{- |
This module provides a solver for exact set cover problems.
<http://en.wikipedia.org/wiki/Exact_cover>
-}
module Math.SetCover.Exact (
   Assign(..), assign,
   bitVectorFromSetAssigns, intSetFromSetAssigns,
   partitions, search, step,
   State(..), initState, updateState,
   Set(..),
   Tree(..), decisionTree, completeTree,
   Choose(..),
   ) where

import qualified Math.SetCover.BitMap as BitMap
import qualified Math.SetCover.BitSet as BitSet
import qualified Math.SetCover.Bit as Bit
import Math.SetCover.EnumMap (constMap)

import Control.Applicative ((<$>), (<$))

import qualified Data.IntSet as IntSet
import qualified Data.Map as Map
import qualified Data.Set as Set
import qualified Data.List.Match as Match
import qualified Data.List as List
import qualified Data.Foldable as Fold
import Data.Function.HT (compose2)
import Data.Maybe.HT (toMaybe)
import Data.Tuple.HT (mapFst, mapSnd)
import Data.Bits (setBit)

import Prelude hiding (null)


{- |
This class provides all operations needed for the set cover algorithm.
It allows to use the same algorithm
both for @containers@' 'Set' and for sets represented by bit vectors.
-}
class Set set where
   null :: set -> Bool
   disjoint :: set -> set -> Bool
   unions :: [set] -> set
   difference :: set -> set -> set
   {- |
   @minimize free assigns@ finds a set element @x@ from @free@
   that is contained in the least number of sets in @assigns@.
   Then it returns the assigns where @x@ is contained in the associated set.
   This formulation allows us not to name @x@
   and thus we do not need a second type variable for @set@ elements
   and no type family from @set@ to its element type.

   Unchecked preconditions:
   @free@ must be a superset of all sets in the assign list.
   @free@ must be non-empty.
   The @assigns@ list may be empty.
   The output of assigns must be a subsequence of the input assigns,
   that is, it must be a subset of the input and it must be in the same order.
   This requirement was originally needed by 'minimize' for 'Map.Map',
   but currently it is not utilized anywhere.
   -}
   minimize :: set -> [Assign label set] -> [Assign label set]

class Set set => Choose set where
   {-
   Compute a set containing one element
   that is contained in a minimal number of assignment sets.
   -}
   chooseMinimize :: set -> [Assign label set] -> (set, [Assign label set])

instance (Ord a) => Set (Set.Set a) where
   null :: Set a -> Bool
null = Set a -> Bool
forall a. Set a -> Bool
Set.null
   disjoint :: Set a -> Set a -> Bool
disjoint Set a
x Set a
y = Set a -> Bool
forall a. Set a -> Bool
Set.null (Set a -> Bool) -> Set a -> Bool
forall a b. (a -> b) -> a -> b
$ Set a -> Set a -> Set a
forall a. Ord a => Set a -> Set a -> Set a
Set.intersection Set a
x Set a
y
   unions :: [Set a] -> Set a
unions = [Set a] -> Set a
forall (f :: * -> *) a. (Foldable f, Ord a) => f (Set a) -> Set a
Set.unions
   difference :: Set a -> Set a -> Set a
difference = Set a -> Set a -> Set a
forall a. Ord a => Set a -> Set a -> Set a
Set.difference
   minimize :: forall label.
Set a -> [Assign label (Set a)] -> [Assign label (Set a)]
minimize Set a
free = ([Assign label (Set a)] -> [Assign label (Set a)] -> Ordering)
-> Map a [Assign label (Set a)] -> [Assign label (Set a)]
forall (t :: * -> *) a.
Foldable t =>
(a -> a -> Ordering) -> t a -> a
Fold.minimumBy [Assign label (Set a)] -> [Assign label (Set a)] -> Ordering
forall a b. [a] -> [b] -> Ordering
Match.compareLength (Map a [Assign label (Set a)] -> [Assign label (Set a)])
-> ([Assign label (Set a)] -> Map a [Assign label (Set a)])
-> [Assign label (Set a)]
-> [Assign label (Set a)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Set a -> [Assign label (Set a)] -> Map a [Assign label (Set a)]
forall k label.
Ord k =>
Set k -> [Assign label (Set k)] -> Map k [Assign label (Set k)]
histogramSet Set a
free

instance (Ord a) => Choose (Set.Set a) where
   chooseMinimize :: forall label.
Set a -> [Assign label (Set a)] -> (Set a, [Assign label (Set a)])
chooseMinimize Set a
free =
      (a -> Set a)
-> (a, [Assign label (Set a)]) -> (Set a, [Assign label (Set a)])
forall a c b. (a -> c) -> (a, b) -> (c, b)
mapFst a -> Set a
forall a. a -> Set a
Set.singleton ((a, [Assign label (Set a)]) -> (Set a, [Assign label (Set a)]))
-> ([Assign label (Set a)] -> (a, [Assign label (Set a)]))
-> [Assign label (Set a)]
-> (Set a, [Assign label (Set a)])
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
      ((a, [Assign label (Set a)])
 -> (a, [Assign label (Set a)]) -> Ordering)
-> [(a, [Assign label (Set a)])] -> (a, [Assign label (Set a)])
forall (t :: * -> *) a.
Foldable t =>
(a -> a -> Ordering) -> t a -> a
List.minimumBy (([Assign label (Set a)] -> [Assign label (Set a)] -> Ordering)
-> ((a, [Assign label (Set a)]) -> [Assign label (Set a)])
-> (a, [Assign label (Set a)])
-> (a, [Assign label (Set a)])
-> Ordering
forall b c a. (b -> b -> c) -> (a -> b) -> a -> a -> c
compose2 [Assign label (Set a)] -> [Assign label (Set a)] -> Ordering
forall a b. [a] -> [b] -> Ordering
Match.compareLength (a, [Assign label (Set a)]) -> [Assign label (Set a)]
forall a b. (a, b) -> b
snd) ([(a, [Assign label (Set a)])] -> (a, [Assign label (Set a)]))
-> ([Assign label (Set a)] -> [(a, [Assign label (Set a)])])
-> [Assign label (Set a)]
-> (a, [Assign label (Set a)])
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
      Map a [Assign label (Set a)] -> [(a, [Assign label (Set a)])]
forall k a. Map k a -> [(k, a)]
Map.toList (Map a [Assign label (Set a)] -> [(a, [Assign label (Set a)])])
-> ([Assign label (Set a)] -> Map a [Assign label (Set a)])
-> [Assign label (Set a)]
-> [(a, [Assign label (Set a)])]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Set a -> [Assign label (Set a)] -> Map a [Assign label (Set a)]
forall k label.
Ord k =>
Set k -> [Assign label (Set k)] -> Map k [Assign label (Set k)]
histogramSet Set a
free

