Maximum height of triangular arrangement of array values

Given an array, we need to find the maximum height of the triangle which we can form, from the array values such that every (i+1)th level contain more elements with the larger sum from the previous level.

Examples:

Input : a = { 40, 100, 20, 30 }
Output : 2
Explanation : We can have 100 and 20 at the bottom level and either 40 or 30 at the upper level of the pyramid

Input : a = { 20, 20, 10, 10, 5, 2 }
Output : 3


First, at a glance, it looks like that we may have to look at the array values. But it’s not so. This is the tricky part of this problem. Here we don’t have to care about the array values because we can arrange any elements of the array in the triangular value satisfying these condition. Even if all the elements are equal like array = { 3,, 3, 3, 3, 3}, we can have solution. We can place two 3’s at the bottom and one 3’s at the top or three 3’s at the bottom and two 3’s at the top. You may take any example of your own and you will always find a solution of arranging them at a configuration. So, if our maximum height will be 2 then we should have at least 2 elements at the bottom and one element at the top, which means we should have minimum 3 elements (2*(2+1)/2). Similarly, for 3 as a height, we should have minimum 6 elements in the array.
Thus our final solution just lies on the logic that if we have maximum height h possible for our pyramid then (h*(h+1))/2 elements must be present in the array.

Below is the implementation of the above approach :

C++

// C++ program to find the maximum height 
// of Pyramidal Arrangement of array values
#include <bits/stdc++.h>
using namespace std;
  
int MaximumHeight(int a[], int n)
{
    int result = 1;
    for (int i = 1; i <= n; ++i) {
  
        // Just checking whether ith level
        // is possible or not if possible
        // then we must have atleast
        // (i*(i+1))/2 elements in the
        // array
        long long y = (i * (i + 1)) / 2;
  
        // updating the result value
        // each time
        if (y < n) 
            result = i;
          
        // otherwise we have exceeded n value
        else 
            break;
    }
    return result;
}
  
int main()
{
    int arr[] = { 40, 100, 20, 30 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << MaximumHeight(arr, n);
    return 0;
}

Java

// Java program to find  
// the maximum height of
// Pyramidal Arrangement 
// of array values
import java.io.*;
  
class GFG
{
static int MaximumHeight(int []a, 
                         int n)
{
          
    int result = 1;
    for (int i = 1; i <= n; ++i) 
    {
  
        // Just checking whether
        // ith level is possible
        // or not if possible then 
        // we must have atleast 
        // (i*(i+1))/2 elements 
        // in the array
        int y = (i * (i + 1)) / 2;
  
        // updating the result 
        // value each time
        if (y < n) 
            result = i;
          
        // otherwise we have 
        // exceeded n value
        else
            break;
    }
    return result;
}
  
// Driver Code 
public static void main (String[] args)
{
    int []arr = { 40, 100, 20, 30 };
    int n = arr.length;
    System.out.println(MaximumHeight(arr, n));
}
}
  
// This code is contributed by ajit

C#

// C# program to find 
// the maximum height of
// Pyramidal Arrangement 
// of array values
using System;
  
class GFG
{
static int MaximumHeight(int []a,
                         int n)
{
    int result = 1;
    for (int i = 1; i <= n; ++i) 
    {
  
        // Just checking whether
        // ith level is possible
        // or not if possible then 
        // we must have atleast 
        // (i*(i+1))/2 elements 
        // in the array
        int y = (i * (i + 1)) / 2;
  
        // updating the result 
        // value each time
        if (y < n) 
            result = i;
          
        // otherwise we have 
        // exceeded n value
        else
            break;
    }
    return result;
}
  
// Driver Code 
static public void Main ()
{
    int []arr = {40, 100, 20, 30};
    int n = arr.Length;
    Console.WriteLine(MaximumHeight(arr, n));
}
}
  
// This code is contributed 
// by m_kit

PHP

<?php
// PHP program to find the maximum height 
// of Pyramidal Arrangement of array values
  
function MaximumHeight($a, $n)
{
    $result = 1;
    for ($i = 1; $i <= $n; ++$i)
    {
  
        // Just checking whether ith level
        // is possible or not if possible
        // then we must have atleast
        // (i*(i+1))/2 elements in the
        // array
        $y = ($i * ($i + 1)) / 2;
  
        // updating the result value
        // each time
        if ($y < $n
            $result = $i;
          
        // otherwise we have 
        // exceeded n value
        else
            break;
    }
    return $result;
}
  
    // Driver Code
    $arr = array(40, 100, 20, 30);
    $n = count($arr);
    echo MaximumHeight($arr, $n);
  
// This code is contributed by anuj_67.
?>

Output:

2

Time Complexity : O(n)
Space Complexity : O(1)

We can solve this problem in O(1) time. We simple need to find the maximum i such that i*(i+1)/2 <= n. If we solve the equation, we get floor((-1+sqrt(1+(8*n)))/2)

C++

// CPP program to find the maximum height 
// of Pyramidal Arrangement of array values
#include <bits/stdc++.h>
using namespace std;
  
int MaximumHeight(int a[], int n)
{
    return floor((-1+sqrt(1+(8*n)))/2);
}
  
int main()
{
    int arr[] = { 40, 100, 20, 30 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << MaximumHeight(arr, n);
    return 0;
}

Java

// Java program to find the maximum height 
// of Pyramidal Arrangement of array values
import java.lang.*;
  
class GFG {
      
    static int MaximumHeight(int a[], int n)
    {
        return (int)Math.floor((-1
                Math.sqrt(1 + (8 * n))) / 2);
    }
      
    public static void main(String[] args)
    {
        int arr[] = new int[]{ 40, 100, 20, 30 };
        int n = arr.length;
          
        System.out.println(MaximumHeight(arr, n));
    }
}
  
// This code is contributed by Smitha

C#

// C# program to find the maximum height 
// of Pyramidal Arrangement of array values
using System;
  
class GFG {
      
    static int MaximumHeight(int[]a, int n)
    {
        return (int)Math.Floor((-1 +
               Math.Sqrt(1 + (8 * n))) / 2);
    }
      
    public static void Main()
    {
        int []arr = new int[]{ 40, 100, 20, 30 };
        int n = 4;
          
        Console.Write(MaximumHeight(arr, n));
    }
}
  
// This code is contributed by Smitha

PHP

<?php
// PHP program to find
// the maximum height 
// of Pyramidal Arrangement
// of array values
  
function MaximumHeight( $a, $n)
{
    return floor((-1 + sqrt(1 + 
               (8 * $n))) / 2);
}
      
    // Driver Code
    $arr = array(40, 100, 20, 30);
    $n = count($arr);
    echo MaximumHeight($arr, $n);
  
// This code is contributed by anuj_67.
?>

Output:

2

Time Complexity : O(1)
Space Complexity : O(1)



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