{-# language LambdaCase #-} ----------------------------------------------------------------------------- -- | -- Module : Algebra.Graph.AdjacencyMap.Algorithm -- Copyright : (c) Andrey Mokhov 2016-2018 -- License : MIT (see the file LICENSE) -- Maintainer : 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 basic graph algorithms, such as /depth-first search/, -- implemented for the "Algebra.Graph.AdjacencyMap" data type. ----------------------------------------------------------------------------- module Algebra.Graph.AdjacencyMap.Algorithm ( -- * Algorithms bfsForest, bfs, dfsForest, dfsForestFrom, dfs, reachable, topSort, isAcyclic, scc, -- * Correctness properties isDfsForestOf, isTopSortOf, -- * Type synonyms Cycle ) where import Control.Monad import Control.Monad.Cont import Control.Monad.State.Strict import Data.Either import Data.List.NonEmpty (NonEmpty(..),(<|)) import Data.Maybe import Data.Tree import Algebra.Graph.AdjacencyMap import Algebra.Graph.Internal import qualified Algebra.Graph.NonEmpty.AdjacencyMap as NonEmpty import qualified Data.Array as Array import qualified Data.List as List import qualified Data.Map.Strict as Map import qualified Data.Set as Set -- | Compute the /breadth-first search/ forest of a graph, such that -- adjacent vertices are explored in increasing order with respect -- to their 'Ord' instance. The search is seeded by a list of -- argument vertices that will be the roots of the resulting -- forest. Duplicates in the list will have their first occurrence -- expanded and subsequent ones ignored. Argument vertices not in -- the graph are also ignored. -- -- Let /L/ be the number of seed vertices. Complexity: -- /O((L+m)*log n)/ time and /O(n)/ space. -- -- @ -- 'forest' (bfsForest [1,2] $ 'edge' 1 2) == 'vertices' [1,2] -- 'forest' (bfsForest [2] $ 'edge' 1 2) == 'vertex' 2 -- 'forest' (bfsForest [3] $ 'edge' 1 2) == 'empty' -- 'forest' (bfsForest [2,1] $ 'edge' 1 2) == 'vertices' [1,2] -- 'isSubgraphOf' ('forest' $ bfsForest vs x) x == True -- bfsForest ('vertexList' g) g == 'map' (\v -> Node v []) ('nub' $ 'vertexList' g) -- bfsForest [] x == [] -- bfsForest [1,4] (3 * (1 + 4) * (1 + 5)) == [ Node { rootLabel = 1 -- , subForest = [ Node { rootLabel = 5 -- , subForest = [] }]} -- , Node { rootLabel = 4 -- , subForest = [] }] -- 'forest' (bfsForest [3] ('circuit' [1..5] + 'circuit' [5,4..1])) == 'path' [3,2,1] + 'path' [3,4,5] -- -- @ bfsForest :: Ord a => [a] -> AdjacencyMap a -> Forest a bfsForest vs g = evalState (explore [ v | v <- vs, hasVertex v g ]) Set.empty where explore = unfoldForestM_BF walk <=< filterM discovered walk v = (v,) <$> adjacentM v adjacentM v = filterM discovered $ Set.toList (postSet v g) discovered v = do new <- gets (not . Set.member v) when new $ modify' (Set.insert v) return new -- | This is 'bfsForest' with the resulting forest converted to a -- level structure. Adjacent vertices are explored in increasing -- order with respect to their 'Ord' instance. Flattening the result -- via @'concat' . 