  We give algorithms for geometric graph problems in the modern parallel models
inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem
over a set of points in the two-dimensional space, our algorithm computes a
$(1+\epsilon)$-approximate MST. Our algorithms work in a constant number of
rounds of communication, while using total space and communication proportional
to the size of the data (linear space and near linear time algorithms). In
contrast, for general graphs, achieving the same result for MST (or even
connectivity) remains a challenging open problem, despite drawing significant
attention in recent years.
  We develop a general algorithmic framework that, besides MST, also applies to
Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic
framework has implications beyond the MapReduce model. For example it yields a
new algorithm for computing EMD cost in the plane in near-linear time,
$n^{1+o_\epsilon(1)}$. We note that while recently Sharathkumar and Agarwal
developed a near-linear time algorithm for $(1+\epsilon)$-approximating EMD,
our algorithm is fundamentally different, and, for example, also solves the
transportation (cost) problem, raised as an open question in their work.
Furthermore, our algorithm immediately gives a $(1+\epsilon)$-approximation
algorithm with $n^{\delta}$ space in the streaming-with-sorting model with
$1/\delta^{O(1)}$ passes. As such, it is tempting to conjecture that the
parallel models may also constitute a concrete playground in the quest for
efficient algorithms for EMD (and other similar problems) in the vanilla
streaming model, a well-known open problem.

Data Structures and Algorithms;Distributed, Parallel, and Cluster Computing