fukuda, Reverse Search for Enumeration, Report, U. Tsukuba. Discrete Applied Math. (1992) (to appear).Google Scholar
  5. [Berlekamp 82]
    E. Berlekamp, J.H. Conway and R.K. Guy, Winning Ways for your Mathematical Plays, Academic Press, London, 1982.Google Scholar
  6. [Culberson 94]
    J. Culberson and J. Schaeffer, Efficiently Searching the 15-Puzzle, internal report, University of Alberta, Edmonton, Canada, 1994.Google Scholar
  7. [Gasser 90]
    R. Gasser, Applying Retrograde Analysis to Nine Men's Morris, in: Heuristic Programming in Artificial Intelligence 2: the Second Computer Olympiad (eds. D.N.L. Levy and D.F. Beal), Ellis Horwood, Chichester, England. 1990, pp. 161–173.Google Scholar
  8. [Gasser 91]
    R. Gasser, Endgame Database Compression for Humans and Machines, in: Heuristic Programming in Artificial Intelligence 3: the Third Computer Olympiad (eds. H.J. van den Herik and L.V. Allis), Ellis Horwood, Chichester, England. 1990, pp. 180–191.Google Scholar
  9. [Gasser 94]
    R. Gasser, J. Nievergelt, Es ist entschieden: Das Muehlespiel ist unentschieden, Informatik Spektrum 17:5 (1994) 314–317.Google Scholar
  10. [Gasser 95]
    R. Gasser, Harnessing Computational Resources for Efficient Exhaustive Search, Doctoral dissertation, ETH Zürich, 1995.Google Scholar
  11. [Gasser, Waldvogel 95]
    R. Gasser, J. Waldvogel, Primes in Intervals of Fixed Length, in preparation (1995).Google Scholar
  12. [Hansson 92]
    O. Hansson, A. Mayer and M. Yung, Criticizing Solutions to Relaxed Models Yields Powerful Admissible Heuristics, Information Sciences 63 (1992) 207–227.CrossRefGoogle Scholar
  13. [Herik 86]
    H.J. van den Herik and I.S. Herschberg, A Data Base on Data Bases, ICCA Journal 9:1 (1986) 29.Google Scholar
  14. [Horgan 93]
    J. Horgan, The Death of Proof, Scientific American (1993) 74–82.Google Scholar
  15. [Knuth 75]
    D. E. Knuth and R. W. Moore, An analysis of Alpha-Beta Pruning, Artificial Intelligence 6 (1975) 293–326.CrossRefGoogle Scholar
  16. [Kociemba 92]
    H. Kociemba, Close to God's Algorithm, Cubism for Fun 28 (1992) 10–13.Google Scholar
  17. [Korf 85]
    R.E. Korf, Depth-first Iterative Deepening, An Optimal Admissible Tree Search, Artificial Intelligence 27 (1985) 97–109.CrossRefMathSciNetGoogle Scholar
  18. [Lake 94]
    R. Lake, J. Schaeffer and P. Lu, Solving Large Retrograde Analysis Problems Using a Network of Workstations, internal report, University of Alberta, Edmonton, Canada, 1994.Google Scholar
  19. [Lawler 85]
    E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, The Traveling Salesman Problem — A Guided Tour of Combinatorial Optimization, Wiley, 1985.Google Scholar
  20. [Lehmer 53]
    D. H. Lehmer, The Sieve Problem for All-Purpose Computers, Mathematical Tables and Aids to Computation 7 (1953) 6–14.Google Scholar
  21. [Lehmer 64]
    D. H. Lehmer, The Machine Tools of Combinatorics, in: E. F. Beckenbach (ed.), Applied Combinatorial Mathematics, Wiley, 1964, Chapter 1, pp. 5–31.Google Scholar
  22. [Levy 91]
    D. Levy, Nine Men's Morris, Heuristic Programming in Artificial Intelligence 2: the Second Computer Olympiad (eds. D.N.L. Levy and D.F. Beal), Ellis Horwood, Chichester, England. 1991, 55–57.Google Scholar
