1{-# LANGUAGE CPP #-} 2#if __GLASGOW_HASKELL__ 3{-# LANGUAGE Rank2Types #-} 4{-# LANGUAGE DeriveDataTypeable #-} 5{-# LANGUAGE StandaloneDeriving #-} 6#endif 7#if __GLASGOW_HASKELL__ >= 703 8{-# LANGUAGE Trustworthy #-} 9#endif 10#if __GLASGOW_HASKELL__ >= 702 11{-# LANGUAGE DeriveGeneric #-} 12{-# LANGUAGE StandaloneDeriving #-} 13#endif 14 15#include "containers.h" 16 17----------------------------------------------------------------------------- 18-- | 19-- Module : Data.Graph 20-- Copyright : (c) The University of Glasgow 2002 21-- License : BSD-style (see the file libraries/base/LICENSE) 22-- 23-- Maintainer : libraries@haskell.org 24-- Portability : portable 25-- 26-- A version of the graph algorithms described in: 27-- 28-- /Structuring Depth-First Search Algorithms in Haskell/, 29-- by David King and John Launchbury. 30-- 31----------------------------------------------------------------------------- 32 33module Data.Graph( 34 35 -- * External interface 36 37 -- At present the only one with a "nice" external interface 38 stronglyConnComp, stronglyConnCompR, SCC(..), flattenSCC, flattenSCCs, 39 40 -- * Graphs 41 42 Graph, Table, Bounds, Edge, Vertex, 43 44 -- ** Building graphs 45 46 graphFromEdges, graphFromEdges', buildG, transposeG, 47 -- reverseE, 48 49 -- ** Graph properties 50 51 vertices, edges, 52 outdegree, indegree, 53 54 -- * Algorithms 55 56 dfs, dff, 57 topSort, 58 components, 59 scc, 60 bcc, 61 -- tree, back, cross, forward, 62 reachable, path, 63 64 module Data.Tree 65 66 ) where 67 68#if __GLASGOW_HASKELL__ 69# define USE_ST_MONAD 1 70#endif 71 72-- Extensions 73#if USE_ST_MONAD 74import Control.Monad.ST 75import Data.Array.ST (STArray, newArray, readArray, writeArray) 76#else 77import Data.IntSet (IntSet) 78import qualified Data.IntSet as Set 79#endif 80import Data.Tree (Tree(Node), Forest) 81 82-- std interfaces 83import Control.Applicative 84#if !MIN_VERSION_base(4,8,0) 85import qualified Data.Foldable as F 86import Data.Traversable 87#else 88import Data.Foldable as F 89#endif 90import Control.DeepSeq (NFData(rnf)) 91import Data.Maybe 92import Data.Array 93import Data.List 94#if MIN_VERSION_base(4,9,0) 95import Data.Functor.Classes 96import Data.Semigroup (Semigroup (..)) 97#endif 98#if __GLASGOW_HASKELL__ >= 706 99import GHC.Generics (Generic, Generic1) 100#elif __GLASGOW_HASKELL__ >= 702 101import GHC.Generics (Generic) 102#endif 103#ifdef __GLASGOW_HASKELL__ 104import Data.Data (Data) 105#endif 106import Data.Typeable 107 108 109------------------------------------------------------------------------- 110-- - 111-- External interface 112-- - 113------------------------------------------------------------------------- 114 115-- | Strongly connected component. 116data SCC vertex = AcyclicSCC vertex -- ^ A single vertex that is not 117 -- in any cycle. 118 | CyclicSCC [vertex] -- ^ A maximal set of mutually 119 -- reachable vertices. 120#if __GLASGOW_HASKELL__ >= 802 121 deriving ( Eq -- ^ @since 0.5.9 122 , Show -- ^ @since 0.5.9 123 , Read -- ^ @since 0.5.9 124 ) 125#else 126 deriving (Eq, Show, Read) 127#endif 128 129INSTANCE_TYPEABLE1(SCC) 130 131#ifdef __GLASGOW_HASKELL__ 132-- | @since 0.5.9 133deriving instance Data vertex => Data (SCC vertex) 134#endif 135 136#if __GLASGOW_HASKELL__ >= 706 137-- | @since 0.5.9 138deriving instance Generic1 SCC 139#endif 140 141#if __GLASGOW_HASKELL__ >= 702 142-- | @since 0.5.9 143deriving instance Generic (SCC vertex) 144#endif 145 146#if MIN_VERSION_base(4,9,0) 147-- | @since 0.5.9 148instance Eq1 SCC where 149 liftEq eq (AcyclicSCC v1) (AcyclicSCC v2) = eq v1 v2 150 liftEq eq (CyclicSCC vs1) (CyclicSCC vs2) = liftEq