{-# LANGUAGE DeriveDataTypeable #-}

{- |

Module : ./Common/Lib/Rel.hs

Description : Relations, based on maps

Copyright : (c) Uni Bremen 2003-2005

License : GPLv2 or higher, see LICENSE.txt

Maintainer : Christian.Maeder@dfki.de

Stability : provisional

Portability : portable

supply a simple data type for (precedence or subsort) relations. A

relation is conceptually a set of (ordered) pairs,

but the hidden implementation is based on a map of sets.

An alternative view is that of a directed Graph possibly

with isolated nodes.

'Rel' is a directed graph with elements (Ord a) as (uniquely labelled) nodes

and (unlabelled) edges (with a multiplicity of at most one).

Usage: start with an 'empty' relation, 'insert' edges, and test for

an edge 'member' (before or after calling 'transClosure').

It is possible to insert self edges or bigger cycles. But rather than

inserting self edges and element can be mapped to the empty set.

Checking for a 'path' corresponds to checking for a member in the

transitive (possibly non-reflexive) closure. A further 'insert', however,

may destroy the closedness property of a relation.

-}

module Common.Lib.Rel

( Rel, empty, nullKeys, rmNullSets

, insertPair, insertDiffPair, insertKeyOrPair

, member, toMap, map

, noPairs, insertKey, deleteKey, memberKey, keysSet

, fromKeysSet

, reflexive

, getCycles

, union, intersection, isSubrelOf, difference, path

, delete, succs, predecessors, irreflex, sccOfClosure

, transClosure, fromList, toList, toPrecMap

, intransKernel, mostRight, restrict, delSet

, toSet, fromSet, topSort, depSort, nodes, collaps

, transpose, transReduce

, fromMap, locallyFiltered

, flatSet, partSet, partList, leqClasses

) where

import Prelude hiding (map, null)

import Data.Data

import qualified Data.Map as Map

import qualified Data.Set as Set

import qualified Data.List as List

import qualified Common.Lib.MapSet as MapSet

-- | no invariant is ensured for relations!

newtype Rel a = Rel { toMap :: Map.Map a (Set.Set a) }

deriving (Eq, Ord, Typeable, Data)

instance Show a => Show (Rel a) where

show = show . toMap

instance (Ord a, Read a) => Read (Rel a) where

readsPrec d = List.map (\ (m, r) -> (fromMap m , r)) . readsPrec d

fromMap :: Map.Map a (Set.Set a) -> Rel a

fromMap = Rel

-- | the empty relation

empty :: Rel a

empty = Rel Map.empty

-- | test for 'empty'

nullKeys :: Rel a -> Bool

nullKeys (Rel m) = Map.null m

-- | keys of the relation

keysSet :: Rel a -> Set.Set a

keysSet = Map.keysSet . toMap

rmNullSets :: Ord a => Rel a -> Rel a

rmNullSets = Rel . MapSet.rmNullSets . toMap

-- | test for 'empty'

