djinn-2008.1.18: Djinn/Util/Digraph.hs
{- |
Module : Util.Digraph
Copyright :
Maintainer : lib@galois.com
Stability :
Portability :
Functional graph algorithms; code taken from King
and Launchbury's POPL paper (via GHC sources.)
-}
module Util.Digraph(
-- At present the only one with a "nice" external interface
stronglyConnComp, stronglyConnCompR, SCC(..),
Graph, Vertex,
graphFromEdges, buildG, transposeG, reverseE, outdegree, indegree,
Tree(..), Forest,
showTree, showForest,
dfs, dff,
topSort,
components,
scc,
back, cross, forward,
reachable, path,
bcc
) where
------------------------------------------------------------------------------
-- A version of the graph algorithms described in:
--
-- ``Lazy Depth-First Search and Linear Graph Algorithms in Haskell''
-- by David King and John Launchbury
--
-- Also included is some additional code for printing tree structures ...
------------------------------------------------------------------------------
import Util.Sort ( sortLe ) -- merge sosrt
import Control.Monad.ST
import Data.Array.ST ( STArray, newArray, writeArray, readArray )
-- std interfaces
import Data.Maybe
import Data.Array
import Data.List ( (\\) )
{-
%************************************************************************
%* *
%* External interface
%* *
%************************************************************************
-}
data SCC vertex = AcyclicSCC vertex
| CyclicSCC [vertex] deriving Show
stronglyConnComp
:: Ord key
=> [(node, key, [key])] -- The graph; its ok for the
-- out-list to contain keys which arent
-- a vertex key, they are ignored
-> [SCC node]
stronglyConnComp edges1
= map get_node (stronglyConnCompR edges1)
where
get_node (AcyclicSCC (n, _, _)) = AcyclicSCC n
get_node (CyclicSCC triples) = CyclicSCC [n | (n,_,_) <- triples]
-- The "R" interface is used when you expect to apply SCC to
-- the (some of) the result of SCC, so you dont want to lose the dependency info
stronglyConnCompR
:: Ord key
=> [(node, key, [key])] -- The graph; its ok for the
-- out-list to contain keys which arent
-- a vertex key, they are ignored
-> [SCC (node, key, [key])]
stronglyConnCompR [] = [] -- added to avoid creating empty array in graphFromEdges -- SOF
stronglyConnCompR edges1
= map decode forest
where
(graph, vertex_fn) = graphFromEdges edges1
forest = scc graph
decode (Node v []) | mentions_itself v = CyclicSCC [vertex_fn v]
| otherwise = AcyclicSCC (vertex_fn v)
decode other = CyclicSCC (dec other [])
where
dec (Node v ts) vs = vertex_fn v : foldr dec vs ts
mentions_itself v = v `elem` (graph ! v)
{-
%************************************************************************
%* *
%* Graphs
%* *
%************************************************************************
-}
type Vertex = Int
type Table a = Array Vertex a
type Graph = Table [Vertex]
type Bounds = (Vertex, Vertex)
type Edge = (Vertex, Vertex)
vertices :: Graph -> [Vertex]
vertices = indices
edges :: Graph -> [Edge]
edges g = [ (v, w) | v <- vertices g, w <- g!v ]
mapT :: (Vertex -> a -> b) -> Table a -> Table b
mapT f t = array (bounds t) [ (,) v (f v (t!v)) | v <- indices t ]
buildG :: Bounds -> [Edge] -> Graph
buildG bounds1 edges1
= accumArray (flip (:)) [] bounds1 [(,) k v | (k,v) <- edges1]
transposeG :: Graph -> Graph
transposeG g = buildG (bounds g) (reverseE g)
reverseE :: Graph -> [Edge]
reverseE g = [ (w, v) | (v, w) <- edges g ]
outdegree :: Graph -> Table Int
outdegree = mapT numEdges
where numEdges _ ws = length ws
indegree :: Graph -> Table Int
indegree = outdegree . transposeG
graphFromEdges
:: Ord key
