Count of subsequences with a sum in range [L, R] and difference between max and min element at least X

Last Updated : 13 Jul, 2021

Given an array arr[] consisting of N positive integers and 3 integers L, R, and X, the task is to find the number of subsequences of size atleast 2 with a sum in the range [L, R], and the difference between the maximum and minimum element is at least X. (N≤15)

Examples:

Input: arr[] = {1 2 3}, L = 5, R = 6, X = 1
Output: 2
Explanation:
There are two subsequences possible i.e. {2, 3} and {1, 2, 3}.

Input: arr[] = {10, 20, 30, 25}, L = 40, R = 50, X = 10
Output: 2

Approach: Since N is small, this problem can be solved using bitmasking. There are total 2ⁿsubsequences possible. So, every subsequence can be represented by a binary string i.e. mask where if the ith bit is set i.e 1 then the element is considered in the subsequences otherwise not. Follow the steps below to solve the problem:

Iterate in the range [0, 2ⁿ - 1] using the variable i and perform the following steps:
- Initialize a variable say, cnt as 0 and sum as 0 to store the sum of selected elements.
- Initialize a variable say, minVal as INT_MAX and maxVal as INT_MIN to store minimum value and maximum value in the subsequence.
- Iterate in the range [0, N-1] using the variable j and perform the following steps:
  - If the jth bit of the ith mask is on, then Increment cnt by 1, add arr[j] to sum and update maxVal as the maximum of maxVal and a[j] and minVal as the minimum of minVal and a[j].
- If cnt >= 2 and sum is in the range [L, R] and the difference of maxVal and minVal is greater than equal to X, then increment ans by 1.
After completing the above steps, print the value of ans as the answer.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

// Function to find the number of subsequences
// of the given array with a sum in range [L, R]
// and the difference between the maximum and
// minimum element is at least X
int numberofSubsequences(int a[], int L,
                         int R, int X, int n)
{
    // Initialize answer as 0
    int ans = 0;

    // Creating mask from [0, 2^n-1]
    for (int i = 0; i < (1 << n); i++) {

        // Stores the count and sum of
        // selected elements respectively
        int cnt = 0, sum = 0;

        // Variables to store the value of
        // Minimum and maximum element
        int minVal = INT_MAX, maxVal = INT_MIN;

        // Traverse the array
        for (int j = 0; j < n; j++) {
            // If the jth bit of the ith
            // mask is on
            if ((i & (1 << j))) {

                cnt += 1;

                // Add the selected element
                sum += a[j];

                // Update maxVal and minVal value
                maxVal = max(maxVal, a[j]);
                minVal = min(minVal, a[j]);
            }
        }

        // Check if the given conditions are
        // true, increment ans by 1.
        if (cnt >= 2 && sum >= L && sum <= R
            && (maxVal - minVal >= X)) {
            ans += 1;
        }
    }
    return ans;
}

// Driver Code
int main()
{
    // Given Input
    int a[] = { 10, 20, 30, 25 };
    int L = 40, R = 50, X = 10;
    int N = sizeof(a) / sizeof(a[0]);

    // Function Call
    cout << numberofSubsequences(a, L, R, X, N)
         << endl;

    return 0;
}

Java

// Java program for the above approach
public class GFG {

    // Function to find the number of subsequences
    // of the given array with a sum in range [L, R]
    // and the difference between the maximum and
    // minimum element is at least X
    static int numberofSubsequences(int a[], int L, int R,
                                    int X, int n)
    {
        // Initialize answer as 0
        int ans = 0;

        // Creating mask from [0, 2^n-1]
        for (int i = 0; i < (1 << n); i++) {

            // Stores the count and sum of
            // selected elements respectively
            int cnt = 0, sum = 0;

            // Variables to store the value of
            // Minimum and maximum element
            int minVal = Integer.MAX_VALUE,
                maxVal = Integer.MIN_VALUE;

            // Traverse the array
            for (int j = 0; j < n; j++) {
                // If the jth bit of the ith
                // mask is on
                if ((i & (1 << j)) == 0) {

                    cnt += 1;

                    // Add the selected element
                    sum += a[j];

                    // Update maxVal and minVal value
                    maxVal = Math.max(maxVal, a[j]);
                    minVal = Math.min(minVal, a[j]);
                }
            }

            // Check if the given conditions are
            // true, increment ans by 1.
            if (cnt >= 2 && sum >= L && sum <= R
                && (maxVal - minVal >= X)) {
                ans += 1;
            }
        }
        return ans;
    }

    // Driver code
    public static void main(String[] args)
    {
        int a[] = { 10, 20, 30, 25 };
        int L = 40, R = 50, X = 10;
        int N = a.length;

        // Function Call
        System.out.println(
            numberofSubsequences(a, L, R, X, N));
    }
}

