Tetravex solving algorithm

Question

Well, i was thinking of making a Tetravex solving program in order to practice my code writing skills (language will propably be Visual Basic) and I need help finding an algorithm for solving it. For those that don't know what tetravex is see this http://en.wikipedia.org/wiki/TetraVex . The only algorithm I can come up with is the brute force way, place a tile randomly in one corner and try every possible tile next to it and continue the same process, if it reaches a dead end revert to a previous state and place a different tile. So can anyone come up with a better algorithm? Thank you for your time.

Answer 1

here some ideas.

A vanilla brute force algorithm would try to fill out the grid recursively by enumerating the grid positions in a fixed order (e.g. row major) and always trying to fit every possible piece in the current position and then recursing. This is what you mentioned and it is very inefficient.

An improvement is to always count for every free position the number of pieces that fit there, and then recurse on the position that has least fits; if one has zero fitting pieces, backtrack immediately; if there is one where only one piece fits fill that and continue (no branch created); otherwise select the one that has least fitting pieces (≥ 2) and continue from there.

Once you have this algorithm in place, the next question is how you can prune the search space more. If have, say, A pieces with "1" on the top position and B pieces with "1" on the bottom position, and A > B, then you know that at least A - B of the "1 at top position" pieces must be actually placed on the top row, so you can exclude them from any other position. This helps to reduce the branching factor and to spot dead-ends earlier.

You should also check at every recursion step that every piece has at least one spot where it fits (do this check after verifying that there is no piece that fits in only one place for speed). If there is a piece that doesn't fit anywhere you need to backtrack immediately. You can extend this to checking that every pair of pieces fits for a potentially better earlier dead-lock checking capability.

There is a also a strategy called "non-chronological backtracking" or "backjumping" which originates from research into SAT solving. This helps you to backtrack more than one level at a time when you reach a dead-end; if you want, you can google for these terms to find more, but you need to do some mental work to map the concept into your problem space.