Let's break down the asymptotic bounds for insertion sort in different scenarios:

1. Best Case Runtime:

• Asymptotic Bound: When the input is already sorted, insertion sort has a best-case runtime where each element only needs to be compared once.
• Applicable Bounds:
  • O(N): The algorithm runs in linear time.
  • Θ(N): The best-case runtime is also tightly bounded by Θ(N), meaning the runtime grows linearly with the size of the input.
  • Ω(N): The best-case runtime has a lower bound of Ω(N), indicating that the minimum time required is linear.

2. Worst Case Runtime:

• Asymptotic Bound: When the input is in reverse order, insertion sort has the worst-case runtime, where each new element must be compared with all previously sorted elements.
• Applicable Bounds:
  • O(N²): The algorithm runs in quadratic time.
  • Θ(N²): The worst-case runtime is tightly bounded by Θ(N²), indicating that the runtime grows quadratically with the size of the input.
  • Ω(N²): The worst-case runtime has a lower bound of Ω(N²), meaning the minimum time required is quadratic.

3. Overall Runtime:

• Asymptotic Bound: Considering all cases, the most general description of insertion sort’s runtime is the worst-case scenario, as it encompasses all possible cases.
• Applicable Bounds:
  • O(N²): The overall time complexity is quadratic in the worst case.
  • Θ(N²): The overall time complexity is quadratically bounded.
  • Ω(N): The minimum possible runtime is linear in the best case, so the overall lower bound is Ω(N).

Summary of Applicable Bounds:

• Best Case: O(N), Θ(N), Ω(N)
• Worst Case: O(N²), Θ(N²), Ω(N²)
• Overall: O(N²), Θ(N²), Ω(N)

Would you like any further details or explanations on any of these bounds?

Related Questions:

1. What is the difference between O, Θ, and Ω notation?
2. How does insertion sort compare to other sorting algorithms in terms of asymptotic bounds?
3. What is the average-case runtime of insertion sort?
4. How can the best-case scenario be practically achieved for insertion sort?
5. Why is the worst-case scenario quadratic for insertion sort?
6. Can insertion sort be optimized to improve its worst-case runtime?
7. How does insertion sort perform on partially sorted data?
8. What are some real-world applications where insertion sort is preferable?

Tip: Remember that Θ notation gives the most precise characterization of an algorithm's runtime because it provides both an upper and lower bound.