Shell sort is an efficient sorting algorithm developed by Donald Shell in 1959, improving on insertion sort by dividing the input into smaller subarrays that are sorted using predefined gap sequences. The algorithm's time complexity varies, being O(n^2) in the worst case and as efficient as O(n log^2 n) in the best case, with advantages including in-place sorting and effectiveness on small to medium-sized lists, while its limitations include the effect of gap sequence choice on performance and lack of stability. Overall, Shell sort is a viable option for specific sorting needs despite its drawbacks.

1. Shell sort
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2. OUTLINE
• 1 Introduction to Shell Sort
• 2 How Shell Sort Works
• 3 Steps of Shell Sort
• 4 Example
• 5 Time Complexity of shell sort
• 6 Advantages and Disadvantages
• 7 Conclusion

3. INTRODUCATION
• Shell sort is an efficient sorting algorithm.
• Also known as Shell's method or the diminishing increment sort.
• Invented by Donald Shell in 1959.
• It's an improvement over insertion sort.

4. How Shell Sort Works
• Shell Sort divides the input into smaller subarrays.
• It sorts the subarrays using the insertion sort algorithm.
• The subarrays are chosen based on a predefined sequence of gaps.

5. Steps of Shell Sort
1.Define a sequence of gaps.
2.Start with the largest gap and divide the list into subarrays.
3.Sort each subarray using the insertion sort algorithm.
4.Reduce the gap and repeat steps 2 and 3 until the gap is 1.
5.Finally, perform a final insertion sort on the entire array.

6. Example
• Let's take an example to understand Shell Sort better.
• Consider the following unsorted array: [9, 5, 7, 3, 2, 1, 8, 4, 6].
• We'll use the sequence of gaps: [5, 3, 1].
•

7. Example (continued)
• With a gap of 5: [1, 5], [2, 1], [8, 4], [7, 3], [9, 6].
• Using insertion sort on each subarray:
• [1, 5] -> [1, 5]
• [2, 1] -> [1, 2]
• [8, 4] -> [4, 8]
• [7, 3] -> [3, 7]
• [9, 6] -> [6, 9]

8. Example (continued)
• With a gap of 3: [1, 2, 4, 3, 6, 5], [5, 7, 8, 9].
• Using insertion sort on each subarray:
• [1, 2, 4, 3, 6, 5] -> [1, 2, 3, 4, 5, 6]
• [5, 7, 8, 9] -> [5, 7, 8, 9]
• With a gap of 1, perform a final insertion sort on the entire array:
[1, 2, 3, 4, 5, 6, 7, 8, 9].
• The array is now sorted.

9. Time Complexity
• The time complexity of Shell Sort depends on the chosen gap
sequence.
• In the worst-case scenario, the time complexity is O(n^2).
• In the best-case scenario, it can be as low as O(n log^2 n).

10. Advantages and Disadvantages
• Advantages:
• Shell Sort is an in-place sorting algorithm.
• It performs well on small to medium-sized lists.
• Disadvantages:
• The choice of gap sequence affects the efficiency of the
algorithm.
• It is not stable, meaning it may change the relative order of equal
elements

11. Conclusion
• Shell Sort is an efficient sorting algorithm based on the insertion
sort technique.
• It divides the input into smaller subarrays and sorts them using
insertion sort.
• Although it has some limitations, Shell Sort can be a good choice
for small to medium-sized lists.