MapReduce as a monad

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1 Introduction
2 Initial Approach
- 2.1 Why a monad?
- 2.2 Details
3 The monad transformer approach
4 Future Directions

1 Introduction

MapReduce is a general technique for massively parallel programming developed by Google. It takes its inspiration from ideas in functional programming, but has moved away from that paradigm to a more imperative approach. I have noticed that MapReduce can be expressed naturally, using functional programming techniques, as a form of monad. The standard implementation of MapReduce is the JAVA-based HADOOP framework, which is very complex and somewhat temperamental. Moreover, it is necessary to write HADOOP-specific code into mappers and reducers. My prototype library takes about 100 lines of code and can wrap generic mapper / reducer functions.

Having shown that we can implement MapReduce as a generalised monad, it transpires that in fact, we can generalise this still further and define a

MapReduceT

monad transformer, so there is a MapReduce type and operation associated to any monad. In particular, it turns out that the

State

monad is just the MapReduce type of the monad

Hom a

of maps

h -> a

where

h

is some fixed type.

2 Initial Approach

2.1 Why a monad?

What the monadic implementation lets us do is the following:

Map and reduce look the same.
You can write a simple wrapper function that takes a mapper / reducer and wraps it in the monad, so authors of mappers / reducers do not need to know anything about the MapReduce framework: they can concentrate on their algorithms.
All of the guts of MapReduce are hidden in the monad's
bind
function
The implementation is naturally parallel
Making a MapReduce program is trivial:

... >>= wrapMR mapper >>= wrapMR reducer >>= ...

2.2 Details

Full details of the implementation and sample code can be found here. I'll just give highlights here.

2.2.1 Generalised mappers / reducers

One can generalise MapReduce a bit, so that each stage (map, reduce, etc) becomes a function of signature

a -> ([(s,a)] -> [(s',b)])

where

s

and

s'

are data types and

a

and

b

are key values.

2.2.2 Generalised Monad

Now, this is suggestive of a monad, but we can't use a monad per se, because the transformation changes the key and value types, and we want to be able to access them separately. Therefore we do the following.

Let

m

be a

Monad'

, a type with four parameters:

m s a s' b

. Generalise the monadic

bind

operation to:

m s a s' b -> ( b -> m s' b s'' c ) -> m s a s'' c

See Parametrized monads.

Then clearly the generalised mapper/reducer above can be written as a

Monad'

, meaning that we can write MapReduce as

... >>= mapper >>= reducer >>= mapper' >>= reducer' >>= ...

2.2.3 Implementation details

class Monad' m where
        return :: a -> m s x s a
        (>>=)  :: (Eq b) => m s a s' b -> ( b -> m s' b s'' c ) -> m s a s'' c

newtype MapReduce s a s' b = MR { runMR :: ([(s,a)] -> [(s',b)]) }

retMR :: a -> MapReduce s x s a
retMR k = MR (\ss -> [(s,k) | s <- fst <$> ss])

bindMR :: (Eq b,NFData s'',NFData c) => MapReduce s a s' b -> (b -> MapReduce s' b s'' c) -> MapReduce s a s'' c
bindMR f g = MR (\s ->
        let
                fs = runMR f s
                gs = P.map g $ nub $ snd <$> fs
        in
        concat $ map (\g' -> runMR g' fs) gs)

The key point here is that

P.map

is a parallel version of the simple

map

function.

Now we can write a wrapper function

wrapMR :: (Eq a) => ([s] -> [(s',b)]) -> (a -> MapReduce s a s' b)
wrapMR f = (\k -> MR (g k))
        where
        g k ss = f $ fst <$> filter (\s -> k == snd s) ss

which takes a conventional mapper / reducer and wraps it in the

Monad'

. Note that this means that the mapper / reducer functions do not need to know anything about the way MapReduce is implemented. So a standard MapReduce job becomes

mapReduce :: [String] -> [(String,Int)]
mapReduce state = runMapReduce mr state
        where
        mr = return () >>= wrapMR mapper >>= wrapMR reducer

I have tested the implementation with the standard word-counter mapper and reducer, and it works perfectly (full code is available via the link above).

3 The monad transformer approach

Define the monad transformer type

MapReduceT

by:

newtype (Monad m) => MapReduceT m t u = MR {run :: m t -> m u}

with operations

lift :: (Monad m) => m t -> MapReduceT m t t
lift x = MR (const x)

return :: (Monad m) => t -> MapReduceT m t t
return x = lift (return x)

bind :: (Monad m) => MapReduceT m u u -> MapReduceT m t u -> (u -> MapReduceT m u v) -> MapReduceT m t v
bind p f g = MR (\ xs -> ps xs >>= gs xs)
        where
            ps xs = (f >>> p) -< xs
            gs xs x = (f >>> g x) -< xs

where

>>>

and

-<

are the obvious arrow operations on

MapeduceT

types.

Then we show in this paper that:

MapReduce = MapReduceT []
with
(>>=) = bind nub
For a suitable choice of
p
the standard
State
monad is
MapReduceT Hom
where

data Hom a b = H {run :: (a ->  b)}

return x = H (const x)
f >>= g = H (\ x -> g' (f' x) x)
    where 
        f' = run f 
        g' x y = run (g x) y

4 Future Directions

My code so far runs concurrently and in multiple threads within a single OS image. It won't work on clustered systems. I have started work in this, see here.
Currently all of the data is sent to all of the mappers / reducers at each iteration. This is okay on a single machine, but may be prohibitive on a cluster.

