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Added DFS solver section to the post

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Abhinav Sarkar 2013-03-13 02:27:35 +05:30
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2
sass/base/_typography.scss
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@ -78,7 +78,7 @@ strong { font-weight: bold; }
em { font-style: italic; }
sup, sub { font-size: 0.8em; position: relative; display: inline-block; }
sup, sub { font-size: 0.7em; position: relative; display: inline-block; }
sup { top: -.5em; }
sub { bottom: -.5em; }

135
source/_posts/2013-03-11-sudoku-and-haskell-a-sudoku-solver.markdown
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@ -22,6 +22,9 @@ In this post we look at how to solve a Sudoku with Haskell.
The code in this post has dependencies on the [`split`][1] package from Hackage.
Basic setup
-----------
```haskell
{-# LANGUAGE BangPatterns, RecordWildCards #-}
@ -30,15 +33,13 @@ module Sudoku where
import qualified Data.Set as S
import qualified Data.Map as M
import Data.Char (digitToInt, intToDigit)
import Data.List (foldl', intersperse, intercalate)
import Data.List (foldl', intersperse, intercalate, sortBy, nub)
import Data.List.Split (chunksOf)
import Data.Map ((!))
import Data.Maybe (fromJust)
import Data.Ord (comparing)
import Control.Monad (foldM, guard)
```
Basic setup
-----------
>
Now that the imports are out of the way let's setup the basic functionalities.
@ -52,17 +53,28 @@ instance Show Digit where
allDigits = S.fromList [ONE .. NIN]
data Cell = Cell { cellIdx :: Int, cellVals :: S.Set Digit }
deriving (Eq)
deriving (Eq, Ord)
instance Show Cell where
show Cell{..} = "<" ++ show cellIdx ++ " " ++ show (S.toList cellVals) ++">"
type Board = M.Map Int Cell
newtype Board = Board (M.Map Int Cell)
deriving (Eq, Ord)
boardCells :: Board -> [Cell]
boardCells (Board ixMap) = M.elems ixMap
cellAt :: Board -> Int -> Maybe Cell
cellAt (Board ixMap) idx = M.lookup idx ixMap
updateBoard :: Board -> Cell -> Board
updateBoard (Board ixMap) cell@Cell{..} = Board (M.insert cellIdx cell ixMap)
emptyBoard :: Board
emptyBoard = foldl' (\m i -> M.insert i (Cell i allDigits) m)
M.empty [0 .. 80]
emptyBoard =
Board $ foldl' (\m i -> M.insert i (Cell i allDigits) m) M.empty [0 .. 80]
```
>
A `Digit` is just one of the nine possible values. It derives `Eq`, `Ord` and `Enum`. We use the
fact the `Digit` is enumerable to create a custom `Show` instance which is just the `Digit`'s ordinal
@ -73,8 +85,12 @@ number between 0 to 80 inclusive. The cell values denote the possible values the
without violating the rules of Sudoku. If a cell is filled, it holds only one value. We create a
custom `Show` instance of `Cell` to pretty print it.
A `Board` is just a map from the cell index to the corresponding cell for faster lookups than a
simple list of `Cell`s.
A `Board` is just a wrapper over a map from the cell index to the corresponding cell. We use a map
instead of a simple list of `Cell`s for faster lookups.
`boardCells`, `cellAt` and `updateBoard` are some convenience functions to manipulate a board.
`boardCells` returns a list of all the cells in a board, `cellAt` returns a cell in a board at
a given index and `updateBoard` update a given cell in a board.
`emptyBoard` creates an empty board, with all the cells, unfilled by folding over all the
index list and inserting a cell with all possible digits in the map corresponding to each index.
@ -87,9 +103,9 @@ can start playing with the actual examples. The board is represented as a single
digit for each cell if it is filled otherwise a dot `.`. The cells are read row first, left to
right column. An example:
<pre>
```
6..3.2....4.....1..........7.26............543.........8.15........4.2........7..
</pre>
```
```haskell
readBoard :: String -> Maybe Board
@ -100,10 +116,11 @@ readBoard str = do
let cellVals = if chr == '.'
then allDigits
else S.singleton $ toEnum $ digitToInt chr - 1
return $ M.insert i (Cell i cellVals) board)
return $ updateBoard board (Cell i cellVals))
emptyBoard
$ zip [0 .. 80 ] str
```
>
`readBoard` converts a string to a `Board`. It returns `Just Board` if the string represents a valid
Sudoku board, otherwise it returns `Nothing`. We use the `Monad` nature of `Maybe` to guard against the
@ -118,7 +135,7 @@ showBoard =
if S.size cellVals == 1
then intToDigit . (+ 1) . fromEnum . head . S.toList $ cellVals
else '.')
