Sorting, Searching, and Simulation in the MapReduce Framework

In this paper, we study the MapReduce framework from an algorithmic standpoint and demonstrate the usefulness of our approach by designing and analyzing efficient MapReduce algorithms for fundamental sorting, searching, and simulation problems. This study is motivated by a goal of ultimately putting the MapReduce framework on an equal theoretical footing with the well-known PRAM and BSP parallel models, which would benefit both the theory and practice of MapReduce algorithms. We describe efficient MapReduce algorithms for sorting, multi-searching, and simulations of parallel algorithms specified in the BSP and CRCW PRAM models. We also provide some applications of these results to problems in parallel computational geometry for the MapReduce framework, which result in efficient MapReduce algorithms for sorting, 2- and 3-dimensional convex hulls, and fixed-dimensional linear programming. For the case when mappers and reducers have a memory/message-I/O size of $M=\Theta(N^\epsilon)$, for a small constant $\epsilon>0$, all of our MapReduce algorithms for these applications run in a constant number of rounds.