----------------------------------------------------------------------------- -- | -- Module : Algebra.Graph.Relation -- 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 'Relation' data type, as well as associated -- operations and algorithms. 'Relation' is an instance of the 'C.Graph' type -- class, which can be used for polymorphic graph construction and manipulation. ----------------------------------------------------------------------------- module Algebra.Graph.Relation ( -- * Data structure Relation, domain, relation, -- * 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, tree, forest, -- * Graph transformation removeVertex, removeEdge, replaceVertex, mergeVertices, transpose, gmap, induce, induceJust, -- * Relational operations compose, closure, reflexiveClosure, symmetricClosure, transitiveClosure, -- * Miscellaneous consistent ) where import Control.DeepSeq import Data.Bifunctor import Data.Set (Set, union) import Data.String import Data.Tree import Data.Tuple import qualified Data.IntSet as IntSet import qualified Data.Maybe as Maybe import qualified Data.Set as Set import qualified Data.Tree as Tree import Algebra.Graph.Internal import qualified Algebra.Graph as G import qualified Algebra.Graph.AdjacencyIntMap as AIM import qualified Algebra.Graph.AdjacencyMap as AM import qualified Algebra.Graph.ToGraph as T {-| The 'Relation' data type represents a graph as a /binary relation/. 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 :: Relation Int) == "empty" show (1 :: Relation Int) == "vertex 1" show (1 + 2 :: Relation Int) == "vertices [1,2]" show (1 * 2 :: Relation Int) == "edge 1 2" show (1 * 2 * 3 :: Relation Int) == "edges [(1,2),(1,3),(2,3)]" show (1 * 2 + 3 :: Relation 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@ -} data Relation a = Relation { -- | The /domain/ of the relation. Complexity: /O(1)/ time and memory. domain :: Set a, -- | The set of pairs of elements that are /related/. It is guaranteed that -- each element belongs to the domain. Complexity: /O(1)/ time and memory. relation :: Set (a, a) } deriving Eq instance (Ord a, Show a) => Show (Relation a) where showsPrec p (Relation d r) | Set.null d = showString "empty" | Set.null r = showParen (p > 10) $ vshow (Set.toAscList d) | d == used = showParen (p > 10) $ eshow (Set.toAscList r) | otherwise = showParen (p > 10) $ showString "overlay (" . vshow (Set.toAscList $ Set.difference d used) . showString ") (" . eshow (Set.toAscList r) . showString ")" where 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 = referredToVertexSet r instance Ord a => Ord (Relation 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 NFData a => NFData (Relation a) where rnf (Relation d r) = rnf d `seq` rnf r -- | __Note:__ this does not satisfy the usual ring laws; see 'Relation' for -- more details. instance (Ord a, Num a) => Num (Relation a) where fromInteger = vertex . fromInteger (+) = overlay (*) = connect signum = const empty abs = id negate = id instance IsString a => IsString (Relation a) where fromString = vertex . fromString -- | Defined via 'overlay'. instance Ord a => Semigroup (Relation a) where (<>) = overlay -- | Defined via 'overlay' and 'empty'. instance Ord a => Monoid (Relation a) where mempty = empty instance Ord a => T.ToGraph (Relation a) where type ToVertex (Relation a) = a toGraph r = G.vertices (Set.toList $ domain r) `G.overlay` G.edges (Set.toList $ relation r) isEmpty = isEmpty hasVertex = hasVertex hasEdge = hasEdge vertexCount = vertexCount edgeCount = edgeCount vertexList = vertexList vertexSet = vertexSet vertexIntSet = IntSet.fromAscList . vertexList edgeList = edgeList edgeSet = edgeSet adjacencyList = adjacencyList toAdjacencyMap = AM.stars . adjacencyList toAdjacencyIntMap = AIM.stars . adjacencyList toAdjacencyMapTranspose = AM.transpose . T.toAdjacencyMap toAdjacencyIntMapTranspose = AIM.transpose . T.toAdjacencyIntMap -- | Construct the /empty graph/. -- -- @ -- 'isEmpty' empty == True -- 'hasVertex' x empty == False -- 'vertexCount' empty == 0 -- 'edgeCount' empty == 0 -- @ empty :: Relation a empty = Relation Set.empty Set.