A Super-Logarithmic Lower Bound for Hypercubic Sorting Networks

Hypercubic sorting networks are a class of comparator networks whose structure maps eeciently to the hypercube and any of its bounded degree variants. Recently, n-input hypercubic sorting networks with depth 2 O(p lg lg n) lg n have been discovered. These networks are the only known sorting networks of depth o(lg 2 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 hypercubic sorting network. In this paper, we resolve this question by establishing an ? lg n lg lg n lg lg lg n lower bound on the depth of any n-input hypercubic sorting network. Our lower bound can be extended to certain restricted classes of non-oblivious sorting algorithms on hypercubic machines.