Tetromino space-filling: need to check if it's possible

Question

I'm writing a program that needs to quickly check whether a contiguous region of space is fillable by tetrominoes (any type, any orientation). My first attempt was to simply check if the number of squares was divisible by 4. However, situations like this can still come up:

As you can see, even though these regions have 8 squares each, they are impossible to tile with tetrominoes.

I've been thinking for a bit and I'm not sure how to proceed. It seems to me that the "hub" squares, or squares that lead to more than two "tunnels", are the key to this. It's easy in the above examples, since you can quickly count the spaces in each such tunnel — 3, 1, and 3 in the first example, and 3, 1, 1, and 2 in the second — and determine that it's impossible to proceed due to the fact that each tunnel needs to connect to the hub square to fit a tetromino, which can't happen for all of them. However, you can have more complicated examples like this:

...where a simple counting technique just doesn't work. (At least, as far as I can tell.) And that's to say nothing of more open spaces with a very small number of hub squares. Plus, I don't have any proof that hub squares are the only trick here. For all I know, there may be tons of other impossible cases.

Is some sort of search algorithm (A*?) the best option for solving this? I'm very concerned about performance with hundreds, or even thousands, of squares. The algorithm needs to be very efficient, since it'll be used for real-time tiling (more or less), and in a browser at that.

Answer 1

Perfect matching on a perfect matching

[EDIT 28/10/2014: As noticed by pix, this approach never tries to use T-tetrominoes, so it is even more likely to give an incorrect "No" answer than I thought...]

This will not guarantee a solution on an arbitrary shape, but it will work quickly and well most of the time.

Imagine a graph in which there is a vertex for each white square, and an edge between two vertices if and only if their corresponding white squares are adjacent. (Each vertex can therefore touch at most 4 edges.) A perfect matching in this graph is a subset of edges such that every vertex touches exactly one edge in the subset. In other words, it is a way of pairing up adjacent vertices -- or in yet other words, a domino tiling of the white squares. Later I'll explain how to find a nicely random-looking perfect matching; for now, let's just assume that it can be done.

Then, starting from this domino tiling, we can just repeat the matching process, gluing dominos together into tetrominos! The only differences the second time around are that instead of having a vertex per white square, we have a vertex per domino; and because we must add an edge whenever two dominos are adjacent, a vertex can now have as many as 6 edges.

The first step (domino tiling) step cannot fail: if a domino tiling for the given shape exists, then one will be found. However, it is possible for the second step (gluing dominos together into tetrominos) to fail, because it has to work with the already-decided domino tiling, and this may limit its options. Here is an example showing how different domino tilings of the same shape can enable or spoil the tetromino tiling:

AABCDD   -->   XXXYYY   Success :)
  BC             XY

AABBDD   -->   Failure.
  CC

Solving the maximum matching problems

In order to generate a random pattern of dominos, the edges in the graph can be given random weights, and the maximum weighted matching problem can be solved. The weights should be in the range [1, V/(V-2)), to guarantee that it is never possible to achieve a higher score by leaving some vertices unpaired. The graph is in fact bipartite as it contains no odd-length cycles, meaning that the faster O(V^2*E) algorithm for the maximum weighted bipartite matching problem can be used for this step. (This is not true for the second matching problem: one domino can touch two other dominos that touch each other.)

If the second step fails to find a complete set of tetrominos, then either no solution is possible*, or a solution is possible using a different set of dominos. You can try randomly reweighting the graph used to find the domino tiling, and then rerunning the first step. Alternatively, instead of completely reweighting it from scratch, you could just increase the weights for the problematic dominos, and try again.

* For a plain square with even side lengths, we know that a solution is always possible: just fill it with 2x2 square tetrominos.