eq vs1 vs2 151 liftEq _ _ _ = False 152-- | @since 0.5.9 153instance Show1 SCC where 154 liftShowsPrec sp _sl d (AcyclicSCC v) = showsUnaryWith sp "AcyclicSCC" d v 155 liftShowsPrec _sp sl d (CyclicSCC vs) = showsUnaryWith (const sl) "CyclicSCC" d vs 156-- | @since 0.5.9 157instance Read1 SCC where 158 liftReadsPrec rp rl = readsData $ 159 readsUnaryWith rp "AcyclicSCC" AcyclicSCC <> 160 readsUnaryWith (const rl) "CyclicSCC" CyclicSCC 161#endif 162 163-- | @since 0.5.9 164instance F.Foldable SCC where 165 foldr c n (AcyclicSCC v) = c v n 166 foldr c n (CyclicSCC vs) = foldr c n vs 167 168-- | @since 0.5.9 169instance Traversable SCC where 170 -- We treat the non-empty cyclic case specially to cut one 171 -- fmap application. 172 traverse f (AcyclicSCC vertex) = AcyclicSCC <$> f vertex 173 traverse _f (CyclicSCC []) = pure (CyclicSCC []) 174 traverse f (CyclicSCC (x : xs)) = 175 liftA2 (\x' xs' -> CyclicSCC (x' : xs')) (f x) (traverse f xs) 176 177instance NFData a => NFData (SCC a) where 178 rnf (AcyclicSCC v) = rnf v 179 rnf (CyclicSCC vs) = rnf vs 180 181-- | @since 0.5.4 182instance Functor SCC where 183 fmap f (AcyclicSCC v) = AcyclicSCC (f v) 184 fmap f (CyclicSCC vs) = CyclicSCC (fmap f vs) 185 186-- | The vertices of a list of strongly connected components. 187flattenSCCs :: [SCC a] -> [a] 188flattenSCCs = concatMap flattenSCC 189 190-- | The vertices of a strongly connected component. 191flattenSCC :: SCC vertex -> [vertex] 192flattenSCC (AcyclicSCC v) = [v] 193flattenSCC (CyclicSCC vs) = vs 194 195-- | The strongly connected components of a directed graph, topologically 196-- sorted. 197stronglyConnComp 198 :: Ord key 199 => [(node, key, [key])] 200 -- ^ The graph: a list of nodes uniquely identified by keys, 201 -- with a list of keys of nodes this node has edges to. 202 -- The out-list may contain keys that don't correspond to 203 -- nodes of the graph; such edges are ignored. 204 -> [SCC node] 205 206stronglyConnComp edges0 207 = map get_node (stronglyConnCompR edges0) 208 where 209 get_node (AcyclicSCC (n, _, _)) = AcyclicSCC n 210 get_node (CyclicSCC triples) = CyclicSCC [n | (n,_,_) <- triples] 211 212-- | The strongly connected components of a directed graph, topologically 213-- sorted. The function is the same as 'stronglyConnComp', except that 214-- all the information about each node retained. 215-- This interface is used when you expect to apply 'SCC' to 216-- (some of) the result of 'SCC', so you don't want to lose the 217-- dependency information. 218stronglyConnCompR 219 :: Ord key 220 => [(node, key, [key])] 221 -- ^ The graph: a list of nodes uniquely identified by keys, 222 -- with a list of keys of nodes this node has edges to. 223 -- The out-list may contain keys that don't correspond to 224 -- nodes of the graph; such edges are ignored. 225 -> [SCC (node, key, [key])] -- ^ Topologically sorted 226 227stronglyConnCompR [] = [] -- added to avoid creating empty array in graphFromEdges -- SOF 228stronglyConnCompR edges0 229 = map decode forest 230 where 231 (graph, vertex_fn,_) = graphFromEdges edges0 232 forest = scc graph 233 decode (Node v []) | mentions_itself v = CyclicSCC [vertex_fn v] 234 | otherwise = AcyclicSCC (vertex_fn v) 235 decode other = CyclicSCC (dec other []) 236 where 237 dec (Node v ts) vs = vertex_fn v : foldr dec vs ts 238 mentions_itself v = v `elem` (graph ! v) 239 240------------------------------------------------------------------------- 241-- - 242-- Graphs 243-- - 244------------------------------------------------------------------------- 245 246-- | Abstract representation of vertices. 