# Zipper monad

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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 in binary trees, using the concept of TheZipper.

## Contents |

## 1 Definition

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

TravelTree

Loc a

Tree a

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)

Cxt

Loc

## 2 Functions

### 2.1 Moving around

There are four main functions for stringing togetherTravelTree

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.

### 2.2 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

get

put

modify

### 2.3 Exit points

To get out of the monad, usetraverse

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

evalState

(tt, Top)

tt

TravelTree

## 3 Examples

The following examples use as the example tree:

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

### 3.1 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

### 3.2 Tree reverser

This is a more in-depth example showinggetTree

putTree

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)

#### 3.2.1 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

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

revTree

Branch l r

l

r

l

r

l

r

Branch

Leaf

l

Leaf

r

Branch

heapise :: Ord a => Tree a -> Tree a heapise = 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> heapise t Branch (Branch (Leaf 3) (Branch (Leaf 1) (Leaf 2))) (Branch (Leaf 4) (Leaf 5))

## 4 Code

{-# GHC_OPTION -fglasgow-exts #-} 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) left :: TravelTree a left = modify left' >> liftM fst get where left' (Branch l r, c) = (l, L c r) right :: TravelTree a right = modify right' >> liftM fst get where right' (Branch l r, c) = (r, R l c) 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) top :: TravelTree a top = modify (second $ const Top) >> liftM fst get modifyTree :: (Tree a -> Tree a) -> TravelTree a modifyTree f = modify (first f) >> liftM fst get putTree :: Tree a -> TravelTree a putTree t = modifyTree $ const t getTree :: TravelTree a getTree = modifyTree id -- works because modifyTree returns the 'new' tree traverse :: Tree a -> TravelTree a -> Tree a traverse t tt = evalState (unT tt) (t, Top)