-----------------------------------------------------------------------------
-- |
-- Module     : Data.Graph.Typed
-- Copyright  : (c) Anton Lorenzen, Andrey Mokhov 2016-2021
-- License    : MIT (see the file LICENSE)
-- Maintainer : anfelor@posteo.de, andrey.mokhov@gmail.com
-- Stability  : unstable
--
-- __Alga__ is a library for algebraic construction and manipulation of graphs
-- in Haskell. See <https://github.com/snowleopard/alga-paper this paper> for the
-- motivation behind the library, the underlying theory, and implementation details.
--
-- This module provides primitives for interoperability between this library and
-- the "Data.Graph" module of the containers library. It is for internal use only
-- and may be removed without notice at any point.
-----------------------------------------------------------------------------
module Data.Graph.Typed (
    -- * Data type and construction
    GraphKL(..), fromAdjacencyMap, fromAdjacencyIntMap,

    -- * Basic algorithms
    dfsForest, dfsForestFrom, dfs, topSort, scc
    ) where

import Data.Tree
import Data.Maybe
import Data.Foldable

import qualified Data.Graph as KL

import qualified Algebra.Graph.AdjacencyMap          as AM
import qualified Algebra.Graph.NonEmpty.AdjacencyMap as NonEmpty
import qualified Algebra.Graph.AdjacencyIntMap       as AIM

import qualified Data.Map.Strict                     as Map
import qualified Data.Set                            as Set

-- | 'GraphKL' encapsulates King-Launchbury graphs, which are implemented in
-- the "Data.Graph" module of the @containers@ library.
data GraphKL a = GraphKL {
    -- | Array-based graph representation (King and Launchbury, 1995).
    GraphKL a -> Graph
toGraphKL :: KL.Graph,
    -- | A mapping of "Data.Graph.Vertex" to vertices of type @a@.
    -- This is partial and may fail if the vertex is out of bounds.
    GraphKL a -> Vertex -> a
fromVertexKL :: KL.Vertex -> a,
    -- | A mapping from vertices of type @a@ to "Data.Graph.Vertex".
    -- Returns 'Nothing' if the argument is not in the graph.
    GraphKL a -> a -> Maybe Vertex
toVertexKL :: a -> Maybe KL.Vertex }

-- | Build 'GraphKL' from an 'AM.AdjacencyMap'. If @fromAdjacencyMap g == h@
-- then the following holds:
--
-- @
-- map ('fromVertexKL' h) ('Data.Graph.vertices' $ 'toGraphKL' h)                               == 'AM.vertexList' g
-- map (\\(x, y) -> ('fromVertexKL' h x, 'fromVertexKL' h y)) ('Data.Graph.edges' $ 'toGraphKL' h) == 'AM.edgeList' g
-- 'toGraphKL' (fromAdjacencyMap (1 * 2 + 3 * 1))                                == 'array' (0,2) [(0,[1]), (1,[]), (2,[0])]
-- 'toGraphKL' (fromAdjacencyMap (1 * 2 + 2 * 1))                                == 'array' (0,1) [(0,[1]), (1,[0])]
-- @
fromAdjacencyMap :: Ord a => AM.AdjacencyMap a -> GraphKL a
fromAdjacencyMap :: AdjacencyMap a -> GraphKL a
fromAdjacencyMap AdjacencyMap a
am = GraphKL :: forall a.
Graph -> (Vertex -> a) -> (a -> Maybe Vertex) -> GraphKL a
GraphKL
    { toGraphKL :: Graph
toGraphKL    = Graph
g
    , fromVertexKL :: Vertex -> a
fromVertexKL = \Vertex
u -> case Vertex -> ((), a, [a])
r Vertex
u of (()
_, a
v, [a]
_) -> a
v
    , toVertexKL :: a -> Maybe Vertex
toVertexKL   = a -> Maybe Vertex
t }
  where
    (Graph
g, Vertex -> ((), a, [a])
r, a -> Maybe Vertex
t) = [((), a, [a])]
-> (Graph, Vertex -> ((), a, [a]), a -> Maybe Vertex)
forall key node.
Ord key =>
[(node, key, [key])]
-> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)
KL.graphFromEdges [ ((), a
x, [a]
ys) | (a
x, [a]
ys) <- AdjacencyMap a -> [(a, [a])]
forall a. AdjacencyMap a -> [(a, [a])]
AM.adjacencyList AdjacencyMap a
am ]