histogramSet ::
   Ord k =>
   Set.Set k ->
   [Assign label (Set.Set k)] ->
   Map.Map k [Assign label (Set.Set k)]
histogramSet :: forall k label.
Ord k =>
Set k -> [Assign label (Set k)] -> Map k [Assign label (Set k)]
histogramSet Set k
free =
   (Map k [Assign label (Set k)]
 -> Map k [Assign label (Set k)] -> Map k [Assign label (Set k)])
-> Map k [Assign label (Set k)]
-> [Map k [Assign label (Set k)]]
-> Map k [Assign label (Set k)]
forall a b. (a -> b -> b) -> b -> [a] -> b
forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (([Assign label (Set k)]
 -> [Assign label (Set k)] -> [Assign label (Set k)])
-> Map k [Assign label (Set k)]
-> Map k [Assign label (Set k)]
-> Map k [Assign label (Set k)]
forall k a. Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
Map.unionWith [Assign label (Set k)]
-> [Assign label (Set k)] -> [Assign label (Set k)]
forall a. [a] -> [a] -> [a]
(++)) ([Assign label (Set k)] -> Set k -> Map k [Assign label (Set k)]
forall a b. Ord a => b -> Set a -> Map a b
constMap [] Set k
free) ([Map k [Assign label (Set k)]] -> Map k [Assign label (Set k)])
-> ([Assign label (Set k)] -> [Map k [Assign label (Set k)]])
-> [Assign label (Set k)]
-> Map k [Assign label (Set k)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   (Assign label (Set k) -> Map k [Assign label (Set k)])
-> [Assign label (Set k)] -> [Map k [Assign label (Set k)]]
forall a b. (a -> b) -> [a] -> [b]
map (\Assign label (Set k)
a -> [Assign label (Set k)] -> Set k -> Map k [Assign label (Set k)]
forall a b. Ord a => b -> Set a -> Map a b
constMap [Assign label (Set k)
a] (Set k -> Map k [Assign label (Set k)])
-> Set k -> Map k [Assign label (Set k)]
forall a b. (a -> b) -> a -> b
$ Assign label (Set k) -> Set k
forall label set. Assign label set -> set
labeledSet Assign label (Set k)
a)


{- |
This instance supports Maps of Sets.
This way you can structure your sets hierarchically.
You may also use it to combine several low-level bitsets.
A Map must not contain empty subsets.
-}
instance (Ord k, Set set) => Set (Map.Map k set) where
   null :: Map k set -> Bool
null = Map k set -> Bool
forall k a. Map k a -> Bool
Map.null
   disjoint :: Map k set -> Map k set -> Bool
disjoint Map k set
x Map k set
y = Map k Bool -> Bool
forall (t :: * -> *). Foldable t => t Bool -> Bool
Fold.and (Map k Bool -> Bool) -> Map k Bool -> Bool
forall a b. (a -> b) -> a -> b
$ (set -> set -> Bool) -> Map k set -> Map k set -> Map k Bool
forall k a b c.
Ord k =>
(a -> b -> c) -> Map k a -> Map k b -> Map k c
Map.intersectionWith set -> set -> Bool
forall set. Set set => set -> set -> Bool
disjoint Map k set
x Map k set
y
   unions :: [Map k set] -> Map k set
unions =
      ([set] -> set) -> Map k [set] -> Map k set
forall a b. (a -> b) -> Map k a -> Map k b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap [set] -> set
forall set. Set set => [set] -> set
unions (Map k [set] -> Map k set)
-> ([Map k set] -> Map k [set]) -> [Map k set] -> Map k set
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Map k [set] -> Map k [set] -> Map k [set])
-> Map k [set] -> [Map k [set]] -> Map k [set]
forall a b. (a -> b -> b) -> b -> [a] -> b
forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (([set] -> [set] -> [set])
-> Map k [set] -> Map k [set] -> Map k [set]
forall k a. Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
Map.unionWith [set] -> [set] -> [set]
forall a. [a] -> [a] -> [a]
(++)) Map k [set]
forall k a. Map k a
Map.empty ([Map k [set]] -> Map k [set])
-> ([Map k set] -> [Map k [set]]) -> [Map k set] -> Map k [set]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Map k set -> Map k [set]) -> [Map k set] -> [Map k [set]]
forall a b. (a -> b) -> [a] -> [b]
map ((set -> [set]) -> Map k set -> Map k [set]
forall a b. (a -> b) -> Map k a -> Map k b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (set -> [set] -> [set]
forall a. a -> [a] -> [a]
:[]))
   difference :: Map k set -> Map k set -> Map k set
difference =
      (set -> set -> Maybe set) -> Map k set -> Map k set -> Map k set
forall k a b.
Ord k =>
(a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a
Map.differenceWith
         (\set
x set
y -> let z :: set
z = set -> set -> set
forall set. Set set => set -> set -> set
difference set
x set
y in Bool -> set -> Maybe set
forall a. Bool -> a -> Maybe a
toMaybe (Bool -> Bool
not (Bool -> Bool) -> Bool -> Bool
forall a b. (a -> b) -> a -> b
$ set -> Bool
forall set. Set set => set -> Bool
null set
z) set
z)
   minimize :: forall label.
Map k set
-> [Assign label (Map k set)] -> [Assign label (Map k set)]
minimize Map k set
free =
      (Assign (Assign label (Map k set)) set -> Assign label (Map k set))
-> [Assign (Assign label (Map k set)) set]
-> [Assign label (Map k set)]
forall a b. (a -> b) -> [a] -> [b]
map Assign (Assign label (Map k set)) set -> Assign label (Map k set)
forall label set. Assign label set -> label
label ([Assign (Assign label (Map k set)) set]
 -> [Assign label (Map k set)])
-> ([Assign label (Map k set)]
    -> [Assign (Assign label (Map k set)) set])
-> [Assign label (Map k set)]
-> [Assign label (Map k set)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ([Assign (Assign label (Map k set)) set]
 -> [Assign (Assign label (Map k set)) set] -> Ordering)
-> Map k [Assign (Assign label (Map k set)) set]
-> [Assign (Assign label (Map k set)) set]
forall (t :: * -> *) a.
Foldable t =>
(a -> a -> Ordering) -> t a -> a
Fold.minimumBy [Assign (Assign label (Map k set)) set]
-> [Assign (Assign label (Map k set)) set] -> Ordering
forall a b. [a] -> [b] -> Ordering
Match.compareLength (Map k [Assign (Assign label (Map k set)) set]
 -> [Assign (Assign label (Map k set)) set])
-> ([Assign label (Map k set)]
    -> Map k [Assign (Assign label (Map k set)) set])
-> [Assign label (Map k set)]
-> [Assign (Assign label (Map k set)) set]
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
      (set
 -> [Assign (Assign label (Map k set)) set]
 -> [Assign (Assign label (Map k set)) set])
-> Map k set
-> Map k [Assign (Assign label (Map k set)) set]
-> Map k [Assign (Assign label (Map k set)) set]
forall k a b c.
Ord k =>
(a -> b -> c) -> Map k a -> Map k b -> Map k c
Map.intersectionWith set
-> [Assign (Assign label (Map k set)) set]
-> [Assign (Assign label (Map k set)) set]
forall label. set -> [Assign label set] -> [Assign label set]
forall set label.
Set set =>
set -> [Assign label set] -> [Assign label set]
minimize Map k set
free (Map k [Assign (Assign label (Map k set)) set]
 -> Map k [Assign (Assign label (Map k set)) set])
-> ([Assign label (Map k set)]
    -> Map k [Assign (Assign label (Map k set)) set])
-> [Assign label (Map k set)]
-> Map k [Assign (Assign label (Map k set)) set]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map k set
-> [Assign label (Map k set)]
-> Map k [Assign (Assign label (Map k set)) set]
forall k set label.
(Ord k, Set set) =>
Map k set
-> [Assign label (Map k set)]
-> Map k [Assign (Assign label (Map k set)) set]
histogramMap Map k set
free

