A super-logarithmic lower bound for hypercubic sorting networks

Abstract

Hypercubic 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 hypercubic 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 hypercubic 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 hypercubic sorting network. Our lower bound can be extended to certain restricted classes of non-oblivious sorting algorithms on hypercubic machines.

Supported by NSF Research Initiation Award CCR-9111591, and the Texas Advanced Research Program under Grant Nos. 003658-480 and 003658-461.

Supported by the Texas Advanced Research Program under Grant Nos. 003658-480 and 003658-461, and by a Schlumberger Graduate Fellowship.