noPairs :: Ord a => Rel a -> Bool

noPairs = nullKeys . rmNullSets

-- | difference of two relations

difference :: Ord a => Rel a -> Rel a -> Rel a

difference (Rel m) = Rel . Map.differenceWith MapSet.setDifference m . toMap

-- | union of two relations

union :: Ord a => Rel a -> Rel a -> Rel a

union (Rel m) = Rel . Map.unionWith Set.union m . toMap

-- | intersection of two relations

intersection :: Ord a => Rel a -> Rel a -> Rel a

intersection (Rel m) = Rel . Map.intersectionWith Set.intersection m . toMap

-- | is the first relation a sub-relation of the second

isSubrelOf :: Ord a => Rel a -> Rel a -> Bool

isSubrelOf (Rel m) = Map.isSubmapOfBy Set.isSubsetOf m . toMap

-- | insert an ordered pair

insertPair :: Ord a => a -> a -> Rel a -> Rel a

insertPair a b = Rel . MapSet.setInsert a b . toMap

-- | insert a pair only if both sides are different

insertDiffPair :: Ord a => a -> a -> Rel a -> Rel a

insertDiffPair a b = if a == b then id else insertPair a b

-- | insert a pair only if both sides are different

insertKeyOrPair :: Ord a => a -> a -> Rel a -> Rel a

insertKeyOrPair a b = if a == b then insertKey a else insertPair a b

-- | insert an unrelated node

insertKey :: Ord a => a -> Rel a -> Rel a

insertKey k r@(Rel m) = if Map.member k m then r else

Rel $ Map.insert k Set.empty m

-- | delete an ordered pair

delete :: Ord a => a -> a -> Rel a -> Rel a

delete a b (Rel m) =

let t = Set.delete b $ MapSet.setLookup a m in

Rel $ if Set.null t then Map.delete a m else Map.insert a t m

-- | delete a node and all its relations

deleteKey :: Ord a => a -> Rel a -> Rel a

deleteKey k = Rel . Map.delete k . toMap

-- | test for an (previously inserted) ordered pair

member :: Ord a => a -> a -> Rel a -> Bool

member a b r = Set.member b $ succs r a

memberKey :: Ord a => a -> Rel a -> Bool

memberKey k = Map.member k . toMap

-- | get direct successors

succs :: Ord a => Rel a -> a -> Set.Set a

succs (Rel m) a = Map.findWithDefault Set.empty a m

-- | get all transitive successors

reachable :: Ord a => Rel a -> a -> Set.Set a

reachable r a = Set.fold reach Set.empty $ succs r a where

reach e s = if Set.member e s then s

else Set.fold reach (Set.insert e s) $ succs r e

-- | predecessors of one node in the given set of a nodes

preds :: Ord a => Rel a -> a -> Set.Set a -> Set.Set a

preds r a = Set.filter ( \ s -> member s a r)

-- | get direct predecessors

predecessors :: Ord a => Rel a -> a -> Set.Set a

predecessors (Rel m) a = Map.keysSet $ Map.filter (Set.member a) m

-- | test for 'member' or transitive membership (non-empty path)

path :: Ord a => a -> a -> Rel a -> Bool

path a b r = Set.member b $ reachable r a

-- | compute transitive closure (make all transitive members direct members)

transClosure :: Ord a => Rel a -> Rel a

transClosure r@(Rel m) = Rel $ Map.mapWithKey ( \ k _ -> reachable r k) m

-- | get transposed relation (losing unrelated keys)

transpose :: Ord a => Rel a -> Rel a

transpose = Rel . MapSet.toMap . MapSet.transpose . MapSet.fromMap . toMap

-- | make relation irreflexive

irreflex :: Ord a => Rel a -> Rel a

irreflex = Rel . Map.mapWithKey Set.delete . toMap

-- | add all keys as reflexive elements

reflexive :: Ord a => Rel a -> Rel a

reflexive = Rel . Map.mapWithKey Set.insert . toMap

-- | get entries that contain the key as element

getCycles :: Ord a => Rel a -> Rel a

getCycles = Rel . Map.filterWithKey Set.member . toMap

-- | compute strongly connected components for a transitively closed relation

sccOfClosure :: Ord a => Rel a -> [Set.Set a]

sccOfClosure r@(Rel m) =

if Map.null m then []

else let ((k, v), p) = Map.deleteFindMin m in

if Set.member k v then -- has a cycle

let c = preds r k v in -- get the cycle

c : sccOfClosure (delSet c r)

else sccOfClosure (Rel p)

