Minimum gcd operations to make all array elements one

Given an array A[] of size N. You can replace any number in the array with the gcd of that element with any of its adjacent elements. Find the minimum number of such operation to make the element of whole array equal to 1. If its not possible print -1.

Examples:

Input : A[] = {4, 8, 9}
Output : 3
Explanation:
In the first move we choose (8, 9) 
gcd(8, 9) = 1. Now the array becomes {4, 1, 9}.
After second move, the array becomes {1, 1, 9}. 
After third move the array becomes {1, 1, 1}.

Input : A[] = { 5, 10, 2, 6 }
Output : 5
Explanation:
There is no pair with GCD equal one. We first
consider (5, 10) and replace 10 with 5. Now array
becomes { 5, 5, 2, 6 }. Now we consider pair (5, 2)
and replace 5 with 1, array becomes { 5, 1, 2, 6 }.
We have a 1, so further steps are simple.

Input : A[] = {8, 10, 12}
Output : -1
Explanation:
Its not possible to make all the element equal to 1.

Input : A[] = { 8, 10, 12, 6, 3 }
Output : 7 



  • If initially the array contains 1, our answer is the difference between the size of the array and count of ones in the array.
  • If the array has no element equal to 1. We need to find the smallest sub array with GCD equal to one. Our result is N + (length of the minimum subarray with GCD 1) – 1. Example cases are { 5, 10, 2, 6 } and { 8, 10, 12, 6, 3 }.

We can find all the sub array in O(N^2) and GCD can be calculated in O(Log N) using Euclidean algorithms.
The overall complexity will be O(N^2 Log N).

Here is the implementation of the above idea.

C++

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// CPP program to find minimum GCD operations
// to make all array elements one.
#include <bits/stdc++.h>
using namespace std;
  
// Function to count number of moves.
int minimumMoves(int A[], int N)
{
   // Counting Number of ones.
    int one = 0;
    for (int i = 0; i < N; i++) 
        if (A[i] == 1)
            one++; 
  
    // If there is a one
    if (one != 0) 
        return N - one; 
      
    // Find smallest subarray with GCD equals
    // to one.
    int minimum = INT_MAX;
    for (int i = 0; i < N; i++) {
        int g = A[i]; // to calculate GCD
        for (int j = i + 1; j < N; j++) {
            g = __gcd(A[j], g); 
            if (g == 1) {
                minimum = min(minimum, j - i); 
                break;
            }
        }
    }
  
    if (minimum == INT_MAX) // Not Possible
        return -1;
    else
        return N + minimum - 1; // Final answer
}
  
// Driver code
int main()
{
    int A[] = { 2, 4, 3, 9 };
    int N = sizeof(A) / sizeof(A[0]);
    cout << minimumMoves(A, N);
    return 0;
}

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Java

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// Java program to find minimum GCD operations
// to make all array elements one.
import java.util.*;
  
class GFG {
      
//__gcd function
static int __gcd(int a, int b) 
{
    if (a == 0)
    return b;
    return __gcd(b % a, a);
}
  
// Function to count number of moves.
static int minimumMoves(int A[], int N) 
{
    // Counting Number of ones.
    int one = 0;
    for (int i = 0; i < N; i++)
    if (A[i] == 1)
        one++;
  
    // If there is a one
    if (one != 0)
    return N - one;
  
    // Find smallest subarray with 
    // GCD equals to one.
    int minimum = Integer.MAX_VALUE;
    for (int i = 0; i < N; i++) {
          
    // to calculate GCD
    int g = A[i]; 
    for (int j = i + 1; j < N; j++) {
        g = __gcd(A[j], g);
        if (g == 1) {
        minimum = Math.min(minimum, j - i);
        break;
        }
    }
    }
  
    if (minimum == Integer.MAX_VALUE) // Not Possible
    return -1;
    else
    return N + minimum - 1; // Final answer
}
  
// Driver code
public static void main(String[] args) 
{
    int A[] = {2, 4, 3, 9};
    int N = A.length;
    System.out.print(minimumMoves(A, N));
}
}
  
// This code is contributed by Anant Agarwal.

