How many passes are required in Shell Sort?

Shell sort, also known as the diminishing increment sort, is a sorting algorithm that improves the efficiency of insertion sort by using a sequence of gaps to sort the elements. The number of passes required in shell sort depends on the gap sequence used.

Gap Sequence:

The gap sequence determines the number of passes required in shell sort. A gap sequence is a series of numbers that define the intervals at which elements are compared and swapped. The most common gap sequence used in shell sort is the Knuth sequence, which is defined as follows:

h = 3h + 1, where h is the current gap value.

Number of Passes:

The number of passes required in shell sort is equal to the number of gaps in the gap sequence. For example, if the Knuth sequence is used, the number of passes will be:

h = 3h + 1
h = 9h + 4
h = 27h + 13
h = 81h + 40
...

As you can see, the number of gaps in the Knuth sequence is infinite. However, in practice, the sorting process is stopped when the gap value becomes 1. This is because when the gap value is 1, shell sort becomes equivalent to insertion sort.

Optimal Number of Passes:

The optimal number of passes in shell sort is not known. However, based on studies have shown that the Knuth sequence provides good performance in most cases. The number of passes required using the Knuth sequence is approximately:

log2(n), where n is the number of elements to be sorted.

Conclusion:

The number of passes required in shell sort depends on the gap sequence used. The most common gap sequence used is the Knuth sequence, which provides good performance in most cases. The optimal number of passes is not known, but based on studies have shown that the Knuth sequence requires approximately log₂₍n) passes.