The Complexity of Flood Filling Games

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.