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Euler problems/181 to 190

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(add problem 181)
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Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_181 = undefined
+
import Data.Map ((!),Map)
  +
import qualified Data.Map as M
  +
import Data.List
  +
import Control.Monad
  +
  +
main :: IO ()
  +
main = do
  +
let es = [40,60]
  +
dg = sum es
  +
mon = Mon dg es
  +
Poly mp = partitionPol mon
  +
print $ mp!mon
  +
  +
data Monomial
  +
= Mon
  +
{ degree :: !Int
  +
, expos :: [Int]
  +
}
  +
  +
infixl 7 <*>, *>
  +
  +
(<*>) :: Monomial -> Monomial -> Monomial
  +
(Mon d1 e1) <*> (Mon d2 e2)
  +
= Mon (d1+d2) (zipWithZ (+) e1 e2)
  +
  +
unit :: Monomial
  +
unit = Mon 0 []
  +
  +
(<<) :: Monomial -> Monomial -> Bool
  +
(Mon d1 e1) << (Mon d2 e2)
  +
= d1 <= d2 && and (zipWithZ (<=) e1 e2)
  +
  +
upTo :: Monomial -> [Monomial]
  +
upTo (Mon 0 _) = [unit]
  +
upTo (Mon d es) =
  +
sort $ go 0 [] es
  +
where
  +
go dg acc [] = return (Mon dg $ reverse acc)
  +
go dg acc (n:ns) = do
  +
k <- [0 .. n]
  +
go (dg+k) (k:acc) ns
  +
  +
newtype Polynomial =
  +
Poly { mapping :: (Map Monomial Integer) }
  +
deriving (Eq, Ord)
  +
  +
(*>) :: Integer -> Monomial -> Polynomial
  +
n *> m = Poly $ M.singleton m n
  +
  +
----------------------------------------------------------------------------
  +
-- The hard stuff --
  +
----------------------------------------------------------------------------
  +
  +
one :: Map Monomial Integer
  +
one = M.singleton unit 1
  +
  +
reciprocal :: Monomial -> Polynomial
  +
reciprocal m =
  +
Poly . foldl' extend one . reverse . drop 1 . upTo $ m
  +
where
  +
extend mp mon =
  +
M.filter (/= 0) $
  +
foldl' (flip (uncurry $ M.insertWith' (+))) mp list
  +
where
  +
list = filter ((<< m) . fst) [(mon <*> mn, -c) |
  +
(mn,c) <- M.assocs mp]
  +
  +
partitionPol :: Monomial -> Polynomial
  +
partitionPol m =
  +
Poly . foldl' update one $ sliced m
  +
where
  +
Poly rec = reciprocal m
  +
sliced mon = sortBy (comparing expos) . drop 1 $ upTo mon
  +
comparing f x y = compare (f x) (f y)
  +
update mp mon@(Mon d es)
  +
| es /= ses = M.insert mon (mp!(Mon d ses)) mp
  +
| otherwise = M.insert mon (negate clc) mp
  +
where
  +
ses = sort es
  +
clc = sum $ do
  +
mn@(Mon dg xs) <- sliced mon
  +
let cmn = Mon (d-dg) (zipWithZ (-) es xs)
  +
case M.lookup mn rec of
  +
Nothing -> []
  +
Just c -> return $ c*(mp!(Mon (d-dg)
  +
(zipWithZ (-) es xs)))
  +
  +
----------------------------------------------------------------------------
  +
-- Auxiliary Functions --
  +
----------------------------------------------------------------------------
  +
  +
zipWithZ :: (Int -> Int -> a) -> [Int] -> [Int] -> [a]
  +
zipWithZ _ [] [] = []
  +
zipWithZ f [] ys = map (f 0) ys
  +
zipWithZ f xs [] = map (flip f 0) xs
  +
zipWithZ f (x:xs) (y:ys) = f x y:zipWithZ f xs ys
  +
  +
unknowns :: [String]
  +
unknowns = ['X':show i | i <- [1 .. ]]
  +
  +
instance Show Monomial where
  +
showsPrec _ (Mon 0 _) = showString "1"
  +
showsPrec _ (Mon _ es) = foldr (.) id $ intersperse (showString "*") us
  +
where
  +
ps = filter ((/= 0) . snd) $ zip unknowns es
  +
us = map (\(s,e) -> showString s . showString "^"
  +
. showParen (e < 0) (shows e)) ps
  +
  +
instance Eq Monomial where
  +
(Mon d1 e1) == (Mon d2 e2)
  +
= d1 == d2 && (d1 == 0 || e1 == e2)
  +
  +
instance Ord Monomial where
  +
compare (Mon d1 e1) (Mon d2 e2)
  +
= case compare d1 d2 of
  +
EQ | d1 == 0 -> EQ
  +
| otherwise -> compare e2 e1
  +
other -> other
  +
  +
instance Show Polynomial where
  +
showsPrec p (Poly m)
  +
= showP p . filter ((/= 0) . snd) $ M.assocs m
  +
  +
showP :: Int -> [(Monomial,Integer)] -> ShowS
  +
showP _ [] = showString "0"
  +
showP p cs =
  +
showParen (p > 6) showL
  +
where
  +
showL = foldr (.) id $ intersperse (showString " + ") ms
  +
ms = map (\(m,c) -> showParen (c < 0) (shows c)
  +
. showString "*" . shows m) cs
  +
  +
instance Num Polynomial where
  +
(Poly m1) + (Poly m2) = Poly (M.filter (/= 0) $ addM m1 m2)
  +
p1 - p2 = p1 + (negate p2)
  +
(Poly m1) * (Poly m2) = Poly (mulM (M.assocs m1) (M.assocs m2))
  +
negate (Poly m) = Poly $ M.map negate m
  +
abs = id
  +
signum = id
  +
fromInteger n
  +
| n == 0 = Poly (M.empty)
  +
| otherwise = Poly (M.singleton unit n)
  +
  +
addM :: Map Monomial Integer -> Map Monomial Integer -> Map Monomial Integer
  +
addM p1 p2 =
  +
foldl' (flip (uncurry (M.insertWith' (+)))) p1 $
  +
M.assocs p2
  +
  +
mulM :: [(Monomial,Integer)] -> [(Monomial,Integer)] -> Map Monomial Integer
  +
mulM p1 p2 =
  +
M.filter (/= 0) .
  +
foldl' (flip (uncurry (M.insertWith' (+)))) M.empty $
  +
liftM2 (\(e1,c1) (e2,c2) -> (e1 <*> e2,c1*c2)) p1 p2
  +
problem_181 = main
 
