Abstract
We study the complexity of the popular one player combinatorial game known as Flood-It. In this game the player is given an n ×n board of tiles, each of which is allocated one of c colours. The goal is to fill the whole board with the same colour via the shortest possible sequence of flood filling operations from the top left. We show that Flood-It is NP-hard for c ≥ 3, as is a variant where the player can flood fill from any position on the board. We present deterministic (c − 1) and randomised 2c/3 approximation algorithms and show that no polynomial time constant factor approximation algorithm exists unless P=NP. We then demonstrate that the number of moves required for the ‘most difficult’ boards grows like \(\Theta(\sqrt{c}\, n)\). Finally, we prove that for random boards with c ≥ 3, the number of moves required to flood the whole board is Ω(n) with high probability.
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References
Arthur, D., Clifford, R., Jalsenius, M., Montanaro, A., Sach, B.: The complexity of flood filling games (2010), http://arxiv.org/abs/1001.4420
Biedl, T.C., Demaine, E.D., Demaine, M.L., Fleischer, R., Jacobsen, L., Munro, J.I.: The complexity of Clickomania. In: More games of no chance. MSRI Publications, vol. 42, pp. 389–404. Cambridge University Press, Cambridge (2002)
Demaine, E.D., Hohenberger, S., Liben-Nowell, D.: Tetris is hard, even to approximate. In: Computing and Combinatorics, pp. 351–363 (2003)
Iwama, K., Miyano, E., Ono, H.: Drawing Borders Efficiently. Theory of Computing Systems 44(2), 230–244 (2009)
Jiang, T., Li, M.: On the approximation of shortest common supersequences and longest common subsequences. SIAM Journal of Computing 24(5), 1122–1139 (1995)
Kaye, R.: Minesweeper is NP-complete. The Mathematical Intelligencer 22(2), 9–15 (2000)
Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhäuser, Basel (1996)
Maier, D.: The complexity of some problems on subsequences and supersequences. Journal of the ACM 25(2), 322–336 (1978)
Munz, P., Hudea, I., Imad, J., Smith, R.J.: When zombies attack!: Mathematical modelling of an outbreak of zombie infection. In: Infectious Disease Modelling Research Progress, pp. 133–150. Nova Science, Bombay (2009)
Räihä, K.-J., Ukkonen, E.: The shortest common supersequence problem over binary alphabet is NP-complete. Theoretical Computer Science 16, 187–198 (1981)
Is this game NP-hard? (May 2009), http://valis.cs.uiuc.edu/blog/?p=2005
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Arthur, D., Clifford, R., Jalsenius, M., Montanaro, A., Sach, B. (2010). The Complexity of Flood Filling Games. In: Boldi, P., Gargano, L. (eds) Fun with Algorithms. FUN 2010. Lecture Notes in Computer Science, vol 6099. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13122-6_30
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DOI: https://doi.org/10.1007/978-3-642-13122-6_30
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