247type Vertex = Int 248-- | Table indexed by a contiguous set of vertices. 249type Table a = Array Vertex a 250-- | Adjacency list representation of a graph, mapping each vertex to its 251-- list of successors. 252type Graph = Table [Vertex] 253-- | The bounds of a 'Table'. 254type Bounds = (Vertex, Vertex) 255-- | An edge from the first vertex to the second. 256type Edge = (Vertex, Vertex) 257 258-- | All vertices of a graph. 259vertices :: Graph -> [Vertex] 260vertices = indices 261 262-- | All edges of a graph. 263edges :: Graph -> [Edge] 264edges g = [ (v, w) | v <- vertices g, w <- g!v ] 265 266mapT :: (Vertex -> a -> b) -> Table a -> Table b 267mapT f t = array (bounds t) [ (,) v (f v (t!v)) | v <- indices t ] 268 269-- | Build a graph from a list of edges. 270buildG :: Bounds -> [Edge] -> Graph 271buildG bounds0 edges0 = accumArray (flip (:)) [] bounds0 edges0 272 273-- | The graph obtained by reversing all edges. 274transposeG :: Graph -> Graph 275transposeG g = buildG (bounds g) (reverseE g) 276 277reverseE :: Graph -> [Edge] 278reverseE g = [ (w, v) | (v, w) <- edges g ] 279 280-- | A table of the count of edges from each node. 281outdegree :: Graph -> Table Int 282outdegree = mapT numEdges 283 where numEdges _ ws = length ws 284 285-- | A table of the count of edges into each node. 286indegree :: Graph -> Table Int 287indegree = outdegree . transposeG 288 289-- | Identical to 'graphFromEdges', except that the return value 290-- does not include the function which maps keys to vertices. This 291-- version of 'graphFromEdges' is for backwards compatibility. 292graphFromEdges' 293 :: Ord key 294 => [(node, key, [key])] 295 -> (Graph, Vertex -> (node, key, [key])) 296graphFromEdges' x = (a,b) where 297 (a,b,_) = graphFromEdges x 298 299-- | Build a graph from a list of nodes uniquely identified by keys, 300-- with a list of keys of nodes this node should have edges to. 301-- The out-list may contain keys that don't correspond to 302-- nodes of the graph; they are ignored. 303graphFromEdges 304 :: Ord key 305 => [(node, key, [key])] 306 -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex) 307graphFromEdges edges0 308 = (graph, \v -> vertex_map ! v, key_vertex) 309 where 310 max_v = length edges0 - 1 311 bounds0 = (0,max_v) :: (Vertex, Vertex) 312 sorted_edges = sortBy lt edges0 313 edges1 = zipWith (,) [0..] sorted_edges 314 315 graph = array bounds0 [(,) v (mapMaybe key_vertex ks) | (,) v (_, _, ks) <- edges1] 316 key_map = array bounds0 [(,) v k | (,) v (_, k, _ ) <- edges1] 317 vertex_map = array bounds0 edges1 318 319 (_,k1,_) `lt` (_,k2,_) = k1 `compare` k2 320 321 -- key_vertex :: key -> Maybe Vertex 322 -- returns Nothing for non-interesting vertices 323 key_vertex k = findVertex 0 max_v 324 where 325 findVertex a b | a > b 326 = Nothing 327 findVertex a b = case compare k (key_map ! mid) of 328 LT -> findVertex a (mid-1) 329 EQ -> Just mid 330 GT -> findVertex (mid+1) b 331 where 332 mid = a + (b - a) `div` 2 333 334------------------------------------------------------------------------- 335-- - 336-- Depth first search 337-- - 338------------------------------------------------------------------------- 339 340-- | A spanning forest of the graph, obtained from a depth-first search of 341-- the graph starting from each vertex in an unspecified order. 342dff :: Graph -> Forest Vertex 343dff g = dfs g (vertices g) 344 345-- | A spanning forest of the part of the graph reachable from the listed 346-- vertices, obtained from a depth-first search of the graph starting at 347-- each of the listed vertices in order. 