```--------------------------------------------------------------------------
-- |
-- Module               : Data.Parition
-- Copyright            : (c) Luke Palmer, 2013
-- License              : BSD3
--
-- Maintainer           : Luke Palmer <lrpalmer@gmail.com>
-- Stability            : experimental
-- Portability          : portable
--
-- Disjoint set data structure -- @Partition a@ maintains a collection of
-- disjoint sets of type @a@, with the ability to find which set a particular
-- item belongs to and the ability to merge any two such sets into one.
---------------------------------------------------------------------------

module Data.Partition
( Partition, discrete, empty, fromSets, nontrivialSets, join, find, rep )
where

import qualified Data.Map as Map
import qualified Data.Set as Set
import Data.Maybe (fromMaybe)

-- | A Partition of @a@: represents a collection of disjoint sets of @a@ whose
-- union includes every element of @a@.  Semantics: @[[Partition a]] = P(P(a))@
-- where @P@ is the power set operation.
data Partition a
= Partition { forwardMap :: Map.Map a a, backwardMap :: Map.Map a (Set.Set a) }
deriving (Eq, Ord)  -- Since the representative is always the least element,
-- we have a canonical representation and Eq is meaningful.
-- Ord may not mean anything, but at least there some
-- computable total ordering on Partitions, and that is helpful
-- sometimes.

instance (Show a) => Show (Partition a) where
show p = "fromSets " ++ show (nontrivialSets p)

-- | A partition in which every element of @a@ is in its own set.  Semantics:
-- @[[discrete]] = { { x } | x in a }@
discrete :: Partition a
discrete = Partition Map.empty Map.empty

-- | Synonym for @discrete@.
empty :: Partition a
empty = discrete

-- | Takes a list of disjoint sets and constructs a partition containing those sets,
-- with every remaining element being given its own set.
fromSets :: (Ord a) => [Set.Set a] -> Partition a
fromSets sets = Partition {
forwardMap = Map.fromList [ (x, Set.findMin s) | s <- sets, x <- Set.toList s ],
backwardMap = Map.fromList [ (Set.findMin s, s) | s <- sets ]
}

-- | Returns a list of all nontrivial sets (sets with more than one element) in the
-- partition.
nontrivialSets :: Partition a -> [Set.Set a]
nontrivialSets = Map.elems . backwardMap

-- | @join x y@ merges the two sets containing @x@ and @y@ into a single set.  Semantics:
-- @[[join x y p]] = (p \`minus\` find x \`minus\` find y) \`union\` { find x \`union\` find y }@
join :: (Ord a) => a -> a -> Partition a -> Partition a
join x y p = case compare x' y' of
LT -> go x' y'
EQ -> p
GT -> go y' x'
where
x' = rep p x
y' = rep p y

go into other = Partition {
forwardMap = compose [ Map.insert o into | o <- Set.toList otherSrc ] (forwardMap p),
backwardMap = Map.insert into (Set.union (repFind p into) otherSrc)
. Map.delete other
\$ backwardMap p
}
where
otherSrc = repFind p other

-- | @find p x@ finds the set that the element @x@ is associated with.  Semantics:
-- @[[find p x]] = the unique s in p such that x in s@.
find :: (Ord a) => Partition a -> a -> Set.Set a
find p = repFind p . rep p

-- | @rep p x@ finds the minimum element in the set containing @x@.
rep :: (Ord a) => Partition a -> a -> a
rep p x = fromMaybe x (Map.lookup x (forwardMap p))

-- Find the set that x is in given that x is already a representative element.
repFind :: (Ord a) => Partition a -> a -> Set.Set a
repFind p x = fromMaybe (Set.singleton x) (Map.lookup x (backwardMap p))

compose :: [a -> a] -> a -> a
compose = foldr (.) id
```