'bfs' vs@ gives an enumeration of vertices -- reachable from @vs@ in breadth first order. -- -- Let /L/ be the number of seed vertices. Complexity: -- /O((L+m)*log n)/ time and /O(n)/ space. -- -- @ -- bfs vs 'empty' == [] -- bfs [] g == [] -- bfs [1] ('edge' 1 1) == [[1]] -- bfs [1] ('edge' 1 2) == [[1],[2]] -- bfs [2] ('edge' 1 2) == [[2]] -- bfs [1,2] ('edge' 1 2) == [[1,2]] -- bfs [2,1] ('edge' 1 2) == [[2,1]] -- bfs [3] ('edge' 1 2) == [] -- bfs [1,2] ( (1*2) + (3*4) + (5*6) ) == [[1,2]] -- bfs [1,3] ( (1*2) + (3*4) + (5*6) ) == [[1,3],[2,4]] -- bfs [3] (3 * (1 + 4) * (1 + 5)) == [[3],[1,4,5]] -- bfs [2] ('circuit' [1..5] + 'circuit' [5,4..1]) == [[2],[1,3],[5,4]] -- 'concat' (bfs [3] $ 'circuit' [1..5] + 'circuit' [5,4..1]) == [3,2,4,1,5] -- bfs vs == 'map' 'concat' . 'List.transpose' . 'map' 'levels' . 'bfsForest' vs -- @ bfs :: Ord a => [a] -> AdjacencyMap a -> [[a]] bfs vs = map concat . List.transpose . map levels . bfsForest vs -- | Compute the /depth-first search/ forest of a graph, where -- adjacent vertices are expanded in increasing order with respect -- to their 'Ord' instance. -- -- Complexity: /O((n+m)*log n)/ time and /O(n)/ space. -- -- @ -- dfsForest 'empty' == [] -- 'forest' (dfsForest $ 'edge' 1 1) == 'vertex' 1 -- 'forest' (dfsForest $ 'edge' 1 2) == 'edge' 1 2 -- 'forest' (dfsForest $ 'edge' 2 1) == 'vertices' [1,2] -- 'isSubgraphOf' ('forest' $ dfsForest x) x == True -- 'isDfsForestOf' (dfsForest x) x == True -- dfsForest . 'forest' . dfsForest == dfsForest -- dfsForest ('vertices' vs) == 'map' (\\v -> Node v []) ('Data.List.nub' $ 'Data.List.sort' vs) -- 'dfsForestFrom' ('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 = [] }]}] -- 'forest' (dfsForest $ 'circuit' [1..5] + 'circuit' [5,4..1]) == 'path' [1,2,3,4,5] -- @ dfsForest :: Ord a => AdjacencyMap a -> Forest a dfsForest g = dfsForestFrom' (vertexList g) g -- | Compute the /depth-first search/ forest of a graph from the given -- vertices, where adjacent vertices are expanded in increasing -- order with respect to their 'Ord' instance. Note that the -- resulting forest does not necessarily span the whole graph, as -- some vertices may be unreachable. Any of the given vertices which -- are not in the graph are ignored. -- -- Let /L/ be the number of seed vertices. Complexity: /O((L+m)*log n)/ -- time and /O(n)/ space. -- -- @ -- dfsForestFrom vs 'empty' == [] -- 'forest' (dfsForestFrom [1] $ 'edge' 1 1) == 'vertex' 1 -- 'forest' (dfsForestFrom [1] $ 'edge' 1 2) == 'edge' 1 2 -- 'forest' (dfsForestFrom [2] $ 'edge' 1 2) == 'vertex' 2 -- 'forest' (dfsForestFrom [3] $ 'edge' 1 2) == 'empty' -- 'forest' (dfsForestFrom [2,1] $ 'edge' 1 2) == 'vertices' [1,2] -- 'isSubgraphOf' ('forest' $ dfsForestFrom vs x) x == True -- 'isDfsForestOf' (dfsForestFrom ('vertexList' x) x) x == True -- dfsForestFrom ('vertexList' x) x == 'dfsForest' x -- dfsForestFrom vs ('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 = [] }] -- 'forest' (dfsForestFrom [3] $ 'circuit' [1..5] + 'circuit' [5,4..1]) == 'path' [3,2,1,5,4] -- @ dfsForestFrom :: Ord a => [a] -> AdjacencyMap a -> Forest a dfsForestFrom vs g = dfsForestFrom' [ v | v <- vs, hasVertex v g ] g dfsForestFrom' :: Ord a => [a] -> AdjacencyMap a -> Forest a dfsForestFrom' vs g = evalState (explore vs) Set.empty where explore (v:vs) = discovered v >>= \case True -> (:) <$> walk v <*> explore vs False -> explore vs explore [] = return [] walk v = Node v <$> explore (adjacent v) adjacent v = Set.toList (postSet v g) discovered v = do new <- gets (not . Set.member v) when new $ modify' (Set.insert v) return new -- | Compute the vertices visited by /depth-first search/ in a graph -- from the given vertices. Adjacent vertices are expanded in -- increasing order with respect to their 'Ord' instance. -- -- Let /L/ be the number of seed vertices. Complexity: /O((L+m)*log n)/ -- time and /O(n)/ space. -- -- @ -- dfs vs $ 'empty' == [] -- dfs [1] $ 'edge' 1 1 == [1] -- dfs [1] $ 'edge' 1 2 == [1,2] -- dfs [2] $ 'edge' 1 2 == [2] -- dfs [3] $ 'edge' 1 2 == [] -- dfs [1,2] $ 'edge' 1 2 == [1,2] -- dfs [2,1] $ 'edge' 1 2 == [2,1] -- dfs [] $ x == [] -- dfs [1,4] $ 3 * (1 + 4) * (1 + 5) == [1,5,4] -- 'isSubgraphOf' ('vertices' $ dfs vs x) x == True -- dfs [3] $ 'circuit' [1..5] + 'circuit' [5,4..1] == [3,2,1,5,4] -- @ dfs :: Ord a => [a] -> AdjacencyMap a -> [a] dfs vs = dfsForestFrom vs >=> flatten -- | Compute the list of vertices that are /reachable/ from a given -- source vertex in a graph. The vertices in the resulting list -- appear in /depth-first order/. -- -- Complexity: /O(m*log n)/ time and /O(n)/ space. -- -- @ -- reachable x $ 'empty' == [] -- reachable 1 $ 'vertex' 1 == [1] -- reachable 1 $ 'vertex' 2 == [] -- reachable 1 $ 'edge' 1 1 == [1] -- reachable 1 $ 'edge' 1 2 == [1,2] -- reachable 4 $ 'path' [1..8] == [4..8] -- reachable 4 $ 'circuit' [1..8] == [4..8] ++ [1..3] -- reachable 8 $ 'clique' [8,7..1] == [8] ++ [1..7] -- 'isSubgraphOf' ('vertices' $ reachable x y) y == True -- @ reachable :: Ord a => a -> AdjacencyMap a -> [a] reachable x = dfs [x] type Cycle = NonEmpty data NodeState = Entered | Exited data S a = S { parent :: Map.Map a a , entry :: Map.Map a NodeState , order :: [a] } topSort' :: (Ord a, MonadState (S a) m, MonadCont m) => AdjacencyMap a -> m (Either (Cycle a) [a]) topSort' g = callCC $ \cyclic -> do let vertices = map fst $ Map.toDescList $ adjacencyMap g adjacent = Set.toDescList . flip postSet g dfsRoot x = nodeState x >>= \case Nothing -> enterRoot x >> dfs x >> exit x _ -> return () dfs x = forM_ (adjacent x) $ \y -> nodeState y >>= \case Nothing -> enter x y >> dfs y >> exit y Just Exited -> return () Just Entered -> cyclic . Left . retrace x y =<< gets parent forM_ vertices dfsRoot Right <$> gets order where nodeState v = gets (Map.lookup v . entry) enter u v = modify' (\(S m n vs) -> S (Map.insert v u