348dfs :: Graph -> [Vertex] -> Forest Vertex 349dfs g vs = prune (bounds g) (map (generate g) vs) 350 351generate :: Graph -> Vertex -> Tree Vertex 352generate g v = Node v (map (generate g) (g!v)) 353 354prune :: Bounds -> Forest Vertex -> Forest Vertex 355prune bnds ts = run bnds (chop ts) 356 357chop :: Forest Vertex -> SetM s (Forest Vertex) 358chop [] = return [] 359chop (Node v ts : us) 360 = do 361 visited <- contains v 362 if visited then 363 chop us 364 else do 365 include v 366 as <- chop ts 367 bs <- chop us 368 return (Node v as : bs) 369 370-- A monad holding a set of vertices visited so far. 371#if USE_ST_MONAD 372 373-- Use the ST monad if available, for constant-time primitives. 374 375newtype SetM s a = SetM { runSetM :: STArray s Vertex Bool -> ST s a } 376 377instance Monad (SetM s) where 378 return = pure 379 {-# INLINE return #-} 380 SetM v >>= f = SetM $ \s -> do { x <- v s; runSetM (f x) s } 381 {-# INLINE (>>=) #-} 382 383instance Functor (SetM s) where 384 f `fmap` SetM v = SetM $ \s -> f `fmap` v s 385 {-# INLINE fmap #-} 386 387instance Applicative (SetM s) where 388 pure x = SetM $ const (return x) 389 {-# INLINE pure #-} 390 SetM f <*> SetM v = SetM $ \s -> f s >>= (`fmap` v s) 391 -- We could also use the following definition 392 -- SetM f <*> SetM v = SetM $ \s -> f s <*> v s 393 -- but Applicative (ST s) instance is present only in GHC 7.2+ 394 {-# INLINE (<*>) #-} 395 396run :: Bounds -> (forall s. SetM s a) -> a 397run bnds act = runST (newArray bnds False >>= runSetM act) 398 399contains :: Vertex -> SetM s Bool 400contains v = SetM $ \ m -> readArray m v 401 402include :: Vertex -> SetM s () 403include v = SetM $ \ m -> writeArray m v True 404 405#else /* !USE_ST_MONAD */ 406 407-- Portable implementation using IntSet. 408 409newtype SetM s a = SetM { runSetM :: IntSet -> (a, IntSet) } 410 411instance Monad (SetM s) where 412 return x = SetM $ \s -> (x, s) 413 SetM v >>= f = SetM $ \s -> case v s of (x, s') -> runSetM (f x) s' 414 415instance Functor (SetM s) where 416 f `fmap` SetM v = SetM $ \s -> case v s of (x, s') -> (f x, s') 417 {-# INLINE fmap #-} 418 419instance Applicative (SetM s) where 420 pure x = SetM $ \s -> (x, s) 421 {-# INLINE pure #-} 422 SetM f <*> SetM v = SetM $ \s -> case f s of (k, s') -> case v s' of (x, s'') -> (k x, s'') 423 {-# INLINE (<*>) #-} 424 425run :: Bounds -> SetM s a -> a 426run _ act = fst (runSetM act Set.empty) 427 428contains :: Vertex -> SetM s Bool 429contains v = SetM $ \ m -> (Set.member v m, m) 430 431include :: Vertex -> SetM s () 432include v = SetM $ \ m -> ((), Set.insert v m) 433 434#endif /* !USE_ST_MONAD */ 435 436------------------------------------------------------------------------- 437-- - 438-- Algorithms 439-- - 440------------------------------------------------------------------------- 441 442------------------------------------------------------------ 443-- Algorithm 1: depth first search numbering 444------------------------------------------------------------ 445 446preorder' :: Tree a -> [a] -> [a] 447preorder' (Node a ts) = (a :) . preorderF' ts 448 449preorderF' :: Forest a -> [a] -> [a] 450preorderF' ts = foldr (.) id $ map preorder' ts 451 452preorderF :: Forest a -> [a] 453preorderF ts = preorderF' ts [] 454 455tabulate :: Bounds -> [Vertex] -> Table Int 456tabulate bnds vs = array bnds (zipWith (,) vs [1..]) 457 458preArr :: Bounds -> Forest Vertex -> Table Int 459preArr bnds = tabulate bnds . preorderF 460 461------------------------------------------------------------ 462-- Algorithm 2: topological sorting 463------------------------------------------------------------ 464 465postorder :: Tree a -> [a] -> [a] 466postorder (Node a ts) = postorderF ts . (a :) 467 468postorderF :: Forest a -> [a] -> [a] 469postorderF ts = foldr (.) id $ map postorder ts 470 471postOrd :: Graph -> [Vertex] 472postOrd g = postorderF (dff g) [] 473 474-- | A topological sort of the graph. 