On distributing symmetric streaming computations

  • Authors:

  • Jon Feldman;S. Muthukrishnan;Anastasios Sidiropoulos;Cliff Stein;Zoya Svitkina

  • Affiliations:

  • Google, Inc., New York, NY;Google, Inc., New York, NY;Computer Science and Artificial Intelligence Laboratory (CSAIL) at MIT, Cambridge, MA;Columbia University;Cornell University

  • Venue:
  • Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2008

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Abstract

A common approach for dealing with large data sets is to stream over the input in one pass, and perform computations using sublinear resources. For truly massive data sets, however, even making a single pass over the data is prohibitive. Therefore, streaming computations must be distribued over many machines. In practice, obtaining significant speedups using distributed computations has numerous challenges including synchronization, load balancing, overcoming processor failures, and data distribution. Successful Systems in practice such as Google's MapReduce and Apache's Hadoop address these problems by only allowing a certain class of highly distributable tasks defined by local computations that can be applied in any order to the input. The fundamental question that arises is: How does the class of computational tasks supported by these systems differ from the class for which streaming solutions exist? We introduce a simple algorithmic model for massive, unordered, distributed (mud) computation, as implemented by these systems. We show that in principle, mud algorithms are equivalent in power to symmetric streaming algorithms. More precisely, we show that any symmetric (order-invariant) function that can be computed by a steraming algorithm can also be computed by a mud algorithym, with comparable space and communication complexity. Our simulation uses Savitch's theorem and therefore has superpolynomial time complexity. We extend our simulation result to some natural classes of approximate and randomized steraming algorithms. We also give negative results, using communication complexity arguments to prove that extensions to private randomness, promise problems and indeterminate functions are impossible. We also introduce an extension of the mud model to multiple keys and multiple rounds.