Sudoku_Solver_in_Haskell - Pure Functional Programming App

Implemented some human-like algorithms to solve Sudoku in Haskell. 纯函数式数独解题器

For details, read Report.pdf

Algorithm Overview:

• Solving Algorithm 1: Depth-first Search (Brute Force/Backtracking)
• Solving Algorithm 2: Prioritize the Guessing Order (Extension 1)
• Solving Algorithm 3: Constraint Propagation (Extension 2)
  • Hidden Singles (Solver level 1 in sudoku.hs)
  • Prune Naked Pairs (Solver level 2 in sudoku.hs)
  • Prune Naked Triples (Solver level 3 in sudoku.hs)
• Generating Algorithm: Las Vegas Algorithm (Extension 3)

Test & Analysis

Test Result for easy.txt

	Level 0: Brute Force	Level 1: Hidden Singles	Level 2: Naked Pairs	Level 3: Naked Triples
expand	real 5m45.574s	real 5m53.522s	NULL	NULL
expandFirst	NULL	real 0m1.175s	real 0m0.695s	real 0m0.789s

Test Result for hard.txt

	Level 0: Brute Force	Level 1: Hidden Singles	Level 2: Naked Pairs	Level 3: Naked Triples
expand	Killed	24min27s	NULL	NULL
expandFirst	NULL	real 4m13.680s	real 3m58.542s	real 5m13.327s

Since “Naked triple” is not very general so Solver-level 3 performs a lot of invalid checks that leads to the relatively slower than Solver_level 2. But we can obviously find the dramatic decline in time if we replace the naïve expand by expandFirst. The guessing order makes a huge difference indeed!

Very Fast Solver for sudoku17.txt

To deal with this huge file, I found a very fast solver online and I tested it for the Sudoku17.txt, only taking 43.898s to print all the output, approximately 50000 Sudoku!).

Data.vector is imported in this solution and this library has a wide range of built-in functions and the complexity of the majority is merely O(1)! Indexing, update, imap, ifilter are all built-in. It is so awesome that this Data.vector library seems to be born to solve Sudoku. Besides, the solution provided applies the Monad in an extremely skilled manner, which also enables to boost the solver.

Reasoning:

• EXTENSION 1: Optimizing the guessing order
  • We are required to find the block with fewest candidates to reduce redundancy
  • In my implementation, I use num of choice to represent the levels of difficulty to work out that single position by guessing. Obviously, if the num of choice equals to min_choice(>1), it will have least redundancy.
  • expandFirst filters the best position to fill first according to the number of possibilities, which is exactly what we want in the extension 1.
  • If you want to test expanding without optimization, replace 'expandFirst' in 'search' by 'expand Performs the expansion beginning with the smallest number of choices (>1) breaks up a matrix is broken into five pieces: mc = rows1 ++ [row1 ++ cs : row2] ++ rows2
• Extension 2 : Constraint Propagation to prune the huge search space (3 levels implemented)
  • levels
    • level_1: prune singles; level_2: prune naked pairs; level_3: prune naked triples
  • Reasoning
    • Obviously, if there is only one element in the Choices list, the only value must be the solution for that position. This means, other positions in the blocks related to this known position cannot choose that value, so we have to reduce this value from the choices list of all related positions (sharing row/col/box) LEVEL 1 for Constraint Propagation is to "remove singles"
  • Solver - LEVEL 1 Prune the single choice:
    • Prune the single choice (a value occurs only once for a block) minus will remove the second set from the first.
  • Solver - LEVEL 2 : Prune the naked pairs
    • In a Block Choices, we find exactly two positions with the choices [2,3]. Then obviously, the value 2 and 3 can only occur in those two places. All other occurrences of the value 1 and 9 can eliminated eliminate other occurrences if there exist dup pairs
    • prop> reduce_samepair [[Just 2,Just 3, Just 4, Just 5], [Just 2,Just 3,Just 4],[Just 2,Just 3],[Just 4,Just 3,Just 5,Just 1],[Just 2,Just 3],[Just 2,Just 5]]
    • [[Just 4,Just 5],[Just 4],[Just 2,Just 3],[Just 4,Just 5,Just 1],[Just 2,Just 3],[Just 5]]
    • dup_pair = [Just 2, Just 3]
  • Solver - LEVEL 3 : Prune the naked triples
    • Resembles Solver - LEVEL 2
    • In a Block Choices, we find exactly 3 positions with the choices [2,3,9]. All other simultaneous occurrences of the value [2,3,9] in sharing blocks can eliminated
    • note: higher levels can also be constructed similarly but there is no sense to filter not frequent occasions eliminate other simultaneous occurrences if there exist 3 same triples

4 Sep 2017