A superlogarithmic lower bound for shuffle-unshuffle sorting networks

Shuffle-unshuffle sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, n-input shuffle-unshuffle sorting networks with depth 2^{O(√lg lg n)} lg n have been discovered. These networks are the only known sorting networks of depth o(lg² n) that are not based on expanders, and their existence raises the question of whether a depth of O (lg n) can be achieved by any shuffle-unshuffle sorting network. In this paper we resolve this question by establishing an Ω (lg n lg lg n/lg lg lg n) lower bound on the depth of any n-input shuffle-unshuffle sorting network. Our lower bound can be extended to certain restricted classes of nonoblivious sorting algorithms on hypercubic machines.