{- | restrict strongly connected components to its minimal representative

(input sets must be non-null). Direct cycles may remain. -}

collaps :: Ord a => [Set.Set a] -> Rel a -> Rel a

collaps = delSet . Set.unions . List.map Set.deleteMin

{- | transitive reduction (minimal relation with the same transitive closure)

of a transitively closed DAG (i.e. without cycles)! -}

transReduce :: Ord a => Rel a -> Rel a

transReduce (Rel m) = Rel

-- keep all (i, j) in rel for which no c with (i, c) and (c, j) in rel

$ Map.mapWithKey ( \ i s -> let d = MapSet.setToMap $ Set.delete i s in

Set.filter ( \ j ->

Map.null $ Map.filter (Set.member j)

$ Map.intersection m $ Map.delete j d) s) m

-- | convert a list of ordered pairs to a relation

fromList :: Ord a => [(a, a)] -> Rel a

fromList = foldr (uncurry insertPair) empty

-- | convert a relation to a list of ordered pairs (this loses isolated keys!)

toList :: Rel a -> [(a, a)]

toList (Rel m) = concatMap (\ (a , bs) -> List.map ( \ b -> (a, b) )

(Set.toList bs)) $ Map.toList m

-- | map the values of a relation

map :: (Ord a, Ord b) => (a -> b) -> Rel a -> Rel b

map f = Rel . Map.mapKeysWith Set.union f . Map.map (Set.map f) . toMap

-- | Restriction of a relation under a set

restrict :: Ord a => Rel a -> Set.Set a -> Rel a

restrict r s = delSet (nodes r Set.\\ s) r

-- | restrict to elements not in the input set

delSet :: Ord a => Set.Set a -> Rel a -> Rel a

delSet s (Rel m) = Rel $ Map.map (Set.\\ s) $ m Map.\\ MapSet.setToMap s

-- | convert a relation to a set of ordered pairs

toSet :: Ord a => Rel a -> Set.Set (a, a)

toSet = Set.fromDistinctAscList . toList

-- | convert a set of ordered pairs to a relation

fromSet :: Ord a => Set.Set (a, a) -> Rel a

fromSet = fromAscList . Set.toList

-- | convert a plain node set to a relation

fromKeysSet :: Ord a => Set.Set a -> Rel a

fromKeysSet = Rel . Set.fold (`Map.insert` Set.empty) Map.empty

-- | convert a sorted list of ordered pairs to a relation

fromAscList :: Ord a => [(a, a)] -> Rel a

fromAscList = Rel . Map.fromDistinctAscList

. List.map ( \ l -> (fst (head l),

Set.fromDistinctAscList $ List.map snd l))

. List.groupBy ( \ (a, _) (b, _) -> a == b)

-- | all nodes of the edges

nodes :: Ord a => Rel a -> Set.Set a

nodes (Rel m) = Set.union (Map.keysSet m) $ MapSet.setElems m

{- | Construct a precedence map from a closed relation. Indices range

between 1 and the second value that is output. -}

toPrecMap :: Ord a => Rel a -> (Map.Map a Int, Int)

toPrecMap = foldl ( \ (m1, c) s -> let n = c + 1 in

(Set.fold (`Map.insert` n) m1 s, n))

(Map.empty, 0) . topSort

topSortDAG :: Ord a => Rel a -> [Set.Set a]

topSortDAG r@(Rel m) = if Map.null m then [] else

let es = MapSet.setElems m

ml = Map.keysSet m Set.\\ es -- most left

Rel m2 = delSet ml r

rs = es Set.\\ Map.keysSet m2 -- re-insert loose ends

in ml : topSortDAG (Rel $ Set.fold (`Map.insert` Set.empty) m2 rs)

-- | topologically sort a closed relation (ignore isolated cycles)

topSort :: Ord a => Rel a -> [Set.Set a]

topSort r = let cs = sccOfClosure r in

List.map (expandCycle cs) $ topSortDAG $ irreflex $ collaps cs r

-- | find the cycle and add it to the result set

expandCycle :: Ord a => [Set.Set a] -> Set.Set a -> Set.Set a

expandCycle cs s = case cs of

[] -> s

c : r ->

if Set.null $ Set.intersection c s then expandCycle r s else Set.union c s

-- dependency sort

depSort :: Ord a => Rel a -> [Set.Set a]

depSort r = let cs = sccOfClosure r in

List.concatMap (List.map (depCycle cs) . Set.toList)