</haskell>
 
</haskell>

Revision as of 02:01, 12 February 2008

Problem 181

Investigating in how many ways objects of two different colours can be grouped.

Solution:

import Data.Map ((!),Map)
import qualified Data.Map as M
import Data.List
import Control.Monad
 
main :: IO ()
main = do
    let es = [40,60]
        dg = sum es
        mon = Mon dg es
        Poly mp = partitionPol mon
    print $ mp!mon
 
data Monomial
    = Mon
    { degree :: !Int
    , expos :: [Int]
    }
 
infixl 7 <*>, *>
 
(<*>) :: Monomial -> Monomial -> Monomial
(Mon d1 e1) <*> (Mon d2 e2)
    = Mon (d1+d2) (zipWithZ (+) e1 e2)
 
unit :: Monomial
unit = Mon 0 []
 
(<<) :: Monomial -> Monomial -> Bool
(Mon d1 e1) << (Mon d2 e2)
    = d1 <= d2 && and (zipWithZ (<=) e1 e2)
 
upTo :: Monomial -> [Monomial]
upTo (Mon 0 _) = [unit]
upTo (Mon d es) = 
    sort $ go 0 [] es
    where
    go dg acc [] = return (Mon dg $ reverse acc)
    go dg acc (n:ns) = do
        k <- [0 .. n]
        go (dg+k) (k:acc) ns
 
newtype Polynomial = 
    Poly { mapping :: (Map Monomial Integer) }
        deriving (Eq, Ord)
 
