Zipper monad

The TravelTree Monad is a monad proposed and designed by Paolo Martini (xerox), and coded by David House (davidhouse). It is based on the State monad and is used for navigating around data structures, using the concept of TheZipper.

As the only zipper currently available is for binary trees, this is what most of the article will be centred around.

Definition

newtype Travel t a = Travel { unT :: State t a }
     deriving (Functor, Monad, MonadState t)
type TravelTree a = Travel (Loc a) (Tree a) -- for trees

Computations in TravelTree are stateful. Loc a and Tree a are defined as follows:

data Tree a = Leaf a | Branch (Tree a) (Tree a)

data Cxt a = Top
           | L (Cxt a) (Tree a)
           | R (Tree a) (Cxt a)
  deriving (Show)

type Loc a = (Tree a, Cxt a)

See TheZipper for an explanation of the Cxt and Loc concepts.

Functions

Moving around

There are four main functions for stringing together TravelTree computations:

left,  -- moves down a level, through the left branch
right, -- moves down a level, through the right branch
up,    -- moves to the node's parent
top    -- moves to the top node
:: TravelTree a

All four return the subtree at the new location.

Mutation

There are also functions available for changing the tree:

getTree    :: TravelTree a
putTree    :: Tree a -> TravelTree a
modifyTree :: (Tree a -> Tree a) -> TravelTree a

These are direct front-doors for State's get, put and modify, and all three return the subtree after any applicable modifications.

Exit points

To get out of the monad, use traverse:

traverse :: Tree a -> TravelTree a -> Tree a

Again, this is just a front-door for evalState, with an initial state of (tt, Top) where tt is the TravelTree passed in.

Examples

The following examples use as the example tree:

t = Branch (Branch (Branch (Leaf 1) (Leaf 2))
                   (Leaf 3))
           (Branch (Leaf 4)
                   (Leaf 5))

The example tree

A simple path

This is a very simple example showing how to use the movement functions:

leftLeftRight :: TravelTree a
leftLeftRight = do left
                   left
                   right

Result of evaluation:

*Tree> t `traverse` leftLeftRight
Leaf 2

Tree reverser

This is a more in-depth example showing getTree and putTree, but is still rather contrived as it's easily done without the zipper (the zipper-less version is shown below).

The algorithm reverses the tree, in the sense that at every branch, the two subtrees are swapped over.

revTree :: Tree a -> Tree a
revTree t = t `traverse` revTree' where
    revTree' :: TravelTree a
    revTree' = do t <- getTree
                        case t of
                          Branch _ _ -> do left
                                           l' <- revTree'
                                           up
                                           right
                                           r' <- revTree'
                                           up
                                           putTree $ Branch r' l'
                          Leaf x     -> return $ Leaf x

-- without using the zipper:
revTreeZipless :: Tree a -> Tree a
revTreeZipless (Leaf x)       = Leaf x
revTreeZipless (Branch xs ys) = Branch (revTreeZipless ys) (revTreeZipless xs)

Result of evaluation:

*Tree> revTree $ Branch (Leaf 1) (Branch (Branch (Leaf 2) (Leaf 3)) (Leaf 4))
Branch (Branch (Leaf 4) (Branch (Leaf 3) (Leaf 2))) (Leaf 1)

Generalisation

Einar Karttunen (musasabi) suggested generalising this to a recursive tree combinator:

treeComb :: (a -> Tree a)                -- what to put at leaves
         -> (Tree a -> Tree a -> Tree a) -- what to put at branches
         -> (Tree a -> Tree a)           -- combinator function
treeComb leaf branch = \t -> t `traverse` treeComb' where
    treeComb' = do t <- getTree
                   case t of
                     Branch _ _ -> do left
                                      l' <- treeComb'
                                      up
                                      right
                                      r' <- treeComb'
                                      up
                                      putTree $ branch l' r'
                     Leaf x     -> return $ leaf x

revTree is then easy:

revTreeZipper :: Tree a -> Tree a
revTreeZipper = treeComb Leaf (flip Branch)

It turns out this is a fairly powerful combinator. As with revTree, it can change the structure of a tree. Here's another example which turns a tree into one where siblings are sorted, i.e. given a Branch l r, if l and r are leaves, then the value of l is less than or equal to that of r. Also, if one of l or r is a Branch and the other a Leaf, then l is the Leaf and r the Branch:

sortSiblings :: Ord a => Tree a -> Tree a
sortSiblings = treeComb Leaf minLeaves where
    minLeaves l@(Branch _ _) r@(Leaf _       ) = Branch r l
    minLeaves l@(Leaf _)     r@(Branch _ _   ) = Branch l r
    minLeaves l@(Branch _ _) r@(Branch _ _   ) = Branch l r
    minLeaves l@(Leaf x)     r@(Leaf y       ) = Branch (Leaf $ min x y)
                                                        (Leaf $ max x y)

Result of evaluation:

*Tree> sortSiblings t
Branch (Branch (Leaf 3) (Branch (Leaf 1) (Leaf 2))) (Branch (Leaf 4) (Leaf 5))

Code

Here's the Zipper Monad in full:

module Zipper where

-- A monad implementing The Zipper. 
-- http://haskell.org/haskellwiki/ZipperMonad
--------------------------------------------------------------------------------

import Control.Monad.State
import Control.Arrow (first, second)

data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving (Show, Eq)

data Cxt a = Top
           | L (Cxt a) (Tree a)
           | R (Tree a) (Cxt a)
  deriving (Show)

type Loc a = (Tree a, Cxt a)

newtype Travel t a = Travel { unT :: State t a }
     deriving (Functor, Monad, MonadState t)
type TravelTree a = Travel (Loc a) (Tree a)

-- Movement around the tree
--

-- move down a level, through the left branch
left :: TravelTree a
left = modify left' >> liftM fst get where
    left' (Branch l r, c) = (l, L c r)

-- move down a level, through the left branch
right :: TravelTree a
right = modify right' >> liftM fst get where
    right' (Branch l r, c) = (r, R l c)

-- move to a node's parent
up :: TravelTree a
up = modify up' >> liftM fst get where 
    up' (t, L c r) = (Branch t r, c)
    up' (t, R l c) = (Branch l t, c)

-- move to the top node
top :: TravelTree a
top = modify (second $ const Top) >> liftM fst get

-- Mutation of the tree
--

-- modify the subtree at the current node
modifyTree :: (Tree a -> Tree a) -> TravelTree a
modifyTree f = modify (first f) >> liftM fst get

-- put a new subtree at the current node
putTree :: Tree a -> TravelTree a
putTree t = modifyTree $ const t

-- get the current node and its descendants
getTree :: TravelTree a
getTree = modifyTree id -- works because modifyTree returns the 'new' tree

-- Exit points
--

-- get out of the monad
traverse :: Tree a -> TravelTree a -> Tree a
traverse t tt = evalState (unT tt) (t, Top)