When we look at how fast different data structures work, there are several helpful methods we can use. These methods help us understand the best, worst, and average case scenarios in simpler ways.

• Big O Notation: This is the main way we talk about how long an algorithm takes to run. It helps us compare algorithms based on how their running time grows when we increase the amount of data. For example, if we have a loop that runs n times, we say it has a time complexity of $O(n)$.

• Amortized Analysis: This method is useful for data structures where the time it takes for different actions can vary. Let’s say we have a dynamic array that doubles in size when it gets too full. Most of the time, adding things to this array takes $O(1)$ time, but when it needs to resize, it takes $O(n)$. However, if we look at many actions together, we see that on average, it still takes $O(1)$ for each action.

• Recursion Trees: When we study algorithms that call themselves (this is called recursion), creating a recursion tree can help us figure out the time it takes to complete everything. Each point in the tree shows the time for a function call, and we can add up these times to find the total. The height of the tree can show us important information about how some algorithms behave in a logarithmic way.

• Master Theorem: For algorithms that split problems into smaller pieces, the Master Theorem gives us a straightforward method to find time complexities without doing a lot of math. If we have a problem of size n that splits into a few smaller problems, we can often write the time complexity as $T(n) = aT(n/b) + f(n)$, where $f(n)$ is the time it takes to split and combine the parts.

• Profiling and Empirical Analysis: Sometimes, looking at real data can help us understand how fast an algorithm works. By running algorithms on typical datasets, we can see how they behave and find patterns that we might not expect.

• Data Structure Properties: Knowing the basic features of certain data structures, like balanced trees, heaps, and hash tables, can make it easier to discuss their complexity. For example, a balanced binary search tree allows us to search, insert, and delete items in $O(\log n)$ time.

By using these methods, students and others can more easily understand and analyze the speed of different algorithms and their complexities.