----------------------------------------------------------------------------- -- | -- Module : Algebra.Graph.AdjacencyMap -- 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 'AdjacencyMap' data type and associated functions. -- See "Algebra.Graph.AdjacencyMap.Algorithm" for basic graph algorithms. -- 'AdjacencyMap' is an instance of the 'C.Graph' type class, which can be used -- for polymorphic graph construction and manipulation. -- "Algebra.Graph.AdjacencyIntMap" defines adjacency maps specialised to graphs -- with @Int@ vertices. ----------------------------------------------------------------------------- module Algebra.Graph.AdjacencyMap ( -- * Data structure AdjacencyMap, adjacencyMap, -- * Basic graph construction primitives empty, vertex, edge, overlay, connect, vertices, edges, overlays, connects, -- * Relations on graphs isSubgraphOf, -- * Graph properties isEmpty, hasVertex, hasEdge, vertexCount, edgeCount, vertexList, edgeList, adjacencyList, vertexSet, edgeSet, preSet, postSet, -- * Standard families of graphs path, circuit, clique, biclique, star, stars, fromAdjacencySets, tree, forest, -- * Graph transformation removeVertex, removeEdge, replaceVertex, mergeVertices, transpose, gmap, induce, induceJust, -- * Graph composition compose, box, -- * Relational operations closure, reflexiveClosure, symmetricClosure, transitiveClosure, -- * Miscellaneous consistent ) where import Control.DeepSeq import Data.List ((\\)) import Data.Map.Strict (Map) import Data.Monoid import Data.Set (Set) import Data.String import Data.Tree import GHC.Generics import qualified Data.Map.Strict as Map import qualified Data.Maybe as Maybe import qualified Data.Set as Set {-| The 'AdjacencyMap' data type represents a graph by a map of vertices to their adjacency sets. We define a 'Num' instance as a convenient notation for working with graphs: @ 0 == 'vertex' 0 1 + 2 == 'overlay' ('vertex' 1) ('vertex' 2) 1 * 2 == 'connect' ('vertex' 1) ('vertex' 2) 1 + 2 * 3 == 'overlay' ('vertex' 1) ('connect' ('vertex' 2) ('vertex' 3)) 1 * (2 + 3) == 'connect' ('vertex' 1) ('overlay' ('vertex' 2) ('vertex' 3)) @ __Note:__ the 'Num' instance does not satisfy several "customary laws" of 'Num', which dictate that 'fromInteger' @0@ and 'fromInteger' @1@ should act as additive and multiplicative identities, and 'negate' as additive inverse. Nevertheless, overloading 'fromInteger', '+' and '*' is very convenient when working with algebraic graphs; we hope that in future Haskell's Prelude will provide a more fine-grained class hierarchy for algebraic structures, which we would be able to utilise without violating any laws. The 'Show' instance is defined using basic graph construction primitives: @show (empty :: AdjacencyMap Int) == "empty" show (1 :: AdjacencyMap Int) == "vertex 1" show (1 + 2 :: AdjacencyMap Int) == "vertices [1,2]" show (1 * 2 :: AdjacencyMap Int) == "edge 1 2" show (1 * 2 * 3 :: AdjacencyMap Int) == "edges [(1,2),(1,3),(2,3)]" show (1 * 2 + 3 :: AdjacencyMap Int) == "overlay (vertex 3) (edge 1 2)"@ The 'Eq' instance satisfies all axioms of algebraic graphs: * 'overlay' is commutative and associative: > x + y == y + x > x + (y + z) == (x + y) + z * 'connect' is associative and has 'empty' as the identity: > x * empty == x > empty * x == x > x * (y * z) == (x * y) * z * 'connect' distributes over 'overlay': > x * (y + z) == x * y + x * z > (x + y) * z == x * z + y * z * 'connect' can be decomposed: > x * y * z == x * y + x * z + y * z The following useful theorems can be proved from the above set of axioms. * 'overlay' has 'empty' as the identity and is idempotent: > x + empty == x > empty + x == x > x + x == x * Absorption and saturation of 'connect': > x * y + x + y == x * y > x * x * x == x * x When specifying the time and memory complexity of graph algorithms, /n/ and /m/ will denote the number of vertices and edges in the graph, respectively. The total order on graphs is