# The Complexity of Flood Filling Games

David Arthur, Rapha?, Markus Jalsenius, Ashley Montanaro, Benjamin Sach, The Complexity of Flood Filling Games. *Proceedings of the 5th International Conference on FUN WITH ALGORITHMS (FUN)*, pp. 307–318. June 2010. No electronic version available. External information
## 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 by 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 > 2, 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>2, the number of moves required to flood the whole board is Omega(n) with high probability.

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