What is meant by "Constant Amortized Time" when talking about time complexity of an algorithm?

"Constant Amortized Time" refers to the average time complexity of an operation in an algorithm over a sequence of operations, where the average time per operation remains constant, even though individual operations may occasionally take longer. This concept is particularly useful for algorithms that have infrequent but expensive operations.

Explanation

In amortized analysis, we consider the total cost of a sequence of operations and spread this cost evenly across all operations. This approach provides a more realistic measure of an algorithm's performance compared to worst-case analysis, which might be overly pessimistic.

Example: Dynamic Array

Consider a dynamic array that doubles its size when it runs out of space. The insertion operation in such an array can be broken down into two cases:

1. Normal Insertion: When there is space available, the insertion takes constant time, $$O(1)$$.
2. Resizing Insertion: When the array is full, a new array of double the size is created, and all elements are copied to the new array. This operation takes $$O(n)$$ time, where $$n$$ is the number of elements in the array.

Although resizing is expensive, it happens infrequently. If we analyze the cost over a sequence of $$n$$ insertions, the total cost of all operations is $$O(n)$$. Therefore, the average cost per insertion, or the amortized cost, is $$O(1)$$.

Methods of Amortized Analysis

1. Aggregate Method: Calculate the total cost of $$n$$ operations and divide by $$n$$ to find the average cost per operation.
2. Accounting Method: Assign a cost to each operation such that the total assigned cost is an upper bound on the actual total cost.
3. Potential Method: Use a potential function to represent the "stored work" that can be used to pay for future expensive operations.

Conclusion

Constant Amortized Time means that while some operations may occasionally take longer, the aver...