Maximum volume of cube for every person when edge of N cubes are given

Given an array of N integers which denotes the edges of N cubical structures respectively. Also given are M integers which denotes the number of peoples. The task is to find the maximum amount of volume of a cube that can be given to every person.

Note: Cubes can be cut of any shape from any of the N cubes.

Examples:

Input: a[] = {1, 1, 1, 2, 2}, m = 3
Output: 4
All three person get a slice of volume 4 each
Person 1 gets a slice of volume 4 from the last cube.
Person 2 gets a slice of volume 4 from the last cube.
Person 3 gets a slice of volume 4 from the second last cube.

Input: a[] = {2, 2, 2, 2, 2}, m = 4
Output: 8



Naive Approach: A naive approach is to first calculate the volume of all of the cubes and then linearly check for every volume that it can be distributed among all M people or not and find the maximum volume among all such volumes.

Time Complexity: O(N2)

Efficient Approach: An efficient approach is to use binary search to find the answer. Since the edge lengths are given in the array, convert them to the volume of the respective cubes.

Find the maximum volume among volumes of all of the cubes. Say, the maximum volume is maxVolume. Now, perform binary search on the range [0, maxVolume].

  • Calculate the middle value of the range, say mid.
  • Now, calculate the total number of cubes that can be cut of all of the cubes of volume mid.
  • If the total cubes that can be cut exceed the number of persons, then that amount of volume of cubes can be cut for every person, hence we check for a larger value in the range [mid+1, maxVolume].
  • If the total cubes do not exceed the number of persons, then we check for an answer in the range [low, mid-1].

Below is the implementation of the above approach:

C++

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// C++ program to implement the above approach
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to get the maximum volume that
// every person can get
int getMaximumVloume(int a[], int n, int m)
{
    int maxVolume = 0;
  
    // Convert the length to respective volumes
    // and find the maximum volumes
    for (int i = 0; i < n; i++) {
        a[i] = a[i] * a[i] * a[i];
  
        maxVolume = max(a[i], maxVolume);
    }
  
    // Apply binary search with intiial
    // low as 0 and high as the maximm most
    int low = 0, high = maxVolume;
  
    // Initial answer is 0 slices
    int maxVol = 0;
  
    // Apply binary search
    while (low <= high) {
  
        // Get the mid element
        int mid = (low + high) >> 1;
  
        // Count the slices of volume mid
        int cnt = 0;
        for (int i = 0; i < n; i++) {
            cnt += a[i] / mid;
        }
  
        // If the slices of volume
        // exceeds the number of persons
        // then every person can get volume mid
        if (cnt >= m) {
  
            // Then check for larger in the right half
            low = mid + 1;
  
            // Replace tbe answer with
            // current maximum i.e., mid
            maxVol = max(maxVol, mid);
        }
  
        // else traverse in the left half
        else
            high = mid - 1;
    }
  
    return maxVol;
}
  
// Driver code
int main()
{
    int a[] = { 1, 1, 1, 2, 2 };
    int n = sizeof(a) / sizeof(a[0]);
    int m = 3;
  
    cout << getMaximumVloume(a, n, m);
  
    return 0;
}

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Java

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// Java program to implement the above approach 
class GFG 
{
  
    // Function to get the maximum volume that 
    // every person can get 
    static int getMaximumVloume(int a[], int n, int m) 
    {
        int maxVolume = 0;
  
        // Convert the length to respective volumes 
        // and find the maximum volumes 
        for (int i = 0; i < n; i++) 
        {
            a[i] = a[i] * a[i] * a[i];
  
            maxVolume = Math.max(a[i], maxVolume);
        }
  
        // Apply binary search with intiial 
        // low as 0 and high as the maximm most 
        int low = 0, high = maxVolume;
  
        // Initial answer is 0 slices 
        int maxVol = 0;
  
        // Apply binary search 
        while (low <= high) 
        {
  
            // Get the mid element 
            int mid = (low + high) >> 1;
  
            // Count the slices of volume mid 
            int cnt = 0;
            for (int i = 0; i < n; i++) 
            {
                cnt += a[i] / mid;
            }
  
            // If the slices of volume 
            // exceeds the number of persons 
            // then every person can get volume mid 
            if (cnt >= m)
            {
  
                // Then check for larger in the right half 
                low = mid + 1;
  
                // Replace tbe answer with 
                // current maximum i.e., mid 
                maxVol = Math.max(maxVol, mid);
            
