Sort an array after applying the given equation

We have an integer array that is sorted in ascending order. We also have 3 integers A, B and C. We need to apply A*x*x + B*x + C for each element x in the array and sort the modified array. 

Example:

Input : arr[] = {-1, 0, 1, 2, 3, 4} 
       A = -1, B = 2, C = -1
Output : {-9, -4, -4, -1, -1, 0}  
Expalnation:
Input array is {-1, 0, 1, 2, 3, 4}. After
applying the equation A*x*x + B*x + C on
every element x we get, {-4,-1, 0, -1, -4, -9}
After sorting, we get {-9, -4, -4, -1, -1, 0}

Method 1 (Simple) : 

1. Apply the given equation to all the elements. O(n) 
2. Sort the modified array. O(n log n)

The time complexity of O(n log n) 

Algorithm:

1.    Initialize an array arr of size n with given elements.
2.    Initialize three integers A, B, and C.
3.    For each element x in arr, apply the given equation Axx + B*x + C and store the result in arr.
4.    Sort the array arr using a sorting algorithm.
5.    Print the sorted array arr.

Below is the implementation of the approach:

C++

// C++ code for the approach

#include<bits/stdc++.h>
using namespace std;

// Function to apply the given equation on each element of the array
void applyEquation(int arr[], int n, int a, int b, int c) {
      // traversing array and updating each value
      // as per equation
    for(int i=0; i<n; i++) {
        arr[i] = a*arr[i]*arr[i] + b*arr[i] + c;
    }
}

// Driver's code
int main() {
      // Input array
    int arr[] = { -21 ,-15, 12, 13, 14 };
    int n = sizeof(arr)/sizeof(arr[0]);

    int a = -6, b = -7, c = 2;

    // Funciton call
    applyEquation(arr, n, a, b, c);

    // Sorting the modified array
    sort(arr, arr+n);

    // Printing the sorted array
      cout << "Array after sorting is : ";
    for(int i=0; i<n; i++) {
        cout << arr[i] << " ";
    }

    return 0;
}

Python3

# Function to apply the given equation on each element of the array
def applyEquation(arr, n, a, b, c):
    # Traversing array and updating each value as per equation
    for i in range(n):
        arr[i] = a * arr[i] * arr[i] + b * arr[i] + c

# Driver's code
if __name__ == '__main__':
    # Input array
    arr = [-21, -15, 12, 13, 14]
    n = len(arr)

    a = -6
    b = -7
    c = 2

    # Function call
    applyEquation(arr, n, a, b, c)

    # Sorting the modified array
    arr.sort()

    # Printing the sorted array
    print("Array after sorting is:", arr)

Output

Array after sorting is : -2497 -1272 -1243 -1103 -946 

Time Complexity: O(n * log₂n) as sort function has been called. Here, n is the size of the input array.
Auxiliary Space: O(1) as no extra space has been used.

Method 2(Efficient): Parabolic Property:

The equation given is parabolic. So the result of applying it to a sorted array will result in an array that will have a maximum/minimum with the sub-arrays to its left and right sorted.
In the above example, maximum is 0 and the subarray to its left {-4, -1} is sorted in ascending order and the sub-array to its right {-1, -4, -9} is sorted in descending order. 
All we need to do is merge these sorted arrays which are linear in time.

So the algorithm is: 

1. Apply equation on each element.
2. Find maximum/minimum.
3. Merge sub-arrays.

Note: The below code assumes that the modified array is first increasing then decreasing. 

Implementation:

C++

// C program to sort an array after applying equation
// A*x*x + B*x + C
#include<bits/stdc++.h>
using namespace std;

// Function to sort an array after applying given
// equation.
void sortArray(int arr[], int n, int A, int B, int C)
{
   // Apply equation on all elements
    for (int i = 0; i < n; i++)
        arr[i] = A*arr[i]*arr[i] + B*arr[i] + C;

    // Find maximum element in resultant array
    int index, maximum = INT_MIN;
    for (int i = 0; i< n; i++)
    {
        if (maximum < arr[i])
        {
            index = i;
            maximum = arr[i];
        }
    }

    // Use maximum element as a break point
    // and merge both subarrays using simple
    // merge function of merge sort
    int i = 0, j = n-1;
    int new_arr[n], k = 0;
    while (i < index && j > index)
    {
        if (arr[i] < arr[j])
            new_arr[k++] = arr[i++];
        else
            new_arr[k++] = arr[j--];
    }

