Algorithm to identify a unique free polyomino (or polyomino hash)

Question

In short: How to hash a free polyomino?

This could be generalized into: How to efficiently hash an arbitrary collection of 2D integer coordinate

Answer 1

You can set up something like a trie to uniquely identify (and not just hash) your polyomino. Take your normalized polyomino and set up a binary search tree, where the root branches on whether (0,0) is has a set pixel, the next level branches on whether (0,1) has a set pixel, and so on. When you look up a polyomino, simply normalize it and then walk the tree. If you find it in the trie, then you're done. If not, assign that polyomino a unique id (just increment a counter), generate all 8 possible rotations and flips, then add those 8 to the trie.

On a trie miss, you'll have to generate all the rotations and reflections. But on a trie hit it should cost less (O(k^2) for k-polyominos).

To make lookups even more efficient, you could use a couple bits at a time and use a wider tree instead of a binary tree.