Shellsort

Shell Sort is the first sorting algorithm we'll look at that requires fewer than O(N2) comparisons and exchanges, on average. Although it is easy to develop an intuitive sense of how this algorithm works, it is very difficult to analyze its execution time.

Shell Sort is really just an extension of Insertion Sort, with two observations in mind:

Insertion Sort is efficient if the input is "almost sorted". Insertion Sort is inefficient, on average, because it moves values just one position at a time. Shell Sort is similar to Insertion Sort, but it works by taking "bigger steps" as it rearranges values, gradually decreasing the step size down towards one. In the end, Shell Sort performs a regular Insertion Sort but by then, the array of data is guaranteed to be "almost sorted".

Consider a small value that is initially stored in the wrong end of the array. Using Insertion Sort, it will take roughly N comparisons and exchanges to move this value all the way to the other end of the array. With Shell Sort, we'll first move values using giant step sizes, so a small value will move a long way towards it's final position, with just a few comparisons and exchanges.