Getting Unique Paths from list of tuple - haskell

Question

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.

Answer 1

{-# 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".

Answer 2

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.