(*>) :: Integer -> Monomial -> Polynomial
n *> m = Poly $ M.singleton m n
 
----------------------------------------------------------------------------
--                             The hard stuff                             --
----------------------------------------------------------------------------
 
one :: Map Monomial Integer
one = M.singleton unit 1
 
reciprocal :: Monomial -> Polynomial
reciprocal m =
    Poly . foldl' extend one . reverse . drop 1 . upTo $ m
    where
    extend mp mon =
        M.filter (/= 0) $
        foldl' (flip (uncurry $ M.insertWith' (+))) mp list
        where
        list = filter ((<< m) . fst) [(mon <*> mn, -c) |
                                      (mn,c) <- M.assocs mp]
 
partitionPol :: Monomial -> Polynomial
partitionPol m =
    Poly . foldl' update one $ sliced m
    where
    Poly rec = reciprocal m
    sliced mon = sortBy (comparing expos) . drop 1 $ upTo mon
    comparing f x y = compare (f x) (f y)
    update mp mon@(Mon d es)
        | es /= ses = M.insert mon (mp!(Mon d ses)) mp
        | otherwise = M.insert mon (negate clc) mp
        where
        ses = sort es
        clc = sum $ do
            mn@(Mon dg xs) <- sliced mon
            let cmn = Mon (d-dg) (zipWithZ (-) es xs)
            case M.lookup mn rec of
                Nothing -> []
                Just c  -> return $ c*(mp!(Mon (d-dg)
                                        (zipWithZ (-) es xs)))
 
----------------------------------------------------------------------------
--                          Auxiliary Functions                           --
----------------------------------------------------------------------------
 
zipWithZ :: (Int -> Int -> a) -> [Int] -> [Int] -> [a]
zipWithZ _ [] [] = []
zipWithZ f [] ys = map (f 0) ys
zipWithZ f xs [] = map (flip f 0) xs
zipWithZ f (x:xs) (y:ys) = f x y:zipWithZ f xs ys
 
unknowns :: [String]
unknowns = ['X':show i | i <- [1 .. ]]
 
instance Show Monomial where
    showsPrec _ (Mon 0 _)  = showString "1"
    showsPrec _ (Mon _ es) = foldr (.) id $ intersperse (showString "*") us
        where
        ps = filter ((/= 0) . snd) $ zip unknowns es
        us = map (\(s,e) -> showString s . showString "^"
                       . showParen (e < 0) (shows e)) ps
 
instance Eq Monomial where
    (Mon d1 e1) == (Mon d2 e2)
        = d1 == d2 && (d1 == 0 || e1 == e2)
 
instance Ord Monomial where
    compare (Mon d1 e1) (Mon d2 e2)
        = case compare d1 d2 of
            EQ | d1 == 0   -> EQ
               | otherwise -> compare e2 e1
            other          -> other
 
instance Show Polynomial where
    showsPrec p (Poly m) 
        = showP p . filter ((/= 0) . snd) $ M.assocs m
 
showP :: Int -> [(Monomial,Integer)] -> ShowS
showP _ [] = showString "0"
showP p cs = 
    showParen (p > 6) showL
    where
    showL = foldr (.) id $ intersperse (showString " + ") ms
    ms = map (\(m,c) -> showParen (c < 0) (shows c)
                         . showString "*" . shows m) cs
 
instance Num Polynomial where
    (Poly m1) + (Poly m2) = Poly (M.filter (/= 0) $ addM m1 m2)
    p1 - p2 = p1 + (negate p2)
    (Poly m1) * (Poly m2) = Poly (mulM (M.assocs m1) (M.assocs m2))
    negate (Poly m) = Poly $ M.map negate m
    abs = id
    signum = id
    fromInteger n
        | n == 0    = Poly (M.empty)
        | otherwise = Poly (M.singleton unit n)
 
addM :: Map Monomial Integer -> Map Monomial Integer -> Map Monomial Integer
addM p1 p2 = 
    foldl' (flip (uncurry (M.insertWith' (+)))) p1 $
    M.assocs p2
 
mulM :: [(Monomial,Integer)] -> [(Monomial,Integer)] -> Map Monomial Integer
mulM p1 p2 = 
    M.filter (/= 0) .
    foldl' (flip (uncurry (M.insertWith' (+)))) M.empty $
    liftM2 (\(e1,c1) (e2,c2) -> (e1 <*> e2,c1*c2)) p1 p2
problem_181 = main