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. 

Implementation:

// 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 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

# Python program to find the 
# maximum height of Pyramidal 
# Arrangement of array values

def MaximumHeight(a, n):
    result = 1

    for i in range(1, n):
        
        # 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 = [40, 100, 20, 30]
n = len(arr)
print(MaximumHeight(arr, n))

# This code is contributed by
# Sanjit_Prasad

// 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 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.
?>

<script>
// Javascript program to find  
// the maximum height of
// Pyramidal Arrangement 
// of array values
    function MaximumHeight( a ,n)
    {

        let 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
            let 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 
    let arr = [ 40, 100, 20, 30 ];
    let n = arr.length;
    document.write(MaximumHeight(arr, n));

// This code is contributed by Rajput-Ji
</script>

2

Complexity Analysis:

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

Implementation: 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)

// 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 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

# Python program to find the 
# maximum height of Pyramidal 
# Arrangement of array values
import math

def MaximumHeight(a, n):
    return (-1 + int(math.sqrt(1 + 
                    (8 * n)))) // 2

# Driver Code
arr = [40, 100, 20, 30]
n = len(arr)
print(MaximumHeight(arr, n))

# This code is contributed by
# Sanjit_Prasad

// 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 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.
?>

<script>
// javascript program to find the maximum height 
// of Pyramidal Arrangement of array values
function MaximumHeight(a, n)
{
    return Math.floor((-1 + Math.sqrt(1 + (8 * n))) / 2);
}

// Driver code
    let arr = [ 40, 100, 20, 30 ];
    let n = arr.length;
   document.write(MaximumHeight(arr, n));
   
    // This code is contributed by gauravrajput1

</script>

2

Complexity Analysis:

• Time Complexity: O(logn) as it is using inbuilt sqrt function
• Space Complexity: O(1)