              
            // else traverse in the left half 
            else
            {
                high = mid - 1;
            }
        }
  
        return maxVol;
    }
  
    // Driver code 
    public static void main(String[] args)
    {
        int a[] = {1, 1, 1, 2, 2};
        int n = a.length;
        int m = 3;
  
        System.out.println(getMaximumVloume(a, n, m));
    }
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python 3 program to implement 
# the above approach
  
# Function to get the maximum volume
# that every person can get
def getMaximumVloume(a, n, m):
    maxVolume = 0
  
    # Convert the length to respective 
    # volumes and find the maximum volumes
    for i in range(n):
        a[i] = a[i] * a[i] * a[i]
  
        maxVolume = max(a[i], maxVolume)
  
    # Apply binary search with intiial
    # low as 0 and high as the maximm most
    low = 0
    high = maxVolume
  
    # Initial answer is 0 slices
    maxVol = 0
  
    # Apply binary search
    while (low <= high):
          
        # Get the mid element
        mid = (low + high) >> 1
  
        # Count the slices of volume mid
        cnt = 0
        for i in range(n):
            cnt += int(a[i] / mid)
  
        # If the slices of volume
        # exceeds the number of persons
        # then every person can get volume mid
        if (cnt >= m):
              
            # Then check for larger in the right half
            low = mid + 1
  
            # Replace tbe answer with
            # current maximum i.e., mid
            maxVol = max(maxVol, mid)
  
        # else traverse in the left half
        else:
            high = mid - 1
  
    return maxVol
  
# Driver code
if __name__ == '__main__':
    a = [1, 1, 1, 2, 2]
    n = len(a)
    m = 3
  
    print(getMaximumVloume(a, n, m))
  
# This code is contributed
# by Surendra_Gangwar

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C#

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// C# program to implement the above approach 
using System;
  
class GFG 
{
  
    // Function to get the maximum volume that 
    // every person can get 
    static int getMaximumVloume(int []a, int n, int m) 
    {
        int maxVolume = 0;
  
        // Convert the length to respective volumes 
        // and find the maximum volumes 
        for (int i = 0; i < n; i++) 
        {
            a[i] = a[i] * a[i] * a[i];
  
            maxVolume = Math.Max(a[i], maxVolume);
        }
  
        // Apply binary search with intiial 
        // low as 0 and high as the maximm most 
        int low = 0, high = maxVolume;
  
        // Initial answer is 0 slices 
        int maxVol = 0;
  
        // Apply binary search 
        while (low <= high) 
        {
  
            // Get the mid element 
            int mid = (low + high) >> 1;
  
            // Count the slices of volume mid 
            int cnt = 0;
            for (int i = 0; i < n; i++) 
            {
                cnt += a[i] / mid;
            }
  
            // If the slices of volume 
            // exceeds the number of persons 
            // then every person can get volume mid 
            if (cnt >= m)
            {
  
                // Then check for larger in the right half 
                low = mid + 1;
  
                // Replace tbe answer with 
                // current maximum i.e., mid 
                maxVol = Math.Max(maxVol, mid);
            
              
            // else traverse in the left half 
            else
            {
                high = mid - 1;
            }
        }
  
        return maxVol;
    }
  
    // Driver code 
    public static void Main(String[] args)
    {
        int []a = {1, 1, 1, 2, 2};
        int n = a.Length;
        int m = 3;
  
        Console.WriteLine(getMaximumVloume(a, n, m));
    }
}
  
/* This code contributed by PrinciRaj1992 */

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PHP

> 1;

// Count the slices of volume mid
$cnt = 0;
for ($i = 0; $i < $n; $i++) { $cnt += (int)($a[$i] / $mid); } // If the slices of volume // exceeds the number of persons // then every person can get volume mid if ($cnt >= $m)
{

// Then check for larger in the right half
$low = $mid + 1;

// Replace tbe answer with
// current maximum i.e., mid
$maxVol = max($maxVol, $mid);
}

// else traverse in the left half
else
$high = $mid – 1;
}

return $maxVol;
}

// Driver code
$a = array(1, 1, 1, 2, 2);
$n = sizeof($a);
$m = 3;

echo getMaximumVloume($a, $n, $m);

// This code is contributed by Akanksha Rai
?>

Output:

4

Time Complexity: O(N * log (maxVolume))



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