How to optimally solve the flood fill puzzle?

Question

I like playing the puzzle game Flood-It, which can be played online at:

http://floodit.appspot.com/

It's also available as an iGoogle gadget. The aim is to fill the whole board with the least number of successive flood-fills.

I'm trying to write a program which can solve this puzzle optimally. What's the best way to approach this problem? Ideally I want to use the A* algorithm, but I have no idea what should be the function estimating the number of steps left. I did write a program which conducted a depth-4 brute force search to maximize the filled area. It worked reasonably well and beat me in solving the puzzle, but I'm not completely satisfied with that algorithm.

Any suggestions? Thanks in advance.

Answer 1

I think you could consider the number of squares that match or don't match the current color. So, your heuristic measure of "distance" would be the number of squares on the board that are -not- the same color as your chosen color, rather than the number of steps.

Answer 2

A naive heuristic could be to use the number of colours left (minus 1) - this is admissible because it will take at least that many clicks to clear off the board.

Answer 3

As a heuristic, you could construct a graph where each node represents a set of contiguous, same-colour squares, and each node is connected to those it touches. (Each edge weighted as 1). You could then use a path-finding algorithm to calculate the "distance" from the top left to all other nodes. Then, by looking the results of flood-filling using each of the other 5 colours, determine which one minimizes the distance to the "furthest" node, since that will likely be your bottleneck.

Add the result of that calculation to the number of fills done so far, and use that as your A* heuristic.

Answer 4

A* is just a prioritized graph search. Each node is a game state, you rank nodes based on some heuristic, and always expand the lowest-expected-final-cost node. As long as your heuristic doesn't underestimate costs, the first solution you find is guaranteed to be optimal.

After playing the games a few times, I found that trying to drill to the opposite corner then all corners tended to result in a win. So a good starting cost estimate would be (cost so far) + a sufficient number of fills to reach the opposite corner [note: not minimum, just sufficient. Just greedily fill towards the corner to compute the heuristic].

Answer 5

A naive 'greedy' algorithm is to pick the next step that maximizes the overall perimeter of the main region.

(A couple of smart friends of mine were thinking about this the other day and decided the optimium may be NP-hard (e.g. you must brute force it) - I do not know if they're correct (wasn't around to hear the reasoning and haven't thought through it myself).)

Note that for computing steps, I presume the union-find algorithm is your friend, it makes computing 'one step' very fast (see e.g. this blog post).

Answer 6

I'm not certain, but I'm fairly sure that this could be solved greedily. You're trying to reduce the number of color fields to 1, so reducing more color fields earlier shouldn't be any less efficient than reducing fewer earlier.

1) Define a collection of existing like-colored groups.

2) For each collection, count the number of neighboring collections by color. The largest count of neighboring collections with a single color is the weight of this collection.

3) Take the collection with the highest count of neighbors with a single color, and fill it to that color. Merge the collections, and update the sort for all the collections affected by the merge (all the new neighbors of the merged collection).

Overall, I think this should actually compute in O(n log n) time, where n is the number of pixels and the log(n) only comes from maintaining the sorted list of weights.

I'm not sure if there needs to be a tie-breaker for when multiple fields have the same weight though. Maybe the tie-breaker goes to the color that's common to the most groups on the map.

Anyway, note that the goal of the game is to reduce the number of distinct color fields and not to maximize the perimeter, as different color schemes can occasionally make a larger field a sub-optimal choice. Consider the field:

3 3 3 3 3

1 1 1 1 1

1 1 1 1 1

2 2 2 2 2

1 2 2 2 2

The color 1 has the largest perimeter by any measure, but the color 2 is the optimal choice.

EDIT>

Scratch that. The example:

3 1 3 1 3

1 1 1 1 1

1 1 1 1 1

2 2 2 2 2

1 2 2 2 2

Invalidates my own greedy algorithm. But I'm not convinced that this is a simple graph traversal, since changing to a color shared by 2 neighbors visits 2 nodes, and not 1.

Color elimination should probably play some role in the heuristic.

1) It is never correct to fill with a color that is not already on the graph.

2) If there is one color field with a unique color, at least one fill will be required for it. It cannot be bundled with any other fills. I think this means that it's safe to fill it sooner rather than later.

3) The greedy algorithm for neighbor field count makes sense for a 2 color map.

Answer 7

After playing the game a few times, I noticed that a good strategy is to always go "deep", to go for the colour which goes farthest into the unflooded territory.

Answer 8

i jus wanna know how to beat the game every time

Answer 9

Here's an idea for implementing the graph to support Smashery's heuristic.

Represent each group of contiguous, same-colour squares in a disjoint set, and a list of adjacent groups of squares. A flood fill merges a set to all its adjacent sets, and merges the adjacency lists. This implicit graph structure will let you find the distance from the upper left corner to the farthest node.