```module Data.Compression.Huffman
( HuffmanTree(..)
, Bit(..)
, Code

, huffman
, huffmanSorted
, codewords
, ppCode
) where

import Data.List (intercalate)
import Control.Arrow (first,second)
import qualified Data.PriorityQueue.FingerTree as PQ
import Data.Sequence as S

data Bit = Zero | One

instance Show Bit where
show Zero = "0"
show One  = "1"

data HuffmanTree a = Empty
| Node (HuffmanTree a) (HuffmanTree a)
| Leaf a
deriving Show

type Code a = [(a,[Bit])]

-- Simple implementation, O(n log n).
huffman :: (Ord w, Num w) => [(a,w)] -> HuffmanTree a
huffman = build . prepare
where
prepare  = PQ.fromList . map (\(x,w) -> (w, Leaf x))
build pq =
case PQ.minViewWithKey pq of
Nothing -> Empty
Just ((w,x), pq') ->
case PQ.minViewWithKey pq' of
Nothing -> x
Just ((w',y), pq'') -> build \$ PQ.insert (w+w') (Node x y) pq''

-- More efficient implementation, O(n).  Requires that the input
-- list of symbols and weight is sorted by increasing weight.
huffmanSorted :: (Ord w, Num w) => [(a,w)] -> HuffmanTree a
huffmanSorted = build S.empty . prepare
where
prepare = S.fromList . map (first Leaf)
dequeue s t =
case (viewl s, viewl t) of
(EmptyL, EmptyL)    -> Nothing
(EmptyL, (x :< ts)) -> Just (x,s,ts)
((x :< ss), EmptyL) -> Just (x,ss,t)
(((x,w) :< ss), ((y,w') :< ts))
| w < w'    -> Just ((x,w),ss,t)
| otherwise -> Just ((y,w'),s,ts)
build s t =
case dequeue s t of
Nothing -> Empty
Just ((x,w),s',t') ->
case dequeue s' t' of
Nothing -> x
Just ((y,w'),s'',t'') -> build (s'' |> (Node x y, w+w')) t''

-- Derive the prefix-free binary code from a huffman tree.
codewords :: HuffmanTree a -> Code a
codewords = code' []
where code' _    Empty      = []
code' bits (Leaf x)   = [(x,bits)]
code' bits (Node l r) = map (second (Zero:)) (code' bits l) ++
map (second (One:)) (code' bits r)

-- Pretty-print a binary code.  Mostly useful for debugging.
ppCode :: Show a => Code a -> String
ppCode = intercalate "\n" .
map (\(x,bits) -> show x ++ ": " ++ concat (map show bits))
```