On the Computational Complexity of MapReduce

On the Computational Complexity of MapReduce by Jeremy Kun.

From the post:

I recently wrapped up a fun paper with my coauthors Ben Fish, Adam Lelkes, Lev Reyzin, and Gyorgy Turan in which we analyzed the computational complexity of a model of the popular MapReduce framework. Check out the preprint on the arXiv.

As usual I’ll give a less formal discussion of the research here, and because the paper is a bit more technically involved than my previous work I’ll be omitting some of the more pedantic details. Our project started after Ben Moseley gave an excellent talk at UI Chicago. He presented a theoretical model of MapReduce introduced by Howard Karloff et al. in 2010, and discussed his own results on solving graph problems in this model, such as graph connectivity. You can read Karloff’s original paper here, but we’ll outline his model below.

Basically, the vast majority of the work on MapReduce has been algorithmic. What I mean by that is researchers have been finding more and cleverer algorithms to solve problems in MapReduce. They have covered a huge amount of work, implementing machine learning algorithms, algorithms for graph problems, and many others. In Moseley’s talk, he posed a question that caught our eye:

Is there a constant-round MapReduce algorithm which determines whether a graph is connected?

After we describe the model below it’ll be clear what we mean by “solve” and what we mean by “constant-round,” but the conjecture is that this is impossible, particularly for the case of sparse graphs. We know we can solve it in a logarithmic number of rounds, but anything better is open.

In any case, we started thinking about this problem and didn’t make much progress. To the best of my knowledge it’s still wide open. But along the way we got into a whole nest of more general questions about the power of MapReduce. Specifically, Karloff proved a theorem relating MapReduce to a very particular class of circuits. What I mean is he proved a theorem that says “anything that can be solved in MapReduce with so many rounds and so much space can be solved by circuits that are yae big and yae complicated, and vice versa.

But this question is so specific! We wanted to know: is MapReduce as powerful as polynomial time, our classical notion of efficiency (does it equal P)? Can it capture all computations requiring logarithmic space (does it contain L)? MapReduce seems to be somewhere in between, but it’s exact relationship to these classes is unknown. And as we’ll see in a moment the theoretical model uses a novel communication model, and processors that never get to see the entire input. So this led us to a host of natural complexity questions:

  1. What computations are possible in a model of parallel computation where no processor has enough space to store even one thousandth of the input?
  2. What computations are possible in a model of parallel computation where processor’s can’t request or send specific information from/to other processors?
  3. How the hell do you prove that something can’t be done under constraints of this kind?
  4. How do you measure the increase of power provided by giving MapReduce additional rounds or additional time?

These questions are in the domain of complexity theory, and so it makes sense to try to apply the standard tools of complexity theory to answer them. Our paper does this, laying some brick for future efforts to study MapReduce from a complexity perspective.

Given the prevalence of MapReduce, progress on understanding what is or is not possible is an important topic.

The first two complexity questions strike me as the ones most relevant to topic map processing with map reduce. Depending upon the nature of your merging algorithm.

Enjoy!

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