Count Triplets such that one of the numbers can be written as sum of the other two

Difficulty Level : Hard
Last Updated : 23 Jul, 2019

Given an array A[] of N integers. The task is to find the number of triples (i, j, k) , where i, j, k are indices and (1 <= i < j < k <= N), such that in the set { $A_i$ , $A_j$ , $A_k$ } at least one of the numbers can be written as the sum of the other two.

Examples:

Input : A[] = {1, 2, 3, 4, 5}
Output : 4
The valid triplets are:
(1, 2, 3), (1, 3, 4), (1, 4, 5), (2, 3, 5)

Input : A[] = {1, 1, 1, 2, 2}
Output : 6

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

This is a counting problem. Lets say f(x) represents the frequency of number $x$ in our array.

There exist four cases:

All three numbers are equal to 0. The number of ways = f(0)C3 (where pCq is the number of ways of choosing q numbers from p numbers).
One number is equal to 0, the other two are equal to some x > 0: f(0) * f(x)C2.
Two numbers are equal to some x>0, the third is 2*x: f(x)C2 * f(2 * x).
The three numbers are x, y and x + y, 0 < x, y: f(x) * f(y) * f(x + y).

Below is the implementation of the above approach:

C++

// C++ program to count Triplets such that at  
// least one of the numbers can be written  
// as sum of the other two  
#include<bits/stdc++.h> 
using namespace std; 
  
    // Functoin to count the number of ways  
    // to choose the triples  
    int countWays(int arr[], int n)  
    {  
        // compute the max value in the array  
        // and create frequency array of size  
        // max_val + 1.  
        // We can also use HashMap to store  
        // frequencies. We have used an array  
        // to keep remaining code simple.  
        int max_val = 0;  
        for (int i = 0; i < n; i++)  
            max_val = max(max_val, arr[i]);  
        int freq[max_val + 1]={0};  
        for (int i = 0; i < n; i++)  
            freq[arr[i]]++;  
  
        int ans = 0; // stores the number of ways  
  
        // Case 1: 0, 0, 0  
        ans += freq[0] * (freq[0] - 1) * (freq[0] - 2) / 6;  
  
        // Case 2: 0, x, x  
        for (int i = 1; i <= max_val; i++)  
            ans += freq[0] * freq[i] * (freq[i] - 1) / 2;  
  
        // Case 3: x, x, 2*x  
        for (int i = 1; 2 * i <= max_val; i++)  
            ans += freq[i] * (freq[i] - 1) / 2 * freq[2 * i];  
  
        // Case 4: x, y, x + y  
        // iterate through all pairs (x, y)  
        for (int i = 1; i <= max_val; i++) {  
            for (int j = i + 1; i + j <= max_val; j++)  
                ans += freq[i] * freq[j] * freq[i + j];  
        }  
  
        return ans;  
    }  
  
    // Driver code  
    int main() 
    {  
        int arr[]={ 1, 2, 3, 4, 5 };  
        int n = sizeof(arr)/sizeof(int);  
        cout<<(countWays(arr, n));  
        return 0; 
    }  
  
//contributed by Arnab Kundu 

Java

// Java program to count Triplets such that at 
// least one of the numbers can be written 
// as a sum of the other two 
  
class GFG { 
  
    // Function to count the number of ways 
    // to choose the triples 
    static int countWays(int[] arr, int n) 
    { 
        // compute the max value in the array 
        // and create frequency array of size 
        // max_val + 1. 
        // We can also use HashMap to store 
        // frequencies. We have used an array 
        // to keep remaining code simple. 
        int max_val = 0; 
        for (int i = 0; i < n; i++) 
            max_val = Math.max(max_val, arr[i]); 
        int[] freq = new int[max_val + 1]; 
        for (int i = 0; i < n; i++) 
            freq[arr[i]]++; 
  
        int ans = 0; // stores the number of ways 
  
        // Case 1: 0, 0, 0 
        ans += freq[0] * (freq[0] - 1) * (freq[0] - 2) / 6; 
  
        // Case 2: 0, x, x 
        for (int i = 1; i <= max_val; i++) 
            ans += freq[0] * freq[i] * (freq[i] - 1) / 2; 
  
        // Case 3: x, x, 2*x 
        for (int i = 1; 2 * i <= max_val; i++) 
            ans += freq[i] * (freq[i] - 1) / 2 * freq[2 * i]; 
  
        // Case 4: x, y, x + y 
        // iterate through all pairs (x, y) 
        for (int i = 1; i <= max_val; i++) { 
            for (int j = i + 1; i + j <= max_val; j++) 
                ans += freq[i] * freq[j] * freq[i + j]; 
        } 
  
        return ans; 
    } 
  
    // Driver code 
    public static void main(String[] args) 
    { 
        int[] arr = new int[] { 1, 2, 3, 4, 5 }; 
        int n = arr.length; 
        System.out.println(countWays(arr, n)); 
    } 
} 

Python3

# Python3 program to count Triplets such  
# that at least one of the numbers can be  
# written as sum of the other two  
import math as mt 
  
