Abstract
For \(n\ge 4\), the \((n^2-1)\)-Puzzle is a generalization of the well known 15-Puzzle. Ratner et al. showed that finding a sequence of moves of minimum length for the \((n^2-1)\)-Puzzle is NP-hard, and many researches have been devoted to it. For the \((n^2-1)\)-Puzzle, a real-time algorithm is proposed by Parberry, which completes the puzzle in at most \(5n^3 - 9n^2/2 + 19n/2 -89\) moves and needs O(1) computation time per move, although there is no guarantee that the number of moves is minimum. In this paper, we follow the direction of the research by Parberry, and present an algorithm, which is obtained by modifying Parberry’s algorithm and giving a tight analysis. The number of moves by the new algorithm is smaller; it needs at most \(5n^3 -21n^2/2 + 35n/2 - 141\) moves.
This work was supported by JSPS KAKENHI Grant Number 25330018.
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Utsunomiya, K., Asahiro, Y. (2015). An Improvement of the Greedy Algorithm for the \((n^2-1)\)-Puzzle. In: Gervasi, O., et al. Computational Science and Its Applications -- ICCSA 2015. ICCSA 2015. Lecture Notes in Computer Science(), vol 9156. Springer, Cham. https://doi.org/10.1007/978-3-319-21407-8_33
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DOI: https://doi.org/10.1007/978-3-319-21407-8_33
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