Check if permutation of N exists with product of atleast 1 subarray's size and min as K

Given two integers N and K, the task is to check if it is possible to form a permutation of N integers such that it contains atleast 1 subarray such that the product of length of that subarray with minimum element present in it is K.

A permutation of size N have all the integers from 1 to N present in it only once.

Examples:

Input: N = 5, K = 6
Output: True
Explanation: {4, 2, 1, 3, 5} is a valid array containing integers from 1 to 5. The required subarray is {3, 5}. 
Length of subarray = 2, minimum element in subarray = 3. 
Their product = 2 x 3 = 6, which is equal to K.

Input: N = 4, K = 10
Output: False

Approach: The problem can be solved based  on the following observation:

Suppose in a N size array having integers from 1 to N, there exist a subarray of size L, having minimum element M such that M * L = K. Therefore, M = K / L or K must be divisible by the length of the subarray. Also, M should be minimum element in subarray of size L. 

In a permutation of N integers, there are N - M + 1 elements, which are greater than or equal to M. So, for M to be minimum in subarray of size L, N - M + 1 ≥ L

Follow the steps mentioned below to implement the above idea:

• Iterate the array from i = 1 to N
• Let i be the length of subarray satisfying the required conditions.
  • Calculate the minimum element in the subarray.
  • As, L * M = K, so, M=K / L, (where M is the minimum element in current subarray)
• Check if conditions stated in observation are satisfied or not i.e. M < N - L + 1.
• If so, return true.

Below is the implementation of the above approach.

C++

// C++ code for above approach

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

// Function toCheck if there can be
// a subarray such that product of its
// length with its minimum element is K
bool isPossible(int N, int K)
{
    // Variable to store answer
    bool ans = true;

    for (int i = 1; i <= N; i++) {

        // Variable to store length of
        // current candidate subarray
        int length = i;

        // Since minimum element * length
        // of subarray should be K,
        // that means K should be divisible
        // by length of candidate subarray
        if (K % length == 0) {

            // Candidate for minimum element
            int min_element = K / length;

            // Checking if candidate for
            // minimum element can actually
            // be a minimum element in a
            // sequence on size "length"
            if (min_element < N - length + 1) {
                ans = true;
                break;
            }
        }
    }

    // Returning answer
    return ans;
}

// Driver code
int main()
{
    int N = 5;
    int K = 6;

    // Function call
    bool answer = isPossible(N, K);
    cout << boolalpha << answer;
    return 0;
}

Java

// JAVA code for above approach
import java.util.*;
class GFG 
{

  // Function toCheck if there can be
  // a subarray such that product of its
  // length with its minimum element is K
  public static boolean isPossible(int N, int K)
  {

    // Variable to store answer
    boolean ans = true;

    for (int i = 1; i <= N; i++) {

      // Variable to store length of
      // current candidate subarray
      int length = i;

      // Since minimum element * length
      // of subarray should be K,
      // that means K should be divisible
      // by length of candidate subarray
      if (K % length == 0) {

        // Candidate for minimum element
        int min_element = K / length;

        // Checking if candidate for
        // minimum element can actually
        // be a minimum element in a
        // sequence on size "length"
        if (min_element < N - length + 1) {
          ans = true;
          break;
        }
      }
    }

    // Returning answer
    return ans;
  }

  // Driver code
  public static void main(String[] args)
  {
    int N = 5;
    int K = 6;

    // Function call
    boolean answer = isPossible(N, K);
    System.out.print(answer);
  }
}

// This code is contributed by Taranpreet

Python3

# Python3 code for above approach

# Function toCheck if there can be
# a subarray such that product of its
# length with its minimum element is K
def isPossible(N, K):
  
    # Variable to store answer
    ans = 1

    for i in range(1, N + 1):

        # Variable to store length of
        # current candidate subarray
        length = i

        # Since minimum element * length
        # of subarray should be K,
        # that means K should be divisible
        # by length of candidate subarray
        if (K % length == 0):

            # Candidate for minimum element
            min_element = K / length

            # Checking if candidate for
            # minimum element can actually
            # be a minimum element in a
            # sequence on size "length"
            if (min_element < N - length + 1):
                ans = 1
                break

    # Returning answer
    return ans

# Driver code
if __name__ == "__main__":
      
    N = 5
    K = 6

    # Function call
    answer = isPossible(N, K)
    print(bool(answer))
    
    # This code is contributed by hrithikgarg03188.

C#

// C# code for above approach
using System;
class GFG {

  // Function toCheck if there can be
  // a subarray such that product of its
  // length with its minimum element is K
  static bool isPossible(int N, int K)
  {

    // Variable to store answer
    bool ans = true;

    for (int i = 1; i <= N; i++) {

      // Variable to store length of
      // current candidate subarray
      int length = i;

      // Since minimum element * length
      // of subarray should be K,
      // that means K should be divisible
      // by length of candidate subarray
      if (K % length == 0) {

        // Candidate for minimum element
        int min_element = K / length;

        // Checking if candidate for
        // minimum element can actually
        // be a minimum element in a
        // sequence on size "length"
        if (min_element < N - length + 1) {
          ans = true;
          break;
        }
      }
    }

    // Returning answer
    return ans;
  }

  // Driver code
  public static void Main()
  {
    int N = 5;
    int K = 6;

    // Function call
    bool answer = isPossible(N, K);
    Console.Write(answer);
  }
}

// This code is contributed by Samim Hossain Mondal.

JavaScript

 <script>
        // JavaScript code for the above approach

        // Function toCheck if there can be
        // a subarray such that product of its
        // length with its minimum element is K
        function isPossible(N, K)
        {
        
            // Variable to store answer
            let ans = true;

            for (let i = 1; i <= N; i++) {

                // Variable to store length of
                // current candidate subarray
                let length = i;

                // Since minimum element * length
                // of subarray should be K,
                // that means K should be divisible
                // by length of candidate subarray
                if (K % length == 0) {

                    // Candidate for minimum element
                    let min_element = Math.floor(K / length);

                    // Checking if candidate for
                    // minimum element can actually
                    // be a minimum element in a
                    // sequence on size "length"
                    if (min_element < N - length + 1) {
                        ans = true;
                        break;
                    }
                }
            }

            // Returning answer
            return ans;
        }

        // Driver code
        let N = 5;
        let K = 6;

        // Function call
        let ans = isPossible(N, K);
        document.write(ans)

      // This code is contributed by Potta Lokesh
    </script>

Output

true

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

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