----------------------------------------------------------------------------- -- | -- Module : Algebra.Graph.NonEmpty.AdjacencyMap.Internal -- Copyright : (c) Andrey Mokhov 2016-2018 -- License : MIT (see the file LICENSE) -- Maintainer : andrey.mokhov@gmail.com -- Stability : experimental -- -- This module exposes the implementation of non-empty adjacency maps. The API -- is unstable and unsafe, and is exposed only for documentation. You should use -- the non-internal module "Algebra.Graph.NonEmpty.AdjacencyMap" instead. ----------------------------------------------------------------------------- module Algebra.Graph.NonEmpty.AdjacencyMap.Internal ( -- * Adjacency map implementation AdjacencyMap (..), consistent ) where import Control.DeepSeq import Data.List import qualified Algebra.Graph.AdjacencyMap as AM import qualified Algebra.Graph.AdjacencyMap.Internal as AM import qualified Data.Map.Strict as Map 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 'signum' method of the type class 'Num' cannot be implemented and will throw an error. Furthermore, 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 (1 :: AdjacencyMap Int) == "vertex 1" show (1 + 2 :: AdjacencyMap Int) == "vertices1 [1,2]" show (1 * 2 :: AdjacencyMap Int) == "edge 1 2" show (1 * 2 * 3 :: AdjacencyMap Int) == "edges1 [(1,2),(1,3),(2,3)]" show (1 * 2 + 3 :: AdjacencyMap Int) == "overlay (vertex 3) (edge 1 2)"@ The 'Eq' instance satisfies the following laws of algebraic graphs: * 'Algebra.Graph.NonEmpty.AdjacencyMap.overlay' is commutative, associative and idempotent: > x + y == y + x > x + (y + z) == (x + y) + z > x + x == x * 'Algebra.Graph.NonEmpty.AdjacencyMap.connect' is associative: > x * (y * z) == (x * y) * z * 'Algebra.Graph.NonEmpty.AdjacencyMap.connect' distributes over 'Algebra.Graph.NonEmpty.AdjacencyMap.overlay': > x * (y + z) == x * y + x * z > (x + y) * z == x * z + y * z * 'Algebra.Graph.NonEmpty.AdjacencyMap.connect' can be decomposed: > x * y * z == x * y + x * z + y * z * 'Algebra.Graph.NonEmpty.AdjacencyMap.connect' satisfies absorption and saturation: > 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: @'Algebra.Graph.NonEmpty.AdjacencyMap.vertex' 1 < 'Algebra.Graph.NonEmpty.AdjacencyMap.vertex' 2 'Algebra.Graph.NonEmpty.AdjacencyMap.vertex' 3 < 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 2 'Algebra.Graph.NonEmpty.AdjacencyMap.vertex' 1 < 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 1 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 1 < 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 2 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 2 < 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 1 + 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 2 2 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 2 < 'Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 3@ Note that the resulting order refines the 'Algebra.Graph.NonEmpty.AdjacencyMap.isSubgraphOf' relation and is compatible with 'Algebra.Graph.NonEmpty.AdjacencyMap.overlay' and 'Algebra.Graph.NonEmpty.AdjacencyMap.connect' operations: @'Algebra.Graph.NonEmpty.AdjacencyMap.isSubgraphOf' x y ==> x <= y@ @x <= x + y x + y <= x * y@ -} newtype AdjacencyMap a = NAM { -- | The /adjacency map/ of a graph: each vertex is associated with a set of -- its direct successors. Complexity: /O(1)/ time and memory. -- -- @ -- adjacencyMap ('vertex' x) == Map.'Map.singleton' x Set.'Set.empty' -- adjacencyMap ('Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 1) == Map.'Map.singleton' 1 (Set.'Set.singleton' 1) -- adjacencyMap ('Algebra.Graph.NonEmpty.AdjacencyMap.edge' 1 2) == Map.'Map.fromList' [(1,Set.'Set.singleton' 2), (2,Set.'Set.empty')] -- @ am :: AM.AdjacencyMap a } deriving (Eq, NFData, Ord) -- | __Note:__ this does not satisfy the usual ring laws; see 'AdjacencyMap' for -- more details. instance (Ord a, Num a) => Num (AdjacencyMap a) where fromInteger = NAM . AM.vertex . fromInteger NAM x + NAM y = NAM (AM.overlay x y) NAM x * NAM y = NAM (AM.connect x y) signum = error "NonEmpty.AdjacencyMap.signum cannot be implemented." abs = id negate = id instance (Ord a, Show a) => Show (AdjacencyMap a) where showsPrec p (NAM (AM.AM m)) | null vs = error "NonEmpty.AdjacencyMap.Show: Graph is 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 = Set.toAscList (Map.keysSet m) es = AM.internalEdgeList m vshow [x] = showString "vertex " . showsPrec 11 x vshow xs = showString "vertices1 " . showsPrec 11 xs eshow [(x, y)] = showString "edge " . showsPrec 11 x . showString " " . showsPrec 11 y eshow xs = showString "edges1 " . showsPrec 11 xs used = Set.toAscList (AM.referredToVertexSet m) -- | Check if the internal graph representation is consistent, i.e. that all -- edges refer to existing vertices, and the graph is non-empty. It should be -- impossible to create an inconsistent adjacency map, and we use this function -- in testing. -- /Note: this function is for internal use only/. -- -- @ -- consistent ('vertex' x) == True -- consistent ('overlay' x y) == True -- consistent ('connect' x y) == True -- consistent ('Algebra.Graph.NonEmpty.AdjacencyMap.edge' x y) == True -- consistent ('Algebra.Graph.NonEmpty.AdjacencyMap.edges' xs) == True -- consistent ('Algebra.Graph.NonEmpty.AdjacencyMap.stars' xs) == True -- @ consistent :: Ord a => AdjacencyMap a -> Bool consistent (NAM x) = AM.consistent x && not (AM.isEmpty x)