algorithm - 在 Haskell 中实现最长路径算法

Question

我需要一些帮助来实现 Haskell 的最长路径算法。我只使用 Haskell 大约两周，之前没有用函数式语言做过任何事情。当您受限于不可变数据和递归时，当我试图用函数式语言实现算法时，我真的很迷茫。

我一直在尝试实现这个算法:http://www.geeksforgeeks.org/find-longest-path-directed-acyclic-graph/

我的图是这样构造的:

data = Graph w = Graph {vertices :: [(Char, w)],
                        edges :: [(Char, Char, w)]} deriving Show

所以我在顶点和边上都有权重，权重可以是任何数据类型。因此，在计算最长路径时，我还需要使用两个函数，f 和 g。从顶点 a 到 b 的最长路径将是 f(w) 和 g(w) 的总和> 对于路径中的所有权重。

我尝试过实现这个，但我总是发现自己在尝试以“命令式”的方式编写代码，这种方式很快就会变得非常丑陋......

请指出正确的方向。

weight_of_longest_path :: (Ord w) => Graph w -> Char -> Char 
                                             -> (w -> w) -> (w -> w) -> w
weight_of_longest_path  (Graph v w) startVert endVert f g =
  let 
    topSort = dropWhile (/= startVert) $ topological_ordering (Graph v w)
    distList = zip topSort $ 
                 (snd $ head $ filter (\(a,b) -> a == startVert) v) 
                 : (repeat (-999999999))
    finalList = getFinalList (Graph v w) topSort distList f g
  in 
    snd $ head $ filter (\(a,b) -> b == endVert) finalList

getFinalList :: (Ord w) => Graph w -> [Char] -> [(Char, w)] 
                                   -> (w -> w) -> (w -> w) -> [(Char, w)]
getFinalList _  [] finalList _ _ = finalList
getFinalList (Graph v w) (firstVert:rest) distList f g =
  let 
    neighbours = secondNodes $ filter (\(a,b,w) -> a == firstVert) w
    finalList = updateList firstVert neighbours distList (Graph v w) f g
  in  
    getFinalList (Graph v w) rest finalList f g

updateList :: (Ord w) => Char -> [Char] -> [(Char, w)] -> Graph w 
                              -> (w -> w) -> (w -> w) -> [(Char, w)]
updateList _ [] updatedList _ _ _ = updatedList
updateList firstVert (neighbour:rest) distList (Graph vertices weights) f g =
  let 
    edgeWeight = selectThird $ head 
          $ filter (\(a,b,w) -> a == firstVert && b == neighbour) weights
    verticeWeight = snd $ head 
          $ filter (\(a,b) -> a == neighbour) vertices
    newDist = calcDist firstVert neighbour verticeWeight edgeWeight 
                       distList f g
    updatedList = replace distList neighbour newDist
  in  
    updateList firstVert rest updatedList (Graph vertices weights) f g


calcDist :: (Ord w) => Char -> Char -> w -> w -> [(Char, w)] 
                            -> (w -> w) -> (w -> w) -> w
calcDist firstVert neighbour verticeWeight edgeWeight distList f g =
  if (compareTo f g 
         (snd $ head $ filter (\(a,b) -> a == neighbour) distList) 
         (snd $ head $ filter (\(a,b) -> a == firstVert) distList) 
         edgeWeight verticeWeight) == True
  then 
     (f (snd $ head $ filter (\(a,b) -> a == firstVert) distList))
     + (g edgeWeight) + (f verticeWeight)
  else 
     (f (snd $ head $ filter (\(a,b) -> a == neighbour) distList))

replace :: [(Char, w)] -> Char -> w -> [(Char, w)]
replace distList vertice value = 
    map (\p@(f, _) -> if f == vertice then (vertice, value) else p) 
        distList

如您所见，对于这样一个简单的算法，它有很多困惑的代码，我相信它可以以更简洁的方式实现。

Answer 1

这是一种采用更“函数式”思维方式的方法。它围绕两个功能展开:

longestPath :: Graph -> Node -> Node -> [Edge]
pathCost :: Graph -> [Edges] -> Int

longestPath 将路径作为最长路径的边列表返回。 pathCost 返回路径的成本。

longestPath 的定义是这样的:

longestPath g start end
  | start == end = []
  | otherwise    = 
    maximumBy (comparing (pathCost g))
              [ e : path | e <- edges of the node start
                           let start' = the other vertex of e,
                           let g' = graph g with node start deleted,
                           let path = longestPath g' start' end ]

(maximumBy 来自Data.List 和comparing 来自Data.Ord)

注意边缘将以相反的顺序生成。

有许多实现细节需要弄清楚，特别是，当没有从 start 到 的路径时，您必须稍微修改它以处理这种情况node(一旦开始删除节点就会发生这种情况)，但这是我要开始的方法。