m) (Map.insert v Entered n) vs) enterRoot v = modify' (\(S m n vs) -> S m (Map.insert v Entered n) vs) exit v = modify' (\(S m n vs) -> S m (Map.alter (fmap leave) v n) (v:vs)) where leave = \case Entered -> Exited Exited -> error "Internal error: dfs search order violated" retrace curr head parent = aux (curr :| []) where aux xs@(curr :| _) | head == curr = xs | otherwise = aux (parent Map.! curr <| xs) -- | Compute a topological sort of a DAG or discover a cycle. -- -- Vertices are expanded in decreasing order with respect to their -- 'Ord' instance. This gives the lexicographically smallest -- topological ordering in the case of success. In the case of -- failure, the cycle is characterized by being the -- lexicographically smallest up to rotation with respect to @Ord -- (Dual a)@ in the first connected component of the graph -- containing a cycle, where the connected components are ordered by -- their largest vertex with respect to @Ord a@. -- -- Complexity: /O((n+m)*log n)/ time and /O(n)/ space. -- -- @ -- topSort (1 * 2 + 3 * 1) == Right [3,1,2] -- topSort ('path' [1..5]) == Right [1..5] -- topSort (3 * (1 * 4 + 2 * 5)) == Right [3,1,2,4,5] -- topSort (1 * 2 + 2 * 1) == Left (2 ':|' [1]) -- topSort ('path' [5,4..1] + 'edge' 2 4) == Left (4 ':|' [3,2]) -- topSort ('circuit' [1..3]) == Left (3 ':|' [1,2]) -- topSort ('circuit' [1..3] + 'circuit' [3,2,1]) == Left (3 ':|' [2]) -- topSort (1*2 + 2*1 + 3*4 + 4*3 + 5*1) == Left (1 ':|' [2]) -- fmap ('flip' 'isTopSortOf' x) (topSort x) /= Right False -- 'isRight' . topSort == 'isAcyclic' -- topSort . 'vertices' == Right . 'nub' . 'sort' -- @ topSort :: Ord a => AdjacencyMap a -> Either (Cycle a) [a] topSort g = runContT (evalStateT (topSort' g) initialState) id where initialState = S Map.empty Map.empty [] -- | Check if a given graph is /acyclic/. -- -- Complexity: /O((n+m)*log n)/ time and /O(n)/ space. -- -- @ -- isAcyclic (1 * 2 + 3 * 1) == True -- isAcyclic (1 * 2 + 2 * 1) == False -- isAcyclic . 'circuit' == 'null' -- isAcyclic == 'isRight' . 'topSort' -- @ isAcyclic :: Ord a => AdjacencyMap a -> Bool isAcyclic = isRight . topSort -- | Compute the /condensation/ of a graph, where each vertex corresponds to a -- /strongly-connected component/ of the original graph. Note that component -- graphs are non-empty, and are therefore of type -- "Algebra.Graph.NonEmpty.AdjacencyMap". -- -- Details about the implementation can be found at -- <https://github.com/jitwit/alga-notes/blob/master/gabow.org gabow-notes>. -- -- Complexity: /O((n+m)*log n)/ time and /O(n+m)/ space. -- -- @ -- scc 'empty' == 'empty' -- scc ('vertex' x) == 'vertex' (NonEmpty.'NonEmpty.vertex' x) -- scc ('vertices' xs) == 'vertices' ('map' 'NonEmpty.vertex' xs) -- scc ('edge' 1 1) == 'vertex' (NonEmpty.'NonEmpty.edge' 1 1) -- scc ('edge' 1 2) == 'edge' (NonEmpty.'NonEmpty.vertex' 1) (NonEmpty.'NonEmpty.vertex' 2) -- scc ('circuit' (1:xs)) == 'vertex' (NonEmpty.'NonEmpty.circuit1' (1 