# Haskell Quiz/Astar/Solution Dolio

### From HaskellWiki

A* requires keeping a priority queue of places to visit. This can be done with a simple sorted list, but I decided to make a PriorityQueue data type for use in the algorithm instead. The implementation uses lazy pairing heaps from Chris Okasaki's **Purely Functional Data Structures**.

module PriorityQueue ( PriorityQueue, empty, singleton, fromList, null, deleteFindMin, deleteMin, findMin, insert, union ) where import Prelude hiding (null) data Ord k => PriorityQueue k a = Nil | Branch k a (PriorityQueue k a) (PriorityQueue k a) empty :: Ord k => PriorityQueue k a empty = Nil singleton :: Ord k => k -> a -> PriorityQueue k a singleton k a = Branch k a Nil Nil fromList :: Ord k => [(k,a)] -> PriorityQueue k a fromList = foldr (\(k,a) q -> singleton k a `union` q) empty null :: Ord k => PriorityQueue k a -> Bool null Nil = True null _ = False deleteFindMin :: Ord k => PriorityQueue k a -> ((k,a), PriorityQueue k a) deleteFindMin Nil = error "Empty heap." deleteFindMin (Branch k a l r) = ((k,a), union l r) deleteMin :: Ord k => PriorityQueue k a -> PriorityQueue k a deleteMin h = snd (deleteFindMin h) findMin :: Ord k => PriorityQueue k a -> (k, a) findMin h = fst (deleteFindMin h) insert :: Ord k => k -> a -> PriorityQueue k a -> PriorityQueue k a insert k a h = union (singleton k a) h union :: Ord k => PriorityQueue k a -> PriorityQueue k a -> PriorityQueue k a union l Nil = l union Nil r = r union l@(Branch kl _ _ _) r@(Branch kr _ _ _) | kl <= kr = link l r | otherwise = link r l link (Branch k a Nil m) r = Branch k a r m link (Branch k a ll lr) r = Branch k a (union ll r) lr

Not all the functions from data structures in the standard library (Data.Map, Data.Set, etc.) are provided; I only wrote those that are needed for the algorithm. However, this could be extended easily.

The rest is just a general A* function, which takes a starting place, and functions for successors, testing for completion, cost of a place, and heuristic estimation from a place to the end, returning the path taken (a list from end to start). The rest of the code deals with the specifics of the ASCII map:

{-# OPTIONS_GHC -fglasgow-exts #-} module Main where import Control.Monad (guard, liftM2) import Control.Monad.Instances import Data.List (findIndex) import qualified Data.Set as S import qualified Data.Map as M import qualified PriorityQueue as Q type Point = (Int, Int) type Map = [[Char]] find :: Char -> Map -> Point find c m = find' 0 m where find' _ [] = error "Can't find tile." find' y (h:t) | Just x <- findIndex (==c) h = (y, x) | otherwise = find' (y+1) t heuristic :: Point -> Point -> Int heuristic (x, y) (u, v) = abs (x - u) `max` abs (y - v) successor :: Map -> Point -> [Point] successor m (x,y) = do u <- [x + 1, x, x - 1] v <- [y + 1, y, y - 1] guard (0 <= u && u < length m) guard (0 <= v && v < length (head m)) guard (u /= x || y /= v) guard (m !! u !! v /= '~') return (u, v) astar start succ end cost heur = astar' (S.singleton start) (Q.singleton (heur start) [start]) where astar' seen q | Q.null q = error "No Solution." | end n = next | otherwise = astar' seen' q' where ((c,next), dq) = Q.deleteFindMin q n = head next succs = filter (`S.notMember` seen) $ succ n costs = map ((+ c) . (subtract $ heur n) . liftM2 (+) cost heur) succs q' = dq `Q.union` Q.fromList (zip costs (map (:next) succs)) seen' = seen `S.union` S.fromList succs path :: [[Char]] -> [Point] -> [[Char]] path m l = iterY m l 0 where iterY [] _ _ = [] iterY (h:t) l n = iterX h l n 0 : iterY t l (n+1) iterX [] _ _ _ = [] iterX (h:t) l n m = (if (n,m) `elem` l then '#' else h) : iterX t l n (m+1) doit s = unlines . path m $ astar start succ (== end) cost h where m = lines s start = find '@' m end = find 'X' m succ = successor m h = heuristic end cost (x, y) = costsM M.! (m !! x !! y) costsM = M.fromList [('@',1),('x',1),('X',1),('.',1),('*',2),('^',3)] main = interact doit