The Complexity of Flood Filling Games

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 where each tile is allocated one of c colours. The goal is to make the colours of all tiles equal via the shortest possible sequence of flooding operations. In the standard version, a flooding operation consists of the player choosing a colour k, which then changes the colour of all the tiles in the monochromatic region connected to the top left tile to k. After this operation has been performed, neighbouring regions which are already of the chosen colour k will then also become connected, thereby extending the monochromatic region of the board. We show that finding an optimal solution for Flood-It is NP-hard for c 3 and that this even holds when the player can perform flooding operations from any position on the board. Next we show how a (c−1) approximation and a randomised 2c/3 approximation algorithm can be derived, and that no polynom...

• Type: Conference
• Year: 2010
• Where: FUN
• Authors: David Arthur, Raphaël Clifford, Markus Jalsenius, Ashley Montanaro, Benjamin Sach
• Keywords: Algorithms, FUN 2010, Monochromatic Region, Player Combinatorial Game, Shortest Possible Sequence