-- | Build 'GraphKL' from an 'AIM.AdjacencyIntMap'. If
-- @fromAdjacencyIntMap g == h@ then the following holds:
--
-- @
-- map ('fromVertexKL' h) ('Data.Graph.vertices' $ 'toGraphKL' h)                               == 'Data.IntSet.toAscList' ('Algebra.Graph.AdjacencyIntMap.vertexIntSet' g)
-- map (\\(x, y) -> ('fromVertexKL' h x, 'fromVertexKL' h y)) ('Data.Graph.edges' $ 'toGraphKL' h) == 'Algebra.Graph.AdjacencyIntMap.edgeList' g
-- 'toGraphKL' (fromAdjacencyIntMap (1 * 2 + 3 * 1))                             == 'array' (0,2) [(0,[1]), (1,[]), (2,[0])]
-- 'toGraphKL' (fromAdjacencyIntMap (1 * 2 + 2 * 1))                             == 'array' (0,1) [(0,[1]), (1,[0])]
-- @
fromAdjacencyIntMap :: AIM.AdjacencyIntMap -> GraphKL Int
fromAdjacencyIntMap :: AdjacencyIntMap -> GraphKL Vertex
fromAdjacencyIntMap AdjacencyIntMap
aim = GraphKL :: forall a.
Graph -> (Vertex -> a) -> (a -> Maybe Vertex) -> GraphKL a
GraphKL
    { toGraphKL :: Graph
toGraphKL    = Graph
g
    , fromVertexKL :: Vertex -> Vertex
fromVertexKL = \Vertex
x -> case Vertex -> ((), Vertex, [Vertex])
r Vertex
x of (()
_, Vertex
v, [Vertex]
_) -> Vertex
v
    , toVertexKL :: Vertex -> Maybe Vertex
toVertexKL   = Vertex -> Maybe Vertex
t }
  where
    (Graph
g, Vertex -> ((), Vertex, [Vertex])
r, Vertex -> Maybe Vertex
t) = [((), Vertex, [Vertex])]
-> (Graph, Vertex -> ((), Vertex, [Vertex]),
    Vertex -> Maybe Vertex)
forall key node.
Ord key =>
[(node, key, [key])]
-> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)
KL.graphFromEdges [ ((), Vertex
x, [Vertex]
ys) | (Vertex
x, [Vertex]
ys) <- AdjacencyIntMap -> [(Vertex, [Vertex])]
AIM.adjacencyList AdjacencyIntMap
aim ]

