The Worst Case in Shellsort and Related Algorithms

We show that sorting a sufficiently long list of length N using Shellsort with m increments (not necessarily decreasing) requires at least N √ m comparisons in the worst case, for some constant c > 0. For m ≤ (log N/ log log N) we obtain an upper bound of the same form. We also prove that Ω(N(log N/ log log N)) comparisons are needed regardless of the number of increments. Our approach is general enough to apply to other sorting algorithms, including Shaker-sort, for which an even stronger result is proved.