GCD 1的最大子集

给定n个整数，我们需要找到GCD等于1的最大子集的大小。
输入约束：

n <= 10^5
A[i] <= 10^5

例子：

Input : A = {2, 3, 5}
Output : 3

Input : A = {3, 18, 12}
Output : -1

天真的解决方案：

我们找到所有可能的子集的GCD，并找到GCD为1的最大子集。总花费的时间将等于评估所有可能的子集的GCD所花费的时间。可能的子集总数为2 ⁿ 。最坏的情况是子集中有n个元素，计算其GCD所需的时间为n * log(n)
容纳当前子集所需的额外空间为O(n)

Time complexity : O(n * log(n) * 2^n)
Space Complexity : O(n)

优化的O(n)解决方案：

假设我们找到了一个GCD 1的子集，如果我们向它添加一个新元素，那么GCD仍然保持1。因此，如果一个子集的GCD 1存在，那么整个集合的GCD也是1。因此，我们首先找到完整的GCD集合，如果它的1，则完整集合是该子集，否则GCD 1不存在任何子集。

C++

// C++ program to find size of the largest subset with GCD 1
#include 
using namespace std;
 
// Function to return gcd of a and b
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b%a, a);
}
 
// Function to find largest subset with GCD 1
int largestGCD1Subset(int A[], int n)
{
    // finding gcd of whole array
    int currentGCD = A[0];
    for (int i=1; i

---

Java

// Java program to find size of the
// largest subset with GCD 1
import java.*;
 
class GFG {
     
    // Function to return gcd of
    // a and b
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
        return gcd(b % a, a);
    }
 
    // Function to find largest
    // subset with GCD 1
    static int largestGCD1Subset(int A[],
                                   int n)
    {
         
        // finding gcd of whole array
        int currentGCD = A[0];
         
        for (int i=1; i

---

Python3

# python program to find size of the
# largest subset with GCD 1
 
# Function to return gcd of a and b
def gcd( a, b):
     
    if (a == 0):
        return b
         
    return gcd(b%a, a)
 
 
# Function to find largest subset
# with GCD 1
def largestGCD1Subset(A, n):
     
    # finding gcd of whole array
    currentGCD = A[0];
    for i in range(1, n):
         
        currentGCD = gcd(currentGCD, A[i])
 
        # If current GCD becomes 1 at
        # any momemnt, then whole
        # array has GCD 1.
        if (currentGCD == 1):
            return n
    return 0
 
# Driver code
A = [2, 18, 6, 3]
n = len(A)
print (largestGCD1Subset(A, n))
 
# This code is Contributed by Sam007.

C#

// C# program to find size of the
// largest subset with GCD 1
using System;
         
public class GFG {
     
    // Function to return gcd of
    // a and b
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
        return gcd(b % a, a);
    }
 
    // Function to find largest subset
    // with GCD 1
    static int largestGCD1Subset(int []A,
                                   int n)
    {
         
        // finding gcd of whole array
        int currentGCD = A[0];
        for (int i = 1; i < n; i++)
        {
            currentGCD =
                    gcd(currentGCD, A[i]);
     
            // If current GCD becomes 1 at
            // any momemnt, then whole
            // array has GCD 1.
            if (currentGCD == 1)
                return n;
        }
     
        return 0;
    }
     
    // Driver method
    public static void Main()
    {
        int []A = {2, 18, 6, 3};
        int n = A.Length;
         
        Console.Write(
              largestGCD1Subset(A, n));
    }
}
 
// This code is contributed by Sam007.

Javascript

 
// Javascript program to find size of the
// largest subset with GCD 1
 
// Function to return gcd of a and b
function gcd(a, b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}
 
// Function to find largest subset
// with GCD 1
function largestGCD1Subset(A, n)
{
     
    // finding gcd of whole array
    let currentGCD = A[0];
    for ( let i = 1; i < n; i++)
    {
        currentGCD =
           gcd(currentGCD, A[i]);
 
        // If current GCD becomes 1
        // at any momemnt, then
        // whole array has GCD 1.
        if (currentGCD == 1)
            return n;
    }
 
    return 0;
}
 
// Driver program
let A = [2, 18, 6, 3];
let n = A.length;
document.write(largestGCD1Subset(A, n));
     
// This code is contributed by _saurabh_jaiswal
 

输出：

4

Time Complexity : O(n* log(n))
Space Complexity : O(1)