Largest Subset with GCD 1
Last Updated :
16 Feb, 2023
Given n integers, we need to find size of the largest subset with GCD equal to 1.
Input Constraint :
n <= 10^5
A[i] <= 10^5
Examples:
Input : A = {2, 3, 5}
Output : 3
Input : A = {3, 18, 12}
Output : -1
Naive Solution :
We find GCD of all possible subsets and find the largest subset whose GCD is 1. Total time taken will be equal to the time taken to evaluate GCD of all possible subsets. Total possible subsets are 2n. In worst case there are n elements in subset and time taken to calculate its GCD will be n * log(n)
Extra space required to hold current subset is O(n)
Time complexity : O(n * log(n) * 2^n)
Space Complexity : O(n)
Optimized O(n) solution :
Let say we find a subset with GCD 1, if we add a new element to it then GCD still remains 1. Hence if a subset exists with GCD 1 then GCD of the complete set is also 1. Hence we first find GCD of the complete set, if its 1 then complete set is that subset else no subset exist with GCD 1.
C++
#include <iostream>
using namespace std;
int gcd( int a, int b)
{
if (a == 0)
return b;
return gcd(b%a, a);
}
int largestGCD1Subset( int A[], int n)
{
int currentGCD = A[0];
for ( int i=1; i<n; i++)
{
currentGCD = gcd(currentGCD, A[i]);
if (currentGCD == 1)
return n;
}
return 0;
}
int main()
{
int A[] = {2, 18, 6, 3};
int n = sizeof (A)/ sizeof (A[0]);
cout << largestGCD1Subset(A, n);
return 0;
}
|
Java
import java.io.*;
class GFG {
static int gcd( int a, int b)
{
if (a == 0 )
return b;
return gcd(b % a, a);
}
static int largestGCD1Subset( int A[],
int n)
{
int currentGCD = A[ 0 ];
for ( int i= 1 ; i<n; i++)
{
currentGCD =
gcd(currentGCD, A[i]);
if (currentGCD == 1 )
return n;
}
return 0 ;
}
public static void main (String[] args)
{
int A[] = { 2 , 18 , 6 , 3 };
int n =A.length;
System.out.println(
largestGCD1Subset(A, n) );
}
}
|
Python3
def gcd( a, b):
if (a = = 0 ):
return b
return gcd(b % a, a)
def largestGCD1Subset(A, n):
currentGCD = A[ 0 ];
for i in range ( 1 , n):
currentGCD = gcd(currentGCD, A[i])
if (currentGCD = = 1 ):
return n
return 0
A = [ 2 , 18 , 6 , 3 ]
n = len (A)
print (largestGCD1Subset(A, n))
|
C#
using System;
public class GFG {
static int gcd( int a, int b)
{
if (a == 0)
return b;
return gcd(b % a, a);
}
static int largestGCD1Subset( int []A,
int n)
{
int currentGCD = A[0];
for ( int i = 1; i < n; i++)
{
currentGCD =
gcd(currentGCD, A[i]);
if (currentGCD == 1)
return n;
}
return 0;
}
public static void Main()
{
int []A = {2, 18, 6, 3};
int n = A.Length;
Console.Write(
largestGCD1Subset(A, n));
}
}
|
PHP
<?php
function gcd( $a , $b )
{
if ( $a == 0)
return $b ;
return gcd( $b % $a , $a );
}
function largestGCD1Subset( $A , $n )
{
$currentGCD = $A [0];
for ( $i = 1; $i < $n ; $i ++)
{
$currentGCD =
gcd( $currentGCD , $A [ $i ]);
if ( $currentGCD == 1)
return $n ;
}
return 0;
}
$A = array (2, 18, 6, 3);
$n = sizeof( $A );
echo largestGCD1Subset( $A , $n );
?>
|
Javascript
<script>
function gcd(a, b)
{
if (a == 0)
return b;
return gcd(b % a, a);
}
function largestGCD1Subset(A, n)
{
let currentGCD = A[0];
for ( let i = 1; i < n; i++)
{
currentGCD =
gcd(currentGCD, A[i]);
if (currentGCD == 1)
return n;
}
return 0;
}
let A = [2, 18, 6, 3];
let n = A.length;
document.write(largestGCD1Subset(A, n));
</script>
|
Time Complexity : O(n* log(n))
Space Complexity : O(1)
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