Count sub-arrays which have elements less than or equal to X

Given an array of n elements and an integer X. Count the number of sub-arrays of this array which have all elements less than or equal to X.

Examples:

Input : arr[] = {1, 5, 7, 8, 2, 3, 9}  
            X = 6
Output : 6
Explanation : Sub-arrays are {1}, {5}, {2}, {3},
{1, 5}, {2, 3}

Input : arr[] =  {1, 10, 12, 4, 5, 3, 2, 7}   
            X = 9
Output : 16



Naive Approach : A simple approach is use two nested loops for generating all sub-arrays of the given a array and a loop to check whether all elements of a sub-array is less than or equal to X or not.

Time Complexity: O(n*n*n)

Efficient Approach: An efficient approach is to observe that we just want the count of those sub-arrays which have all elements less than or equal to X. We can create a binary array of 0s and 1s corresponding to the original array. If an element in the original is less than or equal to X, then the corresponding element in the binary array will be 1 otherwise 0. Now, our problem reduces to count the number of sub-arrays in this binary array which has all 1s. We can also see that for an array which has all 1s all of its sub-arrays will have only 1s and the total number of sub-arrays will be len*(len+1)/2. For example, {1, 1, 1, 1} will have 10 sub-arrays.

Below is the complete algorithm to solve the above problem:

  • Create a corresponding binary array of the original array as described above.
  • Initialize a counter variable to 0 and start traversing the binary array keeping track of the lengths of sub-arrays which has all 1s
  • We can easily calculate the number of sub-arrays of an array which has all 1s by using the formula n*(n+1)/2, where n is the length of the array with all 1s.
  • Calculate length of every sub-array which has all 1s and increment the count variable by length*(length+1)/2. We can do this in O(n) time complexity

Below is the implementation of above approach:

C++

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// C++ program to count all sub-arrays which
// has all elements less than or equal to X
#include <iostream>
using namespace std;
  
// function to count all sub-arrays which
// has all elements less than or equal to X
int countSubArrays(int arr[], int n, int x)
{
    // variable to keep track of length of
    // subarrays with all 1s
    int len = 0;
  
    // variable to keep track of all subarrays
    int count = 0;
  
    // binary array of same size
    int binaryArr[n];
  
    // creating binary array
    for (int i = 0; i < n; i++) {
        if (arr[i] <= x)
            binaryArr[i] = 1;
        else
            binaryArr[i] = 0;
    }
  
    // start traversing the binary array
    for (int i = 0; i < n; i++) {
  
        // once we find the first 1, keep checking
        // for number of consecutive 1s
        if (binaryArr[i] == 1) {
            int j;
  
            for (j = i + 1; j < n; j++) 
                if (binaryArr[j] != 1) 
                    break;
  
            // calculate length of the subarray 
            // with all 1s
            len = j - i;
  
            // increment count
            count += (len) * (len + 1) / 2;
  
            // intialize i to j
            i = j;
        }
    }
  
    return count;
}
  
// Driver code
int main()
{
    int arr[] = { 1, 5, 7, 8, 2, 3, 9 };
    int x = 6;
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << countSubArrays(arr, n, x);
    return 0;
}

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Java

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// Java program to count all sub-arrays which
// has all elements less than or equal to X
import java.io.*;
  
class GFG {
      
    // function to count all sub-arrays which
    // has all elements less than or equal to X
    static int countSubArrays(int arr[], int n, int x)
    {
          
        // variable to keep track of length of
        // subarrays with all 1s
        int len = 0;
      
        // variable to keep track of all subarrays
        int count = 0;
      
        // binary array of same size
        int binaryArr[] = new int[n];
      
        // creating binary array
        for (int i = 0; i < n; i++) {
            if (arr[i] <= x)
                binaryArr[i] = 1;
            else
                binaryArr[i] = 0;
        }
      
        // start traversing the binary array
        for (int i = 0; i < n; i++) {
      
            // once we find the first 1, keep checking
            // for number of consecutive 1s
            if (binaryArr[i] == 1) {
                int j;
      
                for (j = i + 1; j < n; j++) 
                    if (binaryArr[j] != 1
                        break;
      
                // calculate length of the subarray 
                // with all 1s
                len = j - i;
      
                // increment count
                count += (len) * (len + 1) / 2;
      
                // intialize i to j
                i = j;
            }
        }
      
        return count;
    }
      
    // Driver code
    public static void main(String args[])
    {
        int arr[] = { 1, 5, 7, 8, 2, 3, 9 };
        int x = 6;
        int n = arr.length;
          
        System.out.println(countSubArrays(arr, n, x));
    }
}
  
// This code is contributed by Nikita Tiwari.

