Find the minimum value of X for an expression
Given an array arr[]. The task is t find the value of X such that the result of the expression (A[1] – X)^2 + (A[2] – X)^2 + (A[3] – X)^2 + … (A[n-1] – X)^2 + (A[n] – X)^2 is minimum possible.
Examples :
Input : arr[] = {6, 9, 1, 6, 1, 3, 7}
Output : 5
Input : arr[] = {1, 2, 3, 4, 5}
Output : 3
Approach:
We can simplify the expression that we need to minimize. The expression can be written as
(A[1]^2 + A[2]^2 + A[3]^2 + … + A[n]^2) + nX^2 – 2X(A[1] + A[2] + A[3] + … + A[n])
On differentiating the above expression, we get
2nX - 2(A[1] + A[2] + A[3] + … + A[n])
We can denote the term (A[1] + A[2] + A[3] + … + A[n] ) as S. We get
2nX - 2S
Putting 2nX – 2S = 0, we get
X = S/N
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
int valueofX( int ar[], int n)
{
int sum = 0;
for ( int i = 0; i < n; i++) {
sum = sum + ar[i];
}
if (sum % n == 0) {
return sum / n;
}
else {
int A = sum / n, B = sum / n + 1;
int ValueA = 0, ValueB = 0;
for ( int i = 0; i < n; i++) {
ValueA += (ar[i] - A) * (ar[i] - A);
ValueB += (ar[i] - B) * (ar[i] - B);
}
if (ValueA < ValueB) {
return A;
}
else {
return B;
}
}
}
int main()
{
int n = 7;
int arr[7] = { 6, 9, 1, 6, 1, 3, 7 };
cout << valueofX(arr, n) << '\n' ;
return 0;
}
|
Java
class GFG
{
static int valueofX( int ar[], int n)
{
int sum = 0 ;
for ( int i = 0 ; i < n; i++)
{
sum = sum + ar[i];
}
if (sum % n == 0 )
{
return sum / n;
}
else
{
int A = sum / n, B = sum / n + 1 ;
int ValueA = 0 , ValueB = 0 ;
for ( int i = 0 ; i < n; i++)
{
ValueA += (ar[i] - A) * (ar[i] - A);
ValueB += (ar[i] - B) * (ar[i] - B);
}
if (ValueA < ValueB)
{
return A;
}
else
{
return B;
}
}
}
public static void main(String args[])
{
int n = 7 ;
int arr[] = { 6 , 9 , 1 , 6 , 1 , 3 , 7 };
System.out.println(valueofX(arr, n));
}
}
|
Python3
def valueofX(ar, n):
summ = sum (ar)
if (summ % n = = 0 ):
return summ / / n
else :
A = summ / / n
B = summ / / n + 1
ValueA = 0
ValueB = 0
for i in range (n):
ValueA + = (ar[i] - A) * (ar[i] - A)
ValueB + = (ar[i] - B) * (ar[i] - B)
if (ValueA < ValueB):
return A
else :
return B
n = 7
arr = [ 6 , 9 , 1 , 6 , 1 , 3 , 7 ]
print (valueofX(arr, n))
|
C#
using System;
class GFG
{
static int valueofX( int []ar, int n)
{
int sum = 0;
for ( int i = 0; i < n; i++)
{
sum = sum + ar[i];
}
if (sum % n == 0)
{
return sum / n;
}
else
{
int A = sum / n, B = sum / n + 1;
int ValueA = 0, ValueB = 0;
for ( int i = 0; i < n; i++)
{
ValueA += (ar[i] - A) * (ar[i] - A);
ValueB += (ar[i] - B) * (ar[i] - B);
}
if (ValueA < ValueB)
{
return A;
}
else
{
return B;
}
}
}
public static void Main(String []args)
{
int n = 7;
int []arr = { 6, 9, 1, 6, 1, 3, 7 };
Console.WriteLine(valueofX(arr, n));
}
}
|
Javascript
<script>
function valueofX(ar, n)
{
let sum = 0;
for (let i = 0; i < n; i++) {
sum = sum + ar[i];
}
if (sum % n == 0) {
return Math.floor(sum / n);
}
else {
let A = Math.floor(sum / n), B = Math.floor(sum / n + 1);
let ValueA = 0, ValueB = 0;
for (let i = 0; i < n; i++) {
ValueA += (ar[i] - A) * (ar[i] - A);
ValueB += (ar[i] - B) * (ar[i] - B);
}
if (ValueA < ValueB) {
return A;
}
else {
return B;
}
}
}
let n = 7;
let arr = [ 6, 9, 1, 6, 1, 3, 7 ];
document.write(valueofX(arr, n) + "<br>" );
</script>
|
Time Complexity: O(n)
Auxiliary Space: O(1)
Last Updated :
10 Mar, 2022
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