0.6 Sorting  (Page 9/20)

Shell sort improves insertion sort by comparing elements separated by a gap of several positions. This lets an element take "bigger steps" toward its expected position. Multiple passes over the data are taken with smaller and smaller gap sizes. The last step of Shell sort is a plain 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 an O(n2) sort such as bubble sort or insertion sort , it will take roughly n comparisons and exchanges to move this value all the way to the other end of the array. Shell sort first moves values using giant step sizes, so a small value will move a long way towards its final position, with just a few comparisons and exchanges.

One can visualize Shellsort in the following way: arrange the list into a table and sort the columns (using an insertion sort ). Repeat this process, each time with smaller number of longer columns. At the end, the table has only one column. While transforming the list into a table makes it easier to visualize, the algorithm itself does its sorting in-place (by incrementing the index by the step size, i.e. using i += step_size instead of i++).

For example, consider a list of numbers like [ 13 14 94 33 82 25 59 94 65 23 45 27 73 25 39 10 ]. If we started with a step-size of 5, we could visualize this as breaking the list of numbers into a table with 5 columns. This would look like this:

13 14 94 33 82

25 59 94 65 23

45 27 73 25 39

10

We then sort each column, which gives us

10 14 73 25 23

13 27 94 33 39

25 59 94 65 82

45

When read back as a single list of numbers, we get [ 10 14 73 25 23 13 27 94 33 39 25 59 94 65 82 45 ]. Here, the 10 which was all the way at the end, has moved all the way to the beginning. This list is then again sorted using a 3-gap sort, and then 1-gap sort (simple insertion sort).

Gap sequence

The gap sequence is an integral part of the shellsort algorithm. Any increment sequence will work, so long as the last element is 1. The algorithm begins by performing a gap insertion sort, with the gap being the first number in the gap sequence. It continues to perform a gap insertion sort for each number in the sequence, until it finishes with a gap of 1. When the gap is 1, the gap insertion sort is simply an ordinary insertion sort , guaranteeing that the final list is sorted.

The gap sequence that was originally suggested by Donald Shell was to begin with N / 2 and to halve the number until it reaches 1. While this sequence provides significant performance enhancements over the quadratic algorithms such as insertion sort , it can be changed slightly to further decrease the average and worst-case running times. Weiss' textbook [4] demonstrates that this sequence allows a worst case O(n2) sort, if the data is initially in the array as (small_1, large_1, small_2, large_2, ...) - that is, the upper half of the numbers are placed, in sorted order, in the even index locations and the lower end of the numbers are placed similarly in the odd indexed locations.