shellsort

His thesis focused on analysis of the shellsort sorting algorithm and sorting networks.

As the example illustrates, the subarrays that Shellsort operates on are initially short; later they are longer but almost ordered.

It is, however, unknown whether Shellsort can reach this asymptotic order of average-case complexity, which is optimal for comparison sorts.

The graph below shows the average number of element comparisons in various variants of Shellsort, divided by the theoretical lower bound, i.e. logN!

Gonnet and Baeza-Yates observed that Shellsort makes the fewest comparisons on average when the ratios of successive gaps are roughly equal to 2.2.

Analysis of Shellsort and Related Algorithms, Robert Sedgewick, Fourth European Symposium on Algorithms, Barcelona, September 1996.

Therefore Shellsort has prospects of running in an average time that asymptotically grows like NlogN only when using gap sequences whose number of gaps grows in proportion to the logarithm of the array size.

His result was refined by Janson and Knuth: the average number of comparisons made during a Shellsort with three gaps (ch, cg, 1), where h and g are coprime, is in the first pass, in the second pass and in the third pass.