=> [(node, key, [key])]
-> (Graph, Vertex -> (node, key, [key]))
graphFromEdges edgs
= (graph, \v -> vertex_map ! v)
where
max_v = length edgs - 1
bounds1 = (0,max_v) :: (Vertex, Vertex)
sorted_edges = sortLe le edgs
where
(_,k1,_) `le` (_,k2,_) = case k1 `compare` k2 of { GT -> False; _other -> True }
edges1 = zipWith (,) [0..] sorted_edges
graph = array bounds1 [(,) v (mapMaybe key_vertex ks) | (,) v (_, _, ks) <- edges1]
key_map = array bounds1 [(,) v k | (,) v (_, k, _ ) <- edges1]
vertex_map = array bounds1 edges1
-- key_vertex :: key -> Maybe Vertex
-- returns Nothing for non-interesting vertices
key_vertex k = find 0 max_v
where
find a b | a > b
= Nothing
find a b = case compare k (key_map ! mid) of
LT -> find a (mid-1)
EQ -> Just mid
GT -> find (mid+1) b
where
mid = (a + b) `div` 2
{-
%************************************************************************
%* *
%* Trees and forests
%* *
%************************************************************************
-}
data Tree a = Node a (Forest a)
type Forest a = [Tree a]
mapTree :: (a -> b) -> (Tree a -> Tree b)
mapTree f (Node x ts) = Node (f x) (map (mapTree f) ts)
instance Show a => Show (Tree a) where
show t = showTree t
showTree :: Show a => Tree a -> String
showTree = drawTree . mapTree show
showForest :: Show a => Forest a -> String
showForest = unlines . map showTree
drawTree :: Tree String -> String
drawTree = unlines . draw
where
draw (Node x ts) = grp this (space (length this)) (stLoop ts)
where
this = s1 ++ x ++ " "
space n = take n (repeat ' ')
stLoop [] = [""]
stLoop [t] = grp s2 " " (draw t)
stLoop (t:xs) = grp s3 s4 (draw t) ++ [s4] ++ rsLoop xs
rsLoop [] = []
rsLoop [t] = grp s5 " " (draw t)
rsLoop (t:xs) = grp s6 s4 (draw t) ++ [s4] ++ rsLoop xs
grp a rst = zipWith (++) (a:repeat rst)
[s1,s2,s3,s4,s5,s6] = ["- ", "--", "-+", " |", " `", " +"]
{-
%************************************************************************
%* *
%* Depth first search
%* *
%************************************************************************
-}
--type Set s = MutableArray s Vertex Bool
type Set s = STArray s Vertex Bool
mkEmpty :: Bounds -> ST s (Set s)
mkEmpty bnds = newArray bnds False
contains :: Set s -> Vertex -> ST s Bool
contains m v = readArray m v
include :: Set s -> Vertex -> ST s ()
include m v = writeArray m v True
dff :: Graph -> Forest Vertex
dff g = dfs g (vertices g)
dfs :: Graph -> [Vertex] -> Forest Vertex
dfs g vs = prune (bounds g) (map (generate g) vs)
generate :: Graph -> Vertex -> Tree Vertex
generate g v = Node v (map (generate g) (g!v))
prune :: Bounds -> Forest Vertex -> Forest Vertex
prune bnds ts = runST (mkEmpty bnds >>= \m ->
chop m ts)
chop :: Set s -> Forest Vertex -> ST s (Forest Vertex)
chop _ [] = return []
chop m (Node v ts : us)
= contains m v >>= \visited ->
if visited then
chop m us
else
include m v >>= \_ ->
chop m ts >>= \as ->
chop m us >>= \bs ->
return (Node v as : bs)
{-
%************************************************************************
%* *
%* Algorithms
%* *
%************************************************************************
-}
------------------------------------------------------------
-- Algorithm 1: depth first search numbering
------------------------------------------------------------
preorder :: Tree a -> [a]
preorder (Node a ts) = a : preorderF ts
preorderF :: Forest a -> [a]
preorderF ts = concat (map preorder ts)
{- UNUSED:
preOrd :: Graph -> [Vertex]
preOrd = preorderF . dff
-}
tabulate :: Bounds -> [Vertex] -> Table Int
tabulate bnds vs = array bnds (zipWith (,) vs [1..])