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 -> Relation a vertex x = Relation (Set.singleton x) Set.empty -- | 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 -> Relation a edge x y = Relation (Set.fromList [x, y]) (Set.singleton (x, y)) -- | /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 => Relation a -> Relation a -> Relation a overlay x y = Relation (domain x `union` domain y) (relation x `union` relation y) -- | /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 => Relation a -> Relation a -> Relation a connect x y = Relation (domain x `union` domain y) (relation x `union` relation y `union` (domain x `Set.cartesianProduct` domain y)) -- | 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] -> Relation a vertices xs = Relation (Set.fromList xs) Set.empty -- | 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' -- @ edges :: Ord a => [(a, a)] -> Relation a edges es = Relation (Set.fromList $ uncurry (++) $ unzip es) (Set.fromList es) -- | 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 => [Relation a] -> Relation a overlays xs = Relation (Set.unions $ map domain xs) (Set.unions $ map relation xs) -- | 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 => [Relation a] -> Relation a connects = foldr connect empty -- | 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 => Relation a -> Relation a -> Bool isSubgraphOf x y = domain x `Set.isSubsetOf` domain y && relation x `Set.isSubsetOf` relation y -- | Check if a relation 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 :: Relation a -> Bool isEmpty = null . domain -- | 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 -> Relation a -> Bool hasVertex x = Set.member x . domain -- | 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 -> Relation a -> Bool hasEdge x y = Set.member (x, y) . relation -- | 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 :: Relation a -> Int vertexCount = Set.size . domain -- | The number of edges in a graph. -- Complexity: /O(1)/ time. -- -- @ -- edgeCount 'empty' == 0 -- edgeCount ('vertex' x) == 0 -- edgeCount ('edge' x y) == 1 -- edgeCount == 'length' . 'edgeList' -- @ edgeCount :: Relation a -> Int edgeCount = Set.size . relation -- | 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 :: Relation a -> [a] vertexList = Set.toAscList . domain -- | 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 :: Relation a -> [(a, a)] edgeList = Set.toAscList . relation -- | The set of vertices of a given graph. -- Complexity: /O(1)/ time. -- -- @ -- vertexSet 'empty' == Set.'Set.empty' -- vertexSet . 'vertex' == Set.'Set.singleton' -- vertexSet . 'vertices' == Set.'Set.fromList' -- @ vertexSet :: Relation a -> Set.Set a vertexSet = domain -- | The set of edges of a given graph. -- Complexity: /O(1)/ time. -- -- @ -- 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 :: Relation a -> Set.Set (a, a) edgeSet = relation -- | 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 :: Eq a => Relation a -> [(a, [a])] adjacencyList r = go (Set.toAscList $ domain r) (Set.toAscList $ relation r) where go [] _ = [] go vs [] = map (, []) vs go (x:vs) es = let (ys, zs) = span ((==x) . fst) es in (x, map snd ys) : go vs zs -- | The /preset/ of an element @x@ is the set of elements that are related to -- it on the /left/, i.e. @preSet x == { a | aRx }@. In the context of directed -- graphs, this corresponds to the set of /direct predecessors/ of vertex @x@. -- Complexity: /O(n + m)/ 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 -> Relation a -> Set.Set a preSet x = Set.mapMonotonic fst . Set.filter ((== x) . snd) . relation -- | The /postset/ of an element @x@ is the set of elements that are related to -- it on the /right/, i.e. @postSet x == { a | xRa }@. In the context of directed -- graphs, this corresponds to the set of /direct successors/ of vertex @x@. -- Complexity: /O(n + m)/ time and /O(n)/ 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 -> Relation a -> Set.Set a postSet x = Set.mapMonotonic snd . Set.filter ((== x) . fst) . relation -- | 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] -> Relation 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] -> Relation 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] -> Relation a clique xs = Relation (Set.fromList xs) (fst $ go xs) where go [] = (Set.empty, Set.empty) go (x:xs) = (Set.union res (Set.map (x,) set), Set.insert x set) where (res, set) = go xs -- | 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] -> Relation a biclique xs ys = Relation (x `Set.union` y) (x `Set.cartesianProduct` y) where x = Set.fromList xs y = Set.fromList ys -- 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] -> Relation a star x [] = vertex x star x ys = connect (vertex x) (vertices ys) -- | 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])] -> Relation a stars as = Relation (Set.fromList vs) (Set.fromList es) where vs = concatMap (uncurry (:)) as es = [ (x, y) | (x, ys) <- as, y <- ys ] -- | The /tree graph/ constructed from a given 'Tree.