instance (Ord k, Choose set) => Choose (Map.Map k set) where
   chooseMinimize :: forall label.
Map k set
-> [Assign label (Map k set)]
-> (Map k set, [Assign label (Map k set)])
chooseMinimize Map k set
free =
      (\(k
k,(set
minSet,[Assign (Assign label (Map k set)) set]
asns)) -> (k -> set -> Map k set
forall k a. k -> a -> Map k a
Map.singleton k
k set
minSet, (Assign (Assign label (Map k set)) set -> Assign label (Map k set))
-> [Assign (Assign label (Map k set)) set]
-> [Assign label (Map k set)]
forall a b. (a -> b) -> [a] -> [b]
map Assign (Assign label (Map k set)) set -> Assign label (Map k set)
forall label set. Assign label set -> label
label [Assign (Assign label (Map k set)) set]
asns)) ((k, (set, [Assign (Assign label (Map k set)) set]))
 -> (Map k set, [Assign label (Map k set)]))
-> ([Assign label (Map k set)]
    -> (k, (set, [Assign (Assign label (Map k set)) set])))
-> [Assign label (Map k set)]
-> (Map k set, [Assign label (Map k set)])
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
      ((k, (set, [Assign (Assign label (Map k set)) set]))
 -> (k, (set, [Assign (Assign label (Map k set)) set])) -> Ordering)
-> [(k, (set, [Assign (Assign label (Map k set)) set]))]
-> (k, (set, [Assign (Assign label (Map k set)) set]))
forall (t :: * -> *) a.
Foldable t =>
(a -> a -> Ordering) -> t a -> a
List.minimumBy (([Assign (Assign label (Map k set)) set]
 -> [Assign (Assign label (Map k set)) set] -> Ordering)
-> ((k, (set, [Assign (Assign label (Map k set)) set]))
    -> [Assign (Assign label (Map k set)) set])
-> (k, (set, [Assign (Assign label (Map k set)) set]))
-> (k, (set, [Assign (Assign label (Map k set)) set]))
-> Ordering
forall b c a. (b -> b -> c) -> (a -> b) -> a -> a -> c
compose2 [Assign (Assign label (Map k set)) set]
-> [Assign (Assign label (Map k set)) set] -> Ordering
forall a b. [a] -> [b] -> Ordering
Match.compareLength ((set, [Assign (Assign label (Map k set)) set])
-> [Assign (Assign label (Map k set)) set]
forall a b. (a, b) -> b
snd((set, [Assign (Assign label (Map k set)) set])
 -> [Assign (Assign label (Map k set)) set])
-> ((k, (set, [Assign (Assign label (Map k set)) set]))
    -> (set, [Assign (Assign label (Map k set)) set]))
-> (k, (set, [Assign (Assign label (Map k set)) set]))
-> [Assign (Assign label (Map k set)) set]
forall b c a. (b -> c) -> (a -> b) -> a -> c
.(k, (set, [Assign (Assign label (Map k set)) set]))
-> (set, [Assign (Assign label (Map k set)) set])
forall a b. (a, b) -> b
snd)) ([(k, (set, [Assign (Assign label (Map k set)) set]))]
 -> (k, (set, [Assign (Assign label (Map k set)) set])))
-> ([Assign label (Map k set)]
    -> [(k, (set, [Assign (Assign label (Map k set)) set]))])
-> [Assign label (Map k set)]
-> (k, (set, [Assign (Assign label (Map k set)) set]))
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map k (set, [Assign (Assign label (Map k set)) set])
-> [(k, (set, [Assign (Assign label (Map k set)) set]))]
forall k a. Map k a -> [(k, a)]
Map.toList (Map k (set, [Assign (Assign label (Map k set)) set])
 -> [(k, (set, [Assign (Assign label (Map k set)) set]))])
-> ([Assign label (Map k set)]
    -> Map k (set, [Assign (Assign label (Map k set)) set]))
-> [Assign label (Map k set)]
-> [(k, (set, [Assign (Assign label (Map k set)) set]))]
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
      (set
 -> [Assign (Assign label (Map k set)) set]
 -> (set, [Assign (Assign label (Map k set)) set]))
-> Map k set
-> Map k [Assign (Assign label (Map k set)) set]
-> Map k (set, [Assign (Assign label (Map k set)) set])
forall k a b c.
Ord k =>
(a -> b -> c) -> Map k a -> Map k b -> Map k c
Map.intersectionWith set
-> [Assign (Assign label (Map k set)) set]
-> (set, [Assign (Assign label (Map k set)) set])
forall label.
set -> [Assign label set] -> (set, [Assign label set])
forall set label.
Choose set =>
set -> [Assign label set] -> (set, [Assign label set])
chooseMinimize Map k set
free (Map k [Assign (Assign label (Map k set)) set]
 -> Map k (set, [Assign (Assign label (Map k set)) set]))
-> ([Assign label (Map k set)]
    -> Map k [Assign (Assign label (Map k set)) set])
-> [Assign label (Map k set)]
-> Map k (set, [Assign (Assign label (Map k set)) set])
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map k set
-> [Assign label (Map k set)]
-> Map k [Assign (Assign label (Map k set)) set]
forall k set label.
(Ord k, Set set) =>
Map k set
-> [Assign label (Map k set)]
-> Map k [Assign (Assign label (Map k set)) set]
histogramMap Map k set
free

