  We consider the problem of merging two sorted sequences on a comparator
network that is used repeatedly, that is, if the output is not sorted, the
network is applied again using the output as input. The challenging task is to
construct such networks of small depth. The first constructions of merging
networks with a constant period were given by Kuty{\l}owski, Lory\'s and
Oesterdikhoff. They have given $3$-periodic network that merges two sorted
sequences of $N$ numbers in time $12\log N$ and a similar network of period $4$
that works in $5.67\log N$. We present a new family of such networks that are
based on Canfield and Williamson periodic sorter. Our $3$-periodic merging
networks work in time upper-bounded by $6\log N$. The construction can be
easily generalized to larger constant periods with decreasing running time, for
example, to $4$-periodic ones that work in time upper-bounded by $4\log N$.
Moreover, to obtain the facts we have introduced a new proof technique.

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