Copyright  (c) 2018 Oleg Grenrus 

License  BSD3Clause 
Safe Haskell  None 
Language  Haskell2010 
Tools to work with Directed Acyclic Graphs, by taking advantage of topological sorting.
Synopsis
 data G v i = G {
 gVertices :: [i]
 gFromVertex :: i > v
 gToVertex :: v > Maybe i
 gEdges :: i > [i]
 gDiff :: i > i > Int
 gVerticeCount :: Int
 gVertexIndex :: i > Int
 runG :: forall v r. Ord v => Map v (Set v) > (forall i. Ord i => G v i > r) > Either [v] r
 runG' :: forall v r. Ord v => Map v (Set v) > (forall i. Ord i => G v i > r) > Maybe r
 transpose :: forall v i. Ord i => G v i > G v (Down i)
 reduction :: Ord i => G v i > G v i
 closure :: Ord i => G v i > G v i
 dfs :: forall v i. Ord i => G v i > i > [[i]]
 dfsTree :: forall v i. Ord i => G v i > i > Tree i
 allPaths :: forall v i. Ord i => G v i > i > i > [[i]]
 allPaths' :: forall v i. Ord i => G v i > i > i > [i] > [[i]]
 allPathsTree :: forall v i. Ord i => G v i > i > i > Maybe (Tree i)
 shortestPathLengths :: Ord i => G v i > i > [Int]
 longestPathLengths :: Ord i => G v i > i > [Int]
 edgesSet :: Ord i => G v i > Set (i, i)
 adjacencyMap :: Ord v => G v i > Map v (Set v)
 adjacencyList :: Ord v => G v i > [(v, [v])]
 pairs :: [a] > [(a, a)]
 treePairs :: Tree a > [(a, a)]
Graph
Graph used in examples:
>>>
let example :: Map Char (Set Char); example = Map.map Set.fromList $ Map.fromList [('a', "bxde"), ('b', "d"), ('x', "de"), ('d', "e"), ('e', "")]
>>>
:set XRecordWildCards
>>>
import Data.Monoid (All (..))
>>>
import Data.Foldable (traverse_)
>>>
import Data.List (elemIndex)
>>>
import Data.Tree (Tree (..))
Few functions to be used in examples
To make examples slightly shorter:
>>>
let fmap2 = fmap . fmap
>>>
let fmap3 = fmap . fmap2
>>>
let traverse2_ = traverse_ . traverse_
>>>
let traverse3_ = traverse_ . traverse2_
To display trees:
>>>
let dispTree :: Show a => Tree a > IO (); dispTree = go 0 where go i (T.Node x xs) = putStrLn (replicate (i * 2) ' ' ++ show x) >> traverse_ (go (succ i)) xs
Graph representation.
The runG
creates a
structure. Note, that G
v ii
is kept free,
so you cannot construct i
which isn't in the gVertices
.
Therefore operations, like gFromVertex
are total (and fast).
Properties
gVerticeCount
g =length
(gVertices
g)
>>>
runG example $ \G {..} > (length gVertices, gVerticeCount)
Right (5,5)
Just
(gVertexIndex
g x) =elemIndex
x (gVertices
g)
>>>
runG example $ \G {..} > map (`elemIndex` gVertices) gVertices
Right [Just 0,Just 1,Just 2,Just 3,Just 4]
>>>
runG example $ \G {..} > map gVertexIndex gVertices
Right [0,1,2,3,4]
G  

:: Ord v  
=> Map v (Set v)  Adjacency Map 
> (forall i. Ord i => G v i > r)  function on linear indices 
> Either [v] r  Return the result or a cycle in the graph. 
Run action on topologically sorted representation of the graph.
