Amortized analysis helps determine the average time complexity of a sequence of operations, especially in cases where some operations are expensive but happen rarely. Here’s how it works with a dynamic array and how it compares with worst-case and average-case analysis:

Amortized Analysis

• Definition: It averages the time complexity over a sequence of operations, even if some individual operations are costly.
• Dynamic Array Example: When a dynamic array (like an array list) grows, it sometimes needs to resize (copy elements to a new, larger array). This resizing operation is costly, but it doesn’t happen every time you add an element.
• Amortized Cost: By spreading the cost of resizing over many insertions, the average cost per insertion is lower than the worst-case cost. For example, resizing happens log(n) times, so the average cost per insertion remains constant (O(1)).

Comparison

• Worst-Case Analysis: Looks at the maximum time complexity for a single operation. For resizing, this is O(n), which can be misleading because it doesn’t show that resizing happens infrequently.
• Average-Case Analysis: Looks at the expected time complexity over many operations but doesn’t always account for the infrequent expensive operations as clearly as amortized analysis.

Scenario with Different Insights

• Dynamic Array Resizing: Worst-case analysis might suggest that adding an element takes O(n) time due to resizing. Average-case analysis might not clearly show that the resizing cost is spread out. Amortized analysis shows that, on average, each insertion takes O(1) time, providing a clearer picture of the overall efficiency.

In summary, amortized analysis gives a better understanding of performance over time by smoothing out the cost of occasional expensive operations.