# Haskell Quiz/Amazing Mazes/Solution Abhinav

From HaskellWiki

```
{-
A solution to rubyquiz 31 (http://rubyquiz.com/quiz31.html).
Generate a rectangular maze given its width and height. The maze should be
solvable for any start and end positions and there should be only one possible
solution for any pair of start and end positions.
Generate the ASCII output representing the maze.
Find the solution of the maze. Produce ASCII output to visualize the solution.
The maze generation algorithm used is recursive backtracking and the maze
solution algorithm used is A*.
Usage: ./AmazingMazes <width> <height> <start_x> <start_y> <end_x> <end_y>
Coordinates are zero based.
abhinav@xj9:rubyquiz# bin/AmazingMazes 10 10 0 0 9 9
+---+---+---+---+---+---+---+---+---+---+
| s > v | | |
+---+---+ +---+ + +---+ + +---+
| v < < | | | | | |
+ +---+---+---+---+---+ + + + +
| v | | | |
+ +---+---+---+ + +---+---+---+ +
| > > > v | | | > v |
+---+---+---+ + + +---+---+ + +
| | v < | | > > ^ | v |
+ + + +---+---+---+ +---+---+ +
| | | > > > > ^ | | v |
+ +---+---+---+---+---+---+ +---+ +
| | | | v < < |
+ + + + + + +---+ +---+---+
| | | | | | v < | |
+ + +---+---+ +---+ +---+ +---+
| | | | v < | |
+ +---+ + + + +---+---+---+ +
| | | > > > > e |
+---+---+---+---+---+---+---+---+---+---+
Copyright 2012 Abhinav Sarkar <abhinav@abhinavsarkar.net>
-}
{-# LANGUAGE BangPatterns, TupleSections #-}
module Main where
import qualified Data.Map as M
import AStar
import Data.List (nub)
import Data.Maybe (fromMaybe)
import Control.Monad (foldM)
import Control.Monad.State (State, get, put, evalState)
import System.Environment (getArgs)
import System.Random (Random, StdGen, randomR, randomRs, newStdGen, split)
sliding :: Int -> Int -> [a] -> [[a]]
sliding _ _ [] = []
sliding size step xs = take size xs : sliding size step (drop step xs)
-- randomness --
type RandomState = State StdGen
getRandomR :: Random a => (a, a) -> RandomState a
getRandomR limits = do
gen <- get
let (val, gen') = randomR limits gen
put gen'
return val
getRandomRs :: Random a => (a, a) -> RandomState [a]
getRandomRs limits = do
gen <- get
return $ randomRs limits gen
randomShuffle :: [a] -> RandomState [a]
randomShuffle list = do
let len = length list
rs <- getRandomRs (0, len - 1)
g <- get
let (_, g') = split g
put g'
return $ map (list !!) . head . dropWhile ((/= len) . length) . map nub . sliding len 1 $ rs
-- maze --
-- a cell with x and y coordinates
type Cell = (Int, Int)
-- a maze with width, height and a map of cell paths
data Maze = Maze Int Int (M.Map Cell [Cell]) deriving (Show)
-- a solution to a maze with the start and end cells and the path map
data MazeSolution = MazeSolution Cell Cell (M.Map Cell Cell)
-- get the neighbour cells
nextCells :: Int -> Int -> Cell -> [Cell]
nextCells width height (x, y) =
filter (\(x', y') -> and [x' >= 0, x' < width, y' >= 0, y' < height])
. map (\(xd, yd) -> (x + xd, y + yd))
$ [(0,-1), (1,0), (0,1), (-1,0)]
-- generate a random maze given the start cell and an empty maze
generateMaze_ :: Cell -> Maze -> RandomState Maze
generateMaze_ start maze@(Maze width height cellMap) = do
!next <- randomShuffle . filter (not . flip M.member cellMap) $ nextCells width height start
if null next
then return $ Maze width height (M.insertWith' (++) start [] cellMap)
else
foldM (\mz@(Maze _ _ m) n -> M.keys m `seq`
if not . M.member n $ m
then generateMaze_ n
(Maze width height
(M.insertWith' (++) n [start] (M.insertWith' (++) start [n] m)))
else return mz)
maze next
-- generate a random maze given the maze width and height using recursive backtracking
generateMaze :: Int -> Int -> RandomState Maze
generateMaze width height = do
x <- getRandomR (0, width - 1)
y <- getRandomR (0, height - 1)
generateMaze_ (x, y) (Maze width height M.empty)
-- render a maze and its solution as a string
renderMaze :: Maze -> MazeSolution -> String
renderMaze maze@(Maze width height _) solution =
concatMap (renderMazeRow maze solution) [0 .. (height - 1)]
++ concat (replicate width "+---") ++ "+"
-- render a row of a maze and the maze's solution as a string
renderMazeRow :: Maze -> MazeSolution -> Int -> String
renderMazeRow maze@(Maze width height _) solution rowIx =
let (up, side) = unzip . map (renderMazeCell maze solution rowIx) $ [0 .. (width - 1)]
in concat up ++ "+" ++ "\n" ++ concat side ++ "|" ++ "\n"
-- render a cell of a maze and the maze's solution as a pair of strings
renderMazeCell :: Maze -> MazeSolution -> Int -> Int -> (String, String)
renderMazeCell (Maze _ _ cellMap) (MazeSolution start end solution) rowIx colIx = let
cell = (colIx, rowIx)
up = (colIx, rowIx - 1)
side = (colIx - 1, rowIx)
in ("+" ++ if up `elem` next cell then " " else "---",
(if side `elem` next cell then " " else "|") ++ " " ++ mark cell ++ " ")
where
next = fromMaybe [] . flip M.lookup cellMap
mark cell@(x, y)
| cell == start = "s"
| cell == end = "e"
| otherwise = case M.lookup cell solution of
Nothing -> " "
Just (x', y') -> fromMaybe " " $ M.lookup (x' - x, y' - y) marks
-- symbols to mark the solution path
marks = M.fromList [((0,-1), "^"), ((1,0), ">"), ((0,1), "v"), ((-1,0), "<")]
-- solve the maze using A* given the maze and the start and end cells using
-- Manhattan distance as the heuristic
solveMaze :: Maze -> Cell -> Cell -> MazeSolution
solveMaze maze@(Maze _ _ cellMap) start end =
MazeSolution start end
. M.fromList
. map (\a -> (a !! 0, a !! 1))
. filter ((== 2) . length)
. sliding 2 1
. fromMaybe [] . fmap snd
. astar start end (map (,1) . fromMaybe [] . flip M.lookup cellMap)
$ (\(x, y) (x', y') -> abs (x - x') + abs (y - y'))
main = do
(width : height : sx : sy : ex : ey : _) <- fmap (map read) getArgs
g <- newStdGen
let mz = evalState (generateMaze width height) g
putStrLn $ renderMaze mz (solveMaze mz (sx, sy) (ex, ey))
```

**Description:** The program generates the maze using the recursive backtracking algorithm. The maze is represented as a graph of connected cells.
The maze is solved using the A* graph search algorithm with the Manhattan distance as the heuristic.

Source: https://github.com/abhin4v/rubyquiz/blob/master/AmazingMazes.hs