// This code is contributed by abhinavjain194

Python3

# Python3 program for the above approach
import sys

# Function to find the number of subsequences
# of the given array with a sum in range [L, R]
# and the difference between the maximum and
# minimum element is at least X
def numberofSubsequences(a, L, R, X, n):
    
    # Initialize answer as 0
    ans = 0

    # Creating mask from [0, 2^n-1]
    for i in range(0, (1 << n), 1):
        
        # Stores the count and sum of
        # selected elements respectively
        cnt = 0
        sum = 0

        # Variables to store the value of
        # Minimum and maximum element
        minVal = sys.maxsize
        maxVal = -sys.maxsize - 1

        # Traverse the array
        for j in range(n):
            
            # If the jth bit of the ith
            # mask is on
            if ((i & (1 << j))):
                cnt += 1

                # Add the selected element
                sum += a[j]

                # Update maxVal and minVal value
                maxVal = max(maxVal, a[j])
                minVal = min(minVal, a[j])

        # Check if the given conditions are
        # true, increment ans by 1.
        if (cnt >= 2 and sum >= L and 
            sum <= R and (maxVal - minVal >= X)):
            ans += 1
            
    return ans

# Driver Code
if __name__ == '__main__':
    
    # Given Input
    a = [ 10, 20, 30, 25 ]
    L = 40
    R = 50
    X = 10
    N = len(a)

    # Function Call
    print(numberofSubsequences(a, L, R, X, N))

# This code is contributed by bgangwar59

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

class GFG{

// Function to find the number of subsequences
// of the given array with a sum in range [L, R]
// and the difference between the maximum and
// minimum element is at least X
static int numberofSubsequences(int[] a, int L, int R,
                                int X, int n)
{
    
    // Initialize answer as 0
    int ans = 0;

    // Creating mask from [0, 2^n-1]
    for(int i = 0; i < (1 << n); i++)
    {
        
        // Stores the count and sum of
        // selected elements respectively
        int cnt = 0, sum = 0;

        // Variables to store the value of
        // Minimum and maximum element
        int minVal = Int32.MaxValue,
            maxVal = Int32.MinValue;

        // Traverse the array
        for(int j = 0; j < n; j++) 
        {
            
            // If the jth bit of the ith
            // mask is on
            if ((i & (1 << j)) == 0) 
            {
                cnt += 1;

                // Add the selected element
                sum += a[j];

                // Update maxVal and minVal value
                maxVal = Math.Max(maxVal, a[j]);
                minVal = Math.Min(minVal, a[j]);
            }
        }

        // Check if the given conditions are
        // true, increment ans by 1.
        if (cnt >= 2 && sum >= L && sum <= R && 
           (maxVal - minVal >= X))
        {
            ans += 1;
        }
    }
    return ans;
}

// Driver Code
static public void Main()
{
    // Given input
    int[] a = { 10, 20, 30, 25 };
    int L = 40, R = 50, X = 10;
    int N = a.Length;

    // Function Call
    Console.Write(numberofSubsequences(a, L, R, X, N));
}
}

// This code is contributed by avijitmondal1998

JavaScript

<script>
// Javascript program for the above approach

// Function to find the number of subsequences
// of the given array with a sum in range [L, R]
// and the difference between the maximum and
// minimum element is at least X
function numberofSubsequences(a, L, R, X, n)
{

    // Initialize answer as 0
    let ans = 0;

    // Creating mask from [0, 2^n-1]
    for (let i = 0; i < (1 << n); i++) {

        // Stores the count and sum of
        // selected elements respectively
        let cnt = 0, sum = 0;

        // Variables to store the value of
        // Minimum and maximum element
        let minVal = Number.MAX_SAFE_INTEGER, maxVal = Number.MIN_SAFE_INTEGER;

        // Traverse the array
        for (let j = 0; j < n; j++) 
        {
        
            // If the jth bit of the ith
            // mask is on
            if ((i & (1 << j))) 
            {

                cnt += 1;

                // Add the selected element
                sum += a[j];

                // Update maxVal and minVal value
                maxVal = Math.max(maxVal, a[j]);
                minVal = Math.min(minVal, a[j]);
            }
        }

        // Check if the given conditions are
        // true, increment ans by 1.
        if (cnt >= 2 && sum >= L && sum <= R
            && (maxVal - minVal >= X)) {
            ans += 1;
        }
    }
    return ans;
}

// Driver Code

// Given Input
let a = [10, 20, 30, 25];
let L = 40, R = 50, X = 10;
let N = a.length;

// Function Call
document.write(numberofSubsequences(a, L, R, X, N) + "<br>");

// This code is contributed by gfgking.
</script>

Output

Time Complexity: O(N×2^N)
Auxiliary Space: O(1)

Count of subsequences in an array with sum less than or equal to X

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