dom-lt-0.2.3: Data/Graph/Dom.hs
{-# LANGUAGE CPP #-}
{-# LANGUAGE RankNTypes, BangPatterns, FlexibleContexts #-}
#if __GLASGOW_HASKELL__ >= 800
{-# LANGUAGE Strict #-}
#endif
{- |
Module : Data.Graph.Dom
Copyright : (c) Matt Morrow 2009
License : BSD3
Maintainer : <klebinger.andreas@gmx.at>
Stability : stable
Portability : portable
The Lengauer-Tarjan graph dominators algorithm.
\[1\] Lengauer, Tarjan,
/A Fast Algorithm for Finding Dominators in a Flowgraph/, 1979.
\[2\] Muchnick,
/Advanced Compiler Design and Implementation/, 1997.
\[3\] Brisk, Sarrafzadeh,
/Interference Graphs for Procedures in Static Single/
/Information Form are Interval Graphs/, 2007.
* Strictness
Unless stated otherwise all exposed functions might fully evaluate their input
but are not guaranteed to do so.
-}
module Data.Graph.Dom (
Node,Path,Edge
,Graph,Rooted
,idom,ipdom
,domTree,pdomTree
,dom,pdom
,pddfs,rpddfs
,fromAdj,fromEdges
,toAdj,toEdges
,asTree,asGraph
,parents,ancestors
) where
import Data.Monoid(Monoid(..))
import Data.Tuple (swap)
import Data.Tree
import Data.List
import Data.IntMap(IntMap)
import Data.IntSet(IntSet)
import qualified Data.IntSet as IS
import Control.Applicative
import Control.Monad
#if MIN_VERSION_containers(0, 5, 0)
import qualified Data.IntMap.Strict as IM
import Control.Monad.ST.Strict
#else
import qualified Data.IntMap as IM
import Control.Monad.ST
#endif
import Data.Array.ST
import Data.Array.Base
(unsafeNewArray_
,unsafeWrite,unsafeRead)
-----------------------------------------------------------------------------
type Node = Int
type Path = [Node]
type Edge = (Node,Node)
type Graph = IntMap IntSet
type Rooted = (Node, Graph)
-----------------------------------------------------------------------------
-- | /Dominators/.
-- Complexity as for @idom@
dom :: Rooted -> [(Node, Path)]
dom = ancestors . domTree
-- | /Post-dominators/.
-- Complexity as for @idom@.
pdom :: Rooted -> [(Node, Path)]
pdom = ancestors . pdomTree
-- | /Dominator tree/.
-- Complexity as for @idom@.
domTree :: Rooted -> Tree Node
domTree a@(r,_) =
let is = filter ((/=r).fst) (idom a)
tg = fromEdges (fmap swap is)
in asTree (r,tg)
-- | /Post-dominator tree/.
-- Complexity as for @idom@.
pdomTree :: Rooted -> Tree Node
pdomTree a@(r,_) =
let is = filter ((/=r).fst) (ipdom a)
tg = fromEdges (fmap swap is)
in asTree (r,tg)
-- | /Immediate dominators/.
-- /O(|E|*alpha(|E|,|V|))/, where /alpha(m,n)/ is
-- \"a functional inverse of Ackermann's function\".
--
-- This Complexity bound assumes /O(1)/ indexing. Since we're
-- using @IntMap@, it has an additional /lg |V|/ factor
-- somewhere in there. I'm not sure where.
idom :: Rooted -> [(Node,Node)]
idom rg = runST (evalS idomM =<< initEnv (pruneReach rg))
-- | /Immediate post-dominators/.
-- Complexity as for @idom@.
ipdom :: Rooted -> [(Node,Node)]
ipdom rg = runST (evalS idomM =<< initEnv (pruneReach (second predG rg)))
-----------------------------------------------------------------------------
-- | /Post-dominated depth-first search/.
pddfs :: Rooted -> [Node]
pddfs = reverse . rpddfs
-- | /Reverse post-dominated depth-first search/.