. M.elems
. boardCells
asciiShowBoard :: Board -> String
asciiShowBoard =
@ -129,7 +146,11 @@ asciiShowBoard =
. chunksOf 9
. showBoard
where border = "+-------+-------+-------+"
instance Show Board where
show = showBoard
```
>
`showBoard` does the reverse of `readBoard`. It takes a board and creates a valid string
representation of it. It does so by mapping over each cell of the board in the order of their index
@ -139,9 +160,11 @@ and outputting the digit if the cell is filled else `.`.
so by taking the output of `showBoard`, breaking it into chunks corresponding to rows and blocks,
inserting spaces and `|` at appropriate places and then joining them with the borders made of `-`.
Lastly, we add a `Show` instance of `Board` using `showBoard`.
Here is an example run in _ghci_:
<pre>
```haskell
*Sudoku> let boardStr = "6..3.2....4.....1..........7.26............543.........8.15........4.2........7.."
*Sudoku> let (Just board) = readBoard boardStr
*Sudoku> showBoard board
@ -160,7 +183,7 @@ Here is an example run in _ghci_:
| . . . | . 4 . | 2 . . |
| . . . | . . . | 7 . . |
+-------+-------+-------+
</pre>
```
Is it solved yet?
--------------------
@ -174,14 +197,14 @@ data BoardState = SOLVED | INCOMPLETE | INVALID
boardState :: Board -> BoardState
boardState board
| any (\Cell{..} -> S.size cellVals /= 1) $ M.elems board = INCOMPLETE
| any (\Cell{..} -> S.size cellVals /= 1) $ boardCells board = INCOMPLETE
| any isUnitInvalid units = INVALID
| otherwise = SOLVED
where
isUnitInvalid unitCells =
(S.fromList . map (head . S.toList . cellVals) $ unitCells) /= allDigits
units = map (map (board !)) unitIxs
units = map (map (fromJust . cellAt board)) unitIxs
unitIxs = rowIxs ++ columnIxs ++ blockIxs
rowIxs = map (\i -> [i * 9 .. i * 9 + 8]) [0..8]
@ -193,6 +216,7 @@ blockIxs =
row1 row2 row3)
. chunksOf 3 . map (chunksOf 3) $ rowIxs
```
>
We start by defining the board state as an enumeration of three value corresponding to the solved,
incomplete and invalid states. The `boardState` function takes a board and gives its current state.
@ -213,7 +237,7 @@ concatenating them with a function which zips three rows at a time creating the
A run in _ghci_ shows the indexes to be correct:
<pre>
```haskell
*Sudoku> mapM_ print rowIxs
[0,1,2,3,4,5,6,7,8]
[9,10,11,12,13,14,15,16,17]
@ -246,7 +270,7 @@ A run in _ghci_ shows the indexes to be correct:
[54,55,56,63,64,65,72,73,74]
[57,58,59,66,67,68,75,76,77]
[60,61,62,69,70,71,78,79,80]
</pre>
```
See how the row indexes follow the grid indexes as we have taken our grid indexes to be rows first,
left to right. If we take row indexes column-wise we get the column indexes. If we take the row
@ -254,7 +278,7 @@ indexes block-wise we get the block indexes.
Let's do a few sample runs of `boardState` in _ghci_:
<pre>
```haskell
*Sudoku> let (Just board) = readBoard "483921657967345821251876493548132976729564138136798245372689514814253769695417382"
*Sudoku> putStr (asciiShowBoard board)
+-------+-------+-------+
@ -278,13 +302,73 @@ INCOMPLETE
*Sudoku> let (Just board) = readBoard "183921657967345821251876493548132976729564138136798245372689514814253769695417382"
*Sudoku> boardState board
INVALID
</pre>
```
That seems to be working. Now let's move on to actually solving the Sudoku!
Backtracking search
Depth first search
-------------------
One way to solve Sudoku is to think of it as a graph search problem. Each board configuration becomes
a node in the search graph with the moves linking them as edges. A move is filling a particular cell
with a digit. So now we can solve the board just by finding a path from the given board configuration
to a configuration where all cells are filled.
