Question

Given an n-element unsorted array A of n integers and an integer k, describe and implement a recursive algorithm for rearranging the elements in A so that all elements less than or equal to k come before any elements larger than k. What is the running time of your algorithm? Please prove the running time in the comments . in c++ please

Answer 1

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Sample output:

Code to copy:

// Include the required header files.

#include <iostream>

// Use the standard namespace.

using namespace std;

// Define the function to rearrange the elements.

void rearrange(int A[], int k, int start, int end)

{

    // Return if the starting index is equal

    // to the last index.

    if(start == end)

    {

        return;

    }

    // Otherwise, do the following.

    else

    {

        // Swap the element at A[start] with the

        // element at A[end] if the value of A[start]

        // is greater than k.

        if(A[start] > k)

        {

            int temp = A[start];

            A[start] = A[end];

            A[end] = temp;

            // Decrement the value of end by 1

            // and call the function recursively.

            rearrange(A, k, start, end-1);

        }

        // Otherwise, increment the value of start

        // by 1 and call the function recursively.

        else

        {

            rearrange(A, k, start+1, end);

        }

    }

}

// Define the main() function.

int main()

{

    // Declare the required variables.

    int A[] = {100,-1, 4,3,2,0,23,34,6,7,102};

    int n = sizeof(A)/sizeof(A[0]);

    int k = 20;

    cout << "k = " << k << endl;

    cout << "The original array is as follows:" << endl;

    // Start the loop to print the original array.

    for(int i=0; i<n; i++)

    {

        cout << A[i] << " ";

    }

    // Call the function to rearrange the elements.

    rearrange(A, k, 0, n-1);

    cout << endl;

    cout << "\nThe modified array is as follows:" << endl;

    // Start the loop to print the modified array.

    for(int i=0; i<n; i++)

    {

        cout << A[i] << " ";

    }

    // Return 0 and exit the program.

    return 0;

}

The running time of the above code/algorithm is as follows:

• The function rearrange() is called initially with the starting value of start = 0 and end = size of the array – 1.
• In each recursive call, the function is either called by increasing the value of start by 1 or decrement the value of end by 1.
• The base case returns from the recursive calls when the value of start and end is equal.
• Therefore, the total number of recursive calls made by the function is equal to n, where n is the size of the input array.
• All the other operations are performed in constant time.

Hence, the total running time of the algorithm is O(n) + C.