Comparing each pair of elements and sorting according to
[[number less than] minus [number greater than]] is a parallel comparison
sorting algorithm with a depth of $1$ comparison and $O\left(n^2\right)$ total comparisons.
By the AKS network, there is a parallel comparison sorting algorithm with
a depth of $O(\log(n))$ comparisons and $O(n\cdot \log(n))$ total comparisons.
1.
Are there any parallel comparsion sorts with a depth of
$o(\log(n))$ comparisons and $o\left(n^2\right)$ total comparisons?
2.
Are there any parallel comparison sorts with a depth
of $1$ comparison and $O(n\cdot \log(n))$ total comparisons?