We can use [Depth first Search][5] (DSF) algorithm to accomplish this. DFS is a brute force
technique and in worst case it may visit all the nodes in the search graph. In case of Sudoku, this
search graph is very large (approximately 6.67×10<sup>21</sup>) so this is not a very efficient way
of solving Sudoku. For now, we'll add one optimization in DFS: while listing the next possible
configurations for a particular configuration, we start with the cell with smallest number of cell
values. This does not help us in the worst case but it will generally speed up things a little.
We can write this in two parts: a general DFS function and a solver which uses it to solve a Sudoku.
```haskell
dfs :: Ord a => a -> (a -> [a]) -> (a -> Bool) -> [a]
dfs start getNext isGoal = go start S.empty
where
go node visited
| isGoal node = [node]
| S.member node visited = []
| otherwise = concatMap (\nextNode ->
go nextNode (S.insert node visited))
(getNext node)
dfsSolver :: Board -> [Board]
dfsSolver board = dfs board nextBoards ((== SOLVED) . boardState)
where
nextBoards board =
map (updateBoard board)
. concatMap (\Cell{..} -> map (Cell cellIdx . S.singleton) . S.toList $ cellVals)
. sortBy (comparing (S.size . cellVals))
. filter ((/= 1) . S.size . cellVals)
. boardCells
$ board
```
Let's try this out in _ghci_ now:
```haskell
*Sudoku> let boardStr = ".839216579.734582125187649354813297672956413813679824537268951481425376969541738."
*Sudoku> let (Just board) = readBoard boardStr
*Sudoku> :set +s
*Sudoku> (mapM_ print . nub . dfsSolver) board
483921657967345821251876493548132976729564138136798245372689514814253769695417382
(1.75 secs, 356010984 bytes)
```
```haskell
*Sudoku> let boardStr = ".839216579.734582125187.49354813297672956413813679824537268951481425376969541738."
*Sudoku> let (Just board) = readBoard boardStr
*Sudoku> :set +s
*Sudoku> (mapM_ print . nub . dfsSolver) board
483921657967345821251876493548132976729564138136798245372689514814253769695417382
(66.64 secs, 13136374896 bytes)
*Sudoku> (print . head . dfsSolver) board
483921657967345821251876493548132976729564138136798245372689514814253769695417382
(7.32 secs, 1322834576 bytes)
```
Constraint propagation
----------------------
@ -304,3 +388,4 @@ post can be downloaded [here][3] or can be forked [here][4].
[2]: /downloads/code/sudoku1.lhs
[3]: /downloads/code/sudoku1.hs
[4]: https://github.com/abhin4v/abhin4v.github.com/blob/source/source/downloads/code/sudoku1.hs
[5]: http://en.wikipedia.org/wiki/Depth_first_search

55
source/downloads/code/sudoku1.hs
View File

@ -5,9 +5,10 @@ module Sudoku where
import qualified Data.Set as S
import qualified Data.Map as M
import Data.Char (digitToInt, intToDigit)
import Data.List (foldl', intersperse, intercalate)
import Data.List (foldl', intersperse, intercalate, sortBy, nub)
import Data.List.Split (chunksOf)
import Data.Map ((!))
import Data.Maybe (fromJust)
import Data.Ord (comparing)
import Control.Monad (foldM, guard)
data Digit = ONE | TWO | TRE | FOR | FIV | SIX | SVN | EGT | NIN
deriving (Eq, Ord, Enum)
@ -18,16 +19,26 @@ instance Show Digit where
allDigits = S.fromList [ONE .. NIN]
data Cell = Cell { cellIdx :: Int, cellVals :: S.Set Digit }
deriving (Eq)
deriving (Eq, Ord)
instance Show Cell where
show Cell{..} = "<" ++ show cellIdx ++ " " ++ show (S.toList cellVals) ++">"
type Board = M.Map Int Cell
newtype Board = Board (M.Map Int Cell)
deriving (Eq, Ord)
boardCells :: Board -> [Cell]
boardCells (Board ixMap) = M.elems ixMap
cellAt :: Board -> Int -> Maybe Cell
cellAt (Board ixMap) idx = M.lookup idx ixMap
updateBoard :: Board -> Cell -> Board
updateBoard (Board ixMap) cell@Cell{..} = Board (M.insert cellIdx cell ixMap)
emptyBoard :: Board
emptyBoard = foldl' (\m i -> M.insert i (Cell i allDigits) m)
M.empty [0 .. 80]
emptyBoard =
Board $ foldl' (\m i -> M.insert i (Cell i allDigits) m) M.empty [0 .. 80]
readBoard :: String -> Maybe Board
readBoard str = do
guard $ length str == 81
@ -36,7 +47,7 @@ readBoard str = do
let cellVals = if chr == '.'