histogramMap ::
   (Ord k, Set set) =>
   Map.Map k set ->
   [Assign label (Map.Map k set)] ->
   Map.Map k [Assign (Assign label (Map.Map k set)) set]
histogramMap :: forall k set label.
(Ord k, Set set) =>
Map k set
-> [Assign label (Map k set)]
-> Map k [Assign (Assign label (Map k set)) set]
histogramMap Map k set
free =
   (Map k [Assign (Assign label (Map k set)) set]
 -> Map k [Assign (Assign label (Map k set)) set]
 -> Map k [Assign (Assign label (Map k set)) set])
-> Map k [Assign (Assign label (Map k set)) set]
-> [Map k [Assign (Assign label (Map k set)) set]]
-> Map k [Assign (Assign label (Map k set)) set]
forall a b. (a -> b -> b) -> b -> [a] -> b
forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr (([Assign (Assign label (Map k set)) set]
 -> [Assign (Assign label (Map k set)) set]
 -> [Assign (Assign label (Map k set)) set])
-> Map k [Assign (Assign label (Map k set)) set]
-> Map k [Assign (Assign label (Map k set)) set]
-> Map k [Assign (Assign label (Map k set)) set]
forall k a. Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
Map.unionWith [Assign (Assign label (Map k set)) set]
-> [Assign (Assign label (Map k set)) set]
-> [Assign (Assign label (Map k set)) set]
forall a. [a] -> [a] -> [a]
(++)) ([] [Assign (Assign label (Map k set)) set]
-> Map k set -> Map k [Assign (Assign label (Map k set)) set]
forall a b. a -> Map k b -> Map k a
forall (f :: * -> *) a b. Functor f => a -> f b -> f a
<$ Map k set
free) ([Map k [Assign (Assign label (Map k set)) set]]
 -> Map k [Assign (Assign label (Map k set)) set])
-> ([Assign label (Map k set)]
    -> [Map k [Assign (Assign label (Map k set)) set]])
-> [Assign label (Map k set)]
-> Map k [Assign (Assign label (Map k set)) set]
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
   (Assign label (Map k set)
 -> Map k [Assign (Assign label (Map k set)) set])
-> [Assign label (Map k set)]
-> [Map k [Assign (Assign label (Map k set)) set]]
forall a b. (a -> b) -> [a] -> [b]
map (\Assign label (Map k set)
asn -> (Assign (Assign label (Map k set)) set
-> [Assign (Assign label (Map k set)) set]
-> [Assign (Assign label (Map k set)) set]
forall a. a -> [a] -> [a]
:[]) (Assign (Assign label (Map k set)) set
 -> [Assign (Assign label (Map k set)) set])
-> (set -> Assign (Assign label (Map k set)) set)
-> set
-> [Assign (Assign label (Map k set)) set]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Assign label (Map k set)
-> set -> Assign (Assign label (Map k set)) set
forall label set. label -> set -> Assign label set
assign Assign label (Map k set)
asn (set -> [Assign (Assign label (Map k set)) set])
-> Map k set -> Map k [Assign (Assign label (Map k set)) set]
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Assign label (Map k set) -> Map k set
forall label set. Assign label set -> set
labeledSet Assign label (Map k set)
asn)


instance (Bit.C a) => Set (BitSet.Set a) where
   null :: Set a -> Bool
null = Set a -> Bool
forall a. C a => Set a -> Bool
BitSet.null
   disjoint :: Set a -> Set a -> Bool
disjoint = Set a -> Set a -> Bool
forall a. C a => Set a -> Set a -> Bool
BitSet.disjoint
   unions :: [Set a] -> Set a
unions = [Set a] -> Set a
forall m. Monoid m => [m] -> m
forall (t :: * -> *) m. (Foldable t, Monoid m) => t m -> m
Fold.fold
   difference :: Set a -> Set a -> Set a
difference = Set a -> Set a -> Set a
forall a. C a => Set a -> Set a -> Set a
BitSet.difference
   minimize :: forall label.
Set a -> [Assign label (Set a)] -> [Assign label (Set a)]
minimize Set a
free = (Set a, [Assign label (Set a)]) -> [Assign label (Set a)]
forall a b. (a, b) -> b
snd ((Set a, [Assign label (Set a)]) -> [Assign label (Set a)])
-> ([Assign label (Set a)] -> (Set a, [Assign label (Set a)]))
-> [Assign label (Set a)]
-> [Assign label (Set a)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Set a -> [Assign label (Set a)] -> (Set a, [Assign label (Set a)])
forall label.
Set a -> [Assign label (Set a)] -> (Set a, [Assign label (Set a)])
forall set label.
Choose set =>
set -> [Assign label set] -> (set, [Assign label set])
chooseMinimize Set a
free