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Python3

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# python 3 program to count all sub-arrays which
# has all elements less than or equal to X
  
# function to count all sub-arrays which
# has all elements less than or equal to X
def countSubArrays(arr, n, x):
      
    # variable to keep track of length 
    # of subarrays with all 1s
    len = 0
  
    # variable to keep track of 
    # all subarrays
    count = 0
  
    # binary array of same size
    binaryArr = [0 for i in range(n)]
  
    # creating binary array
    for i in range(0, n, 1):
        if (arr[i] <= x):
            binaryArr[i] = 1
        else:
            binaryArr[i] = 0
  
    # start traversing the binary array
    for i in range(0, n, 1):
          
        # once we find the first 1, 
        # keep checking for number 
        # of consecutive 1s
        if (binaryArr[i] == 1):
            for j in range(i + 1, n, 1):
                if (binaryArr[j] != 1):
                    break
  
            # calculate length of the 
            # subarray with all 1s
            len = j - i
  
            # increment count
            count += (len) * (int)((len + 1) / 2)
  
            # intialize i to j
            i = j
  
    return count
  
# Driver code
if __name__ == '__main__':
    arr = [1, 5, 7, 8, 2, 3, 9]
    x = 6
    n = len(arr)
    print(int(countSubArrays(arr, n, x)))
      
# This code is contributed by
# Surendra_Gangwar

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C#

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// C# program to count all sub-arrays which
// has all elements less than or equal to X1
using System;
  
class GFG {
      
    // function to count all sub-arrays which
    // has all elements less than or equal
    // to X
    static int countSubArrays(int []arr, 
                                int n, int x)
    {
          
        // variable to keep track of length
        // of subarrays with all 1s
        int len = 0;
      
        // variable to keep track of all
        // subarrays
        int count = 0;
      
        // binary array of same size
        int []binaryArr = new int[n];
      
        // creating binary array
        for (int i = 0; i < n; i++) {
            if (arr[i] <= x)
                binaryArr[i] = 1;
            else
                binaryArr[i] = 0;
        }
      
        // start traversing the binary array
        for (int i = 0; i < n; i++) {
      
            // once we find the first 1, keep
            // checking for number of
            // consecutive 1s
            if (binaryArr[i] == 1) {
                int j;
      
                for (j = i + 1; j< n; j++) 
                    if (binaryArr[j] != 1) 
                        break;
      
                // calculate length of the 
                // subarray with all 1s
                len = j - i;
      
                // increment count
                count += (len) * (len + 1) / 2;
      
                // intialize i to j
                i = j;
            }
        }
      
        return count;
    }
      
    // Driver code
    public static void Main()
    {
        int []arr = { 1, 5, 7, 8, 2, 3, 9 };
        int x = 6;
        int n = arr.Length;
          
        Console.WriteLine(
                    countSubArrays(arr, n, x));
    }
}
  
// This code is contributed by Sam007.

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PHP

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<?php
// PHP program to count all sub-arrays which
// has all elements less than or equal to X
  
// function to count all sub-arrays which
// has all elements less than or equal to X
function countSubArrays($arr, $n, $x)
{
      
    // variable to keep track of length of
    // subarrays with all 1s
    $len = 0;
  
    // variable to keep track of all subarrays
    $coun = 0;
  
    // binary array of same size
    $binaryArr = array($n);
  
    // creating binary array
    for ($i = 0; $i < $n; $i++)
    {
        if ($arr[$i] <= $x)
            $binaryArr[$i] = 1;
        else
            $binaryArr[$i] = 0;
    }
  
    // start traversing 
    // the binary array
    for ($i = 0; $i < $n; $i++)
    {
  
        // once we find the first 1,
        // keep checking for number 
        // of consecutive 1s
        if ($binaryArr[$i] == 1) {
              
  
            for ($j = $i + 1; $j < $n; $j++) 
                if ($binaryArr[$j] != 1) 
                    break;
  
            // calculate length of 
            // the subarray with all 1s
            $len = $j - $i;
  
            // increment count
            $coun += ($len) * ($len + 1) / 2;
  
            // intialize i to j
            $i = $j;
        }
    }
  
    return $coun;
}
  
    // Driver code
    $arr = array( 1, 5, 7, 8, 2, 3, 9 );
    $x = 6;
    $n = count($arr);
    echo countSubArrays($arr, $n, $x);
  
// This code is contributed by Sam007
?>

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Output:

6

Time Complexity: O(n), where n is the number of elements in the array.
Auxiliary Space: O(n).