    // Merge remaining elements
    while (i < index)
        new_arr[k++] = arr[i++];
    while (j > index)
        new_arr[k++] = arr[j--];

    new_arr[n-1] = maximum;

    // Modify original array
    for (int i = 0; i < n ; i++)
        arr[i] = new_arr[i];
}

// Driver code
int main()
{
    int arr[] = {-21 ,-15, 12, 13, 14 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int A = -6, B =-7, C = 2;

    sortArray(arr, n, A, B, C);

    cout << "Array after sorting is : ";
    for (int i=0; i<n; i++)
       cout << arr[i] << " ";

    return 0;
}

Java

// Java program to sort an array after applying equation
// A*x*x + B*x + C

class Main
{
    // Function to sort an array after applying given
    // equation.
    static void sortArray(int arr[], int n, int A, int B, int C)
    {
       // Apply equation on all elements
        for (int i = 0; i < n; i++)
            arr[i] = A*arr[i]*arr[i] + B*arr[i] + C;

        // Find maximum element in resultant array
        int index=-1;
        int maximum = -999999;
        for (int i = 0; i< n; i++)
        {
            if (maximum < arr[i])
            {
                index = i;
                maximum = arr[i];
            }
        }

        // Use maximum element as a break point
        // and merge both subarrays using simple
        // merge function of merge sort
        int i = 0, j = n-1;
        int[] new_arr = new int[n];
        int k = 0;
        while (i < index && j > index)
        {
            if (arr[i] < arr[j])
                new_arr[k++] = arr[i++];
            else
                new_arr[k++] = arr[j--];
        }

        // Merge remaining elements
        while (i < index)
            new_arr[k++] = arr[i++];
        while (j > index)
            new_arr[k++] = arr[j--];

        new_arr[n-1] = maximum;

        // Modify original array
        for (int p = 0; p < n ; p++)
            arr[p] = new_arr[p];
    }

    // main function
    public static void main (String[] args)
    {
        int arr[] = {-21 ,-15, 12, 13, 14 };
        int n = arr.length;
        int A = -6, B =-7, C = 2;

        sortArray(arr, n, A, B, C);

        System.out.println("Array after sorting is : ");
        for (int i=0; i<n; i++)
           System.out.print(arr[i]+" ");
    }
}

/* This code is contributed by Harsh Agarwal */

Python3

# Python3 program to sort an
# array after applying equation
# A*x*x + B*x + C
import sys

# Function to sort an array
# after applying given equation.
def sortArray(arr, n, A, B, C):

    # Apply equation on all elements
    for i in range(n):
        arr[i] = (A * arr[i] * arr[i] +
                  B * arr[i] + C)
    index = -(sys.maxsize - 1)
    maximum = -(sys.maxsize - 1)

    # Find maximum element in
    # resultant array
    for i in range(n):
        if maximum < arr[i]:
            index = i
            maximum = arr[i]

    # Use maximum element as a break point
    # and merge both subarrays using simple
    # merge function of merge sort
    i = 0; j = n - 1;
    new_arr = [0] * n
    k = 0
    while i < index and j > index:
        if arr[i] < arr[j]:
            new_arr[k] = arr[i]
            k += 1
            i += 1
        else:
            new_arr[k] = arr[j]
            k += 1
            j -= 1

    # Merge remaining elements
    while i < index:
        new_arr[k] = arr[i]
        k += 1
        i += 1

    # Modify original array
    while j > index:
        new_arr[k] = arr[j]
        k += 1
        j -= 1
        new_arr[n - 1] = maximum

    for i in range(n):
        arr[i] = new_arr[i]

# Driver code
arr = [-21, -15, 12, 13, 14]
n = len(arr)
A = -6
B= -7
C = 2
sortArray(arr, n, A, B, C)
print("Array after sorting is:")
for i in range(n):
    print(arr[i], end = " ")

# This code is contributed
# by Shrikant13

C#

// C# program to sort an array after applying equation
// A*x*x + B*x + C

using System;
class MAIN
{
    // Function to sort an array after applying given
    // equation.
    static void sortArray(int[] arr, int n, int A, int B, int C)
    {
       // Apply equation on all elements
        for (int i = 0; i < n; i++)
            arr[i] = A*arr[i]*arr[i] + B*arr[i] + C;

        // Find maximum element in resultant array
        int index=-1;
        int maximum = -999999;
        for (int i = 0; i< n; i++)
        {
            if (maximum < arr[i])
            {
                index = i;
                maximum = arr[i];
            }
        }

