Geodesic
Analysis and optimization of a $k$-chain reduction algorithm for distributed computer systems
Numerical methods and programming, Tome 17 (2016) no. 3, pp. 318-328.

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In the LogP model of parallel computing, an analytical expression of the $k$-chain algorithm's execution time is derived. The optimal value of $k$ in the LogP model is found. A new algorithm based on the optimal value of $k$ is developed. For the reduction of root process's waiting time, an algorithm with an adaptive number of chains is proposed. The dependence of the execution time of the proposed algorithm on the number of processes has a growth rate of O(sqrt(P)), which is more efficient compared to the linear running time of the original $k$-chain algorithm. The proposed algorithms are implemented in the MPI standard and studied on computer clusters with InfiniBand QDR networks.
Keywords: MPI, root reduction, message passing models, MPI, parallel programming, distributed computer systems. 
@article{VMP_2016_17_3_a12,
     author = {M. G. Kurnosov},
     title = {Analysis and optimization of a $k$-chain reduction algorithm for distributed computer systems},
     journal = {Numerical methods and programming},
     pages = {318--328},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2016_17_3_a12/}
}
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M. G. Kurnosov. Analysis and optimization of a $k$-chain reduction algorithm for distributed computer systems. Numerical methods and programming, Tome 17 (2016) no. 3, pp. 318-328. http://geodesic.mathdoc.fr/item/VMP_2016_17_3_a12/