Another Method: We can improve the above solution without using extra space keeping the time complexity O(n). Instead of marking elements as 0 and 1 we can keep track of start and end of each such region and update the count whenever the region ends.

C++

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// C++ program to count all sub-arrays which
// has all elements less than or equal to X
  
#include<bits/stdc++.h>
using namespace std;
  
int countSubArrays(int arr[], int x, int n )
    {
        int count = 0;
        int start = -1, end = -1;
      
        for(int i = 0; i < n; i++)
        {
            if(arr[i] < x) 
            {
                if(start == -1)
                {
                      
                    //create a new subArray
                    start = i;
                    end = i;
                }
                else
                {
                      
                    // append to existing subarray
                    end=i;
                }
            }
            else
            {
                if(start != -1 && end != -1)
                {
                      
                    // given start and end calculate
                    // all subarrays within this range
                    int length = end - start + 1;
                    count = count + ((length * (length + 1)) / 2);
                }
                  
                start = -1;
                end = -1;
            }
  
        }
          
        if(start != -1 && end != -1)
        {
              
            // given start and end calculate all
            // subarrays within this range
            int length = end - start + 1;
            count = count + ((length * (length + 1)) / 2);
        }
          
        return count;
    }
      
    // Driver code 
int main()
    {
        int arr[] = { 1, 5, 7, 8, 2, 3, 9 };
        int x = 6;
        int n = sizeof(arr) / sizeof(arr[0]);
        cout<< countSubArrays(arr, x, n);
          
//This code is contributed by  29AjayKumar
  
    }

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Java

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// Java program to count all sub-arrays which
// has all elements less than or equal to X
  
public class GFG {
  
    public static int countSubArrays(int arr[], int x)
    {
        int count = 0;
        int start = -1, end = -1;
          
        for(int i = 0; i < arr.length; i++)
        {
            if(arr[i] < x) 
            {
                if(start == -1)
                {
                      
                    //create a new subArray
                    start = i;
                    end = i;
                }
                else
                {
                      
                    // append to existing subarray
                    end=i;
                }
            }
            else
            {
                if(start != -1 && end != -1)
                {
                      
                    // given start and end calculate
                    // all subarrays within this range
                    int length = end - start + 1;
                    count = count + ((length * (length + 1)) / 2);
                }
                  
                start = -1;
                end = -1;
            }
  
        }
          
        if(start != -1 && end != -1)
        {
              
            // given start and end calculate all
            // subarrays within this range
            int length = end - start + 1;
            count = count + ((length * (length + 1)) / 2);
        }
          
        return count;
    }
      
    // Driver code 
    public static void main(String[] args)
    {
        int arr[] = { 1, 5, 7, 8, 2, 3, 9 };
        int x = 6;
        System.out.println(countSubArrays(arr, x));
  
    }
}

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C#

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// C# program to count all sub-arrays which
// has all elements less than or equal to X
using System;
  
class GFG 
{
  
    public static int countSubArrays(int []arr, int x)
    {
        int count = 0;
        int start = -1, end = -1;
          
        for(int i = 0; i < arr.Length; i++)
        {
            if(arr[i] < x) 
            {
                if(start == -1)
                {
                      
                    //create a new subArray
                    start = i;
                    end = i;
                }
                else
                {
                      
                    // append to existing subarray
                    end=i;
                }
            }
            else
            {
                if(start != -1 && end != -1)
                {
                      
                    // given start and end calculate
                    // all subarrays within this range
                    int length = end - start + 1;
                    count = count + ((length * (length + 1)) / 2);
                }
                  
                start = -1;
                end = -1;
            }
  
        }
          
        if(start != -1 && end != -1)
        {
              
            // given start and end calculate all
            // subarrays within this range
            int length = end - start + 1;
            count = count + ((length * (length + 1)) / 2);
        }
          
        return count;
    }
      
    // Driver code 
    public static void Main(String[] args)
    {
        int []arr = { 1, 5, 7, 8, 2, 3, 9 };
        int x = 6;
        Console.WriteLine(countSubArrays(arr, x));
    }
}
  
// This code contributed by Rajput-Ji

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Output:

6

Time Complexity: O(n), where n is the number of elements in the array.
Auxiliary Space: O(1).



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