Combining Counting Sort and Bucket Sort can really boost performance in certain situations, especially when you need to sort a lot of items that are within a small range of values. Let’s dive into how these two methods work well together.

First, Counting Sort is super effective when the range of possible values (the difference between the highest and lowest numbers) is small compared to how many items you want to sort, which we refer to as $n$. The time it takes for Counting Sort to finish is $O(n + k)$, where $k$ is the range of the values. This means it can sort things almost instantly for the right types of data.

On the flip side, Bucket Sort shines when the numbers are evenly spread out over a range. This method splits the items into $m$ buckets. Each bucket is then sorted on its own, often using another method like Insertion Sort. The expected time for Bucket Sort is $O(n + m)$.

How They Can Improve Each Other:

1. Using Counting Sort First: Before we use Bucket Sort, we can apply Counting Sort to count how many times each number appears. This helps us figure out how to best put the items into buckets, making the process more efficient.

2. Better Overall Performance: By using the strengths of both sorts, we can sort items better depending on how they are arranged. If the data fits well with what Counting Sort does best, we get to enjoy its fast sorting speed.

3. Saving Memory: Combining these two methods might help us use memory more wisely. Counting Sort can help identify the range of numbers we need to focus on for Bucket Sort, which can reduce the number of buckets we use and save space.

In short, using Counting Sort before Bucket Sort can greatly improve how quickly we can sort items, especially when we know how the data is spread out. This mix takes advantage of what each method does best, making it a smart choice in the toolbox for sorting.