475-- The order is partially specified by the condition that a vertex /i/ 476-- precedes /j/ whenever /j/ is reachable from /i/ but not vice versa. 477topSort :: Graph -> [Vertex] 478topSort = reverse . postOrd 479 480------------------------------------------------------------ 481-- Algorithm 3: connected components 482------------------------------------------------------------ 483 484-- | The connected components of a graph. 485-- Two vertices are connected if there is a path between them, traversing 486-- edges in either direction. 487components :: Graph -> Forest Vertex 488components = dff . undirected 489 490undirected :: Graph -> Graph 491undirected g = buildG (bounds g) (edges g ++ reverseE g) 492 493-- Algorithm 4: strongly connected components 494 495-- | The strongly connected components of a graph. 496scc :: Graph -> Forest Vertex 497scc g = dfs g (reverse (postOrd (transposeG g))) 498 499------------------------------------------------------------ 500-- Algorithm 5: Classifying edges 501------------------------------------------------------------ 502 503{- 504XXX unused code 505 506tree :: Bounds -> Forest Vertex -> Graph 507tree bnds ts = buildG bnds (concat (map flat ts)) 508 where flat (Node v ts') = [ (v, w) | Node w _us <- ts' ] 509 ++ concat (map flat ts') 510 511back :: Graph -> Table Int -> Graph 512back g post = mapT select g 513 where select v ws = [ w | w <- ws, post!v < post!w ] 514 515cross :: Graph -> Table Int -> Table Int -> Graph 516cross g pre post = mapT select g 517 where select v ws = [ w | w <- ws, post!v > post!w, pre!v > pre!w ] 518 519forward :: Graph -> Graph -> Table Int -> Graph 520forward g tree' pre = mapT select g 521 where select v ws = [ w | w <- ws, pre!v < pre!w ] \\ tree' ! v 522-} 523 524------------------------------------------------------------ 525-- Algorithm 6: Finding reachable vertices 526------------------------------------------------------------ 527 528-- | A list of vertices reachable from a given vertex. 529reachable :: Graph -> Vertex -> [Vertex] 530reachable g v = preorderF (dfs g [v]) 531 532-- | Is the second vertex reachable from the first? 533path :: Graph -> Vertex -> Vertex -> Bool 534path g v w = w `elem` (reachable g v) 535 536------------------------------------------------------------ 537-- Algorithm 7: Biconnected components 538------------------------------------------------------------ 539 540-- | The biconnected components of a graph. 541-- An undirected graph is biconnected if the deletion of any vertex 542-- leaves it connected. 543bcc :: Graph -> Forest [Vertex] 544bcc g = (concat . map bicomps . map (do_label g dnum)) forest 545 where forest = dff g 546 dnum = preArr (bounds g) forest 547 548do_label :: Graph -> Table Int -> Tree Vertex -> Tree (Vertex,Int,Int) 549do_label g dnum (Node v ts) = Node (v,dnum!v,lv) us 550 where us = map (do_label g dnum) ts 551 lv = minimum ([dnum!v] ++ [dnum!w | w <- g!v] 552 ++ [lu | Node (_,_,lu) _ <- us]) 553 554bicomps :: Tree (Vertex,Int,Int) -> Forest [Vertex] 555bicomps (Node (v,_,_) ts) 556 = [ Node (v:vs) us | (_,Node vs us) <- map collect ts] 557 558collect :: Tree (Vertex,Int,Int) -> (Int, Tree [Vertex]) 559collect (Node (v,dv,lv) ts) = (lv, Node (v:vs) cs) 560 where collected = map collect ts 561 vs = concat [ ws | (lw, Node ws _) <- collected, lw<dv] 562 cs = concat [ if lw<dv then us else [Node (v:ws) us] 563 | (lw, Node ws us) <- collected ] 564