**Introduction:** Amortized time complexity is a crucial concept in analyzing the performance of data structures. It helps us understand the average time taken per operation over a sequence of operations, considering occasional expensive operations. In this blog post, we’ll explore the amortized time complexity of two fundamental data structures in Java: ArrayList and HashTable (or HashMap).

**ArrayList:** ArrayList is a dynamic array-based implementation of the List interface in Java. Let’s dive into the amortized time complexity analysis for the `add()`

operation in ArrayList.

**Proof:**- When we add an element to the ArrayList, if the internal array is not full, the operation takes O(1) time.
- If the internal array is full and needs resizing, the
`add()`

operation may take O(n) time due to array copying. - Considering a sequence of n
`add()`

operations, resizing occurs log(n) times, and the total time for resizing is proportional to n. Here we have considered the initial size of the array as 1. - The remaining n – log(n)
`add()`

operations take O(1) time each, as they don’t involve resizing. - Therefore, the amortized time complexity per
`add()`

operation is O(1), as the total time is evenly distributed across the n operations.

**HashTable (HashMap):** HashTable (or HashMap) is a key-value pair-based data structure in Java. Let’s explore the amortized time complexity for the `put()`

operation in HashTable.

**Proof:**- In the best-case scenario, the
`put()`

operation takes O(1) time when there are no collisions. - In the worst-case scenario, where all keys hash to the same bucket, the
`put()`

operation may take O(n) time. - Assuming uniform distribution and a suitable resizing strategy, the likelihood of excessive collisions decreases as the number of entries increases.
- Therefore, the amortized time complexity per
`put()`

operation is O(1), as the total time is evenly distributed across the n operations.

- In the best-case scenario, the

**Conclusion:** Understanding amortized time complexity is essential for evaluating the performance of data structures over a sequence of operations. In ArrayList and HashTable (HashMap), the amortized time complexity analysis helps us comprehend the average time taken per operation, considering occasional expensive operations such as resizing or collisions.

By grasping these concepts, developers can make informed decisions when choosing data structures for their applications, ensuring efficient and scalable solutions.

**References:**

- Java Documentation for ArrayList and HashTable (HashMap)
- Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press.