Count the number of unordered triplets with elements in increasing order and product less than or equal to integer X

Given an array A[] and an integer X. Find the number of unordered triplets (i, j, k) such that A[i] < A[j] < A[k] and A[i] * A[j] * A[k] <= X.
Examples: 

Input: A = [3, 2, 5, 7], X = 42 
Output:
Explanation: 
Triplets are : 

  • (1, 0, 2) => 2 < 3 < 5, 2 * 3 * 5 < = 42
  • (1, 0, 3) => 2 < 3 < 7, 2 * 3 * 7 < = 42 
     

Input: A = [3, 1, 2, 56, 21, 8], X = 49 
Output: 5  

Naive Approach:
The naive method to solve the above-mentioned problem is to iterate through all the triplets. For each triplet arrange them in ascending order (since we have to count unordered triplets, therefore rearranging them is allowed), and check the given condition. But this method takes O(N 3) time.

Below is the implementation of the above approach:  



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// C++ implementation to Count the number of
// unordered triplets such that the numbers are
// in increasing order and the product of them is
// less than or equal to integer X
#include <bits/stdc++.h>
using namespace std;
 
// Function to count the number of triplets
int countTriplets(int a[], int n, int x)
{
    int answer = 0;
 
    // Iterate through all the triplets
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
            for (int k = j + 1; k < n; k++) {
                vector<int> temp;
                temp.push_back(a[i]);
                temp.push_back(a[j]);
                temp.push_back(a[k]);
 
                // Rearrange the numbers in ascending order
                sort(temp.begin(), temp.end());
 
                // Check if the necessary conditions satisfy
                if (temp[0] < temp[1] && temp[1] < temp[2]
                    && temp[0] * temp[1] * temp[2] <= x)
 
                    // Increment count
                    answer++;
            }
        }
    }
 
    // Return the answer
    return answer;
}
 
// Driver code
int main()
{
 
    int A[] = { 3, 2, 5, 7 };
 
    int N = sizeof(A) / sizeof(A[0]);
 
    int X = 42;
 
    cout << countTriplets(A, N, X);
 
    return 0;
}
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// Java implementation to count the number of
// unordered triplets such that the numbers are
// in increasing order and the product of them
// is less than or equal to integer X
import java.util.*;
 
class GFG{
     
// Function to count the number of triplets
static int countTriplets(int a[], int n, int x)
{
    int answer = 0;
 
    // Iterate through all the triplets
    for(int i = 0; i < n; i++)
    {
       for(int j = i + 1; j < n; j++)
       {
          for(int k = j + 1; k < n; k++)
          {
              Vector<Integer> temp = new Vector<>();
              temp.add(a[i]);
              temp.add(a[j]);
              temp.add(a[k]);
               
              // Rearrange the numbers in
              // ascending order
              Collections.sort(temp);
               
              // Check if the necessary conditions
              // satisfy
              if (temp.get(0) < temp.get(1) &&
                  temp.get(1) < temp.get(2) &&
                  temp.get(0) * temp.get(1) *
                  temp.get(2) <= x)
                   
                  // Increment count
                  answer++;
          }
       }
    }
     
    // Return the answer
    return answer;
}
 
// Driver code
public static void main(String[] args)
{
    int A[] = { 3, 2, 5, 7 };
    int N = A.length;
    int X = 42;
 
    System.out.println(countTriplets(A, N, X));
}
}
 
// This code is contributed by offbeat
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# Python3 implementation to count the number of
# unordered triplets such that the numbers are
# in increasing order and the product of them is
# less than or equal to integer X
 
# Function to count the number of triplets
def countTriplets(a, n, x):
     
    answer = 0
     
    # Iterate through all the triplets
    for i in range(n):
        for j in range(i + 1, n):
            for k in range(j + 1, n):
                temp = []
                temp.append(a[i])
                temp.append(a[j])
                temp.append(a[k])
                 
                # Rearrange the numbers in
                # ascending order
                temp.sort()
                 
                # Check if the necessary
                # conditions satisfy
                if (temp[0] < temp[1] and
                    temp[1] < temp[2] and
                    temp[0] * temp[1] * temp[2] <= x):
                         
                    # Increment count
                    answer += 1
                     
    # Return the answer               
    return answer
     
# Driver code
A = [ 3, 2, 5, 7 ]
N = len(A)
X = 42
 
print(countTriplets(A, N, X))
 
# This code is contributed by shubhamsingh10
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// C# implementation to count the number of
// unordered triplets such that the numbers are
// in increasing order and the product of them
// is less than or equal to integer X
using System;
 
class GFG{
     
// Function to count the number of triplets
static int countTriplets(int []a, int n, int x)
{
    int answer = 0;
 
    // Iterate through all the triplets
    for(int i = 0; i < n; i++)
    {
        for(int j = i + 1; j < n; j++)
        {
            for(int k = j + 1; k < n; k++)
            {
                int []temp = { a[i], a[j], a[k] };
                 
                // Rearrange the numbers in
                // ascending order
                Array.Sort(temp);
                     
