GYM: A Multiround Join Algorithm In MapReduce
  Multiround algorithms are now commonly used in distributed data processing systems, yet the extent to which algorithms can benefit from running more rounds is not well understood. This paper answers this question for a spectrum of rounds for the problem of computing the equijoin of $n$ relations. Specifically, given any query $Q$ with width $\w$, {\em intersection width} $\iw$, input size $\mathrm{IN}$, output size $\mathrm{OUT}$, and a cluster of machines with $M$ memory available per machine, we show that:   (1) $Q$ can be computed in $O(n)$ rounds with $O(n\frac{(\mathrm{IN}^{\w} + \mathrm{OUT})^2}{M})$ communication cost.   (2) $Q$ can be computed in $O(\log(n))$ rounds with $O(n\frac{(\mathrm{IN}^{\max(\w, 3\iw)} + \mathrm{OUT})^2}{M})$ communication cost. \end{itemize} Intersection width is a new notion of queries and generalized hypertree decompositions (GHDs) of queries we introduce to capture how connected the adjacent cyclic components of the GHDs are.   We achieve our first result by introducing a distributed and generalized version of Yannakakis's algorithm, called GYM. GYM takes as input any GHD of $Q$ with width $\w$ and depth $d$, and computes $Q$ in $O(d + \log(n))$ rounds and $O(n\frac{(\mathrm{IN}^{\w} + \mathrm{OUT})^2}{M})$ communication cost. We achieve our second result by showing how to construct GHDs of $Q$ with width $\max(\w, 3\iw)$ and depth $O(\log(n))$. We describe another technique to construct GHDs with longer widths and shorter depths, demonstrating a spectrum of tradeoffs one can make between communication and the number of rounds. 
Foto Afrati, Manas Joglekar, Christopher R\'e, Semih Salihoglu,   Jeffrey D. Ullman
cs.DB