'Data.List.NonEmpty.:|' xs)) -- scc (3 * 1 * 4 * 1 * 5) == 'edges' [ (NonEmpty.'NonEmpty.vertex' 3 , NonEmpty.'NonEmpty.vertex' 5 ) -- , (NonEmpty.'NonEmpty.vertex' 3 , NonEmpty.'NonEmpty.clique1' [1,4,1]) -- , (NonEmpty.'NonEmpty.clique1' [1,4,1], NonEmpty.'NonEmpty.vertex' 5 ) ] -- 'isAcyclic' . scc == 'const' True -- 'isAcyclic' x == (scc x == 'gmap' NonEmpty.'NonEmpty.vertex' x) -- @ scc :: Ord a => AdjacencyMap a -> AdjacencyMap (NonEmpty.AdjacencyMap a) scc g = condense g $ execState (gabowSCC g) initialState where initialState = SCC 0 0 [] [] Map.empty Map.empty [] [] [] data StateSCC a = SCC { preorder :: {-# unpack #-} !Int , component :: {-# unpack #-} !Int , boundaryStack :: [(Int,a)] , pathStack :: [a] , preorders :: Map.Map a Int , components :: Map.Map a Int , innerGraphs :: [AdjacencyMap a] , innerEdges :: [(Int,(a,a))] , outerEdges :: [(a,a)] } deriving (Show) gabowSCC :: Ord a => AdjacencyMap a -> State (StateSCC a) () gabowSCC g = do let dfs u = do p_u <- enter u forEach (postSet u g) $ \v -> do preorderId v >>= \case Nothing -> do updated <- dfs v if updated then outedge (u,v) else inedge (p_u,(u,v)) Just p_v -> do scc_v <- hasComponent v if scc_v then outedge (u,v) else popBoundary p_v >> inedge (p_u,(u,v)) exit u forM_ (vertexList g) $ \v -> do assigned <- hasPreorderId v unless assigned $ void $ dfs v where -- called when visiting vertex v. assigns preorder number to v, -- adds the (id, v) pair to the boundary stack b, and adds v to -- the path stack s. enter v = do SCC pre scc bnd pth pres sccs gs es_i es_o <- get let pre' = pre+1 bnd' = (pre,v):bnd pth' = v:pth pres' = Map.insert v pre pres put $! SCC pre' scc bnd' pth' pres' sccs gs es_i es_o return pre -- called on back edges. pops the boundary stack while the top -- vertex has a larger preorder number than p_v. popBoundary p_v = modify' (\(SCC pre scc bnd pth pres sccs gs es_i es_o) -> SCC pre scc (dropWhile ((>p_v).fst) bnd) pth pres sccs gs es_i es_o) -- called when exiting vertex v. if v is the bottom of a scc -- boundary, we add a new SCC, otherwise v is part of a larger scc -- being constructed and we continue. exit v = do newComponent <- (v==).snd.head <$> gets boundaryStack when newComponent $ insertComponent v return newComponent insertComponent v = modify' (\(SCC pre scc bnd pth pres sccs gs es_i es_o) -> let (curr,v_pth') = span (/=v) pth pth' = tail v_pth' -- Here we know that v_pth' starts with v (es,es_i') = span ((>=p_v).fst) es_i g_i | null es = vertex v | otherwise = edges (snd <$> es) p_v = fst $ head bnd scc' = scc + 1 bnd' = tail bnd sccs' = List.foldl' (\sccs x -> Map.insert x scc sccs) sccs (v:curr) gs' = g_i:gs in SCC pre scc' bnd' pth' pres sccs' gs' es_i' es_o) inedge uv = modify' (\(SCC pre scc bnd pth pres sccs gs es_i es_o) -> SCC pre scc bnd pth pres sccs gs (uv:es_i) es_o) outedge uv = modify' (\(SCC pre scc bnd pth pres sccs gs es_i es_o) -> SCC pre scc bnd pth pres sccs gs es_i (uv:es_o)) hasPreorderId v = gets (Map.member v . preorders) preorderId v = gets (Map.lookup v . preorders) hasComponent v = gets (Map.member v . components) condense :: Ord a => AdjacencyMap a -> StateSCC a -> AdjacencyMap (NonEmpty.AdjacencyMap a) condense g (SCC _ n _ _ _ assignment inner _ outer) | n == 1 = vertex $ convert g | otherwise = gmap (\c -> inner' Array.! (n-1-c)) outer' where inner' = Array.listArray (0,n-1) (convert <$> inner) outer' = es `overlay` vs vs = vertices [0..n-1] es = edges [ (sccid x, sccid y) | (x,y) <- outer ] sccid v = assignment Map.! v convert = fromJust . NonEmpty.toNonEmpty -- | Check if a given forest is a correct /depth-first search/ forest of a graph. -- The implementation is based on the paper "Depth-First Search and Strong -- Connectivity in Coq" by François Pottier. -- -- @ -- isDfsForestOf [] 'empty' == True -- isDfsForestOf [] ('vertex' 1) == False -- isDfsForestOf [Node 1 []] ('vertex' 1) == True -- isDfsForestOf [Node 1 []] ('vertex' 2) == False -- isDfsForestOf [Node 1 [], Node 1 []] ('vertex' 1) == False -- isDfsForestOf [Node 1 []] ('edge' 1 1) == True -- isDfsForestOf [Node 1 []] ('edge' 1 2) == False -- isDfsForestOf [Node 1 [], Node 2 []] ('edge' 1 2) == False -- isDfsForestOf [Node 2 [], Node 1 []] ('edge' 1 2) == True -- isDfsForestOf [Node 1 [Node 2 []]] ('edge' 1 2) == True -- isDfsForestOf [Node 1 [], Node 2 []] ('vertices' [1,2]) == True -- isDfsForestOf [Node 2 [], Node 1 []] ('vertices' [1,2]) == True -- isDfsForestOf [Node 1 [Node 2 []]] ('vertices' [1,2]) == False -- isDfsForestOf [Node 1 [Node 2 [Node 3 []]]] ('path' [1,2,3]) == True -- isDfsForestOf [Node 1 [Node 3 [Node 2 []]]] ('path' [1,2,3]) == False -- isDfsForestOf [Node 3 [], Node 1 [Node 2 []]] ('path' [1,2,3]) == True -- isDfsForestOf [Node 2 [Node 3 []], Node 1 []] ('path' [1,2,3]) == True -- isDfsForestOf [Node 1 [], Node 2 [Node 3 []]] ('path' [1,2,3]) == False -- @ isDfsForestOf :: Ord a => Forest a -> AdjacencyMap a -> Bool isDfsForestOf f am = case go Set.empty f of Just seen -> seen == vertexSet am Nothing -> False where go seen [] = Just seen go seen (t:ts) = do let root = rootLabel t guard $ root `Set.notMember` seen guard $ and [ hasEdge root (rootLabel subTree) am | subTree <- subForest t ] newSeen <- go (Set.insert root seen) (subForest t) guard $ postSet root am `Set.isSubsetOf` newSeen go newSeen ts -- | Check if a given list of vertices is a correct /topological sort/ of a graph. -- -- @ -- isTopSortOf [3,1,2] (1 * 2 + 3 * 1) == True -- isTopSortOf [1,2,3] (1 * 2 + 3 * 1) == False -- isTopSortOf [] (1 * 2 + 3 * 1) == False -- isTopSortOf [] 'empty' == True -- isTopSortOf [x] ('vertex' x) == True -- isTopSortOf [x] ('edge' x x) == False -- @ isTopSortOf :: Ord a => [a] -> AdjacencyMap a -> Bool isTopSortOf xs m = go Set.empty xs where go seen [] = seen == Map.keysSet (adjacencyMap m) go seen (v:vs) = postSet v m `Set.intersection` newSeen == Set.empty && go newSeen vs where newSeen = Set.insert v seen