{-# LANGUAGE NoImplicitPrelude #-} {- | Copyright : (c) Henning Thielemann 2006 Maintainer : numericprelude@henning-thielemann.de Stability : provisional Portability : Permutation represented by an array of the images. -} module MathObj.Permutation.Table where import qualified MathObj.Permutation as Perm import Data.Set(Set) import qualified Data.Set as Set import Data.Array(Array,(!),(//),Ix) import qualified Data.Array as Array import Data.List ((\\), nub, unfoldr, ) import Data.Tuple.HT (swap, ) import Data.Maybe.HT (toMaybe, ) -- import NumericPrelude (Integer) import PreludeBase hiding (cycle) type T i = Array i i fromFunction :: (Ix i) => (i, i) -> (i -> i) -> T i fromFunction rng f = Array.listArray rng (map f (Array.range rng)) toFunction :: (Ix i) => T i -> (i -> i) toFunction = (!) {- Create a permutation in table form from any other permutation representation. -} fromPermutation :: (Ix i, Perm.C p) => p i -> T i fromPermutation x = let rng = Perm.domain x in Array.listArray rng (map (Perm.apply x) (Array.range rng)) fromCycles :: (Ix i) => (i, i) -> [[i]] -> T i fromCycles rng = foldl (flip cycle) (identity rng) identity :: (Ix i) => (i, i) -> T i identity rng = Array.listArray rng (Array.range rng) cycle :: (Ix i) => [i] -> T i -> T i cycle cyc p = p // zipWith (\i j -> (j,p!i)) cyc (tail (cyc++cyc)) inverse :: (Ix i) => T i -> T i inverse p = let rng = Array.bounds p in Array.array rng (map swap (Array.assocs p)) compose :: (Ix i) => T i -> T i -> T i compose p q = let pRng = Array.bounds p qRng = Array.bounds q in if pRng==qRng then fmap (p!) q else error "compose: ranges differ" -- ++ show pRng ++ " /= " ++ show qRng) {- | Extremely naïve algorithm to generate a list of all elements in a group. Should be replaced by a Schreier-Sims system if this code is ever used for anything bigger than .. say .. groups of order 512 or so. -} {- Alternative to Data.Set.minView in GHC-6.6. -} choose :: Set a -> Maybe (a, Set a) choose set = toMaybe (not (Set.null set)) (Set.deleteFindMin set) closure :: (Ix i) => [T i] -> [T i] closure [] = [] closure generators@(gen:_) = let genSet = Set.fromList generators idSet = Set.singleton (identity (Array.bounds gen)) generate (registered, candidates) = do (cand, remCands) <- choose candidates let newCands = flip Set.difference registered $ Set.map (compose cand) genSet return (cand, (Set.union registered newCands, Set.union remCands newCands)) in unfoldr generate (idSet, idSet) closureSlow :: (Ix i) => [T i] -> [T i] closureSlow [] = [] closureSlow generators@(gen:_) = let addElts grp [] = grp addElts grp cands@(cand:remCands) = let group' = grp ++ [cand] newCands = map (compose cand) generators cands' = nub (remCands ++ newCands) \\ (grp ++ cands) in addElts group' cands' in addElts [] [identity (Array.bounds gen)]