rpddfs :: Rooted -> [Node]
rpddfs = concat . levels . pdomTree
-----------------------------------------------------------------------------
type Dom s a = S s (Env s) a
type NodeSet = IntSet
type NodeMap a = IntMap a
data Env s = Env
{succE :: !Graph
,predE :: !Graph
,bucketE :: !Graph
,dfsE :: {-# UNPACK #-}!Int
,zeroE :: {-# UNPACK #-}!Node
,rootE :: {-# UNPACK #-}!Node
,labelE :: {-# UNPACK #-}!(Arr s Node)
,parentE :: {-# UNPACK #-}!(Arr s Node)
,ancestorE :: {-# UNPACK #-}!(Arr s Node)
,childE :: {-# UNPACK #-}!(Arr s Node)
,ndfsE :: {-# UNPACK #-}!(Arr s Node)
,dfnE :: {-# UNPACK #-}!(Arr s Int)
,sdnoE :: {-# UNPACK #-}!(Arr s Int)
,sizeE :: {-# UNPACK #-}!(Arr s Int)
,domE :: {-# UNPACK #-}!(Arr s Node)
,rnE :: {-# UNPACK #-}!(Arr s Node)}
-----------------------------------------------------------------------------
idomM :: Dom s [(Node,Node)]
idomM = do
dfsDom =<< rootM
n <- gets dfsE
forM_ [n,n-1..1] (\i-> do
w <- ndfsM i
ps <- predsM w
forM_ ps (\v-> do
sw <- sdnoM w
u <- eval v
su <- sdnoM u
when (su < sw)
(store sdnoE w su))
z <- ndfsM =<< sdnoM w
modify(\e->e{bucketE=IM.adjust
(w`IS.insert`)
z (bucketE e)})
pw <- parentM w
link pw w
bps <- bucketM pw
forM_ bps (\v-> do
u <- eval v
su <- sdnoM u
sv <- sdnoM v
let dv = case su < sv of
True-> u
False-> pw
store domE v dv))
forM_ [1..n] (\i-> do
w <- ndfsM i
j <- sdnoM w
z <- ndfsM j
dw <- domM w
when (dw /= z)
(do ddw <- domM dw
store domE w ddw))
fromEnv
-----------------------------------------------------------------------------
eval :: Node -> Dom s Node
eval v = do
n0 <- zeroM
a <- ancestorM v
case a==n0 of
True-> labelM v
False-> do
compress v
a <- ancestorM v
l <- labelM v
la <- labelM a
sl <- sdnoM l
sla <- sdnoM la
case sl <= sla of
True-> return l
False-> return la
compress :: Node -> Dom s ()
compress v = do
n0 <- zeroM
a <- ancestorM v
aa <- ancestorM a
when (aa /= n0) (do
compress a
a <- ancestorM v
aa <- ancestorM a
l <- labelM v
la <- labelM a
sl <- sdnoM l
sla <- sdnoM la
when (sla < sl)
(store labelE v la)
store ancestorE v aa)
-----------------------------------------------------------------------------
link :: Node -> Node -> Dom s ()
link v w = do
n0 <- zeroM
lw <- labelM w
slw <- sdnoM lw
let balance s = do
c <- childM s
lc <- labelM c
slc <- sdnoM lc
case slw < slc of
False-> return s
True-> do
zs <- sizeM s
zc <- sizeM c
cc <- childM c
zcc <- sizeM cc
case 2*zc <= zs+zcc of
True-> do
store ancestorE c s
store childE s cc
balance s
False-> do
store sizeE c zs
store ancestorE s c
balance c
s <- balance w
lw <- labelM w
zw <- sizeM w
store labelE s lw
store sizeE v . (+zw) =<< sizeM v
let follow s = do
when (s /= n0) (do
store ancestorE s v
follow =<< childM s)
zv <- sizeM v
follow =<< case zv < 2*zw of
False-> return s
True-> do
cv <- childM v
store childE v s
return cv
-----------------------------------------------------------------------------
dfsDom :: Node -> Dom s ()
dfsDom i = do
_ <- go i
n0 <- zeroM
r <- rootM
store parentE r n0
where go i = do
n <- nextM
store dfnE i n
store sdnoE i n
store ndfsE n i
store labelE i i
ss <- succsM i
forM_ ss (\j-> do
s <- sdnoM j
case s==0 of
False-> return()
True-> do
store parentE j i
go j)
-----------------------------------------------------------------------------
initEnv :: Rooted -> ST s (Env s)
initEnv (r0,g0) = do
-- Graph renumbered to indices from 1 to |V|
let (g,rnmap) = renum 1 g0
pred = predG g -- reverse graph
root = rnmap IM.! r0 -- renamed root
n = IM.size g
ns = [0..n]
m = n+1
let bucket = IM.fromList
(zip ns (repeat mempty))
rna <- newI m
writes rna (fmap swap
(IM.toList rnmap))
doms <- newI m
sdno <- newI m
size <- newI m
parent <- newI m
ancestor <- newI m
child <- newI m
label <- newI m
ndfs <- newI m
dfn <- newI m
-- Initialize all arrays
forM_ [0..n] (doms.=0)
forM_ [0..n] (sdno.=0)
forM_ [1..n] (size.=1)
forM_ [0..n] (ancestor.=0)
forM_ [0..n] (child.=0)
(doms.=root) root
(size.=0) 0
(label.=0) 0
return (Env
{rnE = rna
,dfsE = 0
,zeroE = 0
,rootE = root
,labelE = label
,parentE = parent
,ancestorE = ancestor
,childE = child
,ndfsE = ndfs
,dfnE = dfn
,sdnoE = sdno
,sizeE = size
,succE = g
,predE = pred
