Maximum difference elements that can added to a set

Given a set containing N elements, you are allowed to add an element Z > 0 to this set only if it can be represented as |X-Y| where X and Y are already present in the set. After adding an element Z, you can use it as an element of the set to add new elements. Find out the maximum number of elements that can be added in this way.

Examples:

Input : set = {2, 3}
Output : 1
The only element that can be added is 1.

Input : set = {4, 6, 10}
Output : 2
The 2 elements that can be added are 
(6-4) = 2 and (10-2) = 8.

This problem is based on the following observations:

  • The maximum element that can be inserted has to be less than the current maximum element in the set, since the difference of 2 integers can not exceed any of the integers.
  • Every number that you insert has to be a multiple of the gcd of the given array. Since at any step, X and Y both are multiples of gcd, (X-Y) will also be a multiple of the gcd.
  • You can insert all multiples of gcd less than the maximum element.
  • The number of terms less than or equal to max divisible by gcd is floor (max/gcd), which will be the total number of elements in the set after performing all insertions, we need to remove the count of original N elements to get our answer.

Below is the implementation of the above approach:

C++

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// CPP program to find the maximum number
// of elements that can be added to a set
// such that it is the absolute difference
// of 2 elements already in the set
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the maximum number
// of elements that can be added to a set
// such that it is the absolute difference
// of 2 elements already in the set
int maxNewElements(int a[], int n)
{
    int gcd = a[0];
  
    int mx = a[0];
  
    for (int i = 1; i < n; i++) {
        gcd = __gcd(gcd, a[i]);
        mx = max(mx, a[i]);
    }
  
    int total_terms = mx / gcd;
  
    return total_terms - n;
}
  
// Driver Code
int main()
{
    int a[] = { 4, 6, 10 };
    int n = sizeof(a) / sizeof(a[0]);
    cout << maxNewElements(a, n);
    return 0;
}

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Java

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// Java program to find the maximum number
// of elements that can be added to a set
// such that it is the absolute difference
// of 2 elements already in the set
   
import java.util.*;
import java.lang.*;
import java.io.*;
  
  
class GFG{
      
static int __gcd(int a, int b) {
   if (b==0) return a;
   return __gcd(b,a%b);
}
// Function to find the maximum number
// of elements that can be added to a set
// such that it is the absolute difference
// of 2 elements already in the set
static int maxNewElements(int []a, int n)
{
    int gcd = a[0];
   
    int mx = a[0];
   
    for (int i = 1; i < n; i++) {
        gcd = __gcd(gcd, a[i]);
        mx = Math.max(mx, a[i]);
    }
   
    int total_terms = mx / gcd;
   
    return total_terms - n;
}
   
// Driver Code
public static void main(String args[])
{
    int a[] = { 4, 6, 10 };
    int n = a.length;
    System.out.print(maxNewElements(a, n));
}
}

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Python 3

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# Python3 program to find the maximum number 
# of elements that can be added to a set 
# such that it is the absolute difference 
# of 2 elements already in the set 
  
# from math lib import gcd method
from math import gcd
  
  
# Function to find the maximum number 
# of elements that can be added to a set 
# such that it is the absolute difference 
# of 2 elements already in the set 
def maxNewElements(a, n) :
  
    __gcd = a[0]
  
    mx = a[0]
  
    for i in range(1, n) :
        __gcd = gcd(__gcd,a[i])
        mx = max(mx, a[i])
  
    total_terms = mx / __gcd
  
    return total_terms - n
  
  
  
  
# Driver code
if __name__ == "__main__" :
  
    a = [ 4, 6, 10 ]
  
    n = len(a)
  
    print(int(maxNewElements(a,n)))
  
# This code is contributed by 
# ANKITRAI1

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C#

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// C# program to find the maximum 
// number of elements that can be 
// added to a set such that it is 
// the absolute difference of 2 
// elements already in the set
class GFG
{
      
static int __gcd(int a, int b) 
{
    if (b == 0) return a;
    return __gcd(b, a % b);
}
  
// Function to find the maximum number
// of elements that can be added to a set
// such that it is the absolute difference
// of 2 elements already in the set
static int maxNewElements(int[] a, int n)
{
    int gcd = a[0];
  
    int mx = a[0];
  
    for (int i = 1; i < n; i++)
    {
        gcd = __gcd(gcd, a[i]);
        mx = System.Math.Max(mx, a[i]);
    }
  
    int total_terms = mx / gcd;
  
    return total_terms - n;
}
  
// Driver Code
static void Main()
{
    int[] a = { 4, 6, 10 };
    int n = a.Length;
    System.Console.WriteLine(maxNewElements(a, n));
}
}
  
// This code is contributed by mits

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PHP

Output:

2

Time Complexity: O(N*LogN)



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