Template Metaprogramming and Puzzle 15 Optimal Solution

Thursday May 10th 2007 by Aliaksei Sanko
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Read about a sample demonstrating how to accelerate finding an optimal Puzzle 15 solution using C++ Template Metaprogramming. It presents an implementation of the IDA* search algorithm.

Here is a sample demonstrating how to accelerate finding an optimal Puzzle 15 solution using C++ Template Metaprogramming [1], [2]. It presents a self-descriptive and straightforward implementation of the IDA* search algorithm (Korf, 1985) [3]. It is interesting here how template meta-programming, along with optimizations made by the compiler, fit within this algorithm and make it work significantly faster. From another point of view, in some ways this implementation looks similar to functional programming.

Manhattan distance heuristics are used here. As soon as each move can cause a Manhattan distance change of either 1 or -1, the number of times the Manhattan distance increased during puzzle solution is used as a cost bound. Steepest Ascent Hill-Climbing order, cutting out, and back and forth move sequences are implemented as well. Of improved admissive heuristics [4], only non-interfering Linear Conflict, Last Moves, and Corner Tiles are implemented here for simplicity. Employing better heuristics would significantly increase search speed as well.

The test run parameters are:

puzzle.exe 15 14 _ 4 11 1 6 13 7 5 8 9 3 2 10 12

The result is:

Solved at 2.202 seconds. Number of moves: 66

and the solution log. The execution time is a couple of seconds, depending on the computer speed.

I have compared the application speed with that of elaborated one implemented by Ken'ichiro Takahashi (takaken) [5]. The solution using template metaprogramming (TM) is slower on the test example from takaken's site (1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 0), 34131 sec. against 20813 sec. on a Pentium D 3.00 GHz. However, using examples from [6] gives quite different results:

Problem/Solution Time, in Seconds proposed (TM) by takaken
Problem 2 13 15 14 0 9 10 11 12 5 6 7 8 1 2 3 4 4128 133545
Problem 4 4 3 2 1 8 7 6 5 12 11 10 9 0 14 15 13 415 2533
Problem 6 15 14 13 12 11 10 9 8 7 6 5 4 3 1 2 0 90 239
Problem 12 0 12 9 13 15 11 10 14 7 8 5 6 4 3 2 1 940 961

It demonstrates that an approach described here is quite promising.

Besides generic console application, an interactive WPF based demo is available (requires Microsoft .NET Framework 3.0 or higher):

References

  1. "Template Metaprograms", Todd Veldhuizen
  2. "Generic<Programming>: Mappings between Types and Values", Andrei Alexandrescu
  3. "Complete Solution of the Eight-Puzzle and the Benefit of Node Ordering in IDA*", Alexander Reinefeld
  4. "Finding Optimal Solutions to the Twenty-Four Puzzle," Richard E. Korf and Larry A. Taylor
  5. "15puzzle Optimal solver" for windows
  6. Problems from The 15 Puzzle by Jerry Slocum
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