preArr :: Bounds -> Forest Vertex -> Table Int
preArr bnds = tabulate bnds . preorderF
------------------------------------------------------------
-- Algorithm 2: topological sorting
------------------------------------------------------------
postorder :: Tree a -> [a]
postorder (Node a ts) = postorderF ts ++ [a]
postorderF :: Forest a -> [a]
postorderF ts = concat (map postorder ts)
postOrd :: Graph -> [Vertex]
postOrd = postorderF . dff
topSort :: Graph -> [Vertex]
topSort = reverse . postOrd
------------------------------------------------------------
-- Algorithm 3: connected components
------------------------------------------------------------
components :: Graph -> Forest Vertex
components = dff . undirected
undirected :: Graph -> Graph
undirected g = buildG (bounds g) (edges g ++ reverseE g)
------------------------------------------------------------
-- Algorithm 4: strongly connected components
------------------------------------------------------------
scc :: Graph -> Forest Vertex
scc g = dfs g (reverse (postOrd (transposeG g)))
------------------------------------------------------------
-- Algorithm 5: Classifying edges
------------------------------------------------------------
{- UNUSED:
tree :: Bounds -> Forest Vertex -> Graph
tree bnds ts = buildG bnds (concat (map flat ts))
where
flat (Node v rs) = [ (v, w) | Node w us <- ts ] ++
concat (map flat ts)
-}
back :: Graph -> Table Int -> Graph
back g post = mapT select g
where select v ws = [ w | w <- ws, post!v < post!w ]
cross :: Graph -> Table Int -> Table Int -> Graph
cross g pre post = mapT select g
where select v ws = [ w | w <- ws, post!v > post!w, pre!v > pre!w ]
forward :: Graph -> Graph -> Table Int -> Graph
forward g tree pre = mapT select g
where select v ws = [ w | w <- ws, pre!v < pre!w ] \\ tree!v
------------------------------------------------------------
-- Algorithm 6: Finding reachable vertices
------------------------------------------------------------
reachable :: Graph -> Vertex -> [Vertex]
reachable g v = preorderF (dfs g [v])
path :: Graph -> Vertex -> Vertex -> Bool
path g v w = w `elem` (reachable g v)
------------------------------------------------------------
-- Algorithm 7: Biconnected components
------------------------------------------------------------
bcc :: Graph -> Forest [Vertex]
bcc g = (concat . map bicomps . map (label g dnum)) forest
where forest = dff g
dnum = preArr (bounds g) forest
label :: Graph -> Table Int -> Tree Vertex -> Tree (Vertex,Int,Int)
label g dnum (Node v ts) = Node (v,dnum!v,lv) us
where us = map (label g dnum) ts
lv = minimum ([dnum!v] ++ [dnum!w | w <- g!v]
++ [lu | Node (_, _, lu) _ <- us])
bicomps :: Tree (Vertex,Int,Int) -> Forest [Vertex]
bicomps (Node (v,_,_) ts)
= [ Node (v:vs) us | (_, Node vs us) <- map collect ts]
collect :: Tree (Vertex,Int,Int) -> (Int, Tree [Vertex])
collect (Node (v,dv,lv) ts) = (lv, Node (v:vs) cs)
where collected = map collect ts
vs = concat [ ws | (lw, Node ws _) <- collected, lw<dv]
cs = concat [ if lw<dv then us else [Node (v:ws) us]
| (lw, Node ws us) <- collected ]