Abstract

The game of Checkers can easily be generalized to be played on an N by N board and the complexity of deciding questions about positions regarded as a function of N. This paper considers mainly the question of whether a particular player can force a win from a given position and also the question of what is the best move in a given position. Each of these problems is shown to be complete in exponential time. This means that any algorithm to solve them must take time which rises exponentially with respect to some power of N and moreover that they are amongst the hardest problems with such a time bound.For instance if there are any problems solvable in exponential time but not in polynomial space, then these two problems are amongst them.

Keywords

  1. checkers
  2. two person game
  3. exponential time complete

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References

1.
A. S. Fraenkel, M. R. Garey, D. S. Johnson, T. Schaefer, Y. Yesha, The complexity of Checkers on an N by N board, preliminary report, Proc. 19th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 1978, 55–64
2.
Aviezri S. Fraenkel, David Lichtenstein, Computing a perfect strategy for $n\times n$ chess requires time exponential in n, J. Combin. Theory Ser. A, 31 (1981), 199–214
3.
Takumi Kasai, Akeo Adachi, Shigeki Iwata, Classes of pebble games and complete problems, SIAM J. Comput., 8 (1979), 574–586
4.
David Lichtenstein, Michael Sipser, GO is polynomial-space hard, J. Assoc. Comput. Mach., 27 (1980), 393–401
5.
J. M. Robson, Optimal storage allocation decisions require exponential time, Australian Computer Science Communications, 3 (1981), 162–172
6.
Larry J. Stockmeyer, Ashok K. Chandra, Provably difficult combinatorial games, SIAM J. Comput., 8 (1979), 151–174

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Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 252 - 267
ISSN (online): 1095-7111

History

Submitted: 25 August 1981
Published online: 13 July 2006

Keywords

  1. checkers
  2. two person game
  3. exponential time complete

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