Examples
Topological sorting
>>>
runG example $ \G {..} > map gFromVertex gVertices
Right "axbde"
Vertices are sorted
>>>
runG example $ \G {..} > map gFromVertex $ sort gVertices
Right "axbde"
Outgoing edges
>>>
runG example $ \G {..} > map (map gFromVertex . gEdges) gVertices
Right ["xbde","de","d","e",""]
Note: target indices are always larger than source vertex' index:
>>>
runG example $ \G {..} > getAll $ foldMap (\a > foldMap (\b > All (a < b)) (gEdges a)) gVertices
Right True
Not DAG
>>>
let loop = Map.map Set.fromList $ Map.fromList [('a', "bx"), ('b', "cx"), ('c', "ax"), ('x', "")]
>>>
runG loop $ \G {..} > map gFromVertex gVertices
Left "abc"
>>>
runG (Map.singleton 'a' (Set.singleton 'a')) $ \G {..} > map gFromVertex gVertices
Left "aa"
Transpose
transpose :: forall v i. Ord i => G v i > G v (Down i) Source #
Graph with all edges reversed.
>>>
runG example $ adjacencyList . transpose
Right [('a',""),('b',"a"),('d',"abx"),('e',"adx"),('x',"a")]
Properties
Commutes with closure
>>>
runG example $ adjacencyList . closure . transpose
Right [('a',""),('b',"a"),('d',"abx"),('e',"abdx"),('x',"a")]
>>>
runG example $ adjacencyList . transpose . closure
Right [('a',""),('b',"a"),('d',"abx"),('e',"abdx"),('x',"a")]
Commutes with reduction
>>>
runG example $ adjacencyList . reduction . transpose
Right [('a',""),('b',"a"),('d',"bx"),('e',"d"),('x',"a")]
>>>
runG example $ adjacencyList . transpose . reduction
Right [('a',""),('b',"a"),('d',"bx"),('e',"d"),('x',"a")]
Transitive reduction
reduction :: Ord i => G v i > G v i Source #
Transitive reduction.
Smallest graph, such that if there is a path from u to v in the original graph, then there is also such a path in the reduction.
The green edges are not in the transitive reduction:
>>>
runG example $ \g > adjacencyList $ reduction g
Right [('a',"bx"),('b',"d"),('d',"e"),('e',""),('x',"d")]
Taking closure first doesn't matter:
>>>
runG example $ \g > adjacencyList $ reduction $ closure g
Right [('a',"bx"),('b',"d"),('d',"e"),('e',""),('x',"d")]
Transitive closure
closure :: Ord i => G v i > G v i Source #
Transitive closure.
A graph, such that if there is a path from u to v in the original graph, then there is an edge from u to v in the closure.
The purple edge is added in a closure:
>>>
runG example $ \g > adjacencyList $ closure g
Right [('a',"bdex"),('b',"de"),('d',"e"),('e',""),('x',"de")]
Taking reduction first, doesn't matter:
>>>
runG example $ \g > adjacencyList $ closure $ reduction g
Right [('a',"bdex"),('b',"de"),('d',"e"),('e',""),('x',"de")]
DFS
dfs :: forall v i. Ord i => G v i > i > [[i]] Source #
Depthfirst paths starting at a vertex.
>>>
runG example $ \g@G{..} > fmap3 gFromVertex $ dfs g <$> gToVertex 'x'
Right (Just ["xde","xe"])
All paths
allPaths :: forall v i. Ord i => G v i > i > i > [[i]] Source #
All paths from a
to b
. Note that every path has at least 2 elements, start and end.
Use allPaths'
for the intermediate steps only.
See dfs
, which returns all paths starting at some vertice.
This function returns paths with specified start and end vertices.
>>>
runG example $ \g@G{..} > fmap3 gFromVertex $ allPaths g <$> gToVertex 'a' <*> gToVertex 'e'
Right (Just ["axde","axe","abde","ade","ae"])
There are no paths from element to itself:
>>>
runG example $ \g@G{..} > fmap3 gFromVertex $ allPaths g <$> gToVertex 'a' <*> gToVertex 'a'
Right (Just [])
allPaths' :: forall v i. Ord i => G v i > i > i > [i] > [[i]] Source #
allPaths
without begin and end elements.