        // Use maximum element as a break point
        // and merge both subarrays using simple
        // merge function of merge sort
        int l = 0, j = n-1;
        int[] new_arr = new int[n];
        int k = 0;
        while (l < index && j > index)
        {
            if (arr[l] < arr[j])
                new_arr[k++] = arr[l++];
            else
                new_arr[k++] = arr[j--];
        }

        // Merge remaining elements
        while (l < index)
            new_arr[k++] = arr[l++];
        while (j > index)
            new_arr[k++] = arr[j--];

        new_arr[n-1] = maximum;

        // Modify original array
        for (int p = 0; p < n ; p++)
            arr[p] = new_arr[p];
    }

    // main function
    public static void Main ()
    {
        int[] arr = {-21 ,-15, 12, 13, 14 };
        int n = arr.Length;
        int A = -6, B =-7, C = 2;

        sortArray(arr, n, A, B, C);

        Console.Write("Array after sorting is : "+"\n");
        for (int i=0; i<n; i++)
           Console.Write(arr[i]+" ");
    }
}

PHP

<?php
// PHP program to sort an array after
// applying equation A*x*x + B*x + C

// Function to sort an array after
// applying given   equation.
function sortArray(&$arr, $n, $A, $B, $C)
{

    // Apply equation on all elements
    for ($i = 0; $i < $n; $i++)
        $arr[$i] = $A * $arr[$i] * $arr[$i] +
                   $B * $arr[$i] + $C;

    // Find maximum element in
    // resultant array
    $index = 0;
    $maximum = PHP_INT_MIN;
    for ($i = 0; $i < $n; $i++)
    {
        if ($maximum < $arr[$i])
        {
            $index = $i;
            $maximum = $arr[$i];
        }
    }

    // Use maximum element as a break point
    // and merge both subarrays using simple
    // merge function of merge sort
    $i = 0;
    $j = $n - 1;
    $new_arr = array();
    while ($i < $index && $j > $index)
    {
        if ($arr[$i] < $arr[$j])
            array_push($new_arr, $arr[$i++]);
        else
            array_push($new_arr, $arr[$j--]);
    }

    // Merge remaining elements
    while ($i < $index)
        array_push($new_arr, $arr[$i++]);
    while ($j > $index)
        array_push($new_arr, $arr[$j--]);

    array_push($new_arr, $maximum);

    // Modify original array
    for ($i = 0; $i < count($new_arr) ; $i++)
        $arr[$i] = $new_arr[$i];
}

// Driver code
$arr = array(-21 ,-15, 12, 13, 14 );
$n = count($arr);
$A = -6;
$B = -7;
$C = 2;

sortArray($arr, $n, $A, $B, $C);

echo "Array after sorting is : \n";
for ($i = 0; $i < $n; $i++)
    echo $arr[$i] . " ";

// This code is contributed by mits
?>

Javascript

<script>

    function sortArray(arr,n,A,B,C)
    {
        // Apply equation on all elements
        for (let i = 0; i < n; i++)
            arr[i] = A*arr[i]*arr[i] + B*arr[i] + C;

        // Find maximum element in resultant array
        let index=-1;
        let maximum = -999999;
        for (let i = 0; i< n; i++)
        {
            if (maximum < arr[i])
            {
                index = i;
                maximum = arr[i];
            }
        }

        // Use maximum element as a break point
        // and merge both subarrays using simple
        // merge function of merge sort
        let i = 0, j = n-1;
        let new_arr = new Array(n);
        for(let i=0;i<n;i++)
        {
            new_arr[i]=0;
        }
        let k = 0;
        while (i < index && j > index)
        {
            if (arr[i] < arr[j])
                new_arr[k++] = arr[i++];
            else
                new_arr[k++] = arr[j--];
        }

        // Merge remaining elements
        while (i < index)
            new_arr[k++] = arr[i++];
        while (j > index)
            new_arr[k++] = arr[j--];

        new_arr[n-1] = maximum;

        // Modify original array
        for (let p = 0; p < n ; p++)
            arr[p] = new_arr[p];
    }

    let arr=[-21 ,-15, 12, 13, 14];
    let n = arr.length;
    let A = -6, B =-7, C = 2;
    sortArray(arr, n, A, B, C);

    document.write("Array after sorting is : <br>");
    for (let i=0; i<n; i++)
        document.write(arr[i]+" ");



    // This code is contributed by avanitrachhadiya2155
</script>

Output

Array after sorting is : n-2497 -1272 -1243 -1103 -946 

Time Complexity: O(n) 
Auxiliary Space: O(n)