-- | Compute the /depth-first search/ forest of a graph.
--
-- In the following examples we will use the helper function:
--
-- @
-- (%) :: (GraphKL Int -> a) -> 'AM.AdjacencyMap' Int -> a
-- a % g = a $ 'fromAdjacencyMap' g
-- @
--
-- for greater clarity.
--
-- @
-- 'AM.forest' (dfsForest % 'AM.edge' 1 1)           == 'AM.vertex' 1
-- 'AM.forest' (dfsForest % 'AM.edge' 1 2)           == 'AM.edge' 1 2
-- 'AM.forest' (dfsForest % 'AM.edge' 2 1)           == 'AM.vertices' [1, 2]
-- 'AM.isSubgraphOf' ('AM.forest' $ dfsForest % x) x == True
-- dfsForest % 'AM.forest' (dfsForest % x)      == dfsForest % x
-- dfsForest % 'AM.vertices' vs                 == 'map' (\\v -> Node v []) ('Data.List.nub' $ 'Data.List.sort' vs)
-- 'AM.dfsForestFrom' ('AM.vertexList' x) % x        == dfsForest % x
-- dfsForest % (3 * (1 + 4) * (1 + 5))     == [ Node { rootLabel = 1
--                                                   , subForest = [ Node { rootLabel = 5
--                                                                        , subForest = [] }]}
--                                            , Node { rootLabel = 3
--                                                   , subForest = [ Node { rootLabel = 4
--                                                                        , subForest = [] }]}]
-- @
dfsForest :: GraphKL a -> Forest a
dfsForest :: GraphKL a -> Forest a
dfsForest (GraphKL Graph
g Vertex -> a
r a -> Maybe Vertex
_) = (Tree Vertex -> Tree a) -> [Tree Vertex] -> Forest a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((Vertex -> a) -> Tree Vertex -> Tree a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Vertex -> a
r) (Graph -> [Tree Vertex]
KL.dff Graph
g)

-- | Compute the /depth-first search/ forest of a graph, searching from each of
-- the given vertices in order. Note that the resulting forest does not
-- necessarily span the whole graph, as some vertices may be unreachable.
--
-- In the following examples we will use the helper function:
--
-- @
-- (%) :: (GraphKL Int -> a) -> 'AM.AdjacencyMap' Int -> a
-- a % g = a $ 'fromAdjacencyMap' g
-- @
--
-- for greater clarity.
--
-- @
-- 'AM.forest' (dfsForestFrom [1]    % 'AM.edge' 1 1)       == 'AM.vertex' 1
-- 'AM.forest' (dfsForestFrom [1]    % 'AM.edge' 1 2)       == 'AM.edge' 1 2
-- 'AM.forest' (dfsForestFrom [2]    % 'AM.edge' 1 2)       == 'AM.vertex' 2
-- 'AM.forest' (dfsForestFrom [3]    % 'AM.edge' 1 2)       == 'AM.empty'
-- 'AM.forest' (dfsForestFrom [2, 1] % 'AM.edge' 1 2)       == 'AM.vertices' [1, 2]
-- 'AM.isSubgraphOf' ('AM.forest' $ dfsForestFrom vs % x) x == True
-- dfsForestFrom ('AM.vertexList' x) % x               == 'dfsForest' % x
-- dfsForestFrom vs               % 'AM.vertices' vs   == 'map' (\\v -> Node v []) ('Data.List.nub' vs)
-- dfsForestFrom []               % x             == []
-- dfsForestFrom [1, 4] % (3 * (1 + 4) * (1 + 5)) == [ Node { rootLabel = 1
--                                                          , subForest = [ Node { rootLabel = 5
--                                                                               , subForest = [] }
--                                                   , Node { rootLabel = 4
--                                                          , subForest = [] }]
-- @
dfsForestFrom :: [a] -> GraphKL a -> Forest a
dfsForestFrom :: [a] -> GraphKL a -> Forest a
dfsForestFrom [a]
vs (GraphKL Graph
g Vertex -> a
r a -> Maybe Vertex
t) = (Tree Vertex -> Tree a) -> [Tree Vertex] -> Forest a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap ((Vertex -> a) -> Tree Vertex -> Tree a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Vertex -> a
r) (Graph -> [Vertex] -> [Tree Vertex]
KL.dfs Graph
g ((a -> Maybe Vertex) -> [a] -> [Vertex]
forall a b. (a -> Maybe b) -> [a] -> [b]
mapMaybe a -> Maybe Vertex
t [a]
vs))

