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If I wanted to find the permutations of a list, I know that the number of permutations is given by the multinomial coefficient. For example, "MISSISSIPPI" has 11 letters, 'S' appears 4 times, 'I' appears 4 times, 'P' appears twice and 'M' appears once. So the number of permutations of "MISSISSIPPI" is equal to 11!/(4!4!2!) = 34650.
If I load up GHCi and write:
ghci> import Data.List
ghci> permutations [1,2,3]
It will return
[[1,2,3],[2,1,3],[3,2,1],[2,3,1],[3,1,2],[1,3,2]]
as expected.
But if I write
ghci> permutations [1,0,0]
it will now return
[[1,0,0],[0,1,0],[0,0,1],[0,0,1],[0,1,0],[1,0,0]]
... which is very disappointing. As there are three elements, and two of them occur twice, one would hope for there only to be 6!/2! = 3 permutations, namely
[[1,0,0],[0,1,0],[0,0,1]]
rather than the six generated by treating each element of the list as distinct.
1) Why does Haskell implement "permutations" in the way described above (i.e. treating all elements of a list as distinct?)
2) Are there any standard library functions that calculate the permutations of a list in the "true" sense of permutations?
Remember that permutations has type
permutations :: [a] -> [[a]]
That means that it satisfies the free theorem
permutations . map f = (map . map) f . permutations
for all functions f. Since you can change the elements of the argument list arbitrarily without affecting the structure of the result list, permutations must really be a function on the indices of the original list, rather than the elements.
So what permutations is really doing --- what it must do --- is calculate all permutations of the indices of the argument list, then apply each of those permutations to the list and return the results. (I.e.,
permutations xn = (map . map) (xn!!) (permutations [0..length xn - 1])
for finite xn).
Mathematical appendix:
Since
xn = map (xn!!) (zipWith const [0..] xn)
for all xn, any function with permutations's type must satisfy
permutations xn
= permutations (map (xn!!) (zipWith const [0..] xn)
= (map . map) (xn!!) (permutations (zipWith const [0..] xn))
by the equation above for xn and the free theorem for permutations. So any function with permutations's type must operate only on the indices of the input list[1].
[1] Technically you can violate this by using seq. But only for input lists that contain undefined as an element, which isn't true in your case.
1 - Why does Haskell implement "permutations" in the way described above (i.e. treating all elements of a list as distinct?)
It is a design question and should be studied in deep. permutation treats the elements of the list as if they were all different from each other. You can do permutations [0, 0, 0] and you'll yet get a list of lists of size 6.
2 - Are there any standard library functions that calculate the permutations of a list in the "true" sense of permutations?
Yes, you have the Math.Combinat.Permutations, but you can easily create your own definition filtering the unique combinations with a complexity of O(n * log n) using sets, taking account that nub is known by being very slow:
module Main where
import Data.List (permutations)
import qualified Data.Set as Set
nubOrd :: (Ord a) => [a] -> [a]
nubOrd xs = go Set.empty xs where
go s (x:xs)
| x `Set.member` s = go s xs
| otherwise = x : go (Set.insert x s) xs
go _ _ = []
permutations' :: (Ord a) => [a] -> [[a]]
permutations' = nubOrd . permutations
Where permutations' [1, 0, 0] gives [[1, 0, 0], [0, 1, 0], [0, 0, 1]].
Why does Haskell implement "permutations" in the way described above (i.e. treating all elements of a list as distinct?)
Because otherwise, the type would have to be:
permutations :: Eq a => [a] -> [[a]]
and then we could permute only things that have an Eq instance. But I remember I had something like
permutations [(+), subtract, (*), (/)]
in some Project Euler code ....
Here is a slightly rearranged Daniel Fischer's solution:
inserts :: [a] -> [a] -> [[a]]
inserts (x:xs) (y:ys) = map (x:) (inserts xs (y:ys)) ++ map (y:) (inserts (x:xs) ys)
inserts xs ys = [xs ++ ys]
uniqPerms :: Ord a => [a] -> [[a]]
uniqPerms = foldM inserts [] . group . sort
I've been trying to wrap my head around this for a while now, but it seems like my lack of Haskell experience just won't get me through it. I couldn't find a similar question here on Stackoverflow (most of them are related to merging all sublists, without any condition)
So here it goes. Let's say I have a list of lists like this:
[[1, 2, 3], [3, 5, 6], [20, 21, 22]]
Is there an efficient way to merge lists if some sort of condition is true? Let's say I need to merge lists that share at least one element. In case of example, result would be:
[[1, 2, 3, 3, 5, 6], [20, 21, 22]]
Another example (when all lists can be merged):
[[1, 2], [2, 3], [3, 4]]
And it's result:
[[1, 2, 2, 3, 3, 4]]
Thanks for your help!