,bucketE = bucket
,domE = doms})
fromEnv :: Dom s [(Node,Node)]
fromEnv = do
dom <- gets domE
rn <- gets rnE
-- r <- gets rootE
(_,n) <- st (getBounds dom)
forM [1..n] (\i-> do
j <- st(rn!:i)
d <- st(dom!:i)
k <- st(rn!:d)
return (j,k))
-----------------------------------------------------------------------------
zeroM :: Dom s Node
zeroM = gets zeroE
domM :: Node -> Dom s Node
domM = fetch domE
rootM :: Dom s Node
rootM = gets rootE
succsM :: Node -> Dom s [Node]
succsM i = gets (IS.toList . (! i) . succE)
predsM :: Node -> Dom s [Node]
predsM i = gets (IS.toList . (! i) . predE)
bucketM :: Node -> Dom s [Node]
bucketM i = gets (IS.toList . (! i) . bucketE)
sizeM :: Node -> Dom s Int
sizeM = fetch sizeE
sdnoM :: Node -> Dom s Int
sdnoM = fetch sdnoE
-- dfnM :: Node -> Dom s Int
-- dfnM = fetch dfnE
ndfsM :: Int -> Dom s Node
ndfsM = fetch ndfsE
childM :: Node -> Dom s Node
childM = fetch childE
ancestorM :: Node -> Dom s Node
ancestorM = fetch ancestorE
parentM :: Node -> Dom s Node
parentM = fetch parentE
labelM :: Node -> Dom s Node
labelM = fetch labelE
nextM :: Dom s Int
nextM = do
n <- gets dfsE
let n' = n+1
modify(\e->e{dfsE=n'})
return n'
-----------------------------------------------------------------------------
type A = STUArray
type Arr s a = A s Int a
infixl 9 !:
infixr 2 .=
-- | arr .= x idx => write x to index
(.=) :: (MArray (A s) a (ST s))
=> Arr s a -> a -> Int -> ST s ()
(v .= x) i = unsafeWrite v i x
(!:) :: (MArray (A s) a (ST s))
=> A s Int a -> Int -> ST s a
a !: i = do
o <- unsafeRead a i
return $! o
new :: (MArray (A s) a (ST s))
=> Int -> ST s (Arr s a)
new n = unsafeNewArray_ (0,n-1)
newI :: Int -> ST s (Arr s Int)
newI = new
-- newD :: Int -> ST s (Arr s Double)
-- newD = new
-- dump :: (MArray (A s) a (ST s)) => Arr s a -> ST s [a]
-- dump a = do
-- (m,n) <- getBounds a
-- forM [m..n] (\i -> a!:i)
writes :: (MArray (A s) a (ST s))
=> Arr s a -> [(Int,a)] -> ST s ()
writes a xs = forM_ xs (\(i,x) -> (a.=x) i)
-- arr :: (MArray (A s) a (ST s)) => [a] -> ST s (Arr s a)
-- arr xs = do
-- let n = length xs
-- a <- new n
-- go a n 0 xs
-- return a
-- where go _ _ _ [] = return ()
-- go a n i (x:xs)
-- | i <= n = (a.=x) i >> go a n (i+1) xs
-- | otherwise = return ()
-----------------------------------------------------------------------------
(!) :: Monoid a => IntMap a -> Int -> a
(!) g n = maybe mempty id (IM.lookup n g)
fromAdj :: [(Node, [Node])] -> Graph
fromAdj = IM.fromList . fmap (second IS.fromList)
fromEdges :: [Edge] -> Graph
fromEdges = collectI IS.union fst (IS.singleton . snd)
toAdj :: Graph -> [(Node, [Node])]
toAdj = fmap (second IS.toList) . IM.toList
toEdges :: Graph -> [Edge]
toEdges = concatMap (uncurry (fmap . (,))) . toAdj
predG :: Graph -> Graph
predG g = IM.unionWith IS.union (go g) g0
where g0 = fmap (const mempty) g
go = flip IM.foldrWithKey mempty (\i a m ->
foldl' (\m p -> IM.insertWith mappend p
(IS.singleton i) m)
m
(IS.toList a))
-- predG :: Graph -> Graph
-- predG g = IM.unionWith IS.union (go g) g0
-- where g0 = fmap (const mempty) g
-- f :: IntMap IntSet -> Int -> IntSet -> IntMap IntSet
-- f m i a = foldl' (\m p -> IM.insertWith mappend p
-- (IS.singleton i) m)
-- m
-- (IS.toList a)
-- go :: IntMap IntSet -> IntMap IntSet
-- go = flip IM.foldlWithKey' mempty f
pruneReach :: Rooted -> Rooted
pruneReach (r,g) = (r,g2)
where is = reachable
(maybe mempty id
. flip IM.lookup g) $ r
g2 = IM.fromList
. fmap (second (IS.filter (`IS.member`is)))
. filter ((`IS.member`is) . fst)
. IM.toList $ g
tip :: Tree a -> (a, [Tree a])
tip (Node a ts) = (a, ts)
parents :: Tree a -> [(a, a)]
parents (Node i xs) = p i xs
++ concatMap parents xs
where p i = fmap (flip (,) i . rootLabel)
ancestors :: Tree a -> [(a, [a])]
ancestors = go []
where go acc (Node i xs)
= let acc' = i:acc
in p acc' xs ++ concatMap (go acc') xs
p is = fmap (flip (,) is . rootLabel)
asGraph :: Tree Node -> Rooted
asGraph t@(Node a _) = let g = go t in (a, fromAdj g)
where go (Node a ts) = let as = (fst . unzip . fmap tip) ts
in (a, as) : concatMap go ts
asTree :: Rooted -> Tree Node
asTree (r,g) = let go a = Node a (fmap go ((IS.toList . f) a))
f = (g !)