instance (Bit.C a) => Choose (BitSet.Set a) where
   chooseMinimize :: forall label.
Set a -> [Assign label (Set a)] -> (Set a, [Assign label (Set a)])
chooseMinimize Set a
free [Assign label (Set a)]
available =
      let singleMin :: Set a
singleMin =
             Set a -> Set a
forall bits. C bits => Set bits -> Set bits
BitSet.keepMinimum (Set a -> Set a) -> Set a -> Set a
forall a b. (a -> b) -> a -> b
$ Set a -> Map a -> Set a
forall bits. C bits => Set bits -> Map bits -> Set bits
BitMap.minimumSet Set a
free (Map a -> Set a) -> Map a -> Set a
forall a b. (a -> b) -> a -> b
$
             (Assign label (Set a) -> Map a) -> [Assign label (Set a)] -> Map a
forall m a. Monoid m => (a -> m) -> [a] -> m
forall (t :: * -> *) m a.
(Foldable t, Monoid m) =>
(a -> m) -> t a -> m
Fold.foldMap (Set a -> Map a
forall bits. C bits => Set bits -> Map bits
BitMap.fromSet (Set a -> Map a)
-> (Assign label (Set a) -> Set a) -> Assign label (Set a) -> Map a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Assign label (Set a) -> Set a
forall label set. Assign label set -> set
labeledSet) [Assign label (Set a)]
available
      in  (Set a
singleMin,
           (Assign label (Set a) -> Bool)
-> [Assign label (Set a)] -> [Assign label (Set a)]
forall a. (a -> Bool) -> [a] -> [a]
filter (Bool -> Bool
not (Bool -> Bool)
-> (Assign label (Set a) -> Bool) -> Assign label (Set a) -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Set a -> Set a -> Bool
forall a. C a => Set a -> Set a -> Bool
BitSet.disjoint Set a
singleMin (Set a -> Bool)
-> (Assign label (Set a) -> Set a) -> Assign label (Set a) -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Assign label (Set a) -> Set a
forall label set. Assign label set -> set
labeledSet) [Assign label (Set a)]
available)


instance Set IntSet.IntSet where
   null :: IntSet -> Bool
null = IntSet -> Bool
IntSet.null
   disjoint :: IntSet -> IntSet -> Bool
disjoint IntSet
x IntSet
y = IntSet -> Bool
IntSet.null (IntSet -> Bool) -> IntSet -> Bool
forall a b. (a -> b) -> a -> b
$ IntSet -> IntSet -> IntSet
IntSet.intersection IntSet
x IntSet
y
   unions :: [IntSet] -> IntSet
unions = [IntSet] -> IntSet
forall (f :: * -> *). Foldable f => f IntSet -> IntSet
IntSet.unions
   difference :: IntSet -> IntSet -> IntSet
difference = IntSet -> IntSet -> IntSet
IntSet.difference
   minimize :: forall label.
IntSet -> [Assign label IntSet] -> [Assign label IntSet]
minimize IntSet
free = (IntSet, [Assign label IntSet]) -> [Assign label IntSet]
forall a b. (a, b) -> b
snd ((IntSet, [Assign label IntSet]) -> [Assign label IntSet])
-> ([Assign label IntSet] -> (IntSet, [Assign label IntSet]))
-> [Assign label IntSet]
-> [Assign label IntSet]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. IntSet -> [Assign label IntSet] -> (IntSet, [Assign label IntSet])
forall label.
IntSet -> [Assign label IntSet] -> (IntSet, [Assign label IntSet])
forall set label.
Choose set =>
set -> [Assign label set] -> (set, [Assign label set])
chooseMinimize IntSet
free

instance Choose IntSet.IntSet where
   chooseMinimize :: forall label.
IntSet -> [Assign label IntSet] -> (IntSet, [Assign label IntSet])
chooseMinimize IntSet
free [Assign label IntSet]
available =
      let singleMin :: Key
singleMin =
            IntSet -> Key
IntSet.findMin (IntSet -> Key) -> IntSet -> Key
forall a b. (a -> b) -> a -> b
$ Set IntSet -> IntSet
forall bits. Set bits -> bits
BitSet.getBits (Set IntSet -> IntSet) -> Set IntSet -> IntSet
forall a b. (a -> b) -> a -> b
$
            Set IntSet -> Map IntSet -> Set IntSet
forall bits. C bits => Set bits -> Map bits -> Set bits
BitMap.minimumSet (IntSet -> Set IntSet
forall bits. bits -> Set bits
BitSet.Set IntSet
free) (Map IntSet -> Set IntSet) -> Map IntSet -> Set IntSet
forall a b. (a -> b) -> a -> b
$
            (Assign label IntSet -> Map IntSet)
-> [Assign label IntSet] -> Map IntSet
forall m a. Monoid m => (a -> m) -> [a] -> m
forall (t :: * -> *) m a.
(Foldable t, Monoid m) =>
(a -> m) -> t a -> m
Fold.foldMap (Set IntSet -> Map IntSet
forall bits. C bits => Set bits -> Map bits
BitMap.fromSet (Set IntSet -> Map IntSet)
-> (Assign label IntSet -> Set IntSet)
-> Assign label IntSet
-> Map IntSet
forall b c a. (b -> c) -> (a -> b) -> a -> c
. IntSet -> Set IntSet
forall bits. bits -> Set bits
BitSet.Set (IntSet -> Set IntSet)
-> (Assign label IntSet -> IntSet)
-> Assign label IntSet
-> Set IntSet
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Assign label IntSet -> IntSet
forall label set. Assign label set -> set
labeledSet) [Assign label IntSet]
available
      in  (Key -> IntSet
IntSet.singleton Key
singleMin,
           (Assign label IntSet -> Bool)
-> [Assign label IntSet] -> [Assign label IntSet]
forall a. (a -> Bool) -> [a] -> [a]
filter (Key -> IntSet -> Bool
IntSet.member Key
singleMin (IntSet -> Bool)
-> (Assign label IntSet -> IntSet) -> Assign label IntSet -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Assign label IntSet -> IntSet
forall label set. Assign label set -> set
labeledSet) [Assign label IntSet]
available)


{- |
'Assign' allows to associate a set with a label.
If a particular set is chosen for a set cover,
then its label is included in the output of 'partitions'.

