99 questions/Solutions/81

(**) Path from one node to another one

Write a function that, given two nodes a and b in a graph, returns all the acyclic paths from a to b.

import List (elem)

paths :: Eq a => a -> a -> [(a,a)] -> [[a]]
paths a b g = paths1 a b g []

paths1 :: Eq a => a -> a -> [(a,a)] -> [a] -> [[a]]
paths1 a b g current = paths2 a b g current [ y | (x,y) <- g, x == a ]

paths2 :: Eq a => a -> a -> [(a,a)] -> [a] -> [a] -> [[a]]
paths2 a b g current []	| a == b = [current++[b]]
			| otherwise = []
paths2 a b g current (x:xs) | a == b = [current++[b]] 
			    | elem a current = []
			    | otherwise = (paths1 x b g (current++[a])) ++ (paths2 a b g current xs)

This solution uses a representation of a (directed) graph as a list of arcs (a,b).

---

Here is another implementation using List's monadic behavior

import Data.List (partition)

pathsImpl :: Eq a => [a] -> a -> a -> [(a, a)] -> [[a]]
pathsImpl trail src dest clauses
    | src == dest = [src:trail]
    | otherwise = do
        let (nexts, rest) = partition ((==src) . fst) clauses
        next <- nexts
        pathsImpl (src:trail) (snd next) dest rest

paths :: Eq a => a -> a -> [(a, a)] -> [[a]]
paths src dest clauses = map reverse (pathsImpl [] src dest clauses)

---

Here is another recursive implementation

paths :: Eq a =>a -> a -> [(a,a)] -> [[a]] 
paths source sink edges 
    | source == sink = [[sink]]
    | otherwise = [
        source:path | edge<-edges, (fst edge) == source,
        path<-(paths (snd edge) sink [e|e<-edges, e/=edge])
    ];

---

yet another solution using monadic behavior of lists

paths :: (Eq a) => a -> a -> [Arc a] -> [[a]]
paths source sink arcs 
    | source == sink = [[source]]
    | otherwise      = map (map fst) $ aux source []
        where 
            aux current pathSoFar = 
                let nextEdges = filter ((== current) . fst) arcs 
                    notCyclic = not . (\(_,t) -> (t == source) || (elem t $ map snd pathSoFar)) 
                    noCycles  = filter notCyclic nextEdges
                in  noCycles >>= \(f,t) -> do 
                                              if (t == sink) then return $ pathSoFar ++ (f,t):[(t,t)]
                                                             else aux t (pathSoFar ++ [(f,t)])