Amortized analysis is a helpful way to understand how much space we use over time, especially with algorithms that change a lot.

Let’s say you’re using a dynamic array. At first, you make space for a certain number of items, let’s call it $n$. But as you add more items, if you need to create a bigger array, you have to copy everything over to the new one. This can seem like it takes a lot of space just for that one time when you expand.

But here’s where amortized analysis comes in. Instead of just thinking about that one big move, we look at the average space used over many actions. If you double your array size every time you run out of room, you might realize that the total space you need across all those insertions is much lower. In fact, this can bring the average space requirement down to something simple like $O(n)$.

Let’s look at a real example. Imagine you start with an empty array. Each time you add an item and fill it up, sure, you’ll need to make a bigger array. But usually, adding items is easy and doesn’t need extra space since they don’t cause a new array to be made.

So, while the worst-case scenario might look like it uses a lot of space, amortized analysis helps us see the bigger picture. It shows us that when we look at the average use over many actions, it’s not as bad as it may seem.

In the end, by understanding space usage through amortized analysis, we get a clearer picture. It highlights why it’s important to think about average costs instead of just focusing on the worst case. This can help us make better choices when designing and analyzing algorithms.