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.Tree a -> Relation 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 'Tree.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 => Tree.Forest a -> Relation a forest = overlays. map tree -- | Remove a vertex from a given graph. -- Complexity: /O(n + m)/ 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 -> Relation a -> Relation a removeVertex x (Relation d r) = Relation (Set.delete x d) (Set.filter notx r) where notx (a, b) = a /= x && b /= x -- | Remove an edge from a given graph. -- Complexity: /O(log(m))/ time. -- -- @ -- removeEdge x y ('AdjacencyMap.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 -> Relation a -> Relation a removeEdge x y (Relation d r) = Relation d (Set.delete (x, y) r) -- | 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 -> Relation a -> Relation 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 -> Relation a -> Relation a mergeVertices p v = gmap $ \u -> if p u then v else u -- | Transpose a given graph. -- Complexity: /O(m * log(m))/ time. -- -- @ -- 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 => Relation a -> Relation a transpose (Relation d r) = Relation d (Set.map swap r) -- | 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 -- 'Relation'. -- 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 b => (a -> b) -> Relation a -> Relation b gmap f (Relation d r) = Relation (Set.map f d) (Set.map (bimap f f) r) -- | 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) -> Relation a -> Relation a induce p (Relation d r) = Relation (Set.filter p d) (Set.filter pp r) where pp (x, y) = p x && p y -- | 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 => Relation (Maybe a) -> Relation a induceJust (Relation d r) = Relation (catMaybesSet d) (catMaybesSet2 r) where catMaybesSet = Set.mapMonotonic Maybe.fromJust . Set.delete Nothing catMaybesSet2 = Set.mapMonotonic (bimap Maybe.fromJust Maybe.fromJust) . Set.filter p p (Nothing, _) = False p (_, Nothing) = False p (_, _) = True -- | 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(m))/ 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 => Relation a -> Relation a -> Relation a compose x y = Relation (referredToVertexSet r) r where vs = Set.toAscList (domain x `Set.union` domain y) r = Set.unions [ preSet v x `Set.cartesianProduct` postSet v y | v <- vs ] -- | Compute the /reflexive and transitive closure/ of a graph. -- Complexity: /O(n * m * log(n) * log(m))/ 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 => Relation a -> Relation a closure = reflexiveClosure . transitiveClosure -- | Compute the /reflexive closure/ of a graph. -- Complexity: /O(n * log(m))/ 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 => Relation a -> Relation a reflexiveClosure (Relation d r) = Relation d $ r `Set.union` Set.fromDistinctAscList [ (a, a) | a <- Set.toAscList d ] -- | Compute the /symmetric closure/ of a graph. -- Complexity: /O(m * log(m))/ 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 => Relation a -> Relation a symmetricClosure (Relation d r) = Relation d $ r `Set.union` Set.map swap r -- | Compute the /transitive closure/ of a graph. -- Complexity: /O(n * m * log(n) * log(m))/ 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 => Relation a -> Relation a transitiveClosure old | old == new = old | otherwise = transitiveClosure new where new = overlay old (old `compose` old) -- | Check that the internal representation of a relation is consistent, i.e. if all -- pairs of elements in the 'relation' refer to existing elements in the 'domain'. -- It should be impossible to create an inconsistent 'Relation', 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 => Relation a -> Bool consistent (Relation d r) = referredToVertexSet r `Set.isSubsetOf` d -- The set of elements that appear in a given set of pairs. referredToVertexSet :: Ord a => Set (a, a) -> Set a referredToVertexSet = Set.fromList . uncurry (++) . unzip . Set.toAscList