1) Sorting the Sequence 3, 1, 4, 1, 5, 9, 2, 6, 5 Using Insertion Sort

Insertion Sort Process:

Insertion sort works by iterating through the list and growing a sorted portion at the beginning of the list. At each step, it takes one element from the unsorted portion and inserts it into its correct position in the sorted portion.

Initial List:
[3, 1, 4, 1, 5, 9, 2, 6, 5]

Let’s go step-by-step through the process:

• Step 1: Start with the second element, 1. Compare it with the first element 3. Since 1 < 3, swap them.
  List after step 1:
  [1, 3, 4, 1, 5, 9, 2, 6, 5]

• Step 2: Move to the third element 4. Compare it with 3 (no swap needed as 4 > 3).
  List after step 2:
  [1, 3, 4, 1, 5, 9, 2, 6, 5]

• Step 3: Move to the fourth element 1. Compare it with 4, 3, and 1. It should be placed at the front.
  List after step 3:
  [1, 1, 3, 4, 5, 9, 2, 6, 5]

• Step 4: Move to the fifth element 5. Compare it with 4 (no swap needed as 5 > 4).
  List after step 4:
  [1, 1, 3, 4, 5, 9, 2, 6, 5]

• Step 5: Move to the sixth element 9. Compare it with 5 (no swap needed as 9 > 5).
  List after step 5:
  [1, 1, 3, 4, 5, 9, 2, 6, 5]

• Step 6: Move to the seventh element 2. Compare it with 9, 5, 4, 3, and 1. It should be placed between 3 and 4.
  List after step 6:
  [1, 1, 2, 3, 4, 5, 9, 6, 5]

• Step 7: Move to the eighth element 6. Compare it with 9 (swap), then compare it with 5 (no swap needed as 6 > 5).
  List after step 7:
  [1, 1, 2, 3, 4, 5, 6, 9, 5]

• Step 8: Move to the ninth element 5. Compare it with 9, 6, and 5. It should be placed between 6 and 9.
  List after step 8:
  [1, 1, 2, 3, 4, 5, 5, 6, 9]

Final Sorted List:

[1, 1, 2, 3, 4, 5, 5, 6, 9]

2) Running Time of Insertion Sort When All Elements Are Equal

When all elements in the array are equal, insertion sort will compare each element with the previous one and find that no swaps are needed. The number of comparisons still happens, but no movement of elements occurs.

In this case:

• For the first element, no comparisons are made (as it is the only element).
• For the second element, it will be compared with the first element, and since they are equal, no shift occurs.
• This continues for all elements.

Therefore, the number of comparisons is still proportional to the size of the array n. In the worst case, insertion sort performs n-1 comparisons for an array of size n. However, the number of shifts will be zero since all elements are equal.

Thus, the time complexity for insertion sort when all elements are equal is O(n), where n is the number of elements in the array.

---

Would you like further details or clarification on the insertion sort process? Feel free to ask!

Here are 5 related questions to expand your understanding:

1. What is the worst-case time complexity of insertion sort?
2. How does insertion sort perform in a nearly sorted list?
3. Can insertion sort be used for sorting large datasets efficiently? Why or why not?
4. What are the advantages and disadvantages of insertion sort compared to other sorting algorithms like merge sort or quicksort?
5. How does the space complexity of insertion sort compare to that of merge sort?

Tip: Insertion sort is highly efficient for small or nearly sorted datasets but performs poorly on larger unsorted datasets due to its quadratic time complexity.