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Added explaination of the DFS solver to the post

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Abhinav Sarkar 2013-03-14 10:14:59 +05:30
parent 5ff3b47ef5
commit af7b2513a2
3 changed files with 101 additions and 16 deletions
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56
source/_posts/2013-03-11-sudoku-and-haskell-a-sudoku-solver.markdown
View File

@ -38,8 +38,8 @@ import Data.List.Split (chunksOf)
import Data.Maybe (fromJust)
import Data.Ord (comparing)
import Control.Monad (foldM, guard)
```
>
Now that the imports are out of the way let's setup the basic functionalities.
@ -73,8 +73,8 @@ updateBoard (Board ixMap) cell@Cell{..} = Board (M.insert cellIdx cell ixMap)
emptyBoard :: Board
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
@ -119,8 +119,8 @@ readBoard str = do
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
@ -149,8 +149,8 @@ asciiShowBoard =
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
@ -188,7 +188,7 @@ Here is an example run in _ghci_:
Is it solved yet?
--------------------
Before we proceed to write a full-fledged Sudoku solver, it is good to have a function which tells us
Before we proceed to write a full-fledged Sudoku solver, we must have a function which tells us
whether a board is filled completely and whether that solution is a valid one.
```haskell
@ -215,8 +215,8 @@ blockIxs =
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.
@ -314,7 +314,7 @@ a node in the search graph with the moves linking them as edges. A move is filli
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
We can use [Depth first Search][5] (DFS) 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
@ -346,7 +346,38 @@ dfsSolver board = dfs board nextBoards ((== SOLVED) . boardState)
```
Let's try this out in _ghci_ now:
The `dfs` functions is a literal translation of the DFS algorithm. It visits the graph node by node,
keeping a track of the nodes visited and the goals seen till now. It first checks if the current
node is a goal and if so it just returns it wrapped in a list. If the current node is not a goal, it
checks if has already been visited and if so just returns an empty list. This is done to avoid getting
stuck into infinite loops, going down the same path again and again. If both of these checks fail then
it gets to the general case in which it just finds all the next nodes for the current node,
recursively maps itself over them accumulating all the goals found (by flattening the lists) and
returns them.
`dfsSolver` uses `dfs` to solve Sudoku as a DFS by supplying the `getNext` and `isGoal` functions.
the `isGoal` function is simple, it just checks if the current board is solved by calling the
`boardState` function. `nextBoards` is a little complicated so let's go over it step by step,
reading from bottom to up:
1. it gets all the cells in the board
2. filters in only the empty cells, the cells with more than one cell values
3. sorts the empty cells comparing the count of their cell values
4. for each cell, it takes each cell value and create a cell containing only that cell value and
flattens this list of list of cells into a list of cells
5. for each cell so created, it creates a board by updating the current board with that cell
So in effect, it goes over all the empty cells in the board in ascending order of their cell value
count, picking each cell value in turn and creating a board where that cell is filled with that cell
value. Hence it outputs all the next board configurations for the current board.
And then `dfs` goes to work; it goes over all the whole graph and finds all the solutions. Note that
since a solution is reachable from more than one path (fill cell 1 first and then cell 2 or do it in
reverse order), the solutions returned are in general not unique. Also, because of the `concatMap` in
`dfs`, it finds the solutions one by one in a lazy fashion. So it is possible to stop the search early
and just get the first solution found.
Let's try this out now in _ghci_:
```haskell
*Sudoku> let boardStr = ".839216579.734582125187649354813297672956413813679824537268951481425376969541738."
@ -369,6 +400,15 @@ Let's try this out in _ghci_ now:
(7.32 secs, 1322834576 bytes)
```
And it solves the Sudoku as expected! Note how we have to use `nub` to find the unique solutions. Also
note how the time take increases from 1.75 secs for 3 empty cells to 66.64 secs for 4 empty cells,
indicating the exponential nature of the problem graph and the brute force nature of the solver. The
first solution in case of 4 empty cells is however found in just 7.32 secs using `head` to stop the
search early.
So we have now successfully written our first Sudoku solver. Too bad it can't be used for solving boards
with more than few empty cells. Let's see a way to improve the solver drastically, in the next section.
