Understanding Counting Sort, Radix Sort, and Bucket Sort

Let’s break down these three sorting methods: Counting Sort, Radix Sort, and Bucket Sort.

1. Counting Sort:

   • How Fast Is It?: It works in $O(n + k)$ time. Here, $n$ means how many items you have, and $k$ is the range of numbers you’re sorting.
   • How Much Space Does It Use?: It needs $O(k)$ space.
   • What’s the Catch?: This method only works for whole numbers and needs you to know the range of numbers in advance. This makes it a bit rigid.

2. Radix Sort:

   • How Fast Is It?: Its speed is $O(d(n + k))$, where $d$ is the number of digits in the largest number.
   • How Much Space Does It Use?: It takes up $O(n + k)$ space.
   • What’s the Catch?: This sort depends on how many digits the numbers have. It’s not very good for really big ranges or for types of numbers that aren’t whole.

3. Bucket Sort:

   • How Fast Is It?: Usually, it runs in $O(n + k)$ time, but the worst-case can be $O(n^2)$.
   • How Much Space Does It Use?: It needs $O(n + k)$ space as well.
   • What’s the Catch?: How well it works really depends on how the numbers are spread out. This makes it hard to predict how it will perform.

Solution: To avoid these problems, choose the right sorting method based on the kind of input you have.