I have decided to separate sets and labels this way,
since it is the easiest way to assign a meaning to a set.
If you really want to know the sets in a partition,
then you can fill the 'label' field with the set.
-}
data Assign label set =
   Assign {
      forall label set. Assign label set -> label
label :: label,
      forall label set. Assign label set -> set
labeledSet :: set
   }

{- |
Construction of a labeled set.
-}
assign :: label -> set -> Assign label set
assign :: forall label set. label -> set -> Assign label set
assign = label -> set -> Assign label set
forall label set. label -> set -> Assign label set
Assign


{- |
You may use this to post-process a set of 'Assign's
in order to speedup the solver considerably.
You must process the whole set of 'Assign's at once,
i.e. do not process only parts of the assignment list.
The output of 'bitVectorFromSetAssigns' should go into the solver as is.
-}
bitVectorFromSetAssigns ::
   (Ord a) =>
   [Assign label (Set.Set a)] -> [Assign label (BitSet.Set Integer)]
bitVectorFromSetAssigns :: forall a label.
Ord a =>
[Assign label (Set a)] -> [Assign label (Set Integer)]
bitVectorFromSetAssigns [Assign label (Set a)]
asns =
   let bitVec :: Set a -> Integer
bitVec = (Integer -> Key -> Integer) -> Integer -> Map a Key -> Integer
forall b a. (b -> a -> b) -> b -> Map a a -> b
forall (t :: * -> *) b a.
Foldable t =>
(b -> a -> b) -> b -> t a -> b
Fold.foldl' Integer -> Key -> Integer
forall a. Bits a => a -> Key -> a
setBit Integer
0 (Map a Key -> Integer) -> (Set a -> Map a Key) -> Set a -> Integer
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Assign label (Set a)] -> Set a -> Map a Key
forall a label.
Ord a =>
[Assign label (Set a)] -> Set a -> Map a Key
mapIntFromSet [Assign label (Set a)]
asns
   in  (Assign label (Set a) -> Assign label (Set Integer))
-> [Assign label (Set a)] -> [Assign label (Set Integer)]
forall a b. (a -> b) -> [a] -> [b]
map ((Set a -> Set Integer)
-> Assign label (Set a) -> Assign label (Set Integer)
forall a b. (a -> b) -> Assign label a -> Assign label b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap (Integer -> Set Integer
forall bits. bits -> Set bits
BitSet.Set (Integer -> Set Integer)
-> (Set a -> Integer) -> Set a -> Set Integer
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Set a -> Integer
bitVec)) [Assign label (Set a)]
asns

{- |
Like 'bitVectorFromSetAssigns' but generates 'IntSet.IntSet'
instead of 'Integer' bitvectors.
Since containers-0.5.5 as shipped with GHC-7.8.4,
'IntSet.IntSet' should usually be more efficient than 'Integer'.
-}
intSetFromSetAssigns ::
   (Ord a) => [Assign label (Set.Set a)] -> [Assign label IntSet.IntSet]
intSetFromSetAssigns :: forall a label.
Ord a =>
[Assign label (Set a)] -> [Assign label IntSet]
intSetFromSetAssigns [Assign label (Set a)]
asns =
   let intSet :: Set a -> IntSet
intSet = [Key] -> IntSet
IntSet.fromList ([Key] -> IntSet) -> (Set a -> [Key]) -> Set a -> IntSet
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map a Key -> [Key]
forall k a. Map k a -> [a]
Map.elems (Map a Key -> [Key]) -> (Set a -> Map a Key) -> Set a -> [Key]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Assign label (Set a)] -> Set a -> Map a Key
forall a label.
Ord a =>
[Assign label (Set a)] -> Set a -> Map a Key
mapIntFromSet [Assign label (Set a)]
asns
   in  (Assign label (Set a) -> Assign label IntSet)
-> [Assign label (Set a)] -> [Assign label IntSet]
forall a b. (a -> b) -> [a] -> [b]
map ((Set a -> IntSet) -> Assign label (Set a) -> Assign label IntSet
forall a b. (a -> b) -> Assign label a -> Assign label b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap Set a -> IntSet
intSet) [Assign label (Set a)]
asns

mapIntFromSet ::
   (Ord a) => [Assign label (Set.Set a)] -> Set.Set a -> Map.Map a Int
mapIntFromSet :: forall a label.
Ord a =>
[Assign label (Set a)] -> Set a -> Map a Key
mapIntFromSet [Assign label (Set a)]
asns =
   let mapToInt :: Map a Key
mapToInt =
         [(a, Key)] -> Map a Key
forall k a. Ord k => [(k, a)] -> Map k a
Map.fromList ([(a, Key)] -> Map a Key) -> [(a, Key)] -> Map a Key
forall a b. (a -> b) -> a -> b
$ [a] -> [Key] -> [(a, Key)]
forall a b. [a] -> [b] -> [(a, b)]
zip (Set a -> [a]
forall a. Set a -> [a]
Set.toList (Set a -> [a]) -> Set a -> [a]
forall a b. (a -> b) -> a -> b
$ [Set a] -> Set a
forall set. Set set => [set] -> set
unions ([Set a] -> Set a) -> [Set a] -> Set a
forall a b. (a -> b) -> a -> b
$ (Assign label (Set a) -> Set a)
-> [Assign label (Set a)] -> [Set a]
forall a b. (a -> b) -> [a] -> [b]
map Assign label (Set a) -> Set a
forall label set. Assign label set -> set
labeledSet [Assign label (Set a)]
asns) [Key
0..]
   in  Map a Key -> Map a () -> Map a Key
forall k a b. Ord k => Map k a -> Map k b -> Map k a
Map.intersection Map a Key
mapToInt (Map a () -> Map a Key)
-> (Set a -> Map a ()) -> Set a -> Map a Key
forall b c a. (b -> c) -> (a -> b) -> a -> c
. () -> Set a -> Map a ()
forall a b. Ord a => b -> Set a -> Map a b
constMap ()

