----------------------------------------------------------------------------- -- | -- Module : Data.Graph.Typed -- Copyright : (c) Anton Lorenzen, Andrey Mokhov 2016-2018 -- 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). 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. 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. 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 am = GraphKL { toGraphKL = g , fromVertexKL = \u -> case r u of (_, v, _) -> v , toVertexKL = t } where (g, r, t) = KL.graphFromEdges [ ((), x, ys) | (x, ys) <- AM.adjacencyList 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 aim = GraphKL { toGraphKL = g , fromVertexKL = \x -> case r x of (_, v, _) -> v , toVertexKL = t } where (g, r, t) = KL.graphFromEdges [ ((), x, ys) | (x, ys) <- AIM.adjacencyList 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 g r _) = fmap (fmap r) (KL.dff 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 vs (GraphKL g r t) = fmap (fmap r) (KL.dfs g (mapMaybe t 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 vs = concatMap flatten . dfsForestFrom 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 g r _) = map r (KL.topSort g) scc :: Ord a => AM.AdjacencyMap a -> AM.AdjacencyMap (NonEmpty.AdjacencyMap a) scc m = AM.gmap (component Map.!) $ removeSelfLoops $ AM.gmap (leader Map.!) m where GraphKL g decode _ = fromAdjacencyMap m sccs = map toList (KL.scc g) leader = Map.fromList [ (decode y, x) | x:xs <- sccs, y <- x:xs ] component = Map.fromList [ (x, expand (x:xs)) | x:xs <- sccs ] expand xs = fromJust $ NonEmpty.toNonEmpty $ AM.induce (`Set.member` s) m where s = Set.fromList (map decode xs) removeSelfLoops :: Ord a => AM.AdjacencyMap a -> AM.AdjacencyMap a removeSelfLoops m = foldr (\x -> AM.removeEdge x x) m (AM.vertexList m)