I don't know what to say about efficiency, but we can break down what's going on and get several different functionalities at least. Particular functionalities might be optimizable, but it's important to clarify exactly what's needed.
Let me rephrase the question: For some set X, some binary relation R, and some binary operation +, produce a set Q = {x+y | x in X, y in X, xRy}. So for your example, we might have X being some set of lists, R being "xRy if and only if there's at least one element in both x and y", and + being ++.
A naive implementation might just copy the set-builder notation itself
shareElement :: Eq a => [a] -> [a] -> Bool
shareElement xs ys = or [x == y | x <- xs, y <- ys]
v1 :: (a -> a -> Bool) -> (a -> a -> b) -> [a] -> [b]
v1 (?) (<>) xs = [x <> y | x <- xs, y <- xs, x ? y]
then p = v1 shareElement (++) :: Eq a => [[a]] -> [[a]] might achieve what you want. Except it probably doesn't.
Prelude> p [[1], [1]]
[[1,1],[1,1],[1,1],[1,1]]
The most obvious problem is that we get four copies: two from merging the lists with themselves, two from merging the lists with each other "in both directions". The problem occurs because List isn't the same as Set so we can't kill uniques. Of course, that's an easy fix, we'll just use Set everywhere
import Data.Set as Set
v2 :: (a -> a -> Bool) -> (a -> a -> b) -> Set.Set a -> Set.Set b
v2 (?) (<>) = Set.fromList . v1 (?) (<>) . Set.toList
So we can try again, p = v2 (shareElementonSet.toList) Set.union with
Prelude Set> p $ Set.fromList $ map Set.fromList [[1,2], [2,1]]
fromList [fromList [1,2]]
which seems to work. Note that we have to "go through" List because Set can't be made an instance of Monad or Applicative due to its Ord constraint.
I'd also note that there's a lot of lost behavior in Set. For instance, we fight either throwing away order information in the list or having to handle both x <> y and y <> x when our relation is symmetric.
Some more convenient versions can be written like
v3 :: Monoid a => (a -> a -> Bool) -> [a] -> [a]
v3 r = v2 r mappend
and more efficient ones can be built if we assume that the relationship is, say, an equality relation since then instead of having an O(n^2) operation we can do it in O(nd) where d is the number of partitions (cosets) of the relation.
Generally, it's a really interesting problem.
I just happened to write something similar here: Finding blocks in arrays
You can just modify it so (although I'm not too sure about the efficiency):
import Data.List (delete, intersect)
example1 = [[1, 2, 3], [3, 5, 6], [20, 21, 22]]
example2 = [[1, 2], [2, 3], [3, 4]]
objects zs = map concat . solve zs $ [] where
areConnected x y = not . null . intersect x $ y
solve [] result = result
solve (x:xs) result =
let result' = solve' xs [x]
in solve (foldr delete xs result') (result':result) where
solve' xs result =
let ys = filter (\y -> any (areConnected y) result) xs
in if null ys
then result
else solve' (foldr delete xs ys) (ys ++ result)
OUTPUT:
*Main> objects example1
[[20,21,22],[3,5,6,1,2,3]]
*Main> objects example2
[[3,4,2,3,1,2]]
Given a tuple of lists, I need to find all unique path from that:
Example I/P: [(1,2),(2,3),(3,4),(9,11),(4,5),(5,6),(6,7),(3,9)]
O/P: [[(1,2),(2,3),(3,4),(4,5),(5,6),(6,7)],[(1,2),(2,3),(3,9),(9,11)]]
Two tuples can connect if the second element of the tuple matches with the first element of the other tuple i.e: One tuple is (_,a) and other tuple is like (a,_).
What is the most efficient implementation for this ? I need to find the best data structure suited for it. Any suggestions ? The number of tuples in which I will execute the algorithm will be like more than 400,000.
{-# LANGUAGE NoMonomorphismRestriction #-}
import Data.List (permutations, nub)
path :: Eq a => [(a, a)] -> [(a, a)]
path [] = []
path [x] = [x]
path (u#(_, a):v#(b, _):xs) = if a == b then u:path (v:xs) else [u]
allPaths = nub . map path . permutations
(you can optimize chain generation but I think this problem has exponential time complexity)
EDITED
In general, you must to define more preciselly what paths you want to return.
