Periodic constant depth sorting networks
Comparator networks of constant depth can be used for sorting in the following way. The computation consists of a number of iterations, say t, each iteration being a single run through the comparator network. The output of iteration j (j < t) is used as the input for iteration j+1. The output of the iteration t is the output of the computation. In such a way, it is possible to apply a network with a small number of comparators for sorting long input sequences. However, it is not clear how to make such a computation fast.
Odd-Even Transposition Sort gives a periodic sorting network of depth 2, that sorts n numbers in n/2 iterations. The network of depth 8 proposed by Schwiegelshohn  sorts n numbers in O(√nlog n) iterations. Krammer
For a fixed but arbitrary k ∃ ℕ, we present a periodic sorting network of depth O(k) that sorts n input numbers in O(k2 · n1/k) steps.
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