Consider three sorting algorithms: Bucket Sort, Radix Sort, and Counting Sort. For

each of the sorting algorithms, answer the following questions:

$1.$ Time Complexity: a$.$ What is the time complexity of each algorithm, both in

the best case and worst case?

b$.$ Under what conditions do these algorithms achieve their best$-$case time

complexity?

$2.$ Space Complexity:

a$.$ Compare the space complexity of Bucket Sort, Radix Sort, and Counting

Sort.

b$.$ Which algorithm requires the most auxiliary space and why?

$3.$ Suitability of Data Types: a$.$ For what types of data $($e$.$g$.,$ integers,

floating$-$point numbers, strings$)$ is each algorithm best suited?

b$.$ Can these algorithms handle negative numbers? If not, explain why and

how they could be adapted to do so$.$

$4.$ Practical Considerations:

a$.$ How does the input size and range of data values affect the performance

of each algorithm?

b$.$ For large datasets with a wide range of values, which sorting algorithm

would you choose and why?

$5.$ Real$-$World Applications:

a$.$ Provide a real$-$world scenario where each of these sorting algorithms

might be preferable over the others.