{-# LANGUAGE ConstrainedClassMethods #-} ----------------------------------------------------------------------------- -- | -- Module : Algebra.Graph.ToGraph -- Copyright : (c) Andrey Mokhov 2016-2022 -- License : MIT (see the file LICENSE) -- Maintainer : andrey.mokhov@gmail.com -- Stability : experimental -- -- __Alga__ is a library for algebraic construction and manipulation of graphs -- in Haskell. See for the -- motivation behind the library, the underlying theory, and implementation details. -- -- This module defines the type class 'ToGraph' for capturing data types that -- can be converted to algebraic graphs. To make an instance of this class you -- need to define just a single method ('toGraph' or 'foldg'), which gives you -- access to many other useful methods for free (although note that the default -- implementations may be suboptimal performance-wise). -- -- This type class is similar to the standard type class 'Data.Foldable.Foldable' -- defined for lists. Furthermore, one can define 'Foldable' methods 'foldMap' -- and 'Data.Foldable.toList' using @ToGraph@.'foldg': -- -- @ -- 'foldMap' f = 'foldg' 'mempty' f ('<>') ('<>') -- 'Data.Foldable.toList' = 'foldg' [] 'pure' ('++') ('++') -- @ -- -- However, the resulting 'Foldable' instance is problematic. For example, -- folding equivalent algebraic graphs @1@ and @1@ + @1@ leads to different -- results: -- -- @ -- 'Data.Foldable.toList' (1 ) == [1] -- 'Data.Foldable.toList' (1 + 1) == [1, 1] -- @ -- -- To avoid such cases, we do not provide 'Foldable' instances for algebraic -- graph datatypes. Furthermore, we require that the four arguments passed to -- 'foldg' satisfy the laws of the algebra of graphs. The above definitions -- of 'foldMap' and 'Data.Foldable.toList' violate this requirement, for example -- @[1] ++ [1] /= [1]@, and are therefore disallowed. ----------------------------------------------------------------------------- module Algebra.Graph.ToGraph ( -- * Type class ToGraph (..), -- * Derived functions adjacencyMap, adjacencyIntMap, adjacencyMapTranspose, adjacencyIntMapTranspose ) where import Data.IntMap (IntMap) import Data.IntSet (IntSet) import Data.Map (Map) import Data.Set (Set) import Data.Tree import qualified Data.IntMap as IntMap import qualified Data.IntSet as IntSet import qualified Data.Map as Map import qualified Data.Set as Set -- Ideally, we would define all instances in the modules where the corresponding -- data types are declared. However, that causes import cycles, so we define a -- few instances here. import qualified Algebra.Graph as G import qualified Algebra.Graph.AdjacencyMap as AM import qualified Algebra.Graph.AdjacencyMap.Algorithm as AM import qualified Algebra.Graph.NonEmpty.AdjacencyMap as NAM import qualified Algebra.Graph.AdjacencyIntMap as AIM import qualified Algebra.Graph.AdjacencyIntMap.Algorithm as AIM -- | The 'ToGraph' type class captures data types that can be converted to -- algebraic graphs. Instances of this type class should satisfy the laws -- specified by the default method definitions. class ToGraph t where {-# MINIMAL toGraph | foldg #-} -- | The type of vertices of the resulting graph. type ToVertex t -- | Convert a value to the corresponding algebraic graph, see "Algebra.Graph". -- -- @ -- toGraph == 'foldg' 'G.Empty' 'G.Vertex' 'G.Overlay' 'G.Connect' -- @ toGraph :: t -> G.Graph (ToVertex t) toGraph = foldg G.Empty G.Vertex G.Overlay G.Connect -- | The method 'foldg' is used for generalised graph folding. It collapses -- a given value by applying the provided graph construction primitives. The -- order of arguments is: empty, vertex, overlay and connect, and it is -- assumed that the arguments satisfy the axioms of the graph algebra. -- -- @ -- foldg == Algebra.Graph.'G.foldg' . 'toGraph' -- @ foldg :: r -> (ToVertex t -> r) -> (r -> r -> r) -> (r -> r -> r) -> t -> r foldg e v o c = G.foldg e v o c . toGraph -- | Check if a graph is empty. -- -- @ -- isEmpty == 'foldg' True ('const' False) (&&) (&&) -- @ isEmpty :: t -> Bool isEmpty = foldg True (const False) (&&) (&&) -- | Check if a graph contains a given vertex. -- -- @ -- hasVertex x == 'foldg' False (==x) (||) (||) -- @ hasVertex :: Eq (ToVertex t) => ToVertex t -> t -> Bool hasVertex x = foldg False (==x) (||) (||) -- | Check if a graph contains a given edge. -- -- @ -- hasEdge x y == Algebra.Graph.'G.hasEdge' x y . 'toGraph' -- @ hasEdge :: Eq (ToVertex t) => ToVertex t -> ToVertex t -> t -> Bool hasEdge x y = G.hasEdge x y . toGraph -- | The number of vertices in a graph. -- -- @ -- vertexCount == Set.'Set.size' . 'vertexSet' -- @ vertexCount :: Ord (ToVertex t) => t -> Int vertexCount = Set.size . vertexSet -- | The number of edges in a graph. -- -- @ -- edgeCount == Set.'Set.size' . 'edgeSet' -- @ edgeCount :: Ord (ToVertex t) => t -> Int edgeCount = AM.edgeCount . toAdjacencyMap -- | The sorted list of vertices of a given graph. -- -- @ -- vertexList == Set.'Set.toAscList' . 'vertexSet' -- @ vertexList :: Ord (ToVertex t) => t -> [ToVertex t] vertexList = Set.toAscList . vertexSet -- | The sorted list of edges of a graph. -- -- @ -- edgeList == Set.'Set.toAscList' . 'edgeSet' -- @ edgeList :: Ord (ToVertex t) => t -> [(ToVertex t, ToVertex t)] edgeList = AM.edgeList . toAdjacencyMap -- | The set of vertices of a graph. -- -- @ -- vertexSet == 'foldg' Set.'Set.empty' Set.'Set.singleton' Set.'Set.union' Set.'Set.union' -- @ vertexSet :: Ord (ToVertex t) => t -> Set (ToVertex t) vertexSet = foldg Set.empty Set.singleton Set.union Set.union -- | The set of vertices of a graph. Like 'vertexSet' but specialised for -- graphs with vertices of type 'Int'. -- -- @ -- vertexIntSet == 'foldg' IntSet.'IntSet.empty' IntSet.'IntSet.singleton' IntSet.'IntSet.union' IntSet.'IntSet.union' -- @ vertexIntSet :: ToVertex t ~ Int => t -> IntSet vertexIntSet = foldg IntSet.empty IntSet.singleton IntSet.union IntSet.union -- | The set of edges of a graph. -- -- @ -- edgeSet == Algebra.Graph.AdjacencyMap.'AM.edgeSet' . 'toAdjacencyMap' -- @ edgeSet :: Ord (ToVertex t) => t -> Set (ToVertex t, ToVertex t) edgeSet = AM.edgeSet . toAdjacencyMap -- | The /preset/ of a vertex is the set of its /direct predecessors/. -- -- @ -- preSet x == Algebra.Graph.AdjacencyMap.'AM.preSet' x . 'toAdjacencyMap' -- @ preSet :: Ord (ToVertex t) => ToVertex t -> t -> Set (ToVertex t) preSet x = AM.postSet x . toAdjacencyMapTranspose -- | The /preset/ (here @preIntSet@) of a vertex is the set of its -- /direct predecessors/. Like 'preSet' but specialised for graphs with -- vertices of type 'Int'. -- -- @ -- preIntSet x == Algebra.Graph.AdjacencyIntMap.'AIM.preIntSet' x . 