Given the worst-case runtime analysis of the function, let's identify all applicable bounds.

Worst-Case Complexity Analysis:

• Worst-case time complexity: $O(N^2)$ when the length of the array $N$ is odd.

Understanding the Bounds:

1. Big-O Notation (Upper Bound):

   • O(1): This does not apply because the worst-case scenario takes much more than constant time.
   • O(N): This does not apply because the worst-case scenario is more than linear time.
   • O(N^2): This applies because $O(N^2)$ correctly describes the upper bound in the worst case.

2. Big-Omega Notation (Lower Bound):

   • Ω(1): This applies because the algorithm will always take at least constant time, even in the worst case.
   • Ω(N): This applies because the algorithm will always take at least linear time in the worst case.
   • Ω(N^2): This applies because in the worst case (odd $N$), the algorithm takes $N^2$ time, making $N^2$ the lower bound.

3. Theta Notation (Tight Bound):

   • θ(1): This does not apply because the worst-case scenario is not constant time.
   • θ(N): This does not apply because the worst-case scenario is not linear time.
   • θ(N^2): This applies because in the worst case, the runtime is exactly $N^2$, making it both the upper and lower bound.

Applicable Bounds:

• Ω(1)
• Ω(N)
• Ω(N^2)
• O(N^2)
• θ(N^2)

These bounds comprehensively describe the worst-case behavior of the algorithm.

Would you like to explore more details, or do you have any other questions?

Follow-up Questions:

1. How would the bounds change if the array length is even?
2. Can you explain why $O(N)$ and $θ(N)$ do not apply in this context?
3. What would happen to the complexity if we added an additional nested loop inside the current loops?
4. How does the choice of $N$ being odd versus even impact real-world performance?
5. Could this function be optimized to improve its worst-case time complexity?

Tip: Remember that $θ$ notation describes both the upper and lower bounds, meaning the runtime will always be within a constant factor of this bound.