Conference: The 46th ACM Symposium on Theory of Computing (STOC 2014).

Abstract:
We give algorithms for geometric graph problems in the modern parallel models such as 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 (STOC 2012) have 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, which they raised as an open question. 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.

Full version: [arXiv]