Topograph

Tools to work with Directed Acyclic Graphs, by taking advantage of topological sorting.

Graph

Initial setup and imports:

>>> :set -XRecordWildCards
>>> import Data.Monoid (All (..))
>>> import Data.Foldable (traverse_)
>>> import Data.List (elemIndex, sort)
>>> import Data.Tree (Tree (..))
>>> import Data.Map (Map)
>>> import Data.Set (Set)
>>> import qualified Data.Tree as T
>>> import qualified Data.Map as Map
>>> import qualified Data.Set as Set

Some compatibility imports

>>> import Control.Applicative
>>> import Data.Foldable (traverse_, foldMap)

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', "")]

Few functions to be used in examples

To make examples slightly shorter:

>>> let fmap2 f = fmap (fmap f)
>>> let fmap3 f = fmap (fmap2 f)
>>> let traverse2_ f = traverse_ (traverse_ f)
>>> let traverse3_ f = traverse_ (traverse2_ f)

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

And fold them (this function is available in recent containers):

>>> let foldTree f = go where go (T.Node x ts) = f x (map go ts)

data G v i

Graph representation.

The runG creates a G v i structure. Note, that i 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]

Constructors

G

Fields

• gVertices :: [i]

  all vertices, in topological order.

• gFromVertex :: i -> v

  O(1). retrieve original vertex data

• gToVertex :: v -> Maybe i

  O(log n).

• gEdges :: i -> [i]

  O(1). Outgoing edges. Note: target indices are larger than source index.

• gDiff :: i -> i -> Int

  O(1). Upper bound of the path length. Negative means there aren't path.

• gVerticeCount :: Int

  O(1). gVerticeCount g = length (gVertices g)

• gVertexIndex :: i -> Int

  O(1). Just (verticeIndex g x) = elemIndex x (gVertices g). Note, there are no efficient way to convert Int into i, conversion back and forth is discouraged on purpose.

runG

Arguments

:: forall v r. 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"

runG'

Arguments

:: forall v r. Ord v

=> Map v (Set v)

Adjacency Map

-> (forall i. Ord i => G v i -> r)

function on linear indices

-> Maybe r

Return the result or Nothing if there is a cycle.

Like runG but returns Maybe

Transpose

transpose :: forall v i. Ord i => G v i -> G v (Down i)

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

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

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]]

Depth-first paths starting at a vertex.

>>> runG example $ \g@G{..} -> fmap3 gFromVertex $ dfs g <$> gToVertex 'x'
Right (Just ["xde","xe"])

dfsTree :: forall v i. Ord i => G v i -> i -> Tree i

like dfs but returns a Tree.

>>> traverse2_ dispTree $ runG example $ \g@G{..} -> fmap2 gFromVertex $ dfsTree g <$> gToVertex 'x'
'x'
  'd'
    'e'
  'e'

All paths

allPaths :: forall v i. Ord i => G v i -> i -> i -> [[i]]

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]]

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)

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 dfs g <$> gToVertex 'a', as in our example graph, all paths from 'a' end up in 'e'.

>>> let t = runG example $ \g@G{..} -> fmap3 gFromVertex $ allPathsTree g <$> gToVertex 'a' <*> gToVertex 'e'
>>> fmap3 (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]

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]

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)

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)

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])]

Adjacency list representation of G.

>>> runG example adjacencyList
Right [('a',"bdex"),('b',"d"),('d',"e"),('e',""),('x',"de")]

Utilities

pairs :: [a] -> [(a, a)]

Consecutive pairs.

>>> pairs [1..10]
[(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10)]

>>> pairs []
[]

treePairs :: Tree a -> [(a, a)]

Like pairs but for Tree.