$ topSortDAG $ irreflex $ collaps cs r

depCycle :: Ord a => [Set.Set a] -> a -> Set.Set a

depCycle cs a = case cs of

[] -> Set.singleton a

c : r ->

if Set.member a c then c else depCycle r a

-- | gets the most right elements of a relation,

mostRightOfCollapsed :: Ord a => Rel a -> Set.Set a

mostRightOfCollapsed r@(Rel m) =

Set.difference (nodes r) . Map.keysSet $ Map.filterWithKey

(\ i s -> not (Set.null s) && s /= Set.singleton i) m

{- |

find s such that x in s => forall y . yRx or not yRx and not xRy

* precondition: (transClosure r == r)

* strongly connected components (cycles) are treated as a compound node

-}

mostRight :: Ord a => Rel a -> Set.Set a

mostRight r = let

cs = sccOfClosure r

in expandCycle cs (mostRightOfCollapsed $ collaps cs r)

{- |

intransitive kernel of a reflexive and transitive closure

* precondition: (transClosure r == r)

* cycles are uniquely represented (according to Ord)

-}

intransKernel :: Ord a => Rel a -> Rel a

intransKernel r =

let cs = sccOfClosure r

in foldr addCycle (transReduce $ collaps cs r) cs

-- add a cycle given by a set in the collapsed node

addCycle :: Ord a => Set.Set a -> Rel a -> Rel a

addCycle c r = if Set.null c then error "Common.Lib.Rel.addCycle" else

let (a, b) = Set.deleteFindMin c

(m, d) = Set.deleteFindMax c

in insertPair m a $ foldr (uncurry insertPair) (delete a a r) $

zip (Set.toList d) (Set.toList b)

{- | calculates if two given elements have a common left element

* if one of the arguments is not present False is returned

-}

haveCommonLeftElem :: Ord a => a -> a -> Rel a -> Bool

haveCommonLeftElem t1 t2 =

Map.fold (\ e -> (|| Set.member t1 e && Set.member t2 e)) False . toMap

{- | partitions a set into a list of disjoint non-empty subsets

determined by the given function as equivalence classes -}

partSet :: Ord a => (a -> a -> Bool) -> Set.Set a -> [Set.Set a]

partSet f = List.map Set.fromList . leqClasses f

{- | partitions a list into a list of disjoint non-empty lists

determined by the given function as equivalence classes -}

partList :: (a -> a -> Bool) -> [a] -> [[a]]

partList f l = case l of

[] -> []

x : r -> let

(ds, es) = List.partition (not . any (f x)) $ partList f r

in (x : concat es) : ds

-- | Divide a Set (List) into equivalence classes w.r.t. eq

leqClasses :: Ord a => (a -> a -> Bool) -> Set.Set a -> [[a]]

leqClasses f = partList f . Set.toList

{- | flattens a list of non-empty sets and uses the minimal element of

each set to represent the set -}

flatSet :: Ord a => [Set.Set a] -> Set.Set a

flatSet = Set.fromList . List.map (\ s -> if Set.null s

then error "Common.Lib.Rel.flatSet"

else Set.findMin s)

{- | checks if a given relation is locally filtered

* precondition: the relation must already be closed by transitive closure

-}

locallyFiltered :: Ord a => Rel a -> Bool

locallyFiltered rel = check . flatSet . partSet iso $ mostRight rel

where iso x y = member x y rel && member y x rel

check s = Set.null s ||

Set.fold (\ y ->

(&& not (haveCommonLeftElem x y rel))) True s'

&& check s'

where (x, s') = Set.deleteFindMin s