defined using /size-lexicographic/ comparison: * Compare the number of vertices. In case of a tie, continue. * Compare the sets of vertices. In case of a tie, continue. * Compare the number of edges. In case of a tie, continue. * Compare the sets of edges. Here are a few examples: @'vertex' 1 < 'vertex' 2 'vertex' 3 < 'edge' 1 2 'vertex' 1 < 'edge' 1 1 'edge' 1 1 < 'edge' 1 2 'edge' 1 2 < 'edge' 1 1 + 'edge' 2 2 'edge' 1 2 < 'edge' 1 3@ Note that the resulting order refines the 'isSubgraphOf' relation and is compatible with 'overlay' and 'connect' operations: @'isSubgraphOf' x y ==> x <= y@ @'empty' <= x x <= x + y x + y <= x * y@ -} newtype AdjacencyMap a = AM { -- | The /adjacency map/ of a graph: each vertex is associated with a set of -- its direct successors. Complexity: /O(1)/ time and memory. -- -- @ -- adjacencyMap 'empty' == Map.'Map.empty' -- adjacencyMap ('vertex' x) == Map.'Map.singleton' x Set.'Set.empty' -- adjacencyMap ('edge' 1 1) == Map.'Map.singleton' 1 (Set.'Set.singleton' 1) -- adjacencyMap ('edge' 1 2) == Map.'Map.fromList' [(1,Set.'Set.singleton' 2), (2,Set.'Set.empty')] -- @ adjacencyMap :: Map a (Set a) } deriving (Eq, Generic) instance Ord a => Ord (AdjacencyMap a) where compare x y = mconcat [ compare (vertexCount x) (vertexCount y) , compare (vertexSet x) (vertexSet y) , compare (edgeCount x) (edgeCount y) , compare (edgeSet x) (edgeSet y) ] instance (Ord a, Show a) => Show (AdjacencyMap a) where showsPrec p am@(AM m) | null vs = showString "empty" | null es = showParen (p > 10) $ vshow vs | vs == used = showParen (p > 10) $ eshow es | otherwise = showParen (p > 10) $ showString "overlay (" . vshow (vs \\ used) . showString ") (" . eshow es . showString ")" where vs = vertexList am es = edgeList am vshow [x] = showString "vertex " . showsPrec 11 x vshow xs = showString "vertices " . showsPrec 11 xs eshow [(x, y)] = showString "edge " . showsPrec 11 x . showString " " . showsPrec 11 y eshow xs = showString "edges " . showsPrec 11 xs used = Set.toAscList (referredToVertexSet m) -- | __Note:__ this does not satisfy the usual ring laws; see 'AdjacencyMap' -- for more details. instance (Ord a, Num a) => Num (AdjacencyMap a) where fromInteger = vertex . fromInteger (+) = overlay (*) = connect signum = const empty abs = id negate = id instance IsString a => IsString (AdjacencyMap a) where fromString = vertex . fromString instance NFData a => NFData (AdjacencyMap a) where rnf (AM a) = rnf a -- | Defined via 'overlay'. instance Ord a => Semigroup (AdjacencyMap a) where (<>) = overlay -- | Defined via 'overlay' and 'empty'. instance Ord a => Monoid (AdjacencyMap a) where mempty = empty -- | Construct the /empty graph/. -- -- @ -- 'isEmpty' empty == True -- 'hasVertex' x empty == False -- 'vertexCount' empty == 0 -- 'edgeCount' empty == 0 -- @ empty :: AdjacencyMap a empty = AM Map.empty {-# NOINLINE [1] empty #-} -- | Construct the graph comprising /a single isolated vertex/. -- -- @ -- 'isEmpty' (vertex x) == False -- 'hasVertex' x (vertex y) == (x == y) -- 'vertexCount' (vertex x) == 1 -- 'edgeCount' (vertex x) == 0 -- @ vertex :: a -> AdjacencyMap a vertex x = AM $ Map.singleton x Set.empty {-# NOINLINE [1] vertex #-} -- | Construct the graph comprising /a single edge/. -- -- @ -- edge x y == 'connect' ('vertex' x) ('vertex' y) -- 'hasEdge' x y (edge x y) == True -- 'edgeCount' (edge x y) == 1 -- 'vertexCount' (edge 1 1) == 1 -- 'vertexCount' (edge 1 2) == 2 -- @ edge :: Ord a => a -> a -> AdjacencyMap a edge x y | x == y = AM $ Map.singleton x (Set.singleton y) | otherwise = AM $ Map.fromList [(x, Set.singleton y), (y, Set.empty)] -- | /Overlay/ two graphs. This is a commutative, associative and idempotent -- operation with the identity 'empty'. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- 'isEmpty' (overlay x y) == 'isEmpty' x && 'isEmpty' y -- 'hasVertex' z (overlay x y) == 'hasVertex' z x || 'hasVertex' z y -- 'vertexCount' (overlay x y) >= 'vertexCount' x -- 'vertexCount' (overlay x y) <= 'vertexCount' x + 'vertexCount' y -- 'edgeCount' (overlay x y) >= 'edgeCount' x -- 'edgeCount' (overlay x y) <= 'edgeCount' x + 'edgeCount' y -- 'vertexCount' (overlay 1 2) == 2 -- 'edgeCount' (overlay 1 2) == 0 -- @ overlay :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a overlay (AM x) (AM y) = AM $ Map.unionWith Set.union x y {-# NOINLINE [1] overlay #-} -- | /Connect/ two graphs. This is an associative operation with the identity -- 'empty', which distributes over 'overlay' and obeys the decomposition axiom. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. Note that the -- number of edges in the resulting graph is quadratic with respect to the number -- of vertices of the arguments: /m = O(m1 + m2 + n1 * n2)/. -- -- @ -- 'isEmpty' (connect x y) == 'isEmpty' x && 'isEmpty' y -- 'hasVertex' z (connect x y) == 'hasVertex' z x || 'hasVertex' z y -- 'vertexCount' (connect x y) >= 'vertexCount' x -- 'vertexCount' (connect x y) <= 'vertexCount' x + 'vertexCount' y -- 'edgeCount' (connect x y) >= 'edgeCount' x -- 'edgeCount' (connect x y) >= 'edgeCount' y -- 'edgeCount' (connect x y) >= 'vertexCount' x * 'vertexCount' y -- 'edgeCount' (connect x y) <= 'vertexCount' x * 'vertexCount' y + 'edgeCount' x + 'edgeCount' y -- 'vertexCount' (connect 1 2) == 2 -- 'edgeCount' (connect 1 2) == 1 -- @ connect :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a connect (AM x) (AM y) = AM $ Map.unionsWith Set.union [ x, y, Map.fromSet (const $ Map.keysSet y) (Map.keysSet x) ] {-# NOINLINE [1] connect #-} -- | Construct the graph comprising a given list of isolated vertices. -- Complexity: /O(L * log(L))/ time and /O(L)/ memory, where /L/ is the length -- of the given list. -- -- @ -- vertices [] == 'empty' -- vertices [x] == 'vertex' x -- vertices == 'overlays' . map 'vertex' -- 'hasVertex' x . vertices == 'elem' x -- 'vertexCount' . vertices == 'length' . 'Data.List.nub' -- 'vertexSet' . vertices == Set.'Set.fromList' -- @ vertices :: Ord a => [a] -> AdjacencyMap a vertices = AM . Map.fromList . map (, Set.empty) {-# NOINLINE [1] vertices #-} -- | Construct the graph from a list of edges. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- edges [] == 'empty' -- edges [(x,y)] == 'edge' x y -- edges == 'overlays' . 'map' ('uncurry' 'edge') -- 'edgeCount' . edges == 'length' . 'Data.List.nub' -- 'edgeList' . edges == 'Data.List.nub' . 'Data.List.sort' -- @ edges :: Ord a => [(a, a)] -> AdjacencyMap a edges = fromAdjacencySets . map (fmap Set.singleton) -- | Overlay a given list of graphs. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- overlays [] == 'empty' -- overlays [x] == x -- overlays [x,y] == 'overlay' x y -- overlays == 'foldr' 'overlay' 'empty' -- 'isEmpty' . overlays == 'all' 'isEmpty' -- @ overlays :: Ord a => [AdjacencyMap a] -> AdjacencyMap a overlays = AM . Map.unionsWith Set.union . map adjacencyMap {-# NOINLINE overlays #-} -- | Connect a given list of graphs. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- connects [] == 'empty' -- connects [x] == x -- connects [x,y] == 'connect' x y -- connects == 'foldr' 'connect' 'empty' -- 'isEmpty' . connects == 'all' 'isEmpty' -- @ connects :: Ord a => [AdjacencyMap a] -> AdjacencyMap a connects = foldr connect empty {-# NOINLINE connects #-} -- | The 'isSubgraphOf' function takes two graphs and returns 'True' if the -- first graph is a /subgraph/ of the second. -- Complexity: /O((n + m) * log(n))/ time. -- -- @ -- isSubgraphOf 'empty' x == True -- isSubgraphOf ('vertex' x) 'empty' == False -- isSubgraphOf x ('overlay' x y) == True -- isSubgraphOf ('overlay' x y) ('connect' x y) == True -- isSubgraphOf ('path' xs) ('circuit' xs) == True -- isSubgraphOf x y ==> x <= y -- @ isSubgraphOf :: Ord a => AdjacencyMap a -> AdjacencyMap a -> Bool isSubgraphOf (AM x) (AM y) = Map.isSubmapOfBy Set.isSubsetOf x y -- | Check if a graph is empty. -- Complexity: /O(1)/ time. -- -- @ -- isEmpty 'empty' == True -- isEmpty ('overlay' 'empty' 'empty') == True -- isEmpty ('vertex' x) == False -- isEmpty ('removeVertex' x $ 'vertex' x) == True -- isEmpty ('removeEdge' x y $ 'edge' x y) == False -- @ isEmpty :: AdjacencyMap a -> Bool isEmpty = Map.null . adjacencyMap -- | Check if a graph contains a given vertex. -- Complexity: /O(log(n))/ time. -- -- @ -- hasVertex x 'empty' == False -- hasVertex x ('vertex' y) == (x == y) -- hasVertex x . 