then allDigits
else S.singleton $ toEnum $ digitToInt chr - 1
return $ M.insert i (Cell i cellVals) board)
return $ updateBoard board (Cell i cellVals))
emptyBoard
$ zip [0 .. 80 ] str
showBoard :: Board -> String
@ -45,7 +56,7 @@ showBoard =
if S.size cellVals == 1
then intToDigit . (+ 1) . fromEnum . head . S.toList $ cellVals
else '.')
. M.elems
. boardCells
asciiShowBoard :: Board -> String
asciiShowBoard =
@ -56,19 +67,22 @@ asciiShowBoard =
. chunksOf 9
. showBoard
where border = "+-------+-------+-------+"
instance Show Board where
show = showBoard
data BoardState = SOLVED | INCOMPLETE | INVALID
deriving (Eq, Show)
boardState :: Board -> BoardState
boardState board
| any (\Cell{..} -> S.size cellVals /= 1) $ M.elems board = INCOMPLETE
| any (\Cell{..} -> S.size cellVals /= 1) $ boardCells board = INCOMPLETE
| any isUnitInvalid units = INVALID
| otherwise = SOLVED
where
isUnitInvalid unitCells =
(S.fromList . map (head . S.toList . cellVals) $ unitCells) /= allDigits
units = map (map (board !)) unitIxs
units = map (map (fromJust . cellAt board)) unitIxs
unitIxs = rowIxs ++ columnIxs ++ blockIxs
rowIxs = map (\i -> [i * 9 .. i * 9 + 8]) [0..8]
@ -79,3 +93,24 @@ blockIxs =
blockRow1 ++ blockRow2 ++ blockRow3)
row1 row2 row3)
. chunksOf 3 . map (chunksOf 3) $ rowIxs
dfs :: Ord a => a -> (a -> [a]) -> (a -> Bool) -> [a]
dfs start getNext isGoal = go start S.empty
where
go node visited
| isGoal node = [node]
| S.member node visited = []
| otherwise = concatMap (\nextNode ->
go nextNode (S.insert node visited))
(getNext node)
dfsSolver :: Board -> [Board]
dfsSolver board = dfs board nextBoards ((== SOLVED) . boardState)
where
nextBoards board =
map (updateBoard board)
. concatMap (\Cell{..} -> map (Cell cellIdx . S.singleton) . S.toList $ cellVals)
. sortBy (comparing (S.size . cellVals))
. filter ((/= 1) . S.size . cellVals)
. boardCells
$ board

123
source/downloads/code/sudoku1.lhs
View File

@ -23,6 +23,9 @@ In this post we look at how to solve a Sudoku with Haskell.
The code in this post has dependencies on the [`split`][1] package from Hackage.
Basic setup
-----------
> {-# LANGUAGE BangPatterns, RecordWildCards #-}
>
> module Sudoku where
@ -30,14 +33,12 @@ The code in this post has dependencies on the [`split`][1] package from Hackage.
> import qualified Data.Set as S
> import qualified Data.Map as M
> import Data.Char (digitToInt, intToDigit)
> import Data.List (foldl', intersperse, intercalate)
> import Data.List (foldl', intersperse, intercalate, sortBy, nub)
> import Data.List.Split (chunksOf)
> import Data.Map ((!))
> import Data.Maybe (fromJust)
> import Data.Ord (comparing)
> import Control.Monad (foldM, guard)
Basic setup
-----------
>
Now that the imports are out of the way let's setup the basic functionalities.
@ -50,16 +51,27 @@ Now that the imports are out of the way let's setup the basic functionalities.