in go r
reachable :: (Node -> NodeSet) -> (Node -> NodeSet)
reachable f a = go (IS.singleton a) a
where go seen a = let s = f a
as = IS.toList (s `IS.difference` seen)
in foldl' go (s `IS.union` seen) as
collectI :: (c -> c -> c)
-> (a -> Int) -> (a -> c) -> [a] -> IntMap c
collectI (<>) f g
= foldl' (\m a -> IM.insertWith (<>)
(f a)
(g a) m) mempty
-- collect :: (Ord b) => (c -> c -> c)
-- -> (a -> b) -> (a -> c) -> [a] -> Map b c
-- collect (<>) f g
-- = foldl' (\m a -> SM.insertWith (<>)
-- (f a)
-- (g a) m) mempty
-- | renum n g: Rename all nodes
--
-- Gives nodes sequential names starting at n.
-- Returns the new graph and a mapping.
-- (renamed, old -> new)
renum :: Int -> Graph -> (Graph, NodeMap Node)
renum from = (\(_,m,g)->(g,m))
. IM.foldrWithKey
(\i ss (!n,!env,!new)->
let (j,n2,env2) = go n env i
(n3,env3,ss2) = IS.fold
(\k (!n,!env,!new)->
case go n env k of
(l,n2,env2)-> (n2,env2,l `IS.insert` new))
(n2,env2,mempty) ss
new2 = IM.insertWith IS.union j ss2 new
in (n3,env3,new2)) (from,mempty,mempty)
where go :: Int
-> NodeMap Node
-> Node
-> (Node,Int,NodeMap Node)
go !n !env i =
case IM.lookup i env of
Just j -> (j,n,env)
Nothing -> (n,n+1,IM.insert i n env)
-----------------------------------------------------------------------------
-- Nothing better than reinvinting the state monad.
newtype S z s a = S {unS :: forall o. (a -> s -> ST z o) -> s -> ST z o}
instance Functor (S z s) where
fmap f (S g) = S (\k -> g (k . f))
instance Monad (S z s) where
return = pure
S g >>= f = S (\k -> g (\a -> unS (f a) k))
instance Applicative (S z s) where
pure a = S (\k -> k a)
(<*>) = ap
-- get :: S z s s
-- get = S (\k s -> k s s)
gets :: (s -> a) -> S z s a
gets f = S (\k s -> k (f s) s)
-- set :: s -> S z s ()
-- set s = S (\k _ -> k () s)
modify :: (s -> s) -> S z s ()
modify f = S (\k -> k () . f)
-- runS :: S z s a -> s -> ST z (a, s)
-- runS (S g) = g (\a s -> return (a,s))
evalS :: S z s a -> s -> ST z a
evalS (S g) = g ((return .) . const)
-- execS :: S z s a -> s -> ST z s
-- execS (S g) = g ((return .) . flip const)
st :: ST z a -> S z s a
st m = S (\k s-> do
a <- m
k a s)
store :: (MArray (A z) a (ST z))
=> (s -> Arr z a) -> Int -> a -> S z s ()
store f i x = do
a <- gets f
st ((a.=x) i)
fetch :: (MArray (A z) a (ST z))
=> (s -> Arr z a) -> Int -> S z s a
fetch f i = do
a <- gets f
st (a!:i)
-- Redefine Data.Bifunctor.second for GHC 7 compatibility
second :: (b -> c) -> (a, b) -> (a, c)
second f (a, b) = (a, f b)