I would be eager for collaborative working on taking this forward.

julianporter 18:10, 31 October 2011 (UTC)

Retrieved from "http://www.haskell.org/haskellwiki/MapReduce_as_a_monad"

@@ Line 4: / Line 4: @@
 MapReduce is a general technique for massively parallel programming developed by Google.  It takes its inspiration from ideas in functional programming, but has moved away from that paradigm to a more imperative approach.
-I have noticed that MapReduce can be expressed naturally, using functional programming techniques, as a form of monad.
+I have noticed that MapReduce can be expressed naturally, using functional programming techniques, as a form of monad.  The standard implementation of MapReduce is the JAVA-based HADOOP framework, which is very complex and somewhat temperamental.  Moreover, it is necessary to write HADOOP-specific code into mappers and reducers.  My prototype library takes about 100 lines of code and can wrap generic mapper / reducer functions.
-The standard implementation of MapReduce is the JAVA-based HADOOP framework, which is very complex and somewhat temperamental.  Moreover, it is necessary to write HADOOP-specific code into mappers and reducers.  My prototype library takes about 100 lines of code and can wrap generic mapper / reducer functions.
+Having shown that we can implement MapReduce as a generalised monad, it transpires that in fact, we can generalise this still further and define a <hask>MapReduceT</hask> monad transformer, so there is a MapReduce type and operation associated to any monad.  In particular, it turns out that the <hask>State</hask> monad is just the MapReduce type of the monad <hask>Hom a</hask> of maps <hask>h -> a</hask> where <hask>h</hask> is some fixed type.
-==Why a monad?==
+==Initial Approach==
+===Why a monad?===
 What the monadic implementation lets us do is the following:
@@ Line 20: / Line 22: @@
 </hask><br/>
-==Details==
+===Details===
 Full details of the implementation and sample code can be found [http://jpembeddedsolutions.wordpress.com/2011/04/02/mapreduce/ here].  I'll just give highlights here.
-===Generalised mappers / reducers===
+====Generalised mappers / reducers====
 One can generalise MapReduce a bit, so that each stage (map, reduce, etc) becomes a function of signature<br/>
 <hask>
@@ Line 30: / Line 32: @@
 where <hask>s</hask> and <hask>s'</hask> are data types and <hask>a</hask> and <hask>b</hask> are key values.
-===Generalised Monad===
+====Generalised Monad====
 Now, this is suggestive of a monad, but we can't use a monad ''per se'', because the transformation changes the key and value types, and we want to be able to access them separately.  Therefore we do the following.
@@ Line 47: / Line 49: @@
 </hask>
-===Implementation details===
+====Implementation details====
 <hask>
@@ Line 84: / Line 86: @@
 </hask><br/>
 I have tested the implementation with the standard word-counter mapper and reducer, and it works perfectly (full code is available via the link above).
+==The monad transformer approach==
+Define the monad transformer type <hask>MapReduceT</hask> by:<br/>
+<hask>
+newtype (Monad m) => MapReduceT m t u = MR {run :: m t -> m u}
+</hask>
+with operations<br/>
+<hask>
+lift :: (Monad m) => m t -> MapReduceT m t t
+lift x = MR (const x)
+return :: (Monad m) => t -> MapReduceT m t t
+return x = lift (return x)
+bind :: (Monad m) => MapReduceT m u u -> MapReduceT m t u -> (u -> MapReduceT m u v) -> MapReduceT m t v
+bind p f g = MR (\ xs -> ps xs >>= gs xs)
+        where
+            ps xs = (f >>> p) -< xs
+            gs xs x = (f >>> g x) -< xs
+</hask>
+where <hask> >>> </hask> and <hask> -< </hask> are the obvious arrow operations on <hask>MapeduceT</hask> types.
+Then we show in [http://media.jpembeddedsolutions.com/pdf/mrmonad.pdf this paper] that:
+* <hask>MapReduce = MapReduceT []</hask> with <hask> (>>=) = bind nub</hask>
+* For a suitable choice of <hask>p</hask> the standard <hask>State</hask> monad is <hask>MapReduceT Hom</hask> where
+:<hask>
+data Hom a b = H {run :: (a ->  b)}
+return x = H (const x)
+f >>= g = H (\ x -> g' (f' x) x)
+    where
+        f' = run f
+        g' x y = run (g x) y
+</hask>
 ==Future Directions==
-*My code so far runs concurrently and in multiple threads within a single OS image.  It won't work on clustered systems.  This is clearly where work should go next.
+*My code so far runs concurrently and in multiple threads within a single OS image.  It won't work on clustered systems.  I have started work in this, see [[MapReduce_with_CloudHaskell|here]].
 *Currently all of the data is sent to all of the mappers / reducers at each iteration.  This is okay on a single machine, but may be prohibitive on a cluster.
 I would be eager for collaborative working on taking this forward.
-[[User:Julianporter|julianporter]] 18:32, 2 April 2011 (UTC)
+[[User:Julianporter|julianporter]] 18:10, 31 October 2011 (UTC)

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MapReduce as a monad

From HaskellWiki

Current revision

Contents

1 Introduction

2 Initial Approach

2.1 Why a monad?

2.2 Details

2.2.1 Generalised mappers / reducers

2.2.2 Generalised Monad

2.2.3 Implementation details

3 The monad transformer approach

4 Future Directions

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