> allDigits = S.fromList [ONE .. NIN]
>
> data Cell = Cell { cellIdx :: Int, cellVals :: S.Set Digit }
> deriving (Eq)
> deriving (Eq, Ord)
>
> instance Show Cell where
> show Cell{..} = "<" ++ show cellIdx ++ " " ++ show (S.toList cellVals) ++">"
>
> type Board = M.Map Int Cell
> newtype Board = Board (M.Map Int Cell)
> deriving (Eq, Ord)
>
> boardCells :: Board -> [Cell]
> boardCells (Board ixMap) = M.elems ixMap
>
> cellAt :: Board -> Int -> Maybe Cell
> cellAt (Board ixMap) idx = M.lookup idx ixMap
>
> updateBoard :: Board -> Cell -> Board
> updateBoard (Board ixMap) cell@Cell{..} = Board (M.insert cellIdx cell ixMap)
>
> emptyBoard :: Board
> emptyBoard = foldl' (\m i -> M.insert i (Cell i allDigits) m)
> M.empty [0 .. 80]
> emptyBoard =
> Board $ foldl' (\m i -> M.insert i (Cell i allDigits) m) M.empty [0 .. 80]
>
A `Digit` is just one of the nine possible values. It derives `Eq`, `Ord` and `Enum`. We use the
fact the `Digit` is enumerable to create a custom `Show` instance which is just the `Digit`'s ordinal
@ -70,8 +82,12 @@ number between 0 to 80 inclusive. The cell values denote the possible values the
without violating the rules of Sudoku. If a cell is filled, it holds only one value. We create a
custom `Show` instance of `Cell` to pretty print it.
A `Board` is just a map from the cell index to the corresponding cell for faster lookups than a
simple list of `Cell`s.
A `Board` is just a wrapper over a map from the cell index to the corresponding cell. We use a map
instead of a simple list of `Cell`s for faster lookups.
`boardCells`, `cellAt` and `updateBoard` are some convenience functions to manipulate a board.
`boardCells` returns a list of all the cells in a board, `cellAt` returns a cell in a board at
a given index and `updateBoard` update a given cell in a board.
`emptyBoard` creates an empty board, with all the cells, unfilled by folding over all the
index list and inserting a cell with all possible digits in the map corresponding to each index.
@ -96,9 +112,10 @@ right column. An example:
> let cellVals = if chr == '.'
> then allDigits
> else S.singleton $ toEnum $ digitToInt chr - 1
> return $ M.insert i (Cell i cellVals) board)
> return $ updateBoard board (Cell i cellVals))
> emptyBoard
> $ zip [0 .. 80 ] str
>
`readBoard` converts a string to a `Board`. It returns `Just Board` if the string represents a valid
Sudoku board, otherwise it returns `Nothing`. We use the `Monad` nature of `Maybe` to guard against the
@ -112,7 +129,7 @@ string contained a digit at the cell index else they have all the digits as cell
> if S.size cellVals == 1
> then intToDigit . (+ 1) . fromEnum . head . S.toList $ cellVals
> else '.')
> . M.elems
> . boardCells
>
> asciiShowBoard :: Board -> String
> asciiShowBoard =
@ -123,6 +140,10 @@ string contained a digit at the cell index else they have all the digits as cell
> . chunksOf 9
> . showBoard
> where border = "+-------+-------+-------+"
>
> instance Show Board where
> show = showBoard
>
`showBoard` does the reverse of `readBoard`. It takes a board and creates a valid string
representation of it. It does so by mapping over each cell of the board in the order of their index
@ -132,9 +153,11 @@ and outputting the digit if the cell is filled else `.`.
so by taking the output of `showBoard`, breaking it into chunks corresponding to rows and blocks,
inserting spaces and `|` at appropriate places and then joining them with the borders made of `-`.
Lastly, we add a `Show` instance of `Board` using `showBoard`.
Here is an example run in _ghci_:
<pre>
<pre>haskell
*Sudoku> let boardStr = "6..3.2....4.....1..........7.26............543.........8.15........4.2........7.."
*Sudoku> let (Just board) = readBoard boardStr
*Sudoku> showBoard board
@ -166,14 +189,14 @@ whether a board is filled completely and whether that solution is a valid one.
>
> boardState :: Board -> BoardState
> boardState board
> | any (\Cell{..} -> S.size cellVals /= 1) $ M.elems board = INCOMPLETE
> | any (\Cell{..} -> S.size cellVals /= 1) $ boardCells board = INCOMPLETE
> | any isUnitInvalid units = INVALID
> | otherwise = SOLVED
> where
> isUnitInvalid unitCells =
> (S.fromList . map (head . S.toList . cellVals) $ unitCells) /= allDigits
>
> units = map (map (board !)) unitIxs
> units = map (map (fromJust . cellAt board)) unitIxs
>
> unitIxs = rowIxs ++ columnIxs ++ blockIxs
> rowIxs = map (\i -> [i * 9 .. i * 9 + 8]) [0..8]
@ -184,6 +207,7 @@ whether a board is filled completely and whether that solution is a valid one.
