Count number of triplets in an array having sum in the range [a, b]

Given an array of distinct integers and a range [a, b], the task is to count the number of triplets having a sum in the range [a, b].

Examples:

Input : arr[] = {8, 3, 5, 2}
        range = [7, 11]
Output : 1
There is only one triplet {2, 3, 5}
having sum 10 in range [7, 11].

Input : arr[] = {2, 7, 5, 3, 8, 4, 1, 9}
        range = [8, 16]
Output : 36

A naive approach is to run three loops to consider all the triplets one by one. Find the sum of each triplet and increment the count if the sum lies in a given range [a, b].



Below is the implementation of above approach:

C++

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// C++ program to count triplets with
// sum that lies in given range [a, b].
#include <bits/stdc++.h>
  
using namespace std;
  
// Function to count triplets
int countTriplets(int arr[], int n, int a, int b)
{
    // Initialize result
    int ans = 0;
  
    // Fix the first element as A[i]
    for (int i = 0; i < n - 2; i++) {
  
        // Fix the second element as A[j]
        for (int j = i + 1; j < n - 1; j++) {
  
            // Now look for the third number
            for (int k = j + 1; k < n; k++)
  
                if (arr[i] + arr[j] + arr[k] >= a
                    && arr[i] + arr[j] + arr[k] <= b)
                    ans++;
        }
    }
  
    return ans;
}
  
// Driver Code
int main()
{
    int arr[] = { 2, 7, 5, 3, 8, 4, 1, 9 };
    int n = sizeof arr / sizeof arr[0];
    int a = 8, b = 16;
    cout << countTriplets(arr, n, a, b) << endl;
    return 0;
}

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Java

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// Java program to count triplets 
// with sum that lies in given 
// range [a, b].
import java.util.*;
  
class GFG
{
      
// Function to count triplets
public static int countTriplets(int []arr, int n,
                                int a, int b)
{
    // Initialize result
    int ans = 0;
  
    // Fix the first 
    // element as A[i]
    for (int i = 0; i < n - 2; i++)
    {
  
        // Fix the second 
        // element as A[j]
        for (int j = i + 1; j < n - 1; j++) 
        {
  
            // Now look for the
            // third number
            for (int k = j + 1; k < n; k++)
            {
                if (arr[i] + arr[j] + arr[k] >= a &&
                    arr[i] + arr[j] + arr[k] <= b)
                    {ans++;}
            }
        }
    }
  
    return ans;
}
  
// Driver Code
public static void main(String[] args)
{
    int[] arr = { 2, 7, 5, 3, 8, 4, 1, 9 };
    int n = arr.length;
    int a = 8, b = 16;
    System.out.println("" + countTriplets(arr, n, 
                                        a, b));
}
}
  
// This code is contributed 
// by Harshit Saini 

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Python3

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# Python3 program to count 
# triplets with sum that 
# lies in given range [a, b].
  
# Function to count triplets
def countTriplets(arr, n, a, b):
      
    # Initialize result
    ans = 0
  
    # Fix the first 
    # element as A[i]
    for i in range(0, n - 2):
          
        # Fix the second 
        # element as A[j]
        for j in range(i + 1, n - 1):
  
            # Now look for 
            # the third number
            for k in range(j + 1, n):
  
                if ((arr[i] + arr[j] + arr[k] >= a) 
                and (arr[i] + arr[j] + arr[k] <= b)):
                        ans += 1
                          
    return ans
  
# Driver code
if __name__ == "__main__":
      
    arr = [ 2, 7, 5, 3, 8, 4, 1, 9 ]
    n = len(arr)
    a = 8; b = 16
    print(countTriplets(arr, n, a, b))
  
# This code is contributed 
# by Harshit Saini

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C#

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// C# program to count triplets 
// with sum that lies in given 
// range [a, b].
using System;
  
class GFG
{
      
// Function to count triplets
public static int countTriplets(int []arr, int n,
                                int a, int b)
{
    // Initialize result
    int ans = 0;
  
