Archive for November, 2013

Sliding Block Puzzles, Part 4 of 3

And you thought it was over! 1 2 3

I was originally planning to finish the series of articles on SBPs at this post, concluding with the announcement of the optimal puzzles for a 4×4 grid. Partially, this was because the SBPFinder algorithm would have taken prohibitive amounts of time to compute the boards for any larger size. For example, for the classic 4×5 grid, it would take somewhere around 128 days! A much worse problem is that it would have used approximately 64 GB of RAM, which, even at its cheapest, is an investment I cannot reasonably justify.

Fortunately, however, I soon realized how utterly stupid I had been when designing the SBPFinder algorithm. In case the reader doesn’t want to read the thousands of words in the previous posts, I’ll quickly describe it: Essentially, the algorithm used dynamic programming and a peculiar way of indexing into a 2D array to, cell by cell, generate all possible positions. The primary flaw with this technique is that it generates lots of duplicates, which both slows it down and requires the use of a hash set or other method of eliminating duplicate positions. (To go into the technical details, this is because certain oddly shaped pieces, which -not coincidentally- appear in the list of hardest simple^3 SBPs, cause the dynamic programming stage to consider too many branches of the tree. This, in turn, is because the computer cannot currently time travel and see what it does before it does it)

    The solution is to place entire pieces (or voids) in undecided spaces, proceeding left-to-right and top-to-bottom. Currently, this has been implemented as a recursive function, as follows:

Given a board B (which may be partially filled):

  1. Find the first cell which has not been decided yet- that is, does not contain a cell of a piece or a void, but rather exists in a state of indecision between the two. (This is implemented as a third cell type)
  2. If there are no undecided cells, the board is a valid placement of pieces and voids. (It may not be justsolved, though, so these are typically placed in a file to be processed further)
  3. Otherwise,
    1. Recurse to the case where that cell is filled with a void,
    2. Find all pieces which can be placed on the board and which cover only undecided cells and which covers the cell determined in step 1.
    3. For each of those pieces, recurse to the case where that piece is placed on the board.

Here’s the search tree for 2×2 SBPs using this method:

Click to enlarge; the search tree is really wide

Click to enlarge; the search tree is really wide. Also, the post continues after the break.

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