MapReduce as a monad

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.

Initial Approach

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 >>= ...

Details

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

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.

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' >>= ...

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).

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

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)