    // Fix the first 
    // element as A[i]
    for (int i = 0; 
            i < n - 2; i++)
    {
  
        // Fix the second 
        // element as A[j]
        for (int j = i + 1; 
                j < n - 1; j++) 
        {
  
            // Now look for the
            // third number
            for (int k = j + 1;
                    k < n; k++)
            {
                if (arr[i] + arr[j] + arr[k] >= a &&
                    arr[i] + arr[j] + arr[k] <= b)
                    {ans++;}
            }
        }
    }
  
    return ans;
}
  
// Driver Code
public static void Main()
{
    int[] arr = {2, 7, 5, 3, 8, 4, 1, 9};
    int n = arr.Length;
    int a = 8, b = 16;
    Console.WriteLine("" + countTriplets(arr, n, 
                                        a, b));
}
}
  
// This code is contributed 
// by Akanksha Rai(Abby_akku) 

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PHP

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<?php
// PHP program to count triplets with
// sum that lies in given range [a, b].
  
// Function to count triplets
function countTriplets($arr, $n, $a, $b)
{
    // Initialize result
    $ans = 0;
  
    // Fix the first element as A[i]
    for ($i = 0; $i < $n - 2; $i++)
    {
  
        // Fix the second element as A[j]
        for ($j = $i + 1; $j < $n - 1; $j++) 
        {
  
            // Now look for the third number
            for ($k = $j + 1; $k < $n; $k++)
  
                if ($arr[$i] + $arr[$j] + $arr[$k] >= $a && 
                    $arr[$i] + $arr[$j] + $arr[$k] <= $b)
                    $ans++;
        }
    }
  
    return $ans;
}
  
// Driver Code
$arr = array( 2, 7, 5, 3, 8, 4, 1, 9 );
$n = sizeof($arr);
$a = 8; $b = 16;
echo countTriplets($arr, $n, $a, $b) . "\n";
  
// This code is contributed 
// by Akanksha Rai(Abby_akku)
?>

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Output:

36

Time complexity: O(n3)

An efficient solution is to first find the count of triplets having a sum less than or equal to upper limit b in the range [a, b]. This count of triplets will also include triplets having a sum less than the lower limit a. Subtract the count of triplets having a sum less than a. The final result is the count of triplets having a sum in the range [a, b].

The algorithm is as follows:

  • Find count of triplets having a sum less than or equal to b. Let this count be x.
  • Find count of triplets having a sum less than a. Let this count be y.
  • Final result is x-y.

To find the count of triplets having a sum less than or equal to given value, refer Count triplets with sum smaller than a given value

Below is the implementation of the above approach:

C++

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// C++ program to count triplets with
// sum that lies in given range [a, b].
#include <bits/stdc++.h>
  
using namespace std;
  
// Function to find count of triplets having
// sum less than or equal to val.
int countTripletsLessThan(int arr[], int n, int val)
{
    // sort the input array.
    sort(arr, arr + n);
  
    // Initialize result
    int ans = 0;
  
    int j, k;
  
    // to store sum
    int sum;
  
    // Fix the first element
    for (int i = 0; i < n - 2; i++) {
  
        // Initialize other two elements as
        // corner elements of subarray arr[j+1..k]
        j = i + 1;
        k = n - 1;
  
        // Use Meet in the Middle concept.
        while (j != k) {
            sum = arr[i] + arr[j] + arr[k];
  
            // If sum of current triplet
            // is greater, then to reduce it
            // decrease k.
            if (sum > val)
                k--;
  
            // If sum is less than or equal
            // to given value, then add
            // possible triplets (k-j) to result.
            else {
                ans += (k - j);
                j++;
            }
        }
    }
  
    return ans;
}
  
// Function to return count of triplets having
// sum in range [a, b].
int countTriplets(int arr[], int n, int a, int b)
{
  
    // to store count of triplets.
    int res;
  