Ignoring cycle invariant ([(1,2),(2,3),(3,1)] == [(2,3),(3,1),(1,3)]) you can generate all paths (without using permutations)
{-# LANGUAGE NoMonomorphismRestriction #-}
import Data.List (permutations, nub, sortBy, isInfixOf)
data Tree a = Node a [Tree a] deriving Show
treeFromList :: Eq a => a -> [(a, a)] -> Tree a
treeFromList a [] = Node a []
treeFromList a xs = Node a $ map subTree $ filter ((a==).fst) xs
where subTree v#(_, b) = treeFromList b $ filter (v/=) xs
treesFromList :: Eq a => [(a, a)] -> [Tree a]
treesFromList xs = map (flip treeFromList xs) $ nub $ map fst xs ++ map snd xs
treeToList :: Tree a -> [[a]]
treeToList (Node a []) = [[a]]
treeToList (Node a xs) = [a:ws | ws <- concatMap treeToList xs]
treesToList :: [Tree a] -> [[a]]
treesToList = concatMap treeToList
uniqTrees :: Eq a => [[a]] -> [[a]]
uniqTrees = f . reverse . sortBy ((.length).compare.length)
where f [] = []
f (x:xs) = x: filter (not.flip isInfixOf x) (f xs)
allPaths = uniqTrees . treesToList . treesFromList
then
*Main> allPaths [(1, 2), (1, 3), (2, 3), (2, 4), (3, 4), (4, 1)]
[[2,4,1,2,3,4],[2,3,4,1,2,4],[1,3,4,1,2,4],[1,3,4,1,2,3],[1,2,4,1,3,4],[1,2,3,4,1,3]]
uniqTrees has poor efficiency and, in general, you can do many optimizations.
If you want to avoid cycle invariant, you can normalize a cycle selecting minimum base10 representation, in previous example ([(1,2),(2,3),(3,1)] == [(2,3),(3,1),(1,3)]) 1231 < 2313 then
normalize [(2,3),(3,1),(1,3)] == [(1,2),(2,3),(3,1)]
you can normalize a path rotating it n-times and taking "head . sortBy toBase10 . rotations".
I think your problem fits on the NP category since:
A Hamiltonian path, also called a Hamilton path, is a path between two
vertices of a graph that visits each vertex exactly once.
In general, the problem of finding a Hamiltonian path is NP-complete
(Garey and Johnson 1983, pp. 199-200), so the only known way to
determine whether a given general graph has a Hamiltonian path is to
undertake an exhaustive search (source)
You problem is even "harder" since you don't know before hand what will be the end node.
In terms of data structure you can try to simulate the hash table structure in Haskell, since this data type is commonly use in graph and you problem can be turn into a graph.
I have two lists. One list contains some random data and other list contains the index of first list which needs to be deleted.
For example, let us consider two lists:
let a = [3,4,5,6,6,7,8]
let b = [1,3]
Then, the resultant output should be [3,5,6,7,8]. The number 4 and 6 are deleted since they are on index positions 1 and 3 respectively.
I'm new to Haskell, so finding it difficult to find the solution.
Update: Following code makes it work
import Data.List
dele :: Eq a => [a] -> [Int] -> [a]
dele [] _ = []
dele x [] = x
dele x (y:ys) = dele (delete (x !! y) x) ys
I was just wondering, is there a way to solve it through map/fold way ?
deleteByIndex :: (Enum a, Eq a, Num a) => [a] -> [b] -> [b]
deleteByIndex r = map snd . filter (\(i, _) -> notElem i r) . zip [0..]
[0..] produces an infinite list [0, 1, 2, 3, ...]
zip constructs a list of pairs with the values of this list and your input list in the form [(0,x), (1, y), ...]
filter takes a function a -> Bool. The lambda checks if the index (first element of the pair) is in your input list r.
map snd returns the second element of each pair of the zip list.
zip,filter, map and notElem are documented here
Off the top of my head:
removeByIndex :: [Integer] -> [a] -> [a]
removeByIndex indices = map snd . filter notInIndices . zip [0..]
where notInIndices (i,_) = i `notElem` indices
An alternative answer using the lens library which has received considerable attention recently
import Control.Lens
>let a = [3,4,5,6,6,7,8]
>let b = [1,3]
>a^..elements (`notElem`b)
[3,5,6,7,8]
(^..) is jus the infix for of toListOf which can be used to traverse a structure and make a list out of its parts. The elements function just lets you choose which ones to include.
Other options are 'traverse' to traverse a traversables, 'both' to traverse a (,) and they compose together with (.) so traverse.both would traverse [(1,2), (3,4)] for example.
