Optimum combinations of sorting and merging

• Authors: G. K. Manacher;T. D. Bui;T. Mai
• Venue: Journal of the ACM (JACM)
• Year: 1989

In 1979, G. K. Manacher showed that the Ford-Johnson sorting algorithm [FJA], acting on t real numbers, can be beaten for an infinite set of values t. These values form a partial cover of constant density not close to 1 over an initial sequence of each band running from uk = ⌊(4/3)2k⌋ to uk+l - 1. This early result depended on showing that the Hwang-Lin merging algorithm [HLA], merging m elements with n, m ≠ n, could be surpassed by cm comparisons, where c is an arbitrary small positive constant.In this paper, it is shown that the FJA can be beaten for a set of integers of asymptotic density 1 under the greatly weakened assumption that the HLA can be surpassed by only (1/2 + &egr;)log m comparisons, with &egr; a small positive constant. The even weaker assumption that no improvement in the HLA exists, but that an isolated value to exists for which the FJA can be surpassed by only (1 + &egr;)log to comparisons yields the same result. Only for a set of “refractory” integers of size about t1/2 in the neighborhood of each uk does the FJA fail to be beaten.All these results depend on a new technique for obtaining optimum sort-merge sequences for best-possible sorting given a merging method. The technique turns out to be amenable to precise asymptotic analysis. When the technique is applied using the most powerful known merging algorithm [Christen's], the density mentioned above is still 1, but islands of refractory points still remain, this time forming sets provably of size &THgr;(log2t) in the neighborhood of each uk.It is shown that if “information theoretic” merging were achievable, the FJA could be beaten for all t u10 = 1365. From these results and a few others, we adduce evidence in support of our main conjecture: that even optimum combinations of optimum merging and Ford-Johnson sorting will not beat the FJA when t = uk, but will instead produce refractory regions of size &THgr;(log2t) in the neighborhood of each &lgr;.