> blockRow1 ++ blockRow2 ++ blockRow3)
> row1 row2 row3)
> . chunksOf 3 . map (chunksOf 3) $ rowIxs
>
We start by defining the board state as an enumeration of three value corresponding to the solved,
incomplete and invalid states. The `boardState` function takes a board and gives its current state.
@ -204,7 +228,7 @@ concatenating them with a function which zips three rows at a time creating the
A run in _ghci_ shows the indexes to be correct:
<pre>
<pre>haskell
*Sudoku> mapM_ print rowIxs
[0,1,2,3,4,5,6,7,8]
[9,10,11,12,13,14,15,16,17]
@ -245,7 +269,7 @@ indexes block-wise we get the block indexes.
Let's do a few sample runs of `boardState` in _ghci_:
<pre>
<pre>haskell
*Sudoku> let (Just board) = readBoard "483921657967345821251876493548132976729564138136798245372689514814253769695417382"
*Sudoku> putStr (asciiShowBoard board)
+-------+-------+-------+
@ -273,9 +297,67 @@ INVALID
That seems to be working. Now let's move on to actually solving the Sudoku!
Backtracking search
Depth first search
-------------------
One way to solve Sudoku is to think of it as a graph search problem. Each board configuration becomes
a node in the search graph with the moves linking them as edges. A move is filling a particular cell
with a digit. So now we can solve the board just by finding a path from the given board configuration
to a configuration where all cells are filled.
We can use [Depth first Search][5] (DSF) algorithm to accomplish this. DFS is a brute force
technique and in worst case it may visit all the nodes in the search graph. In case of Sudoku, this
search graph is very large (approximately 6.67×10<sup>21</sup>) so this is not a very efficient way
of solving Sudoku. For now, we'll add one optimization in DFS: while listing the next possible
configurations for a particular configuration, we start with the cell with smallest number of cell
values. This does not help us in the worst case but it will generally speed up things a little.
We can write this in two parts: a general DFS function and a solver which uses it to solve a Sudoku.
> dfs :: Ord a => a -> (a -> [a]) -> (a -> Bool) -> [a]
> dfs start getNext isGoal = go start S.empty
> where
> go node visited
> | isGoal node = [node]
> | S.member node visited = []
> | otherwise = concatMap (\nextNode ->
> go nextNode (S.insert node visited))
> (getNext node)
>
> dfsSolver :: Board -> [Board]
> dfsSolver board = dfs board nextBoards ((== SOLVED) . boardState)
> where
> nextBoards board =
> map (updateBoard board)
> . concatMap (\Cell{..} -> map (Cell cellIdx . S.singleton) . S.toList $ cellVals)
> . sortBy (comparing (S.size . cellVals))
> . filter ((/= 1) . S.size . cellVals)
> . boardCells
> $ board
>
Let's try this out in _ghci_ now:
<pre>haskell
*Sudoku> let boardStr = ".839216579.734582125187649354813297672956413813679824537268951481425376969541738."
*Sudoku> let (Just board) = readBoard boardStr
*Sudoku> :set +s
*Sudoku> (mapM_ print . nub . dfsSolver) board
483921657967345821251876493548132976729564138136798245372689514814253769695417382
(1.75 secs, 356010984 bytes)
</pre>
<pre>haskell
*Sudoku> let boardStr = ".839216579.734582125187.49354813297672956413813679824537268951481425376969541738."
*Sudoku> let (Just board) = readBoard boardStr
*Sudoku> :set +s
*Sudoku> (mapM_ print . nub . dfsSolver) board
483921657967345821251876493548132976729564138136798245372689514814253769695417382
(66.64 secs, 13136374896 bytes)
*Sudoku> (print . head . dfsSolver) board
483921657967345821251876493548132976729564138136798245372689514814253769695417382
(7.32 secs, 1322834576 bytes)
</pre>
Constraint propagation
----------------------
@ -295,3 +377,4 @@ post can be downloaded [here][3] or can be forked [here][4].
[2]: /downloads/code/sudoku1.lhs
[3]: /downloads/code/sudoku1.hs
[4]: https://github.com/abhin4v/abhin4v.github.com/blob/source/source/downloads/code/sudoku1.hs
[5]: http://en.wikipedia.org/wiki/Depth_first_search