[(1,2), (3,4)]^..traverse.both
[1,2,3,4]
Given a list of tuples like this:
dic = [(1,"aa"),(1,"cc"),(2,"aa"),(3,"ff"),(3,"gg"),(1,"bb")]
How to group items of dic resulting in a list grp where,
grp = [(1,["aa","bb","cc"]), (2, ["aa"]), (3, ["ff","gg"])]
I'm actually a newcomer to Haskell...and seems to be falling in love with it..
Using group or groupBy in Data.List will only group similar adjacent items in a list.
I wrote an inefficient function for this, but it results in memory failures as I need to process a very large coded string list. Hope you would help me find a more efficient way.
Whenever possible, reuse library code.
import Data.Map
sortAndGroup assocs = fromListWith (++) [(k, [v]) | (k, v) <- assocs]
Try it out in ghci:
*Main> sortAndGroup [(1,"aa"),(1,"cc"),(2,"aa"),(3,"ff"),(3,"gg"),(1,"bb")]
fromList [(1,["bb","cc","aa"]),(2,["aa"]),(3,["gg","ff"])]
EDIT In the comments, some folks are worried about whether (++) or flip (++) is the right choice. The documentation doesn't say which way things get associated; you can find out by experimenting, or you can sidestep the whole issue using difference lists:
sortAndGroup assocs = ($[]) <$> fromListWith (.) [(k, (v:)) | (k, v) <- assocs]
-- OR
sortAndGroup = fmap ($[]) . M.fromListWith (.) . map (fmap (:))
These alternatives are about the same length as the original, but they're a bit less readable to me.
Here's my solution:
import Data.Function (on)
import Data.List (sortBy, groupBy)
import Data.Ord (comparing)
myGroup :: (Eq a, Ord a) => [(a, b)] -> [(a, [b])]
myGroup = map (\l -> (fst . head $ l, map snd l)) . groupBy ((==) `on` fst)
. sortBy (comparing fst)
This works by first sorting the list with sortBy:
[(1,"aa"),(1,"cc"),(2,"aa"),(3,"ff"),(3,"gg"),(1,"bb")]
=> [(1,"aa"),(1,"bb"),(1,"cc"),(2,"aa"),(3,"ff"),(3,"gg")]
then grouping the list elements by the associated key with groupBy:
[(1,"aa"),(1,"bb"),(1,"cc"),(2,"aa"),(3,"ff"),(3,"gg")]
=> [[(1,"aa"),(1,"bb"),(1,"cc")],[(2,"aa")],[(3,"ff"),(3,"gg")]]
and then transforming the grouped items to tuples with map:
[[(1,"aa"),(1,"bb"),(1,"cc")],[(2,"aa")],[(3,"ff"),(3,"gg")]]
=> [(1,["aa","bb","cc"]), (2, ["aa"]), (3, ["ff","gg"])]`)
Testing:
> myGroup dic
[(1,["aa","bb","cc"]),(2,["aa"]),(3,["ff","gg"])]
Also you can use TransformListComp extension, for example:
Prelude> :set -XTransformListComp
Prelude> import GHC.Exts (groupWith, the)
Prelude GHC.Exts> let dic = [ (1, "aa"), (1, "bb"), (1, "cc") , (2, "aa"), (3, "ff"), (3, "gg")]
Prelude GHC.Exts> [(the key, value) | (key, value) <- dic, then group by key using groupWith]
[(1,["aa","bb","cc"]),(2,["aa"]),(3,["ff","gg"])]
If the list is not sorted on the first element, I don't think you can do better than O(nlog(n)).
One simple way would be to just sort and then use anything from the answer of second part.
You can use from Data.Map a map like Map k [a] to use first element of tuple as key and keep on adding to the values.
You can write your own complex function, which even after you all the attempts will still take O(nlog(n)).
If list is sorted on the first element as is the case in your example, then the task is trivial for something like groupBy as given in the answer by #Mikhail or use foldr and there are numerous other ways.
An example of using foldr is here:
grp :: Eq a => [(a,b)] -> [(a,[b])]
grp = foldr f []
where
f (z,s) [] = [(z,[s])]
f (z,s) a#((x,y):xs) | x == z = (x,s:y):xs
| otherwise = (z,[s]):a
{-# LANGUAGE TransformListComp #-}
import GHC.Exts
import Data.List
import Data.Function (on)
process :: [(Integer, String)] -> [(Integer, [String])]
process list = [(the a, b) | let info = [ (x, y) | (x, y) <- list, then sortWith by y ], (a, b) <- info, then group by a using groupWith]