Like jelly in the PB&J, or Art in Simon and Garfunkel, the reducer gets second billing in the term MapReduce. But it really is the more mathematicaly interesting function of the two, as I hope to demonstrate.

The secret to big data is of course the ability to do work in parallel. Modern Big Data engines like Hadoop don’t rely on the invention of clever new algorithms or artificial intelligence to produce impressive results; instead, they are based on the idea of taking lots of input, working on little pieces of it in lots of places at the same time, then bringing together the results. Usually, the results are much smaller than the inputs, small enough that human beings can look at them directly.

## Parallel

In order to work on lots of small pieces at the same time, whatever task we’re performing has to be structured to be run in parallel. Many algorithms we’re used to seeing work well sequentially have to be tweaked at least a little to work in parallel, and some have to be discarded as unusable in parallel. For example, some cryptographic hash functions cannot be made to run in parallel, which makes sense because a good cryptographic hash function is one in which the output changes in unpredictable ways if the input has a small change anywhere.

On the other hand, a problem where a parallel algorithm is obvious gets the name “embarassingly parallel”. Generally this means that the relationship of the input to the output obviously maintains independence of each piece. For example, doubling every number in a long array of integers is embarrassingly parallel because it’s easy to see that it can be broken into pieces.

## Mappers

The epiphany behind MapReduce is the fact that many problems (such as search engines) can be broken into two pieces. In the map phase, we just look at each data item in parallel and perform some operation specific to it only. (In practice we group items together. Hadoop calls this a split. But it’s solely for optimization, not for math reasons.) In the reduce phase, we combine the output of the mappers to produce the final output. (Hadoop has another phase it calls a “combiner”, but this is just a reducer for the mappers on a single machine, to save network traffic. Again, just for optimization.)

Most map algorithms are embarassingly parallel. In Hadoop’s word count example, the input text file is tokenized into individual words. The map phase takes the word in and puts out a pair (a simple two valued object) with the word and the number 1. All of the actual counting is done in the reducer. With a search, the map phase might just calculate a score for a particular document given the search terms, while the reducer would do the hard work of putting the results in order.

## Reducer

So the reducer is where the math gets more interesting. We absolutely have to find a way to run our reducer in parallel. Otherwise, the ability to run the mapper in parallel would be wasted, and our total run time would be no better than just running a sequential algorithm. But the reducer is pretty clearly going to operate across lots of data. For the word count example, it might not seem so bad, because we could have a separate reducer for each unique word in the text. That is likely to be more reducers than we could reasonably create given the overhead of creating a new one, so we would expect to be able to optimize to get the best speedup.

However, what about our text search, that needs to find the “top 10”, or an example where we want to sum all of the numbers in a long list of integers, rather than just operate on each of them?

For these examples, we can see pretty clearly that there is a parallel way to handle them. For the top 10, let each reducer find its top 10 for its small piece of data, then find the “top of the top”. You can even add intermediate steps to that if you need to, so in Big O Notation we would say it is `O(log n)`. For the sum of all numbers, each reducer can sum its own small group, then we can add the sums together. Again, this is `O(log n)`.

While this is great for those specific examples, it would be nice if we had a general rule for knowing when we can expect our reducer to work in parallel. It’s not quite as simple as the mapper, because we’re never going to have independence between the parallel steps. But there is a concept we can bring to bear.

## Associativity

The associative property is (still) taught in elementary-level math. It’s the idea that, for some operations, the order in which you apply them doesn’t matter. Addition is of course associative, as `(4 + 5) + 3 = 4 + (5 + 3)`. It turns out that if we can write a function that is associative, then we can make a parallel reducer out of it. This is why our ranking example above works, because the function `max()` is associative: `max(max(a, b), c) == max(a, max(b, c)`.

(Side note: this is assuming that items can’t have identical rank, or that we don’t care if identically ranked items have the same order.)

As it turns out, even this is a little strict, because there are tricks we can play with our reducer in cases where it is not quite all the way associative (i.e. it is “semiassociative”). This topic deserves its own full post, since it needs a more detailed example, so I’ll save it for next time.