                // Check if the necessary conditions
                // satisfy
                if (temp[0] < temp[1] &&
                    temp[1] < temp[2] &&
                    temp[0] * temp[1] *
                    temp[2] <= x)
                         
                    // Increment count
                    answer++;
            }
        }
    }
     
    // Return the answer
    return answer;
}
 
// Driver code
public static void Main()
{
    int []A = { 3, 2, 5, 7 };
    int N = A.Length;
    int X = 42;
 
    Console.WriteLine(countTriplets(A, N, X));
}
}
 
// This code is contributed by Stream_Cipher    
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Output: 
2



 

Efficient Approach:
To optimize the method given above we can use a sorted form of the array since it would not change the answer because the triplets are unordered. Traverse through all the pairs of elements in the sorted array. For a pair (p, q) the problem now reduces to finding the number of elements r in the sorted array such that r <= X/(p*q). To perform this efficiently we will use Binary Search method and find the position of the largest element in the array which is < = X/(p*q). All the elements between the index of q until position will be added to the answer.

Below is the implementation of the above approach:  

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// C++ implementation to Count the number of
// unordered triplets such that the numbers are
// in increasing order and the product of them is
// less than or equal to integer X
#include <bits/stdc++.h>
using namespace std;
 
// Function to count the triplets
int countTriplets(int a[], int n, int x)
{
    int answer = 0;
 
    // Sort the array
    sort(a, a + n);
 
    // Iterate through all the triplets
    for (int i = 0; i < n; i++) {
        for (int j = i + 1; j < n; j++) {
 
            // Apply Binary Search method
            long long limit = (long long)x / a[i];
 
            limit = limit / a[j];
 
            int pos = upper_bound(a, a + n, limit) - a;
 
            // Check if the position is greater than j
            if (pos > j)
                answer = answer + (pos - j - 1);
        }
    }
 
    // Return the answer
    return answer;
}
 
// Driver code
int main()
{
 
    int A[] = { 3, 2, 5, 7 };
 
    int N = sizeof(A) / sizeof(A[0]);
 
    int X = 42;
 
    cout << countTriplets(A, N, X);
 
    return 0;
}
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// Java implementation to count the number
// of unordered triplets such that the
// numbers are in increasing order and
// the product of them is less than or
// equal to integer X
import java.io.*;
import java.util.Arrays;
 
class GFG{
     
// Function to count the triplets
static int countTriplets(int a[], int n, int x)
{
    int answer = 0;
 
    // Sort the array
    Arrays.sort(a);
 
    // Iterate through all the triplets
    for(int i = 0; i < n; i++)
    {
        for(int j = i + 1; j < n; j++)
        {
             
            // Apply Binary Search method
            int limit = x / a[i];
             
            limit = limit / a[j];
 
            int pos = Arrays.binarySearch(a, limit) + 1;
 
            // Check if the position is greater than j
            if (pos > j)
                answer = answer + (pos - j - 1);
        }
    }
 
    // Return the answer
    return answer;
}
 
// Driver Code
public static void main (String[] args)
{
    int A[] = { 3, 2, 5, 7 };
    int N = A.length;
    int X = 42;
     
    System.out.print(countTriplets(A, N, X));
}
}
 
// This code is contributed by math_lover
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# Python3 implementation to Count the number of
# unordered triplets such that the numbers are
# in increasing order and the product of them is
# less than or equal to integer X
import bisect
 
# Function to count the triplets
def countTriplets(a, n, x):
     
    answer = 0
     
    # Sort the array
    a.sort()
     
    # Iterate through all the triplets
    for i in range(n):
        for j in range(i + 1, n):
             
            # Apply Binary Search method
            limit = x / a[i]
             
            limit = limit / a[j]
             
            pos = bisect.bisect_right(a, limit)
             
            # Check if the position is greater than j
            if (pos > j):
                answer = answer + (pos - j - 1)
                 
    # Return the answer
    return answer
 
# Driver code
A = [3, 2, 5, 7]
 
N = len(A)
 
X = 42
 
print(countTriplets(A, N, X))
 
# This code is contributed by shubhamsingh10
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// C# implementation to Count the number
// of unordered triplets such that the
// numbers are in increasing order and
// the product of them is less than or
// equal to integer X
using System;
 
class GFG{
     
// Function to count the triplets
static int countTriplets(int []a, int n, int x)
{
    int answer = 0;
 
    // Sort the array
    Array.Sort(a);
 
    // Iterate through all the triplets
    for(int i = 0; i < n; i++)
    {
        for(int j = i + 1; j < n; j++)
        {
             
            // Apply Binary Search method
            int limit = x / a[i];
 
            limit = limit / a[j];
 
            int pos = Array.BinarySearch(a, limit) + 1;
 
            // Check if the position is greater than j
            if (pos > j)
                answer = answer + (pos - j - 1);
        }
    }
 
    // Return the answer
    return answer;
}
 
// Driver Code
public static void Main (String[] args)
{
    int []A = { 3, 2, 5, 7 };
    int N = A.Length;
    int X = 42;
     
    Console.Write(countTriplets(A, N, X));
}
}
 
// This code is contributed by math_lover
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Output: 
2



 

Time Complexity: O(N2 * log(N))
 

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