'removeVertex' x == 'const' False -- @ hasVertex :: Ord a => a -> AdjacencyMap a -> Bool hasVertex x = Map.member x . adjacencyMap -- | Check if a graph contains a given edge. -- Complexity: /O(log(n))/ time. -- -- @ -- hasEdge x y 'empty' == False -- hasEdge x y ('vertex' z) == False -- hasEdge x y ('edge' x y) == True -- hasEdge x y . 'removeEdge' x y == 'const' False -- hasEdge x y == 'elem' (x,y) . 'edgeList' -- @ hasEdge :: Ord a => a -> a -> AdjacencyMap a -> Bool hasEdge u v (AM m) = case Map.lookup u m of Nothing -> False Just vs -> Set.member v vs -- | The number of vertices in a graph. -- Complexity: /O(1)/ time. -- -- @ -- vertexCount 'empty' == 0 -- vertexCount ('vertex' x) == 1 -- vertexCount == 'length' . 'vertexList' -- vertexCount x \< vertexCount y ==> x \< y -- @ vertexCount :: AdjacencyMap a -> Int vertexCount = Map.size . adjacencyMap -- | The number of edges in a graph. -- Complexity: /O(n)/ time. -- -- @ -- edgeCount 'empty' == 0 -- edgeCount ('vertex' x) == 0 -- edgeCount ('edge' x y) == 1 -- edgeCount == 'length' . 'edgeList' -- @ edgeCount :: AdjacencyMap a -> Int edgeCount = getSum . foldMap (Sum . Set.size) . adjacencyMap -- | The sorted list of vertices of a given graph. -- Complexity: /O(n)/ time and memory. -- -- @ -- vertexList 'empty' == [] -- vertexList ('vertex' x) == [x] -- vertexList . 'vertices' == 'Data.List.nub' . 'Data.List.sort' -- @ vertexList :: AdjacencyMap a -> [a] vertexList = Map.keys . adjacencyMap -- | The sorted list of edges of a graph. -- Complexity: /O(n + m)/ time and /O(m)/ memory. -- -- @ -- edgeList 'empty' == [] -- edgeList ('vertex' x) == [] -- edgeList ('edge' x y) == [(x,y)] -- edgeList ('star' 2 [3,1]) == [(2,1), (2,3)] -- edgeList . 'edges' == 'Data.List.nub' . 'Data.List.sort' -- edgeList . 'transpose' == 'Data.List.sort' . 'map' 'Data.Tuple.swap' . edgeList -- @ edgeList :: AdjacencyMap a -> [(a, a)] edgeList (AM m) = [ (x, y) | (x, ys) <- Map.toAscList m, y <- Set.toAscList ys ] {-# INLINE edgeList #-} -- | The set of vertices of a given graph. -- Complexity: /O(n)/ time and memory. -- -- @ -- vertexSet 'empty' == Set.'Set.empty' -- vertexSet . 'vertex' == Set.'Set.singleton' -- vertexSet . 'vertices' == Set.'Set.fromList' -- @ vertexSet :: AdjacencyMap a -> Set a vertexSet = Map.keysSet . adjacencyMap -- | The set of edges of a given graph. -- Complexity: /O((n + m) * log(m))/ time and /O(m)/ memory. -- -- @ -- edgeSet 'empty' == Set.'Set.empty' -- edgeSet ('vertex' x) == Set.'Set.empty' -- edgeSet ('edge' x y) == Set.'Set.singleton' (x,y) -- edgeSet . 'edges' == Set.'Set.fromList' -- @ edgeSet :: Eq a => AdjacencyMap a -> Set (a, a) edgeSet = Set.fromAscList . edgeList -- | The sorted /adjacency list/ of a graph. -- Complexity: /O(n + m)/ time and memory. -- -- @ -- adjacencyList 'empty' == [] -- adjacencyList ('vertex' x) == [(x, [])] -- adjacencyList ('edge' 1 2) == [(1, [2]), (2, [])] -- adjacencyList ('star' 2 [3,1]) == [(1, []), (2, [1,3]), (3, [])] -- 'stars' . adjacencyList == id -- @ adjacencyList :: AdjacencyMap a -> [(a, [a])] adjacencyList = map (fmap Set.toAscList) . Map.toAscList . adjacencyMap -- | The /preset/ of an element @x@ is the set of its /direct predecessors/. -- Complexity: /O(n * log(n))/ time and /O(n)/ memory. -- -- @ -- preSet x 'empty' == Set.'Set.empty' -- preSet x ('vertex' x) == Set.'Set.empty' -- preSet 1 ('edge' 1 2) == Set.'Set.empty' -- preSet y ('edge' x y) == Set.'Set.fromList' [x] -- @ preSet :: Ord a => a -> AdjacencyMap a -> Set a preSet x = Set.fromAscList . map fst . filter p . Map.toAscList . adjacencyMap where p (_, set) = x `Set.member` set -- | The /postset/ of a vertex is the set of its /direct successors/. -- Complexity: /O(log(n))/ time and /O(1)/ memory. -- -- @ -- postSet x 'empty' == Set.'Set.empty' -- postSet x ('vertex' x) == Set.'Set.empty' -- postSet x ('edge' x y) == Set.'Set.fromList' [y] -- postSet 2 ('edge' 1 2) == Set.'Set.empty' -- @ postSet :: Ord a => a -> AdjacencyMap a -> Set a postSet x = Map.findWithDefault Set.empty x . adjacencyMap -- | The /path/ on a list of vertices. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- path [] == 'empty' -- path [x] == 'vertex' x -- path [x,y] == 'edge' x y -- path . 'reverse' == 'transpose' . path -- @ path :: Ord a => [a] -> AdjacencyMap a path xs = case xs of [] -> empty [x] -> vertex x (_:ys) -> edges (zip xs ys) -- | The /circuit/ on a list of vertices. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- circuit [] == 'empty' -- circuit [x] == 'edge' x x -- circuit [x,y] == 'edges' [(x,y), (y,x)] -- circuit . 'reverse' == 'transpose' . circuit -- @ circuit :: Ord a => [a] -> AdjacencyMap a circuit [] = empty circuit (x:xs) = path $ [x] ++ xs ++ [x] -- | The /clique/ on a list of vertices. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- clique [] == 'empty' -- clique [x] == 'vertex' x -- clique [x,y] == 'edge' x y -- clique [x,y,z] == 'edges' [(x,y), (x,z), (y,z)] -- clique (xs '++' ys) == 'connect' (clique xs) (clique ys) -- clique . 'reverse' == 'transpose' . clique -- @ clique :: Ord a => [a] -> AdjacencyMap a clique = fromAdjacencySets . fst . go where go [] = ([], Set.empty) go (x:xs) = let (res, set) = go xs in ((x, set) : res, Set.insert x set) {-# NOINLINE [1] clique #-} -- | The /biclique/ on two lists of vertices. -- Complexity: /O(n * log(n) + m)/ time and /O(n + m)/ memory. -- -- @ -- biclique [] [] == 'empty' -- biclique [x] [] == 'vertex' x -- biclique [] [y] == 'vertex' y -- biclique [x1,x2] [y1,y2] == 'edges' [(x1,y1), (x1,y2), (x2,y1), (x2,y2)] -- biclique xs ys == 'connect' ('vertices' xs) ('vertices' ys) -- @ biclique :: Ord a => [a] -> [a] -> AdjacencyMap a biclique xs ys = AM $ Map.fromSet adjacent (x `Set.union` y) where x = Set.fromList xs y = Set.fromList ys adjacent v = if v `Set.member` x then y else Set.empty -- TODO: Optimise. -- | The /star/ formed by a centre vertex connected to a list of leaves. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- star x [] == 'vertex' x -- star x [y] == 'edge' x y -- star x [y,z] == 'edges' [(x,y), (x,z)] -- star x ys == 'connect' ('vertex' x) ('vertices' ys) -- @ star :: Ord a => a -> [a] -> AdjacencyMap a star x [] = vertex x star x ys = connect (vertex x) (vertices ys) {-# INLINE star #-} -- | The /stars/ formed by overlaying a list of 'star's. An inverse of -- 'adjacencyList'. -- Complexity: /O(L * log(n))/ time, memory and size, where /L/ is the total -- size of the input. -- -- @ -- stars [] == 'empty' -- stars [(x, [])] == 'vertex' x -- stars [(x, [y])] == 'edge' x y -- stars [(x, ys)] == 'star' x ys -- stars == 'overlays' . 'map' ('uncurry' 'star') -- stars . 'adjacencyList' == id -- 'overlay' (stars xs) (stars ys) == stars (xs '++' ys) -- @ stars :: Ord a => [(a, [a])] -> AdjacencyMap a stars = fromAdjacencySets . map (fmap Set.fromList) -- | Construct a graph from a list of adjacency sets; a variation of 'stars'. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- fromAdjacencySets [] == 'empty' -- fromAdjacencySets [(x, Set.'Set.empty')] == 'vertex' x -- fromAdjacencySets [(x, Set.'Set.singleton' y)] == 'edge' x y -- fromAdjacencySets . 'map' ('fmap' Set.'Set.fromList') == 'stars' -- 'overlay' (fromAdjacencySets xs) (fromAdjacencySets ys) == fromAdjacencySets (xs '++' ys) -- @ fromAdjacencySets :: Ord a => [(a, Set a)] -> AdjacencyMap a fromAdjacencySets ss = AM $ Map.unionWith Set.union vs es where vs = Map.fromSet (const Set.empty) . Set.unions $ map snd ss es = Map.fromListWith Set.union ss -- | The /tree graph/ constructed from a given 'Tree' data structure. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- tree (Node x []) == 'vertex' x -- tree (Node x [Node y [Node z []]]) == 'path' [x,y,z] -- tree (Node x [Node y [], Node z []]) == 'star' x [y,z] -- tree (Node 1 [Node 2 [], Node 3 [Node 4 [], Node 5 []]]) == 'edges' [(1,2), (1,3), (3,4), (3,5)] -- @ tree :: Ord a => Tree a -> AdjacencyMap a tree (Node x []) = vertex x tree (Node x f ) = star x (map rootLabel f) `overlay` forest (filter (not . null . subForest) f) -- | The /forest graph/ constructed from a given 'Forest' data structure. -- Complexity: /O((n + m) * log(n))/ time and /O(n + m)/ memory. -- -- @ -- forest [] == 'empty' -- forest [x] == 'tree' x -- forest [Node 1 [Node 2 [], Node 3 []], Node 4 [Node 5 []]] == 'edges' [(1,2), (1,3), (4,5)] -- forest == 'overlays' . 'map' 'tree' -- @ forest :: Ord a => Forest a -> AdjacencyMap a forest = overlays . map tree -- | Remove a vertex from a given graph. -- Complexity: /O(n*log(n))/ time. -- -- @ -- removeVertex x ('vertex' x) == 'empty' -- removeVertex 1 ('vertex' 2) == 'vertex' 2 -- removeVertex x ('edge' x x) == 'empty' -- removeVertex 1 ('edge' 1 2) == 'vertex' 2 -- removeVertex x . removeVertex x == removeVertex x -- @ removeVertex :: Ord a => a -> AdjacencyMap a -> AdjacencyMap a removeVertex x = AM . Map.map (Set.delete x) . Map.delete x . adjacencyMap -- | Remove an edge from a given graph. -- Complexity: /O(log(n))/ time. -- -- @ -- removeEdge x y ('edge' x y) == 'vertices' [x,y] -- removeEdge x y . removeEdge x y == removeEdge x y -- removeEdge x y . 'removeVertex' x == 'removeVertex' x -- removeEdge 1 1 (1 * 1 * 2 * 2) == 1 * 2 * 2 -- removeEdge 1 2 (1 * 1 * 2 * 2) == 1 * 1 + 2 * 2 -- @ removeEdge :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a removeEdge x y = AM . Map.adjust (Set.delete y) x . adjacencyMap -- | The function @'replaceVertex' x y@ replaces vertex @x@ with vertex @y@ in a -- given 'AdjacencyMap'. If @y@ already exists, @x@ and @y@ will be merged. -- Complexity: /O((n + m) * log(n))/ time. -- -- @ -- replaceVertex x x == id -- replaceVertex x y ('vertex' x) == 'vertex' y -- replaceVertex x y == 'mergeVertices' (== x) y -- @ replaceVertex :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a replaceVertex u v = gmap $ \w -> if w == u then v else w -- | Merge vertices satisfying a given predicate into a given vertex. -- Complexity: /O((n + m) * log(n))/ time, assuming that the predicate takes -- constant time. -- -- @ -- mergeVertices ('const' False) x == id -- mergeVertices (== x) y == 'replaceVertex' x y -- mergeVertices 'even' 1 (0 * 2) == 1 * 1 -- mergeVertices 'odd' 1 (3 + 4 * 5) == 4 * 1 -- @ mergeVertices :: Ord a => (a -> Bool) -> a -> AdjacencyMap a -> AdjacencyMap a mergeVertices p v = gmap $ \u -> if p u then v else u -- | Transpose a given graph. -- Complexity: /O(m * log(n))/ time, /O(n + m)/ memory. -- -- @ -- transpose 'empty' == 'empty' -- transpose ('vertex' x) == 'vertex' x -- transpose ('edge' x y) == 'edge' y x -- transpose . transpose == id -- 'edgeList' . transpose == 'Data.List.sort' . 'map' 'Data.Tuple.swap' . 'edgeList' -- @ transpose :: Ord a => AdjacencyMap a -> AdjacencyMap a transpose (AM m) = AM $ Map.foldrWithKey combine vs m where combine v es = Map.unionWith Set.union (Map.fromSet (const $ Set.singleton v) es) vs = Map.fromSet (const Set.empty) (Map.keysSet m) {-# NOINLINE [1] transpose #-} {-# RULES "transpose/empty" transpose empty = empty "transpose/vertex" forall x. transpose (vertex x) = vertex x "transpose/overlay" forall g1 g2. transpose (overlay g1 g2) = overlay (transpose g1) (transpose g2) "transpose/connect" forall g1 g2. transpose (connect g1 g2) = connect (transpose g2) (transpose g1) "transpose/overlays" forall xs. transpose (overlays xs) = overlays (map transpose xs) "transpose/connects" forall xs. transpose (connects xs) = connects (reverse (map transpose xs)) "transpose/vertices" forall xs. transpose (vertices xs) = vertices xs "transpose/clique" forall xs. transpose (clique xs) = clique (reverse xs) #-} -- | Transform a graph by applying a function to each of its vertices. This is -- similar to @Functor@'s 'fmap' but can be used with non-fully-parametric -- 'AdjacencyMap'. -- Complexity: /O((n + m) * log(n))/ time. -- -- @ -- gmap f 'empty' == 'empty' -- gmap f ('vertex' x) == 'vertex' (f x) -- gmap f ('edge' x y) == 'edge' (f x) (f y) -- gmap 'id' == 'id' -- gmap f . gmap g == gmap (f . g) -- @ gmap :: (Ord a, Ord b) => (a -> b) -> AdjacencyMap a -> AdjacencyMap b gmap f = AM . Map.map (Set.map f) . Map.mapKeysWith Set.union f . adjacencyMap -- | Construct the /induced subgraph/ of a given graph by removing the -- vertices that do not satisfy a given predicate. -- Complexity: /O(n + m)/ time, assuming that the predicate takes constant time. -- -- @ -- induce ('const' True ) x == x -- induce ('const' False) x == 'empty' -- induce (/= x) == 'removeVertex' x -- induce p . induce q == induce (\\x -> p x && q x) -- 'isSubgraphOf' (induce p x) x == True -- @ induce :: (a -> Bool) -> AdjacencyMap a -> AdjacencyMap a induce p = AM . Map.map (Set.filter p) . Map.filterWithKey (\k _ -> p k) . adjacencyMap -- | Construct the /induced subgraph/ of a given graph by removing the vertices -- that are 'Nothing'. -- Complexity: /O(n + m)/ time. -- -- @ -- induceJust ('vertex' 'Nothing') == 'empty' -- induceJust ('edge' ('Just' x) 'Nothing') == 'vertex' x -- induceJust . 'gmap' 'Just' == 'id' -- induceJust . 'gmap' (\\x -> if p x then 'Just' x else 'Nothing') == 'induce' p -- @ induceJust :: Ord a => AdjacencyMap (Maybe a) -> AdjacencyMap a induceJust = AM . Map.map catMaybesSet . catMaybesMap . adjacencyMap where catMaybesSet = Set.mapMonotonic Maybe.fromJust . Set.delete Nothing catMaybesMap = Map.mapKeysMonotonic Maybe.fromJust . Map.delete Nothing -- | Left-to-right /relational composition/ of graphs: vertices @x@ and @z@ are -- connected in the resulting graph if there is a vertex @y@, such that @x@ is -- connected to @y@ in the first graph, and @y@ is connected to @z@ in the -- second graph. There are no isolated vertices in the result. This operation is -- associative, has 'empty' and single-'vertex' graphs as /annihilating zeroes/, -- and distributes over 'overlay'. -- Complexity: /O(n * m * log(n))/ time and /O(n + m)/ memory. -- -- @ -- compose 'empty' x == 'empty' -- compose x 'empty' == 'empty' -- compose ('vertex' x) y == 'empty' -- compose x ('vertex' y) == 'empty' -- compose x (compose y z) == compose (compose x y) z -- compose x ('overlay' y z) == 'overlay' (compose x y) (compose x z) -- compose ('overlay' x y) z == 'overlay' (compose x z) (compose y z) -- compose ('edge' x y) ('edge' y z) == 'edge' x z -- compose ('path' [1..5]) ('path' [1..5]) == 'edges' [(1,3), (2,4), (3,5)] -- compose ('circuit' [1..5]) ('circuit' [1..5]) == 'circuit' [1,3,5,2,4] -- @ compose :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a compose x y = fromAdjacencySets [ (t, ys) | v <- Set.toList vs, let ys = postSet v y, not (Set.null ys) , t <- Set.toList (postSet v tx) ] where tx = transpose x vs = vertexSet x `Set.union` vertexSet y -- | Compute the /Cartesian product/ of graphs. -- Complexity: /O((n + m) * log(n))/ time and O(n + m) memory. -- -- @ -- box ('path' [0,1]) ('path' "ab") == 'edges' [ ((0,\'a\'), (0,\'b\')) -- , ((0,\'a\'), (1,\'a\')) -- , ((0,\'b\'), (1,\'b\')) -- , ((1,\'a\'), (1,\'b\')) ] -- @ -- -- Up to isomorphism between the resulting vertex types, this operation is -- /commutative/, /associative/, /distributes/ over 'overlay', has singleton -- graphs as /identities/ and 'empty' as the /annihilating zero/. Below @~~@ -- stands for equality up to an isomorphism, e.g. @(x,@ @()) ~~ x@. -- -- @ -- box x y ~~ box y x -- box x (box y z) ~~ box (box x y) z -- box x ('overlay' y z) == 'overlay' (box x y) (box x z) -- box x ('vertex' ()) ~~ x -- box x 'empty' ~~ 'empty' -- 'transpose' (box x y) == box ('transpose' x) ('transpose' y) -- 'vertexCount' (box x y) == 'vertexCount' x * 'vertexCount' y -- 'edgeCount' (box x y) <= 'vertexCount' x * 'edgeCount' y + 'edgeCount' x * 'vertexCount' y -- @ box :: (Ord a, Ord b) => AdjacencyMap a -> AdjacencyMap b -> AdjacencyMap (a, b) box (AM x) (AM y) = overlay (AM $ Map.fromAscList xs) (AM $ Map.fromAscList ys) where xs = do (a, as) <- Map.toAscList x b <- Set.toAscList (Map.keysSet y) return ((a, b), Set.mapMonotonic (,b) as) ys = do a <- Set.toAscList (Map.keysSet x) (b, bs) <- Map.toAscList y return ((a, b), Set.mapMonotonic (a,) bs) -- | Compute the /reflexive and transitive closure/ of a graph. -- Complexity: /O(n * m * log(n)^2)/ time. -- -- @ -- closure 'empty' == 'empty' -- closure ('vertex' x) == 'edge' x x -- closure ('edge' x x) == 'edge' x x -- closure ('edge' x y) == 'edges' [(x,x), (x,y), (y,y)] -- closure ('path' $ 'Data.List.nub' xs) == 'reflexiveClosure' ('clique' $ 'Data.List.nub' xs) -- closure == 'reflexiveClosure' . 'transitiveClosure' -- closure == 'transitiveClosure' . 'reflexiveClosure' -- closure . closure == closure -- 'postSet' x (closure y) == Set.'Set.fromList' ('Algebra.Graph.ToGraph.reachable' y x) -- @ closure :: Ord a => AdjacencyMap a -> AdjacencyMap a closure = reflexiveClosure . transitiveClosure -- | Compute the /reflexive closure/ of a graph by adding a self-loop to every -- vertex. -- Complexity: /O(n * log(n))/ time. -- -- @ -- reflexiveClosure 'empty' == 'empty' -- reflexiveClosure ('vertex' x) == 'edge' x x -- reflexiveClosure ('edge' x x) == 'edge' x x -- reflexiveClosure ('edge' x y) == 'edges' [(x,x), (x,y), (y,y)] -- reflexiveClosure . reflexiveClosure == reflexiveClosure -- @ reflexiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a reflexiveClosure (AM m) = AM $ Map.mapWithKey Set.insert m -- | Compute the /symmetric closure/ of a graph by overlaying it with its own -- transpose. -- Complexity: /O((n + m) * log(n))/ time. -- -- @ -- symmetricClosure 'empty' == 'empty' -- symmetricClosure ('vertex' x) == 'vertex' x -- symmetricClosure ('edge' x y) == 'edges' [(x,y), (y,x)] -- symmetricClosure x == 'overlay' x ('transpose' x) -- symmetricClosure . symmetricClosure == symmetricClosure -- @ symmetricClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a symmetricClosure m = overlay m (transpose m) -- | Compute the /transitive closure/ of a graph. -- Complexity: /O(n * m * log(n)^2)/ time. -- -- @ -- transitiveClosure 'empty' == 'empty' -- transitiveClosure ('vertex' x) == 'vertex' x -- transitiveClosure ('edge' x y) == 'edge' x y -- transitiveClosure ('path' $ 'Data.List.nub' xs) == 'clique' ('Data.List.nub' xs) -- transitiveClosure . transitiveClosure == transitiveClosure -- @ transitiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a transitiveClosure old | old == new = old | otherwise = transitiveClosure new where new = overlay old (old `compose` old) -- | Check that the internal graph representation is consistent, i.e. that all -- edges refer to existing vertices. It should be impossible to create an -- inconsistent adjacency map, and we use this function in testing. -- -- @ -- consistent 'empty' == True -- consistent ('vertex' x) == True -- consistent ('overlay' x y) == True -- consistent ('connect' x y) == True -- consistent ('edge' x y) == True -- consistent ('edges' xs) == True -- consistent ('stars' xs) == True -- @ consistent :: Ord a => AdjacencyMap a -> Bool consistent (AM m) = referredToVertexSet m `Set.isSubsetOf` Map.keysSet m -- The set of vertices that are referred to by the edges of an adjacency map. referredToVertexSet :: Ord a => Map a (Set a) -> Set a referredToVertexSet m = Set.fromList $ concat [ [x, y] | (x, ys) <- Map.toAscList m, y <- Set.toAscList ys ]