Data Structures and Algorithms MCQ - Performance Analysis of Algorithms and Recurrences

111 Which one of the following is the recurrence equation for the worst case time complexity of the Quicksort algorithm for sorting n(≥ 2) numbers? In the recurrence equations given in the options below, c is a constant.

The recurrence equation is
T(n) = T(n – 1) + T(1) + cn
In quick sort, worst case divide the list into two list with 1 and (n – 1) elements each time.

112 Suppose you are provided with the following function declaration in the C programming language.
int partition (int a[ ], int n);
The function treats the first element of a[ ] as a pivot, and rearranges the array so that all elements less than or equal to the pivot is in the left part of the array, and all elements greater than the pivot is in the right part. In addition, it moves the pivot so that the pivot is the last element of the left part. The return value is the number of elements in the left part. The following partially given function in the C programming language is used to find the kth smallest element in an array a[ ] of size n using the partition function. We assume k < n.

int kth_smallest (int a[ ], int n, int k)
{
int left_end = partition (a, n);
if(left_end+ 1==k)
{
return a [left_end];
}
if(left end+l>k)
{
return kth_smallest (_______);
}
else
{
return kth_smallest (_______);
}
}
The missing argument lists are respectively

When we put (a, left–End, K) and (a+ left – end + 1, n– left – end –1, k– left–end –1)at the place of fill in the blank algorithm work for partition.

113 The usual Θ(n²) implementation of Insertion Sort to sort an array uses linear search to identify the position where an element is to be inserted into the already sorted part of the array. If instead, we use binary search to identify the position, the worst case running time will

Complexity is remain same because we had swap after finding to position of element so in worst case swapping for each element requires Q (n) time hence complexity is O (n²).

114 Merge sort uses

If we take example, 40, 30, 10, 12, 85, 48, 9, 16
Next step : 30, 40, 10, 12, 35, 48, 9, 16
Next step : 10, 12, 30, 40, 9, 16, 35, 48

115 What would be the worst case time complexity of the insertion sort algorithm, if the inputs are restricted to permutations of l...n with at most n inversions?

If array of size n with n inversion then complexity of insertion sort is Q (n)
Let us take an example.
(6 1 2 3 5 6 4)
total no. of comparision require to sort above array is 2 (n–1) = 2* 5 = 10
so complexity is Q (n).