-- | Compute the list of vertices visited by the /depth-first search/ in a
-- graph, when searching from each of the given vertices in order.
--
-- In the following examples we will use the helper function:
--
-- @
-- (%) :: (GraphKL Int -> a) -> 'AM.AdjacencyMap' Int -> a
-- a % g = a $ 'fromAdjacencyMap' g
-- @
--
-- for greater clarity.
--
-- @
-- dfs [1]   % 'AM.edge' 1 1                 == [1]
-- dfs [1]   % 'AM.edge' 1 2                 == [1,2]
-- dfs [2]   % 'AM.edge' 1 2                 == [2]
-- dfs [3]   % 'AM.edge' 1 2                 == []
-- dfs [1,2] % 'AM.edge' 1 2                 == [1,2]
-- dfs [2,1] % 'AM.edge' 1 2                 == [2,1]
-- dfs []    % x                        == []
-- dfs [1,4] % (3 * (1 + 4) * (1 + 5))  == [1,5,4]
-- 'AM.isSubgraphOf' ('AM.vertices' $ dfs vs x) x == True
-- @
dfs :: [a] -> GraphKL a -> [a]
dfs :: [a] -> GraphKL a -> [a]
dfs [a]
vs = (Tree a -> [a]) -> [Tree a] -> [a]
forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap Tree a -> [a]
forall a. Tree a -> [a]
flatten ([Tree a] -> [a]) -> (GraphKL a -> [Tree a]) -> GraphKL a -> [a]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [a] -> GraphKL a -> [Tree a]
forall a. [a] -> GraphKL a -> Forest a
dfsForestFrom [a]
vs

-- | Compute the /topological sort/ of a graph. Note that this function returns
-- a result even if the graph is cyclic.
--
-- In the following examples we will use the helper function:
--
-- @
-- (%) :: (GraphKL Int -> a) -> 'AM.AdjacencyMap' Int -> a
-- a % g = a $ 'fromAdjacencyMap' g
-- @
--
-- for greater clarity.
--
-- @
-- topSort % (1 * 2 + 3 * 1) == [3,1,2]
-- topSort % (1 * 2 + 2 * 1) == [1,2]
-- @
topSort :: GraphKL a -> [a]
topSort :: GraphKL a -> [a]
topSort (GraphKL Graph
g Vertex -> a
r a -> Maybe Vertex
_) = (Vertex -> a) -> [Vertex] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map Vertex -> a
r (Graph -> [Vertex]
KL.topSort Graph
g)

