Round Compression for Parallel Matching Algorithms
Artur Czumaj, Jakub {\L}\k{a}cki, Aleksander M\k{a}dry, Slobodan, Mitrovi\'c, Krzysztof Onak, Piotr Sankowski

TL;DR
This paper introduces a novel round compression technique that significantly reduces the number of rounds needed for parallel maximum matching algorithms in MPC frameworks, achieving near-exponential improvements over previous bounds.
Contribution
The authors develop a new method called round compression that breaks the longstanding $O( ext{log} n)$ round barrier for approximate maximum matching in MPC models with sublinear memory.
Findings
Achieves $(2+ ext{epsilon})$-approximate maximum matching in $O(( ext{log} ext{log} n)^2)$ rounds.
Breaks the $O( ext{log} n)$ round complexity barrier in MPC.
Demonstrates the power of local computation in reducing parallel algorithm rounds.
Abstract
For over a decade now we have been witnessing the success of {\em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately capture the nature of large-scale computation. In particular, compared to the classic distributed algorithms or PRAM models, these frameworks allow for much more local computation. The fundamental question that arises in this context is though: can we leverage this additional power to obtain even faster parallel algorithms? A prominent example here is the {\em maximum matching} problem---one of the most classic graph problems. It is well known that in the PRAM model one can compute a 2-approximate maximum matching in rounds. However, the exact complexity of this problem in the MPC framework is still far from understood.…
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