# Functoin to count the number of ways  
# to choose the triples  
def countWays(arr, n):  
  
    # compute the max value in the array  
    # and create frequency array of size  
    # max_val + 1.  
    # We can also use HashMap to store  
    # frequencies. We have used an array  
    # to keep remaining code simple.  
    max_val = 0
    for i in range(n):  
        max_val = max(max_val, arr[i]) 
  
    freq = [0 for i in range(max_val + 1)]  
  
    for i in range(n):  
        freq[arr[i]] += 1
  
    ans = 0 # stores the number of ways  
  
    # Case 1: 0, 0, 0  
    ans += (freq[0] * (freq[0] - 1) * 
           (freq[0] - 2) // 6) 
  
    # Case 2: 0, x, x  
    for i in range(1, max_val + 1):  
        ans += (freq[0] * freq[i] *
               (freq[i] - 1) // 2) 
  
    # Case 3: x, x, 2*x  
    for i in range(1, (max_val + 1) // 2):  
        ans += (freq[i] *
               (freq[i] - 1) // 2 * freq[2 * i])  
  
    # Case 4: x, y, x + y  
    # iterate through all pairs (x, y)  
    for i in range(1, max_val + 1):  
        for j in range(i + 1, max_val - i + 1):  
            ans += freq[i] * freq[j] * freq[i + j]  
  
    return ans  
  
# Driver code  
arr = [ 1, 2, 3, 4, 5] 
n = len(arr) 
print(countWays(arr, n))  
  
# This code is contributed by 
# mohit kumar 29 

C#

// C# program to count Triplets  
// such that at least one of the  
// numbers can be written as sum  
// of the other two 
using System; 
  
class GFG 
{ 
  
// Function to count the number  
// of ways to choose the triples 
static int countWays(int[] arr, int n) 
{ 
    // compute the max value in the array 
    // and create frequency array of size 
    // max_val + 1. 
    // We can also use HashMap to store 
    // frequencies. We have used an array 
    // to keep remaining code simple. 
    int max_val = 0; 
    for (int i = 0; i < n; i++) 
        max_val = Math.Max(max_val, arr[i]); 
          
    int[] freq = new int[max_val + 1]; 
    for (int i = 0; i < n; i++) 
        freq[arr[i]]++; 
  
    int ans = 0; // stores the number of ways 
  
    // Case 1: 0, 0, 0 
    ans += freq[0] * (freq[0] - 1) *  
                     (freq[0] - 2) / 6; 
  
    // Case 2: 0, x, x 
    for (int i = 1; i <= max_val; i++) 
        ans += freq[0] * freq[i] *  
                        (freq[i] - 1) / 2; 
  
    // Case 3: x, x, 2*x 
    for (int i = 1;  
             2 * i <= max_val; i++) 
        ans += freq[i] * (freq[i] - 1) /  
                      2 * freq[2 * i]; 
  
    // Case 4: x, y, x + y 
    // iterate through all pairs (x, y) 
    for (int i = 1; i <= max_val; i++) 
    { 
        for (int j = i + 1;  
                 i + j <= max_val; j++) 
            ans += freq[i] * freq[j] *  
                             freq[i + j]; 
    } 
  
    return ans; 
} 
  
// Driver code 
public static void Main() 
{ 
    int[] arr = { 1, 2, 3, 4, 5 }; 
    int n = arr.Length; 
    Console.WriteLine(countWays(arr, n)); 
} 
} 
  
// This code is contributed by shs.. 

PHP

<?php 
// PHP program to count Triplets such that at  
// least one of the numbers can be written  
// as sum of the other two  
  
// Function to count the number of ways  
// to choose the triples  
function countWays($arr, $n)  
{  
    // compute the max value in the array  
    // and create frequency array of size  
    // max_val + 1.  
    // We can also use HashMap to store  
    // frequencies. We have used an array  
    // to keep remaining code simple.  
    $max_val = 0;  
    for ($i = 0; $i < $n; $i++)  
        $max_val = max($max_val, $arr[$i]);  
    $freq = array_fill(0, $max_val + 1, 0);  
    for ($i = 0; $i < $n; $i++)  
        $freq[$arr[$i]]++;  
  
    $ans = 0; // stores the number of ways  
  
    // Case 1: 0, 0, 0  
    $ans += (int)($freq[0] * ($freq[0] - 1) *  
                             ($freq[0] - 2) / 6);  
  
    // Case 2: 0, x, x  
    for ($i = 1; $i <= $max_val; $i++)  
        $ans += (int)($freq[0] * $freq[$i] *  
                     ($freq[$i] - 1) / 2);  
  
    // Case 3: x, x, 2*x  
    for ($i = 1; 2 * $i <= $max_val; $i++)  
        $ans += (int)($freq[$i] * ($freq[$i] - 1) / 2 *  
                                   $freq[2 * $i]);  
  
    // Case 4: x, y, x + y  
    // iterate through all pairs (x, y)  
    for ($i = 1; $i <= $max_val; $i++)  
    {  
        for ($j = $i + 1; $i + $j <= $max_val; $j++)  
            $ans += $freq[$i] * $freq[$j] *  
                                $freq[$i + $j];  
    }  
  
    return $ans;  
}  
  
// Driver code  
$arr = array( 1, 2, 3, 4, 5 );  
$n = count($arr);  
echo countWays($arr, $n);  
  
// This code is contributed by mits 
?> 

Output:

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Related Articles

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

PHP