Constraint propagation
----------------------

5
source/downloads/code/sudoku1.hs
View File

@ -10,6 +10,7 @@ import Data.List.Split (chunksOf)
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)
@ -39,6 +40,7 @@ updateBoard (Board ixMap) cell@Cell{..} = Board (M.insert cellIdx cell ixMap)
emptyBoard :: Board
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
@ -50,6 +52,7 @@ readBoard str = do
return $ updateBoard board (Cell i cellVals))
emptyBoard
$ zip [0 .. 80 ] str
showBoard :: Board -> String
showBoard =
map (\Cell{..} ->
@ -70,6 +73,7 @@ asciiShowBoard =
instance Show Board where
show = showBoard
data BoardState = SOLVED | INCOMPLETE | INVALID
deriving (Eq, Show)
@ -93,6 +97,7 @@ 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

56
source/downloads/code/sudoku1.lhs
View File

@ -38,7 +38,7 @@ Basic setup
> import Data.Maybe (fromJust)
> import Data.Ord (comparing)
> import Control.Monad (foldM, guard)
>
>
Now that the imports are out of the way let's setup the basic functionalities.
@ -71,7 +71,7 @@ Now that the imports are out of the way let's setup the basic functionalities.
> emptyBoard :: Board
> 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
@ -115,7 +115,7 @@ right column. An example:
> 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
@ -143,7 +143,7 @@ string contained a digit at the cell index else they have all the digits as cell
>
> 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
@ -181,7 +181,7 @@ Here is an example run in _ghci_:
Is it solved yet?
--------------------
Before we proceed to write a full-fledged Sudoku solver, it is good to have a function which tells us
Before we proceed to write a full-fledged Sudoku solver, we must have a function which tells us
whether a board is filled completely and whether that solution is a valid one.
> data BoardState = SOLVED | INCOMPLETE | INVALID
@ -207,7 +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.
@ -305,7 +305,7 @@ a node in the search graph with the moves linking them as edges. A move is filli
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
We can use [Depth first Search][5] (DFS) 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
@ -335,7 +335,38 @@ We can write this in two parts: a general DFS function and a solver which uses i
> $ board
>
Let's try this out in _ghci_ now:
The `dfs` functions is a literal translation of the DFS algorithm. It visits the graph node by node,
keeping a track of the nodes visited and the goals seen till now. It first checks if the current
node is a goal and if so it just returns it wrapped in a list. If the current node is not a goal, it
checks if has already been visited and if so just returns an empty list. This is done to avoid getting
stuck into infinite loops, going down the same path again and again. If both of these checks fail then
it gets to the general case in which it just finds all the next nodes for the current node,
recursively maps itself over them accumulating all the goals found (by flattening the lists) and
returns them.
`dfsSolver` uses `dfs` to solve Sudoku as a DFS by supplying the `getNext` and `isGoal` functions.
the `isGoal` function is simple, it just checks if the current board is solved by calling the
`boardState` function. `nextBoards` is a little complicated so let's go over it step by step,
reading from bottom to up:
1. it gets all the cells in the board
2. filters in only the empty cells, the cells with more than one cell values
3. sorts the empty cells comparing the count of their cell values
4. for each cell, it takes each cell value and create a cell containing only that cell value and
flattens this list of list of cells into a list of cells
5. for each cell so created, it creates a board by updating the current board with that cell
So in effect, it goes over all the empty cells in the board in ascending order of their cell value
count, picking each cell value in turn and creating a board where that cell is filled with that cell
value. Hence it outputs all the next board configurations for the current board.
And then `dfs` goes to work; it goes over all the whole graph and finds all the solutions. Note that
since a solution is reachable from more than one path (fill cell 1 first and then cell 2 or do it in
reverse order), the solutions returned are in general not unique. Also, because of the `concatMap` in
`dfs`, it finds the solutions one by one in a lazy fashion. So it is possible to stop the search early
and just get the first solution found.
Let's try this out now in _ghci_:
<pre>haskell
*Sudoku> let boardStr = ".839216579.734582125187649354813297672956413813679824537268951481425376969541738."
@ -358,6 +389,15 @@ Let's try this out in _ghci_ now:
(7.32 secs, 1322834576 bytes)
</pre>
And it solves the Sudoku as expected! Note how we have to use `nub` to find the unique solutions. Also
note how the time take increases from 1.75 secs for 3 empty cells to 66.64 secs for 4 empty cells,
indicating the exponential nature of the problem graph and the brute force nature of the solver. The
first solution in case of 4 empty cells is however found in just 7.32 secs using `head` to stop the
search early.
So we have now successfully written our first Sudoku solver. Too bad it can't be used for solving boards
with more than few empty cells. Let's see a way to improve the solver drastically, in the next section.
Constraint propagation
----------------------