    // Find count of triplets having sum less
    // than or equal to b and subtract count
    // of triplets having sum less than or
    // equal to a-1.
    res = countTripletsLessThan(arr, n, b) - 
        countTripletsLessThan(arr, n, a - 1);
  
    return res;
}
  
// Driver Code
int main()
{
    int arr[] = { 2, 7, 5, 3, 8, 4, 1, 9 };
    int n = sizeof arr / sizeof arr[0];
    int a = 8, b = 16;
    cout << countTriplets(arr, n, a, b) << endl;
    return 0;
}

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Java

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// Java program to count triplets 
// with sum that lies in given 
// range [a, b].
import java.util.*;
  
class GFG
{
// Function to find count of 
// triplets having sum less
// than or equal to val.
public static int countTripletsLessThan(int []arr, 
                                        int n, int val)
{
    // sort the input array.
    Arrays.sort(arr);
  
    // Initialize result
    int ans = 0;
  
    int j, k;
  
    // to store sum
    int sum;
  
    // Fix the first element
    for (int i = 0; i < n - 2; i++)
    {
  
        // Initialize other two elements 
        // as corner elements of subarray
        // arr[j+1..k]
        j = i + 1;
        k = n - 1;
  
        // Use Meet in the
        // Middle concept.
        while (j != k) 
        {
            sum = arr[i] + arr[j] + arr[k];
  
            // If sum of current triplet
            // is greater, then to reduce it
            // decrease k.
            if (sum > val)
                k--;
  
            // If sum is less than or 
            // equal to given value, 
            // then add possible 
            // triplets (k-j) to result.
            else
            {
                ans += (k - j);
                j++;
            }
        }
    }
  
    return ans;
}
  
    // Function to return count 
    // of triplets having sum 
    // in range [a, b].
    public static int countTriplets(int arr[], int n, 
                                    int a, int b)
    {
      
        // to store count
        // of triplets.
        int res;
      
        // Find count of triplets 
        // having sum less than or 
        // equal to b and subtract 
        // count of triplets having
        // sum less than or equal 
        // to a-1.
        res = countTripletsLessThan(arr, n, b) - 
            countTripletsLessThan(arr, n, a - 1);
      
        return res;
    }
  
// Driver Code
public static void main(String[] args)
{
    int[] arr = {2, 7, 5, 3
                8, 4, 1, 9};
    int n = arr.length;
    int a = 8, b = 16;
    System.out.println("" + countTriplets(arr, n, 
                                        a, b));
}
}
  
// This code is contributed 
// by Harshit Saini 

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Python3

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# Python program to count 
# triplets with sum that 
# lies in given range [a, b].
  
# Function to find count of 
# triplets having sum less
# than or equal to val.
def countTripletsLessThan(arr, n, val):
  
    # sort the input array.
    arr.sort()
  
    # Initialize result
    ans = 0
  
    j = 0; k = 0
  
    # to store sum
    sum = 0
  
    # Fix the first element
    for i in range(0,n-2):
  
        # Initialize other two 
        # elements as corner 
        # elements of subarray 
        # arr[j+1..k]
        j = i + 1
        k = n - 1
  
        # Use Meet in the 
        # Middle concept.
        while j != k :
            sum = arr[i] + arr[j] + arr[k]
              
            # If sum of current triplet
            # is greater, then to reduce it
            # decrease k.
            if sum > val:
                k-=1
  
            # If sum is less than or 
            # equal to given value, 
            # then add possible 
            # triplets (k-j) to result.
            else :
                ans += (k - j)
                j += 1
    return ans
  
# Function to return
# count of triplets having
# sum in range [a, b].
def countTriplets(arr, n, a, b):
      
    # to store count of triplets.
    res = 0
  
    # Find count of triplets 
    # having sum less than or 
    # equal to b and subtract 
    # count of triplets having 
    # sum less than or equal to a-1.
    res = (countTripletsLessThan(arr, n, b) -
        countTripletsLessThan(arr, n, a - 1))
  
    return res
  
# Driver code
if __name__ == "__main__":
      
    arr = [ 2, 7, 5, 3, 8, 4, 1, 9 ]
    n = len(arr)
    a = 8; b = 16
    print(countTriplets(arr, n, a, b))
      
