Complexity of Sorting Algorithms Sort a number of n integers. The value of the integers ranges from 0 to n4-1. Please provide the worst-case running time for the following sorting algorithms (1) ~(5) using big-O or big-Θ notation wherever appropriate. (1) Insertion Sort: (2) Merge Sort: (3) Quick Sort: (4) Counting Sort: (5) Radix Sort:

1. Insertion Sort:

Worst case time complexity of insertion Sort is O(n^2), this can be possible when the array is reverse order when there is

n*(n-1)/2 inversion occurs . In best case time complexity is O(N) and in avg case time complexity is O(N^2).

The worst case time complexity is O(N^2)

2. Merge Sort :

Time complexity of Merge Sort is O(n*Log n) in worst case , avg case and best case also. In merge sort the whole array is divided into 2 sub-array and calling the merge sort on these subarray as well as merge these two halves.

T(n)= 2T(n/2)+O(n) solving these recurrence will give T(n)=O(NlogN)

The worst case time complexity is O(NlogN)

3. Quick Sort :

The worst case time complexity QuickSort is O(n²).

The worst case situation occurs when the input array is sorted or reverse given or last or first element is pivot.

The worst-case situation occurs when the picked pivot is always an smallest or largest element .

The worst case time complexity is O(N^2).

4. Counting Sort :.

Counting sort basically works with counting the number of objects having distinct key values

Counting sort uses O(N+K) time in worst case

5. Radix Sort :

This sorting algorithm uses counting sort as a subroutine to sort an array of numbers .

Radix sort uses O(N*K) time in the worst case .