{- |
The state of the search.
@usedSubsets@ contains the partial partition built up so far.
@availableSubsets@ is the list of sets we can still try to put into a partition.
The lists @usedSubsets@ and @availableSubsets@ are disjoint,
but their union is not necessarily equal to the list of initially given sets.
There are sets not contained in the partial partition
that overlap with the partial partition.
Those sets are not available for extending the partition.

@freeElements@ contains the elements that are not covered
by the partial partition in @usedSubsets@.
@unions usedSubset@ and @freeElements@ are disjoint
and their union is the set of all elements.
-}
data State label set =
   State {
      forall label set. State label set -> [Assign label set]
availableSubsets :: [Assign label set],
      forall label set. State label set -> set
freeElements :: set,
      forall label set. State label set -> [label]
usedSubsets :: [label]
   }

instance Functor (Assign label) where
   fmap :: forall a b. (a -> b) -> Assign label a -> Assign label b
fmap a -> b
f (Assign label
lab a
set) = label -> b -> Assign label b
forall label set. label -> set -> Assign label set
Assign label
lab (a -> b
f a
set)

instance Functor (State label) where
   fmap :: forall a b. (a -> b) -> State label a -> State label b
fmap a -> b
f (State [Assign label a]
ab a
fp [label]
pb) =
      [Assign label b] -> b -> [label] -> State label b
forall label set.
[Assign label set] -> set -> [label] -> State label set
State ((Assign label a -> Assign label b)
-> [Assign label a] -> [Assign label b]
forall a b. (a -> b) -> [a] -> [b]
map ((a -> b) -> Assign label a -> Assign label b
forall a b. (a -> b) -> Assign label a -> Assign label b
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> b
f) [Assign label a]
ab) (a -> b
f a
fp) [label]
pb

initState :: Set set => [Assign label set] -> State label set
initState :: forall set label. Set set => [Assign label set] -> State label set
initState [Assign label set]
subsets =
   State {
      availableSubsets :: [Assign label set]
availableSubsets = [Assign label set]
subsets,
      freeElements :: set
freeElements = [set] -> set
forall set. Set set => [set] -> set
unions ([set] -> set) -> [set] -> set
forall a b. (a -> b) -> a -> b
$ (Assign label set -> set) -> [Assign label set] -> [set]
forall a b. (a -> b) -> [a] -> [b]
map Assign label set -> set
forall label set. Assign label set -> set
labeledSet [Assign label set]
subsets,
      usedSubsets :: [label]
usedSubsets = []
   }

{-# INLINE updateState #-}
updateState :: Set set => Assign label set -> State label set -> State label set
updateState :: forall set label.
Set set =>
Assign label set -> State label set -> State label set
updateState (Assign label
attemptLabel set
attemptedSet) State label set
s =
   State {
      availableSubsets :: [Assign label set]
availableSubsets =
         (Assign label set -> Bool)
-> [Assign label set] -> [Assign label set]
forall a. (a -> Bool) -> [a] -> [a]
filter (set -> set -> Bool
forall set. Set set => set -> set -> Bool
disjoint set
attemptedSet (set -> Bool)
-> (Assign label set -> set) -> Assign label set -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Assign label set -> set
forall label set. Assign label set -> set
labeledSet) ([Assign label set] -> [Assign label set])
-> [Assign label set] -> [Assign label set]
forall a b. (a -> b) -> a -> b
$
         State label set -> [Assign label set]
forall label set. State label set -> [Assign label set]
availableSubsets State label set
s,
      freeElements :: set
freeElements = set -> set -> set
forall set. Set set => set -> set -> set
difference (State label set -> set
forall label set. State label set -> set
freeElements State label set
s) set
attemptedSet,
      usedSubsets :: [label]
usedSubsets = label
attemptLabel label -> [label] -> [label]
forall a. a -> [a] -> [a]
: State label set -> [label]
forall label set. State label set -> [label]
usedSubsets State label set
s
   }


{- |
This is the key of the search algorithm.
The search algorithm tries to build partitions
by adding sets to a partition list successively.
A step starts on a partial partition
and looks for new sets that could be added.
The goal is to avoid to check a set again down in a search branch
and to quickly determine search directions that lead to a dead end.
To this end a search step selects a certain set element
and tries all sets that contain that element
and that do not overlap with the partial partition.
Practically, 'step' selects an element with the minimal number
of non-overlapping sets it is contained in.
If this number is zero, then the search can be aborted in this branch.

Most oftenly the power of the algorithm
originates from the formulation of a problem as a set-cover problem
and from the equal treatment of all elements.
E.g. in the Soma cube example
the algorithm chooses whether to do a case analysis on all bricks
that cover a certain position,
or to do a case analysis on all positions that are possible for a certain brick.

The algorithm might not be extraordinarily fast,
but in all cases it consumes only little memory
since it only has to maintain the current state of search.