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Python3

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# Python program to find
# minimum GCD operations
# to make all array elements one.
  
#__gcd function
def __gcd(a,b):
  
    if (a == 0):
        return b
    return __gcd(b % a, a)
   
# Function to count number of moves.
def minimumMoves(A,N):
  
    # Counting Number of ones.
    one = 0
    for i in range(N): 
        if (A[i] == 1):
            one+=1
   
    # If there is a one
    if (one != 0): 
        return N - one 
       
    # Find smallest subarray with GCD equals
    # to one.
    minimum = +2147483647
    for i in range(N):
        g = A[i] # to calculate GCD
        for j in range(i + 1,N):
            g = __gcd(A[j], g) 
            if (g == 1):
                minimum = min(minimum, j - i) 
                break
   
    if (minimum == +2147483647): # Not Possible
        return -1
    else:
        return N + minimum - 1; # Final answer
  
  
# Driver program
A = [ 2, 4, 3, 9 ]
N = len(A)
print(minimumMoves(A, N))
  
# This code is contributed
# by Anant Agarwal.

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C#

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// C# program to find minimum GCD operations
// to make all array elements one.
using System;
  
class GFG {
      
//__gcd function
static int __gcd(int a, int b) 
{
    if (a == 0)
    return b;
    return __gcd(b % a, a);
}
  
// Function to count number of moves.
static int minimumMoves(int []A, int N) 
{
    // Counting Number of ones.
    int one = 0;
    for (int i = 0; i < N; i++)
    if (A[i] == 1)
        one++;
  
    // If there is a one
    if (one != 0)
    return N - one;
  
    // Find smallest subarray with 
    // GCD equals to one.
    int minimum = int.MaxValue;
    for (int i = 0; i < N; i++) {
          
    // to calculate GCD
    int g = A[i]; 
    for (int j = i + 1; j < N; j++) {
        g = __gcd(A[j], g);
        if (g == 1) {
        minimum = Math.Min(minimum, j - i);
        break;
        }
    }
    }
  
    if (minimum == int.MaxValue) // Not Possible
    return -1;
    else
    return N + minimum - 1; // Final answer
}
  
// Driver code
public static void Main() 
{
    int []A = {2, 4, 3, 9};
    int N = A.Length;
    Console.WriteLine(minimumMoves(A, N));
}
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// PHP program to find minimum 
// GCD operations to make all 
// array elements one.
  
// __gcd function
function __gcd($a, $b
{
    if ($a == 0)
    return $b;
    return __gcd($b % $a, $a);
}
  
// Function to count
// number of moves.
function minimumMoves($A, $N
{
    // Counting Number of ones.
    $one = 0;
    for ($i = 0; $i < $N; $i++)
    if ($A[$i] == 1)
        $one++;
  
    // If there is a one
    if ($one != 0)
    return $N - $one;
  
    // Find smallest subarray  
    // with GCD equals to one.
    $minimum = PHP_INT_MAX;
    for ($i = 0; $i < $N; $i++)
    {
          
    // to calculate GCD
    $g = $A[$i]; 
    for ($j = $i + 1; 
         $j < $N; $j++) 
    {
        $g = __gcd($A[$j], $g);
        if ($g == 1) 
        {
            $minimum = min($minimum
                           $j - $i);
            break;
        }
    }
    }
  
    if ($minimum == PHP_INT_MAX) // Not Possible
    return -1;
    else
    return $N + $minimum - 1; // Final answer
}
  
// Driver code
$A = array(2, 4, 3, 9);
$N = sizeof($A);
echo (minimumMoves($A, $N));
  
// This code is contributed 
// by akt_mit.
?>

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Output:

4


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