scc :: Ord a => AM.AdjacencyMap a -> AM.AdjacencyMap (NonEmpty.AdjacencyMap a)
scc :: AdjacencyMap a -> AdjacencyMap (AdjacencyMap a)
scc AdjacencyMap a
m = (Vertex -> AdjacencyMap a)
-> AdjacencyMap Vertex -> AdjacencyMap (AdjacencyMap a)
forall a b.
(Ord a, Ord b) =>
(a -> b) -> AdjacencyMap a -> AdjacencyMap b
AM.gmap (Map Vertex (AdjacencyMap a)
component Map Vertex (AdjacencyMap a) -> Vertex -> AdjacencyMap a
forall k a. Ord k => Map k a -> k -> a
Map.!) (AdjacencyMap Vertex -> AdjacencyMap (AdjacencyMap a))
-> AdjacencyMap Vertex -> AdjacencyMap (AdjacencyMap a)
forall a b. (a -> b) -> a -> b
$ AdjacencyMap Vertex -> AdjacencyMap Vertex
forall a. Ord a => AdjacencyMap a -> AdjacencyMap a
removeSelfLoops (AdjacencyMap Vertex -> AdjacencyMap Vertex)
-> AdjacencyMap Vertex -> AdjacencyMap Vertex
forall a b. (a -> b) -> a -> b
$ (a -> Vertex) -> AdjacencyMap a -> AdjacencyMap Vertex
forall a b.
(Ord a, Ord b) =>
(a -> b) -> AdjacencyMap a -> AdjacencyMap b
AM.gmap (Map a Vertex
leader Map a Vertex -> a -> Vertex
forall k a. Ord k => Map k a -> k -> a
Map.!) AdjacencyMap a
m
  where
    GraphKL Graph
g Vertex -> a
decode a -> Maybe Vertex
_ = AdjacencyMap a -> GraphKL a
forall a. Ord a => AdjacencyMap a -> GraphKL a
fromAdjacencyMap AdjacencyMap a
m
    sccs :: [[Vertex]]
sccs      = (Tree Vertex -> [Vertex]) -> [Tree Vertex] -> [[Vertex]]
forall a b. (a -> b) -> [a] -> [b]
map Tree Vertex -> [Vertex]
forall (t :: * -> *) a. Foldable t => t a -> [a]
toList (Graph -> [Tree Vertex]
KL.scc Graph
g)
    leader :: Map a Vertex
leader    = [(a, Vertex)] -> Map a Vertex
forall k a. Ord k => [(k, a)] -> Map k a
Map.fromList [ (Vertex -> a
decode Vertex
y, Vertex
x)      | Vertex
x:[Vertex]
xs <- [[Vertex]]
sccs, Vertex
y <- Vertex
xVertex -> [Vertex] -> [Vertex]
forall a. a -> [a] -> [a]
:[Vertex]
xs ]
    component :: Map Vertex (AdjacencyMap a)
component = [(Vertex, AdjacencyMap a)] -> Map Vertex (AdjacencyMap a)
forall k a. Ord k => [(k, a)] -> Map k a
Map.fromList [ (Vertex
x, [Vertex] -> AdjacencyMap a
expand (Vertex
xVertex -> [Vertex] -> [Vertex]
forall a. a -> [a] -> [a]
:[Vertex]
xs)) | Vertex
x:[Vertex]
xs <- [[Vertex]]
sccs ]
    expand :: [Vertex] -> AdjacencyMap a
expand [Vertex]
xs = Maybe (AdjacencyMap a) -> AdjacencyMap a
forall a. HasCallStack => Maybe a -> a
fromJust (Maybe (AdjacencyMap a) -> AdjacencyMap a)
-> Maybe (AdjacencyMap a) -> AdjacencyMap a
forall a b. (a -> b) -> a -> b
$ AdjacencyMap a -> Maybe (AdjacencyMap a)
forall a. AdjacencyMap a -> Maybe (AdjacencyMap a)
NonEmpty.toNonEmpty (AdjacencyMap a -> Maybe (AdjacencyMap a))
-> AdjacencyMap a -> Maybe (AdjacencyMap a)
forall a b. (a -> b) -> a -> b
$ (a -> Bool) -> AdjacencyMap a -> AdjacencyMap a
forall a. (a -> Bool) -> AdjacencyMap a -> AdjacencyMap a
AM.induce (a -> Set a -> Bool
forall a. Ord a => a -> Set a -> Bool
`Set.member` Set a
s) AdjacencyMap a
m
      where
        s :: Set a
s = [a] -> Set a
forall a. Ord a => [a] -> Set a
Set.fromList ((Vertex -> a) -> [Vertex] -> [a]
forall a b. (a -> b) -> [a] -> [b]
map Vertex -> a
decode [Vertex]
xs)

removeSelfLoops :: Ord a => AM.AdjacencyMap a -> AM.AdjacencyMap a
removeSelfLoops :: AdjacencyMap a -> AdjacencyMap a
removeSelfLoops AdjacencyMap a
m = (a -> AdjacencyMap a -> AdjacencyMap a)
-> AdjacencyMap a -> [a] -> AdjacencyMap a
forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (\a
x -> a -> a -> AdjacencyMap a -> AdjacencyMap a
forall a. Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a
AM.removeEdge a
x a
x) AdjacencyMap a
m (AdjacencyMap a -> [a]
forall a. AdjacencyMap a -> [a]
AM.vertexList AdjacencyMap a
m)