'toAdjacencyIntMap' -- @ preIntSet :: ToVertex t ~ Int => Int -> t -> IntSet preIntSet x = AIM.postIntSet x . toAdjacencyIntMapTranspose -- | The /postset/ of a vertex is the set of its /direct successors/. -- -- @ -- postSet x == Algebra.Graph.AdjacencyMap.'AM.postSet' x . 'toAdjacencyMap' -- @ postSet :: Ord (ToVertex t) => ToVertex t -> t -> Set (ToVertex t) postSet x = AM.postSet x . toAdjacencyMap -- | The /postset/ (here @postIntSet@) of a vertex is the set of its -- /direct successors/. Like 'postSet' but specialised for graphs with -- vertices of type 'Int'. -- -- @ -- postIntSet x == Algebra.Graph.AdjacencyIntMap.'AIM.postIntSet' x . 'toAdjacencyIntMap' -- @ postIntSet :: ToVertex t ~ Int => Int -> t -> IntSet postIntSet x = AIM.postIntSet x . toAdjacencyIntMap -- | The sorted /adjacency list/ of a graph. -- -- @ -- adjacencyList == Algebra.Graph.AdjacencyMap.'AM.adjacencyList' . 'toAdjacencyMap' -- @ adjacencyList :: Ord (ToVertex t) => t -> [(ToVertex t, [ToVertex t])] adjacencyList = AM.adjacencyList . toAdjacencyMap -- | Compute the /depth-first search/ forest of a graph that corresponds to -- searching from each of the graph vertices in the 'Ord' @a@ order. -- -- @ -- dfsForest == Algebra.Graph.AdjacencyMap.'AM.dfsForest' . toAdjacencyMap -- @ dfsForest :: Ord (ToVertex t) => t -> Forest (ToVertex t) dfsForest = AM.dfsForest . toAdjacencyMap -- | 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. -- -- @ -- dfsForestFrom == Algebra.Graph.AdjacencyMap.'AM.dfsForestFrom' . toAdjacencyMap -- @ dfsForestFrom :: Ord (ToVertex t) => t -> [ToVertex t] -> Forest (ToVertex t) dfsForestFrom = AM.dfsForestFrom . toAdjacencyMap -- | Compute the list of vertices visited by the /depth-first search/ in a -- graph, when searching from each of the given vertices in order. -- -- @ -- dfs == Algebra.Graph.AdjacencyMap.'AM.dfs' . toAdjacencyMap -- @ dfs :: Ord (ToVertex t) => t -> [ToVertex t] -> [ToVertex t] dfs = AM.dfs . toAdjacencyMap -- | Compute the list of vertices that are /reachable/ from a given source -- vertex in a graph. The vertices in the resulting list appear in the -- /depth-first order/. -- -- @ -- reachable == Algebra.Graph.AdjacencyMap.'AM.reachable' . toAdjacencyMap -- @ reachable :: Ord (ToVertex t) => t -> ToVertex t -> [ToVertex t] reachable = AM.reachable . toAdjacencyMap -- | Compute the /topological sort/ of a graph or a @AM.Cycle@ if the -- graph is cyclic. -- -- @ -- topSort == Algebra.Graph.AdjacencyMap.'AM.topSort' . toAdjacencyMap -- @ topSort :: Ord (ToVertex t) => t -> Either (AM.Cycle (ToVertex t)) [ToVertex t] topSort = AM.topSort . toAdjacencyMap -- | Check if a given graph is /acyclic/. -- -- @ -- isAcyclic == Algebra.Graph.AdjacencyMap.'AM.isAcyclic' . toAdjacencyMap -- @ isAcyclic :: Ord (ToVertex t) => t -> Bool isAcyclic = AM.isAcyclic . toAdjacencyMap -- | Convert a value to the corresponding 'AM.AdjacencyMap'. -- -- @ -- toAdjacencyMap == 'foldg' 'AM.empty' 'AM.vertex' 'AM.overlay' 'AM.connect' -- @ toAdjacencyMap :: Ord (ToVertex t) => t -> AM.AdjacencyMap (ToVertex t) toAdjacencyMap = foldg AM.empty AM.vertex AM.overlay AM.connect -- | Convert a value to the corresponding 'AM.AdjacencyMap' and transpose the -- result. -- -- @ -- toAdjacencyMapTranspose == 'foldg' 'AM.empty' 'AM.vertex' 'AM.overlay' ('flip' 'AM.connect') -- @ toAdjacencyMapTranspose :: Ord (ToVertex t) => t -> AM.AdjacencyMap (ToVertex t) toAdjacencyMapTranspose = foldg AM.empty AM.vertex AM.overlay (flip AM.connect) -- | Convert a value to the corresponding 'AIM.AdjacencyIntMap'. -- -- @ -- toAdjacencyIntMap == 'foldg' 'AIM.empty' 'AIM.vertex' 'AIM.overlay' 'AIM.connect' -- @ toAdjacencyIntMap :: ToVertex t ~ Int => t -> AIM.AdjacencyIntMap toAdjacencyIntMap = foldg AIM.empty AIM.vertex AIM.overlay AIM.connect -- | Convert a value to the corresponding 'AIM.AdjacencyIntMap' and transpose -- the result. -- -- @ -- toAdjacencyIntMapTranspose == 'foldg' 'AIM.empty' 'AIM.vertex' 'AIM.overlay' ('flip' 'AIM.connect') -- @ toAdjacencyIntMapTranspose :: ToVertex t ~ Int => t -> AIM.AdjacencyIntMap toAdjacencyIntMapTranspose = foldg AIM.empty AIM.vertex AIM.overlay (flip AIM.connect) -- | Check if a given forest is a valid /depth-first search/ forest of a -- graph. -- -- @ -- isDfsForestOf f == Algebra.Graph.AdjacencyMap.'AM.isDfsForestOf' f . toAdjacencyMap -- @ isDfsForestOf :: Ord (ToVertex t) => Forest (ToVertex t) -> t -> Bool isDfsForestOf f = AM.isDfsForestOf f . toAdjacencyMap -- | Check if a given list of vertices is a valid /topological sort/ of a -- graph. -- -- @ -- isTopSortOf vs == Algebra.Graph.AdjacencyMap.'AM.isTopSortOf' vs . toAdjacencyMap -- @ isTopSortOf :: Ord (ToVertex t) => [ToVertex t] -> t -> Bool isTopSortOf vs = AM.isTopSortOf vs . toAdjacencyMap -- | See "Algebra.Graph". instance Ord a => ToGraph (G.Graph a) where type ToVertex (G.Graph a) = a toGraph = id foldg = G.foldg hasEdge = G.hasEdge -- | See "Algebra.Graph.AdjacencyMap". instance Ord a => ToGraph (AM.AdjacencyMap a) where type ToVertex (AM.AdjacencyMap a) = a toGraph = G.stars . map (fmap Set.toList) . Map.toList . AM.adjacencyMap isEmpty = AM.isEmpty hasVertex = AM.hasVertex hasEdge = AM.hasEdge vertexCount = AM.vertexCount edgeCount = AM.edgeCount vertexList = AM.vertexList vertexSet = AM.vertexSet vertexIntSet = IntSet.fromAscList . AM.vertexList edgeList = AM.edgeList edgeSet = AM.edgeSet adjacencyList = AM.adjacencyList preSet = AM.preSet postSet = AM.postSet dfsForest = AM.dfsForest dfsForestFrom = AM.dfsForestFrom dfs = AM.dfs reachable = AM.reachable topSort = AM.topSort isAcyclic = AM.isAcyclic toAdjacencyMap = id toAdjacencyIntMap = AIM.fromAdjacencyMap toAdjacencyMapTranspose = AM.transpose . toAdjacencyMap toAdjacencyIntMapTranspose = AIM.transpose . toAdjacencyIntMap isDfsForestOf = AM.isDfsForestOf isTopSortOf = AM.isTopSortOf -- | See "Algebra.Graph.AdjacencyIntMap". instance ToGraph AIM.AdjacencyIntMap where type ToVertex AIM.AdjacencyIntMap = Int toGraph = G.stars . map (fmap IntSet.toList) . IntMap.toList . AIM.adjacencyIntMap isEmpty = AIM.isEmpty hasVertex = AIM.hasVertex hasEdge = AIM.hasEdge vertexCount = AIM.vertexCount edgeCount = AIM.edgeCount vertexList = AIM.vertexList vertexSet = Set.fromAscList . IntSet.toAscList . AIM.vertexIntSet vertexIntSet = AIM.vertexIntSet edgeList = AIM.edgeList edgeSet = AIM.edgeSet adjacencyList = AIM.adjacencyList preIntSet = AIM.preIntSet postIntSet = AIM.postIntSet dfsForest = AIM.dfsForest dfsForestFrom = AIM.dfsForestFrom dfs = AIM.dfs reachable = AIM.reachable topSort = AIM.topSort isAcyclic = AIM.isAcyclic toAdjacencyMap = AM.stars . AIM.adjacencyList toAdjacencyIntMap = id toAdjacencyMapTranspose = AM.transpose . toAdjacencyMap toAdjacencyIntMapTranspose = AIM.transpose . toAdjacencyIntMap isDfsForestOf = AIM.isDfsForestOf isTopSortOf = AIM.isTopSortOf -- | See "Algebra.Graph.NonEmpty.AdjacencyMap". instance Ord a => ToGraph (NAM.AdjacencyMap a) where type ToVertex (NAM.AdjacencyMap a) = a toGraph = toGraph . toAdjacencyMap isEmpty _ = False hasVertex = NAM.hasVertex hasEdge = NAM.hasEdge vertexCount = NAM.vertexCount edgeCount = NAM.edgeCount vertexList = vertexList . toAdjacencyMap vertexSet = NAM.vertexSet vertexIntSet = vertexIntSet . toAdjacencyMap edgeList = NAM.edgeList edgeSet = NAM.edgeSet adjacencyList = adjacencyList . toAdjacencyMap preSet = NAM.preSet postSet = NAM.postSet dfsForest = dfsForest . toAdjacencyMap dfsForestFrom = dfsForestFrom . toAdjacencyMap dfs = dfs . toAdjacencyMap reachable = reachable . toAdjacencyMap topSort = topSort . toAdjacencyMap isAcyclic = isAcyclic . toAdjacencyMap toAdjacencyMap = NAM.fromNonEmpty toAdjacencyIntMap = toAdjacencyIntMap . toAdjacencyMap toAdjacencyMapTranspose = toAdjacencyMap . NAM.transpose toAdjacencyIntMapTranspose = toAdjacencyIntMap . NAM.transpose isDfsForestOf f = isDfsForestOf f . toAdjacencyMap isTopSortOf x = isTopSortOf x . toAdjacencyMap -- | The /adjacency map/ of a graph: each vertex is associated with a set of its -- /direct successors/. -- -- @ -- adjacencyMap == Algebra.Graph.AdjacencyMap.'Algebra.Graph.AdjacencyMap.adjacencyMap' . 'toAdjacencyMap' -- @ adjacencyMap :: ToGraph t => Ord (ToVertex t) => t -> Map (ToVertex t) (Set (ToVertex t)) adjacencyMap = AM.adjacencyMap . toAdjacencyMap -- | The /adjacency map/ of a graph: each vertex is associated with a set of its -- /direct successors/. Like 'adjacencyMap' but specialised for graphs with -- vertices of type 'Int'. -- -- @ -- adjacencyIntMap == Algebra.Graph.AdjacencyIntMap.'Algebra.Graph.AdjacencyIntMap.adjacencyIntMap' . 'toAdjacencyIntMap' -- @ adjacencyIntMap :: (ToGraph t, ToVertex t ~ Int) => t -> IntMap IntSet adjacencyIntMap = AIM.adjacencyIntMap . toAdjacencyIntMap -- | The transposed /adjacency map/ of a graph: each vertex is associated with a -- set of its /direct predecessors/. -- -- @ -- adjacencyMapTranspose == Algebra.Graph.AdjacencyMap.'Algebra.Graph.AdjacencyMap.adjacencyMap' . 'toAdjacencyMapTranspose' -- @ adjacencyMapTranspose :: (ToGraph t, Ord (ToVertex t)) => t -> Map (ToVertex t) (Set (ToVertex t)) adjacencyMapTranspose = AM.adjacencyMap . toAdjacencyMapTranspose -- | The transposed /adjacency map/ of a graph: each vertex is associated with a -- set of its /direct predecessors/. Like 'adjacencyMapTranspose' but -- specialised for graphs with vertices of type 'Int'. -- -- @ -- adjacencyIntMapTranspose == Algebra.Graph.AdjacencyIntMap.'Algebra.Graph.AdjacencyIntMap.adjacencyIntMap' . 'toAdjacencyIntMapTranspose' -- @ adjacencyIntMapTranspose :: (ToGraph t, ToVertex t ~ Int) => t -> IntMap IntSet adjacencyIntMapTranspose = AIM.adjacencyIntMap . toAdjacencyIntMapTranspose