no choose all of the options that can apply since there can be multiple options for What asymptotic bounds describe the best case runtime for insertion sort?worst case? OVERALL? out of O(1),  O(N),  O(N^2), THETA(1),  THETA(N),  THETA(N^2), OMEGA(1),  OMEGA(N),  OMEGA(N^2)

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.

Explore the asymptotic bounds (O(N), Θ(N), Ω(N), O(N²), Θ(N²), Ω(N²)) describing the best-case, worst-case, and overall runtime complexities of insertion sort. Learn how insertion sort performs under different scenarios and its implications for algorithmic analysis.

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