We show that if n is sufficiently large then there is a graph G of order n with ⌊n 3 2logn⌋ edges such that the transitive closure of every acyclic orientation of G has at least n 2-n 3 2logn edges. A consequence of this is that with ⌊n 3 2logn⌋ parallel processors n objects may be sorted in two time intervals. This improves considerably some results of Häggkvist and Hell. We prove similar assertions about sorting with only d-step implications. © 1983.
Discrete Applied Mathematics
Bollobás, B., & Thomason, A. (1983). Parallel sorting. Discrete Applied Mathematics, 6 (1), 1-11. https://doi.org/10.1016/0166-218X(83)90095-1