>>>
runG example $ \g@G{..} > fmap3 gFromVertex $ allPaths' g <$> gToVertex 'a' <*> gToVertex 'e' <*> pure []
Right (Just ["xd","x","bd","d",""])
allPathsTree :: forall v i. Ord i => G v i > i > i > Maybe (Tree i) Source #
Like allPaths
but return a Tree
.
All paths from a
to b
. Note that every path has at least 2 elements, start and end,
Unfortunately, this is the same as
,
as in our example graph, all paths from dfs
g <$> gToVertex
'a''a'
end up in 'e'
.
>>>
let t = runG example $ \g@G{..} > fmap3 gFromVertex $ allPathsTree g <$> gToVertex 'a' <*> gToVertex 'e'
>>>
fmap3 (T.foldTree $ \a bs > if null bs then [[a]] else concatMap (map (a:)) bs) t
Right (Just (Just ["axde","axe","abde","ade","ae"]))
>>>
fmap3 (Set.fromList . treePairs) t
Right (Just (Just (fromList [('a','b'),('a','d'),('a','e'),('a','x'),('b','d'),('d','e'),('x','d'),('x','e')])))
>>>
let ls = runG example $ \g@G{..} > fmap3 gFromVertex $ allPaths g <$> gToVertex 'a' <*> gToVertex 'e'
>>>
fmap2 (Set.fromList . concatMap pairs) ls
Right (Just (fromList [('a','b'),('a','d'),('a','e'),('a','x'),('b','d'),('d','e'),('x','d'),('x','e')]))
Tree
paths show how one can explore the paths.
>>>
traverse3_ dispTree t
'a' 'x' 'd' 'e' 'e' 'b' 'd' 'e' 'd' 'e' 'e'
>>>
traverse3_ (putStrLn . T.drawTree . fmap show) t
'a'  + 'x'    + 'd'      ` 'e'    ` 'e' ...
There are no paths from element to itself, but we'll return a
single root node, as Tree
cannot be empty.
>>>
runG example $ \g@G{..} > fmap3 gFromVertex $ allPathsTree g <$> gToVertex 'a' <*> gToVertex 'a'
Right (Just (Just (Node {rootLabel = 'a', subForest = []})))
Path lengths
shortestPathLengths :: Ord i => G v i > i > [Int] Source #
Shortest paths lengths starting from a vertex.
The resulting list is of the same length as gVertices
.
It's quite efficient to compute all shortest (or longest) paths' lengths
at once. Zero means that there are no path.
>>>
runG example $ \g@G{..} > shortestPathLengths g <$> gToVertex 'a'
Right (Just [0,1,1,1,1])
>>>
runG example $ \g@G{..} > shortestPathLengths g <$> gToVertex 'b'
Right (Just [0,0,0,1,2])
longestPathLengths :: Ord i => G v i > i > [Int] Source #
Longest paths lengths starting from a vertex.
The resulting list is of the same length as gVertices
.
>>>
runG example $ \g@G{..} > longestPathLengths g <$> gToVertex 'a'
Right (Just [0,1,1,2,3])
>>>
runG example $ \G {..} > map gFromVertex gVertices
Right "axbde"
>>>
runG example $ \g@G{..} > longestPathLengths g <$> gToVertex 'b'
Right (Just [0,0,0,1,2])
Query
edgesSet :: Ord i => G v i > Set (i, i) Source #
Edges set.
>>>
runG example $ \g@G{..} > map (\(a,b) > [gFromVertex a, gFromVertex b]) $ Set.toList $ edgesSet g
Right ["ax","ab","ad","ae","xd","xe","bd","de"]
adjacencyMap :: Ord v => G v i > Map v (Set v) Source #
Recover adjacency map representation from the G
.
>>>
runG example adjacencyMap
Right (fromList [('a',fromList "bdex"),('b',fromList "d"),('d',fromList "e"),('e',fromList ""),('x',fromList "de")])
adjacencyList :: Ord v => G v i > [(v, [v])] Source #
Adjacency list representation of G
.
>>>
runG example adjacencyList
Right [('a',"bdex"),('b',"d"),('d',"e"),('e',""),('x',"de")]