# Maximum size of subset of given array such that a triangle can be formed by any three integers as the sides of the triangle

• Last Updated : 13 Oct, 2021

Given an array arr[] consisting of N integers, the task is to find the size of the largest subset of the array such that a triangle can be formed from any of the three integers of the subset as the sides of a triangle.

Examples:

Input: arr[] = {1, 4, 7, 4}
Output: 3
Explanation: A possible subsets that follow the given conditions are {1, 4, 4} and {4, 4, 7}. The size of both of these subsets is 3 which is the maximum possible.

Input: arr[] = {2, 7, 4, 1, 6, 9, 5, 3}
Output: 4

Approach: The given problem can be solved with the help of the Greedy Approach using the Sliding Window Technique. It is known that for a triangle having side lengths A, B, and C, A + B > C must hold true where A and B are the sides with smaller lengths. Based on the above observation the given problem can be solved using the following steps:

• Sort the given array arr[] in non-decreasing order.
• Maintain two variables i and j where i keep track of the starting point of the current window and j keep track of the ending point of the current window. Initially i = 0 and j = i + 2.
• Increment the value of j until arr[i] + arr[i+1] > arr[j] and keep track of the maximum value of j – i in a variable maxSize. If arr[i] + arr[i+1] > arr[j], increment the value of i by 1.
• Follow the above step till the whole array has been traversed.
• After completing the above steps, the value stored in maxSize is the required result.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// Function to find the maximum size of``// the subset of the given array such``// that a triangle can be formed from any``// three integers of the subset as sides``int` `maximizeSubset(``int` `arr[], ``int` `N)``{``    ``// Sort arr[] in increasing order``    ``sort(arr, arr + N);` `    ``// Stores the maximum size of a valid``    ``// subset of the given array``    ``int` `maxSize = 0;` `    ``// Stores the starting index of the``    ``// current window``    ``int` `i = 0;` `    ``// Stores the last index of the``    ``// current window``    ``int` `j = i + 2;` `    ``// Iterate over the array arr[]``    ``while` `(i < N - 2) {` `        ``// Increment j till the value``        ``// of arr[i] + arr[i + 1] >``        ``// arr[j] holds true``        ``while` `(arr[i] + arr[i + 1] > arr[j]) {``            ``j++;``        ``}` `        ``// Update the value of maxSize``        ``maxSize = max(maxSize, j - i);` `        ``i++;``        ``j = max(j, i + 2);``    ``}` `    ``// Return Answer``    ``return` `maxSize;``}` `// Driver Code``int` `main()``{``    ``int` `arr[] = { 2, 7, 4, 1, 6, 9, 5, 3 };``    ``int` `N = ``sizeof``(arr) / ``sizeof``(arr);` `    ``cout << maximizeSubset(arr, N) << endl;``    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.util.*;``class` `GFG{` `// Function to find the maximum size of``// the subset of the given array such``// that a triangle can be formed from any``// three integers of the subset as sides``static` `int` `maximizeSubset(``int` `arr[], ``int` `N)``{``  ` `    ``// Sort arr[] in increasing order``    ``Arrays.sort(arr);` `    ``// Stores the maximum size of a valid``    ``// subset of the given array``    ``int` `maxSize = ``0``;` `    ``// Stores the starting index of the``    ``// current window``    ``int` `i = ``0``;` `    ``// Stores the last index of the``    ``// current window``    ``int` `j = i + ``2``;` `    ``// Iterate over the array arr[]``    ``while` `(i < N - ``2``) {` `        ``// Increment j till the value``        ``// of arr[i] + arr[i + 1] >``        ``// arr[j] holds true``        ` `        ``while` `(j arr[j]) {``            ``j++;``        ``}` `        ``// Update the value of maxSize``        ``maxSize = Math.max(maxSize, j - i);` `        ``i++;``        ``j = Math.max(j, i + ``2``);``    ``}` `    ``// Return Answer``    ``return` `maxSize;``}` `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int` `arr[] = { ``2``, ``7``, ``4``, ``1``, ``6``, ``9``, ``5``, ``3` `};``    ``int` `N = arr.length;` `    ``System.out.print(maximizeSubset(arr, N) +``"\n"``);``}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# python program for the above approach` `# Function to find the maximum size of``# the subset of the given array such``# that a triangle can be formed from any``# three integers of the subset as sides`  `def` `maximizeSubset(arr, N):``    ``# Sort arr[] in increasing order``    ``arr.sort()` `    ``# Stores the maximum size of a valid``    ``# subset of the given array``    ``maxSize ``=` `0` `    ``# Stores the starting index of the``    ``# current window``    ``i ``=` `0` `    ``# Stores the last index of the``    ``# current window``    ``j ``=` `i ``+` `2` `    ``# Iterate over the array arr[]``    ``while` `(i < N ``-` `2``):` `                ``# Increment j till the value``                ``# of arr[i] + arr[i + 1] >``                ``# arr[j] holds true``        ``while` `(j < N ``and` `arr[i] ``+` `arr[i ``+` `1``] > arr[j]):``            ``j ``=` `j ``+` `1` `            ``# Update the value of maxSize``        ``maxSize ``=` `max``(maxSize, j ``-` `i)``        ``i ``+``=` `1``        ``j ``=` `max``(j, i ``+` `2``)` `        ``# Return Answer``    ``return` `maxSize`  `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``arr ``=` `[``2``, ``7``, ``4``, ``1``, ``6``, ``9``, ``5``, ``3``]``    ``N ``=` `len``(arr)``    ``print``(maximizeSubset(arr, N))` `    ``# This code is contributed by rakeshsahni`

## C#

 `// C# program for the above approach``using` `System;``using` `System.Collections.Generic;` `class` `GFG{` `// Function to find the maximum size of``// the subset of the given array such``// that a triangle can be formed from any``// three integers of the subset as sides``static` `int` `maximizeSubset(``int` `[]arr, ``int` `N)``{``    ``// Sort arr[] in increasing order``    ``Array.Sort(arr);` `    ``// Stores the maximum size of a valid``    ``// subset of the given array``    ``int` `maxSize = 0;` `    ``// Stores the starting index of the``    ``// current window``    ``int` `i = 0;` `    ``// Stores the last index of the``    ``// current window``    ``int` `j = i + 2;` `    ``// Iterate over the array arr[]``    ``while` `(i < N - 2) {` `        ``// Increment j till the value``        ``// of arr[i] + arr[i + 1] >``        ``// arr[j] holds true``        ``if``(j>=N || i+1 >=N)``           ``break``;``        ``while` `(j arr[j]) {``            ``j++;``        ``}` `        ``// Update the value of maxSize``        ``maxSize = Math.Max(maxSize, j - i);``        ` `        ``i++;``        ``j = Math.Max(j, i + 2);``    ``}` `    ``// Return Answer``    ``return` `maxSize;``}` `// Driver Code``public` `static` `void` `Main()``{``    ``int` `[]arr = { 2, 7, 4, 1, 6, 9, 5, 3 };``    ``int` `N = arr.Length;` `    ``Console.Write(maximizeSubset(arr, N));``}``}` `// This code is contributed by SURENDRA_GANGWAR.`

## Javascript

 ``

Output:

`4`

Time Complexity: O(N*log N)
Auxiliary Space: O(1)

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