# This code is contributed by 
# Harshit Saini

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C#

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// C# program to count triplets 
// with sum that lies in given 
// range [a, b].
using System;
  
class GFG
{
// Function to find count of 
// triplets having sum less
// than or equal to val.
public static int countTripletsLessThan(int[] arr, 
                                        int n, int val)
{
    // sort the input array.
    Array.Sort(arr);
  
    // Initialize result
    int ans = 0;
  
    int j, k;
  
    // to store sum
    int sum;
  
    // Fix the first element
    for (int i = 0; i < n - 2; i++)
    {
  
        // Initialize other two elements 
        // as corner elements of subarray
        // arr[j+1..k]
        j = i + 1;
        k = n - 1;
  
        // Use Meet in the
        // Middle concept.
        while (j != k) 
        {
            sum = arr[i] + arr[j] + arr[k];
  
            // If sum of current triplet
            // is greater, then to reduce it
            // decrease k.
            if (sum > val)
                k--;
  
            // If sum is less than or 
            // equal to given value, 
            // then add possible 
            // triplets (k-j) to result.
            else
            {
                ans += (k - j);
                j++;
            }
        }
    }
  
    return ans;
}
  
    // Function to return count 
    // of triplets having sum 
    // in range [a, b].
    public static int countTriplets(int[] arr, int n, 
                                    int a, int b)
    {
      
        // to store count
        // of triplets.
        int res;
      
        // Find count of triplets 
        // having sum less than or 
        // equal to b and subtract 
        // count of triplets having
        // sum less than or equal 
        // to a-1.
        res = countTripletsLessThan(arr, n, b) - 
            countTripletsLessThan(arr, n, a - 1);
      
        return res;
    }
  
// Driver Code
public static void Main()
{
    int[] arr = {2, 7, 5, 3, 
                8, 4, 1, 9};
    int n = arr.Length;
    int a = 8, b = 16;
    Console.WriteLine("" + countTriplets(arr, n, 
                                        a, b));
}
}
  
// This code is contributed 
// by Akanksha Rai(Abby_akku) 

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PHP

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<?php
// PHP program to count triplets with
// sum that lies in given range [a, b].
  
// Function to find count of triplets 
// having sum less than or equal to val.
function countTripletsLessThan($arr, $n, $val)
{
    // sort the input array.
    sort($arr);
  
    // Initialize result
    $ans = 0;
  
    // to store sum
    $sum;
  
    // Fix the first element
    for ($i = 0; $i < $n - 2; $i++)
    {
  
        // Initialize other two elements as
        // corner elements of subarray arr[j+1..k]
        $j = $i + 1;
        $k = $n - 1;
  
        // Use Meet in the Middle concept.
        while ($j != $k)
        {
            $sum = $arr[$i] + $arr[$j] + $arr[$k];
  
            // If sum of current triplet is greater, 
            // then to reduce it decrease k.
            if ($sum > $val)
                $k--;
  
            // If sum is less than or equal
            // to given value, then add possible
            // triplets (k-j) to result.
            else 
            {
                $ans += ($k - $j);
                $j++;
            }
        }
    }
  
    return $ans;
}
  
// Function to return count of triplets 
// having sum in range [a, b].
function countTriplets($arr, $n, $a, $b)
{
  
    // to store count of triplets.
    $res;
  
    // Find count of triplets having sum less
    // than or equal to b and subtract count
    // of triplets having sum less than or
    // equal to a-1.
    $res = countTripletsLessThan($arr, $n, $b) - 
           countTripletsLessThan($arr, $n, $a - 1);
  
    return $res;
}
  
// Driver Code
$arr = array( 2, 7, 5, 3, 8, 4, 1, 9 );
$n = sizeof($arr);
$a = 8;
$b = 16;
echo countTriplets($arr, $n, $a, $b), "\n";
      
// This code is contributed by Sachin
?>

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Output:

36

Time complexity: O(n2)
Auxiliary space: O(1)



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