Joint Size and Depth Optimization of Sorting Networks

Sorting networks are oblivious sorting algorithms with many interesting theoretical properties and practical applications. One of the related classical challenges is the search of optimal networks respect to size (number of comparators) of depth (number of layers). However, up to our knowledge, the joint size-depth optimality of small sorting networks has not been addressed before. This paper presents size-depth optimality results for networks up to $12$ channels. Our results show that there are sorting networks for $n\leq9$ inputs that are optimal in both size and depth, but this is not the case for $10$ and $12$ channels. For $n=10$ inputs, we were able to proof that optimal-depth optimal sorting networks with $7$ layers require $31$ comparators while optimal-size networks with $29$ comparators need $8$ layers. For $n=11$ inputs we show that networks with $8$ or $9$ layers require at least $35$ comparators (the best known upper bound for the minimal size). And for networks with $n=12$ inputs and $8$ layers we need $40$ comparators, while for $9$ layers the best known size is $39$.