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++
// 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 setint 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 Codeint main(){ int a[] = { 4, 6, 10 }; int n = sizeof(a) / sizeof(a[0]); cout << maxNewElements(a, n); return 0;} |
Java
// 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 setstatic 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 Codepublic static void main(String args[]){ int a[] = { 4, 6, 10 }; int n = a.length; System.out.print(maxNewElements(a, n));}} |
Python 3
# 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 methodfrom 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 codeif __name__ == "__main__" : a = [ 4, 6, 10 ] n = len(a) print(int(maxNewElements(a,n)))# This code is contributed by # ANKITRAI1 |
C#
// 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 setclass 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 setstatic 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 Codestatic void Main(){ int[] a = { 4, 6, 10 }; int n = a.Length; System.Console.WriteLine(maxNewElements(a, n));}}// This code is contributed by mits |
PHP
<?php// PHP 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 setfunction __gcd($a, $b) { if ($b == 0) return $a; return __gcd($b, $a % $b);}// Function to find the maximum number// of elements that can be added to // a set such that it is the absolute // difference of 2 elements already in// the setfunction maxNewElements($a, $n){ $gcd = $a[0]; $mx = $a[0]; for ($i = 1; $i < $n; $i++) { $gcd = __gcd($gcd, $a[$i]); $mx = max($mx, $a[$i]); } $total_terms = $mx / $gcd; return $total_terms - $n;}// Driver Code$a = array(4, 6, 10 );$n = sizeof($a);echo maxNewElements($a, $n);// This code is contributed // by Akanksha Rai |
Javascript
<script> // Javascript 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 function __gcd(a, 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 function maxNewElements(a, n) { let gcd = a[0]; let mx = a[0]; for (let i = 1; i < n; i++) { gcd = __gcd(gcd, a[i]); mx = Math.max(mx, a[i]); } let total_terms = parseInt(mx / gcd, 10); return total_terms - n; } let a = [ 4, 6, 10 ]; let n = a.length; document.write(maxNewElements(a, n)); // This code is contributed by divyesh072019.</script> |
Output:
2
Time Complexity: O(N*LogN)
Auxiliary Space: O(1)
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