When I started learning about sorting algorithms in college, I quickly realized how important it was to understand their time complexities.

Today, let’s talk about three common sorting algorithms: Insertion Sort, Merge Sort, and Quick Sort.

Each of these algorithms works differently and has different speeds. This means you might choose one over the other depending on your needs.

1. Insertion Sort:

• Best Case: This happens when the list is already sorted. In this case, it takes $O(n)$ time since the algorithm only looks at each item once and places them where they belong.

• Average Case: If the list is random, it takes longer, at $O(n^2)$, because every item has to be compared to all the items that were sorted before it.

• Worst Case: This is also $O(n^2)$. It occurs if the list is sorted in the opposite order.

Insertion Sort is simple and works well for small lists or lists that are almost sorted. But when dealing with larger lists, it can slow down a lot.

2. Merge Sort:

• Best, Average, and Worst Case: The great thing about Merge Sort is that it has the same time of $O(n \log n)$ for all cases.

This happens because the algorithm splits the list into halves over and over (the $\log n$ part). Then it puts them back together (the $n$ part).

Even though it needs extra space for the temporary lists during merging, Merge Sort is stable and works well for larger lists. So, many people trust it for different tasks.

3. Quick Sort:

• Best Case: Quick Sort is at its best when it picks a good pivot, which keeps the parts balanced. This again gives it a time of $O(n \log n)$.

• Average Case: For the average situation, Quick Sort also remains at $O(n \log n)$. This is often because picking the pivot randomly helps keep everything balanced.

• Worst Case: The worst-case time is $O(n^2)$. This happens if the smallest (or largest) item is always picked as the pivot, which can lead to uneven parts. But, by choosing pivots wisely—like using the middle value or random choices—you can prevent this problem.

To sum it up, each sorting algorithm has its strengths. The right one to use often depends on what you need.

• Insertion Sort is great for small, nearly sorted lists.

• Merge Sort does a fantastic job with larger lists and stays consistent.

• Quick Sort is usually the go-to for average performance if you can choose your pivots wisely.