Pairwise sorting network

The pairwise sorting network is a sorting network discovered and published by Ian Parberry in 1992 in Parallel Processing Letters.[1] The pairwise sorting network has the same size (number of comparators) and depth as the odd–even mergesort network. At the time of publication, the network was one of several known networks with a depth of ${\displaystyle O(log^{2}n)}$. It requires ${\displaystyle n(\log n)(\log n-1)/4+n-1}$ comparators and has depth ${\displaystyle (\log n)(\log n+1)/2}$.

Pairwise sorting network

Visualization of the Pairwise sorting network with 16 inputs
Class	Sorting algorithm
Data structure	Array
Worst-case performance	${\displaystyle (\log n)(\log n+1)/2}$ parallel time
Worst-case space complexity	${\displaystyle n(\log n)(\log n-1)/4+n-1}$ non-parallel time

The sorting procedure implemented by the network is as follows (guided by the zero-one principle):

1. Sort consecutive pairwise bits of the input (corresponds to the first layer of the diagram)
2. Sort all pairs into lexicographic order by recursively sorting all odd bits and even bits separately (corresponds to the next 14 layers of the diagram)
3. Sort the pairs in nondecreasing order using a specialized network (corresponds to the final layers of the diagram)