# Shell Sort

- December 31, 2020
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To exhibit the value of knowing properties of elementary sorts, we next consider a fast algorithm based on insertion sort. Insertion sort is slow for large unordered arrays because the only exchanges it does involve adjacent entries, so items can move through the array only one place at a time. For example, if the item with the smallest key happens to be at the end of the array, N1 exchanges are needed to get that one item where it belongs. Shellsort is a simple extension of insertion sort that gains speed by allowing exchanges of array entries that are far apart, to produce partially sorted arrays that can be efficiently sorted, eventually by insertion sort.

The idea is to rearrange the array to give it the property that taking every hth entry (starting anywhere) yields a sorted subsequence. Such an array is said to be h-sorted. Put another way, an h-sorted array is h independent sorted subsequences, interleaved together. By h-sorting for some large values of h, we can move items in the array long distances and thus make it easier to h-sort for smaller values of h. Using such a procedure for any sequence of values of h that ends in 1 will produce a sorted array: that is shellsort. The implementation is that starting at the largest increment less than N/3 and decreasing to 1. We refer to such a sequence as an increment sequence. Algorithm 2.3 computes its increment sequence; another alternative is to store an increment sequence in an array.

One way to implement shellsort would be, for each h, to use insertion sort independently on each of the h subsequences. Because the subsequences are independent, we can use an even simpler approach: when h-sorting the array, we insert each item among the previous items in its h-subsequence by exchanging it with those that have larger keys (moving them each one position to the right in the subsequence). We accomplish this task by using the insertion-sort code, but modified to decrement by h instead of 1 when moving through the array. This observation reduces the shellsort implementation to an insertion-sort-like pass through the array for each increment.

Shellsort gains efficiency by making a tradeoff between size and partial order in the subsequences. At the beginning, the subsequences are short; later in the sort, the subsequences are partially sorted. In both cases, insertion sort is the method of choice. The extent to which the subsequences are partially sorted is a variable factor that depends strongly on the increment sequence. Understanding shellsort?s performance is a challenge.