Precondition: 'freeElements' of the input state must not be empty.
-}
{-# INLINE step #-}
step :: Set set => State label set -> [State label set]
step :: forall set label. Set set => State label set -> [State label set]
step State label set
s =
   (Assign label set -> State label set)
-> [Assign label set] -> [State label set]
forall a b. (a -> b) -> [a] -> [b]
map ((Assign label set -> State label set -> State label set)
-> State label set -> Assign label set -> State label set
forall a b c. (a -> b -> c) -> b -> a -> c
flip Assign label set -> State label set -> State label set
forall set label.
Set set =>
Assign label set -> State label set -> State label set
updateState State label set
s) ([Assign label set] -> [State label set])
-> [Assign label set] -> [State label set]
forall a b. (a -> b) -> a -> b
$ set -> [Assign label set] -> [Assign label set]
forall label. set -> [Assign label set] -> [Assign label set]
forall set label.
Set set =>
set -> [Assign label set] -> [Assign label set]
minimize (State label set -> set
forall label set. State label set -> set
freeElements State label set
s) (State label set -> [Assign label set]
forall label set. State label set -> [Assign label set]
availableSubsets State label set
s)

{- |
Start the search for partitions on a certain search state.
This can be an 'initState' or the result of performing some search 'step's.
In the examples we use this for parallelization:
We perform some steps manually
and then run 'search' on the results in parallel.
-}
{-# INLINE search #-}
search :: Set set => State label set -> [[label]]
search :: forall set label. Set set => State label set -> [[label]]
search State label set
s =
   if set -> Bool
forall set. Set set => set -> Bool
null (State label set -> set
forall label set. State label set -> set
freeElements State label set
s)
     then [State label set -> [label]
forall label set. State label set -> [label]
usedSubsets State label set
s]
     else State label set -> [State label set]
forall set label. Set set => State label set -> [State label set]
step State label set
s [State label set] -> (State label set -> [[label]]) -> [[label]]
forall a b. [a] -> (a -> [b]) -> [b]
forall (m :: * -> *) a b. Monad m => m a -> (a -> m b) -> m b
>>= State label set -> [[label]]
forall set label. Set set => State label set -> [[label]]
search

{- |
@partitions [assign '0' set0, assign '1' set1, assign '2' set2]@
computes @unions [set0, set1, set2]@ and tries to partition the union set
using the sets @set0@, @set1@, @set2@.
'partitions' returns all such partitions.
If a set is chosen for a partition,
then its label is included in the output.
E.g. @set0 = Set.fromList [0,1], set1 = Set.fromList [2], set2 = Set.fromList [0,1,2]@,
then 'partitions' returns @[\"01\", \"2\"]@.

The order of partitions and the order of labels
depends on the implementation
and you must not rely on them.

You may use 'Data.Maybe.listToMaybe' in order to select only the first solution.
-}
{-# INLINE partitions #-}
partitions :: Set set => [Assign label set] -> [[label]]
partitions :: forall set label. Set set => [Assign label set] -> [[label]]
partitions = State label set -> [[label]]
forall set label. Set set => State label set -> [[label]]
search (State label set -> [[label]])
-> ([Assign label set] -> State label set)
-> [Assign label set]
-> [[label]]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Assign label set] -> State label set
forall set label. Set set => [Assign label set] -> State label set
initState



data Tree label set = Leaf | Branch set [(label, Tree label set)]
   deriving (Tree label set -> Tree label set -> Bool
(Tree label set -> Tree label set -> Bool)
-> (Tree label set -> Tree label set -> Bool)
-> Eq (Tree label set)
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
forall label set.
(Eq set, Eq label) =>
Tree label set -> Tree label set -> Bool
$c== :: forall label set.
(Eq set, Eq label) =>
Tree label set -> Tree label set -> Bool
== :: Tree label set -> Tree label set -> Bool
$c/= :: forall label set.
(Eq set, Eq label) =>
Tree label set -> Tree label set -> Bool
/= :: Tree label set -> Tree label set -> Bool
Eq)

completeTree :: Choose set => State label set -> Tree label set
completeTree :: forall set label. Choose set => State label set -> Tree label set
completeTree State label set
s =
   if set -> Bool
forall set. Set set => set -> Bool
null (State label set -> set
forall label set. State label set -> set
freeElements State label set
s)
      then Tree label set
forall label set. Tree label set
Leaf
      else
         (set -> [(label, Tree label set)] -> Tree label set)
-> (set, [(label, Tree label set)]) -> Tree label set
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry set -> [(label, Tree label set)] -> Tree label set
forall label set.
set -> [(label, Tree label set)] -> Tree label set
Branch ((set, [(label, Tree label set)]) -> Tree label set)
-> (set, [(label, Tree label set)]) -> Tree label set
forall a b. (a -> b) -> a -> b
$
         ([Assign label set] -> [(label, Tree label set)])
-> (set, [Assign label set]) -> (set, [(label, Tree label set)])
forall b c a. (b -> c) -> (a, b) -> (a, c)
mapSnd ((Assign label set -> (label, Tree label set))
-> [Assign label set] -> [(label, Tree label set)]
forall a b. (a -> b) -> [a] -> [b]
map (\Assign label set
asn -> (Assign label set -> label
forall label set. Assign label set -> label
label Assign label set
asn, State label set -> Tree label set
forall set label. Choose set => State label set -> Tree label set
completeTree (State label set -> Tree label set)
-> State label set -> Tree label set
forall a b. (a -> b) -> a -> b
$ Assign label set -> State label set -> State label set
forall set label.
Set set =>
Assign label set -> State label set -> State label set
updateState Assign label set
asn State label set
s))) ((set, [Assign label set]) -> (set, [(label, Tree label set)]))
-> (set, [Assign label set]) -> (set, [(label, Tree label set)])
forall a b. (a -> b) -> a -> b
$
         set -> [Assign label set] -> (set, [Assign label set])
forall label.
set -> [Assign label set] -> (set, [Assign label set])
forall set label.
Choose set =>
set -> [Assign label set] -> (set, [Assign label set])
chooseMinimize (State label set -> set
forall label set. State label set -> set
freeElements State label set
s) (State label set -> [Assign label set]
forall label set. State label set -> [Assign label set]
availableSubsets State label set
s)

decisionTree :: Choose set => [Assign label set] -> Tree label set
decisionTree :: forall set label.
Choose set =>
[Assign label set] -> Tree label set
decisionTree = State label set -> Tree label set
forall set label. Choose set => State label set -> Tree label set
completeTree (State label set -> Tree label set)
-> ([Assign label set] -> State label set)
-> [Assign label set]
-> Tree label set
forall b c a. (b -> c) -> (a -> b) -> a -> c
. [Assign label set] -> State label set
forall set label. Set set => [Assign label set] -> State label set
initState