Check whether each Array element can be reduced to minimum element by replacing it with remainder with some X

Given an array arr[] of non-negative integers, the task is to check whether all array elements can be reduced to the minimum element in the array by choosing any positive integer X and update an array element arr[i] to arr[i]%X. If it is possible to do so, then print Yes. Otherwise, print No.

Examples:

Input: arr[] = {1, 1, 3, 4}
Output: Yes
Explanation:
Following operations can be performed to reduce all array elements to 1:

1. Choose X = 2 for array element arr[3], and replacing it with arr[3] % 2, modifies the array arr[] to {1, 1, 1, 4}.
2. Choose X = 3 for array element arr[4], and replacing it with arr[4] % 3, modifies the array arr[] to {1, 1, 1, 1}.

After the above operations all array elements have been reduced to the minimum element of arr[]. Therefore, print Yes.

Input: arr[] = {1, 2, 3}
Output: No

Approach: The given problem can be solved by using an observation that any non-negative integer, say num can be changed to the following integers [0, ceil(num/2) – 1] by choosing the value of X as num/2. It is very evident rest of X will have shorter range of changed numbers than X equal to num/2. Follow the steps below to solve the problem:

• Initialize a variable, say mini that stores the minimum element of the array arr[].
• Traverse the given array arr[] and if the value of mini doesn’t lie in the range[0, ceil(arr[i]/2) – 1] for any array element arr[i] then print No. Otherwise, print Yes.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to check if every integer
// in the array can be reduced to the
// minimum array element
string isPossible(int arr[], int n)
{
    // Stores the minimum array element
    int mini = INT_MAX;

    // Find the minimum element
    for (int i = 0; i < n; i++)
        mini = min(mini, arr[i]);

    // Traverse the array arr[]
    for (int i = 0; i < n; i++) {
        if (arr[i] == mini)
            continue;

        // Stores the maximum value
        // in the range
        int Max = (arr[i] + 1) / 2 - 1;

        // Check whether mini lies in
        // the range or not
        if (mini < 0 || mini > Max)
            return "No";
    }

    // Otherwise, return Yes
    return "Yes";
}

// Driver Code
int main()
{
    int arr[] = { 1, 1, 3, 4 };
    int N = sizeof(arr) / sizeof(arr[0]);

    cout << isPossible(arr, N);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.util.*;

class GFG{


// Function to check if every integer
// in the array can be reduced to the
// minimum array element
static String isPossible(int arr[], int n)
{
    // Stores the minimum array element
    int mini = Integer.MAX_VALUE;

    // Find the minimum element
    for (int i = 0; i < n; i++)
        mini = Math.min(mini, arr[i]);

    // Traverse the array arr[]
    for (int i = 0; i < n; i++) {
        if (arr[i] == mini)
            continue;

        // Stores the maximum value
        // in the range
        int Max = (arr[i] + 1) / 2 - 1;

        // Check whether mini lies in
        // the range or not
        if (mini < 0 || mini > Max)
            return "No";
    }

    // Otherwise, return Yes
    return "Yes";
}

// Driver code
public static void main(String args[])
{
    int arr[] = { 1, 1, 3, 4 };
    int N = arr.length;

    System.out.print(isPossible(arr, N));

}
}

// This code is contributed by code_hunt.

Python3

# Python 3 program for the above approach

import sys
# Function to check if every integer
# in the array can be reduced to the
# minimum array element
def isPossible(arr, n):
    # Stores the minimum array element
    mini = sys.maxsize

    # Find the minimum element
    for i in range(n):
        mini = min(mini, arr[i])

    # Traverse the array arr[]
    for i in range(n):
        if (arr[i] == mini):
            continue

        # Stores the maximum value
        # in the range
        Max = (arr[i] + 1) // 2 - 1

        # Check whether mini lies in
        # the range or not
        if (mini < 0 or mini > Max):
            return "No"

    # Otherwise, return Yes
    return "Yes"

# Driver Code
if __name__ == '__main__':
    arr = [1, 1, 3, 4]
    N = len(arr)

    print(isPossible(arr, N))

    # This code is contributed by ipg2016107.

C#

// C# program for the above approach
using System;

public class GFG
{

// Function to check if every integer
// in the array can be reduced to the
// minimum array element
static String isPossible(int []arr, int n)
{

    // Stores the minimum array element
    int mini = int.MaxValue;

    // Find the minimum element
    for (int i = 0; i < n; i++)
        mini = Math.Min(mini, arr[i]);

    // Traverse the array []arr
    for (int i = 0; i < n; i++) {
        if (arr[i] == mini)
            continue;

        // Stores the maximum value
        // in the range
        int Max = (arr[i] + 1) / 2 - 1;

        // Check whether mini lies in
        // the range or not
        if (mini < 0 || mini > Max)
            return "No";
    }

    // Otherwise, return Yes
    return "Yes";
}

// Driver code
public static void Main(String []args)
{
    int []arr = { 1, 1, 3, 4 };
    int N = arr.Length;

    Console.Write(isPossible(arr, N));

}
}

// This code is contributed by Princi Singh

Javascript

<script>
// JavaScript program for the above approach
// Function to check if every integer
// in the array can be reduced to the
// minimum array element
function isPossible(arr, n)
{
    // Stores the minimum array element
    var mini = Number.MAX_VALUE;

    // Find the minimum element
    for (var i = 0; i < n; i++)
        mini = Math.min(mini, arr[i]);

    // Traverse the array arr[]
    for (var i = 0; i < n; i++) {
        if (arr[i] == mini)
            continue;

        // Stores the maximum value
        // in the range
        var Max = (arr[i] + 1) / 2 - 1;

        // Check whether mini lies in
        // the range or not
        if (mini < 0 || mini > Max)
            return "No";
    }

    // Otherwise, return Yes
    return "Yes";
}

// Driver code

    var arr = [ 1, 1, 3, 4 ];
    var N = arr.length;

    document.write(isPossible(arr, N));


// This code is contributed by shivanisinghss2110
</script>

Output:

Yes

Time Complexity: O(N)
Auxiliary Space: O(1)