  23. [Nievergelt 93]
    J. Nievergelt, Experiments in Computational Heuristics, and their Lessons for Software and Knowledge Engineering, Advances in Computers, Vol 37 (M. Yovits, ed.), Academic Press. 1993, pp. 167–205.Google Scholar
  24. [Nievergelt 95]
    J. Nievergelt, N. Deo, Metric Graphs Elastically Embeddable in the Plane, Information Processing Letters (1995) (to appear).Google Scholar
  25. [Nunn 92–95]
    J. Nunn, Secrets of Rook Endings 1992, Secrets of Pawnless Endings 1994, Secrets of Minor Piece Endings, to appear 1995, Batsford Ltd. UK.Google Scholar
  26. [Patashnik 80]
    O. Patashnik, Qubic: 4×4×4 Tic-Tac-Toe, Mathematics Magazine 53:4 (1980) 202–216.Google Scholar
  27. [Pearl 84]
    J. Pearl, Heuristics, Addison-Wesley Publ. Comp., Reading, MA, 1984.Google Scholar
  28. [Pohl 71]
    I. Pohl, Bi-Directional Search, Machine Intelligence, Vol. 6 (eds. B. Metzler and D. Michie), American Elsevier, New York. 1971, pp. 127–140.Google Scholar
  29. [Ratner 90]
    D. Ratner and M. Warmuth, Finding a Shortest Solution for the (N x N)-Extension of the 15-Puzzle is Intractable, J. Symbolic Computation 10 (1990) 111–137.Google Scholar
  30. [Reinelt 95]
    G. Reinelt, “TSPLIB”, available on the Worldwide Web (WWW), http://www.iwr.uni-heidelberg.de/iwr/comopt/soft/TSPLIB95/TSPLIB.htmlGoogle Scholar
  31. [Schaeffer 92]
    J. Schaeffer, J. Culberson, N. Treloar, B. Knight, P. Lu and D. Szafron, A World Championship Caliber Checkers Program, Artificial Intelligence 53 (1992) 273–289.CrossRefGoogle Scholar
  32. [Stiller 91]
    L. Stiller, Group Graphs and Computational Symmetry on Massively Parallel Architecture, The Journal of Supercomputing 5 (1991) 99–117.CrossRefGoogle Scholar
  33. [Stroehlein 70]
    T. Stroehlein, Untersuchungen ueber Kombinatorische Spiele, Doctoral thesis, Technische Hochschule München, München, 1970.Google Scholar
  34. [Sucrow 92]
    B. Sucrow, Algorithmische und Kombinatorische Untersuchungen zum Puzzle von Sam Loyd, Doctoral thesis, University of Essen, 1992.Google Scholar
  35. [Tay 95]
    S. Tay, J. Nievergelt, A Minmax Relationship between Embeddable and Rigid Graphs, in preparation (1995).Google Scholar
  36. [Thompson 86]
    K. Thompson, Retrograde Analysis of Certain Endgames, ICCA Journal 9:3 (1986) 131–139.Google Scholar
  37. [Thompson 91]
    K. Thompson, Chess Endgames Vol. 1, ICCA Journal 14:1 (1991) 22.Google Scholar
  38. [Thompson 92]
    K. Thompson, Chess Endgames Vol. 2, ICCA Journal 15:3 (1992) 149.Google Scholar
  39. [von Neumann 43]
    J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton Univ. Press, Princeton, NJ, 1953.Google Scholar
  40. [Wells 71]
    M. Wells, Elements of Combinatorial Computing, Pergamon Press, Oxford, 1971.Google Scholar
  41. [Wilson 74]
    R.M. Wilson, Graph Puzzles, Homotopy, and the Alternating Group, J. Combinatorial Theory, Series B: 16 (1974) 86–96.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Jürg Nievergelt
    • 1
  • Ralph Gasser
    • 1
  • Fabian Mäser
    • 1
  • Christoph Wirth
    • 1
  1. 1.Department of Computer ScienceETH ZürichZürichSwitzerland

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