Size of smallest square that contains N non-overlapping rectangles of given dimensions

Given two positive integers W and H and N rectangles of dimension W*H, the task is to find the smallest size of the square required such that all the N rectangles can be packed without overlapping. 

Examples:

Input: N = 10, W = 2, H = 3
Output: 9
Explanation:
The smallest size of the square is 9 units to pack the given 10 rectangles of size 2*3 as illustrated in the below image:

Input: N = 1, W = 3, H = 3
Output: 3

Approach: The given problem is based on the following observations:

• It can be shown that one of the optimal spacing of rectangles within a square is given by:

• The maximum number of rectangles of dimension W*H, that can be fitted in the square with sides X is given by ?X/W???X/H?.
• The above function is monotonically increasing. Therefore, the idea is to use the Binary Search to find the smallest side of a square that satisfies the given condition.

Follow the steps below to solve the problem:

• Initialize two variables, say low as 1, and high as W*H*N.
• Iterate until i is less than j and perform the following steps:
  • Find the value of mid as (i + j)/2.
  • Now, if the value (mid/W)*(mid/H) is at most N, then update the value of high as mid.
  • Otherwise, update the value of low as (mid + 1).
• After completing the above steps, print the value of high as the resultant value.

Below is the implementation of the above approach:

// CPP program for the above approach
#include<bits/stdc++.h>
using namespace std;

// Function to check if side of square X
// can pack all the N rectangles or not
bool bound(int w, int h, int N, int x)
{
    
    // Find the number of rectangle
    // it can pack
    int val = (x / w) * (x / h);

    // If val is atleast N,
    // then return true
    if (val >= N)
        return true;

    // Otherwise, return false
    else
        return false;
}

// Function to find the size of the
// smallest square that can contain
// N rectangles of dimensions W * H
int FindSquare(int N, int W, int H)
{
    
    // Stores the lower bound
    int i = 1;

    // Stores the upper bound
    int j = W * H * N;

    // Iterate until i is less than j
    while (i < j) 
    {
        
        // Calculate the mid value
        int mid = i + (j - i) / 2;

        // If the current size of square
        // cam contain N rectangles
        if (bound(W, H, N, mid))
            j = mid;

        // Otherwise, update i
        else
            i = mid + 1;
    }

    // Return the minimum size of the
    // square required
    return j;
}

// Driver code
int main()
{
    int W = 2;
    int H = 3;
    int N = 10;

    // Function Call
    cout << FindSquare(N, W, H);
}

// This code is contributed by ipg2016107.

// Java program for the above approach
class GFG{
    
// Function to check if side of square X
// can pack all the N rectangles or not
static boolean bound(int w, int h, int N, int x)
{
    
    // Find the number of rectangle
    // it can pack
    int val = (x / w) * (x / h);

    // If val is atleast N,
    // then return true
    if (val >= N)
        return true;

    // Otherwise, return false
    else
        return false;
}

// Function to find the size of the
// smallest square that can contain
// N rectangles of dimensions W * H
static int FindSquare(int N, int W, int H)
{
    
    // Stores the lower bound
    int i = 1;

    // Stores the upper bound
    int j = W * H * N;

    // Iterate until i is less than j
    while (i < j) 
    {
        
        // Calculate the mid value
        int mid = i + (j - i) / 2;

        // If the current size of square
        // cam contain N rectangles
        if (bound(W, H, N, mid))
            j = mid;

        // Otherwise, update i
        else
            i = mid + 1;
    }

    // Return the minimum size of the
    // square required
    return j;
}

// Driver code
public static void main(String[] args)
{
    int W = 2;
    int H = 3;
    int N = 10;

    // Function Call
    System.out.print(FindSquare(N, W, H));
}
}

// This code is contributed by sk944795

# Python program for the above approach

# Function to check if side of square X
# can pack all the N rectangles or not
def bound(w, h, N, x):

    # Find the number of rectangle
    # it can pack
    val = (x//w)*(x//h)
    
    # If val is atleast N,
    # then return true
    if(val >= N):
        return True
      
    # Otherwise, return false
    else:
        return False

# Function to find the size of the
# smallest square that can contain
# N rectangles of dimensions W * H
def FindSquare(N, W, H):

    # Stores the lower bound
    i = 1
    
    # Stores the upper bound
    j = W * H*N
    
    # Iterate until i is less than j
    while(i < j):
      
        # Calculate the mid value
        mid = i + (j - i)//2

        # If the current size of square
        # cam contain N rectangles
        if(bound(W, H, N, mid)):
            j = mid
        
        # Otherwise, update i
        else:
            i = mid + 1

    # Return the minimum size of the
    # square required
    return j

# Driver Code

W = 2
H = 3
N = 10

# Function Call
print(FindSquare(N, W, H))

// C# program for the above approach
using System;

class GFG{

// Function to check if side of square X
// can pack all the N rectangles or not
static bool bound(int w, int h, int N, int x)
{
    
    // Find the number of rectangle
    // it can pack
    int val = (x / w) * (x / h);

    // If val is atleast N,
    // then return true
    if (val >= N)
        return true;

    // Otherwise, return false
    else
        return false;
}

// Function to find the size of the
// smallest square that can contain
// N rectangles of dimensions W * H
static int FindSquare(int N, int W, int H)
{

    // Stores the lower bound
    int i = 1;

    // Stores the upper bound
    int j = W * H * N;

    // Iterate until i is less than j
    while (i < j) 
    {
        
        // Calculate the mid value
        int mid = i + (j - i) / 2;

        // If the current size of square
        // cam contain N rectangles
        if (bound(W, H, N, mid))
            j = mid;

        // Otherwise, update i
        else
            i = mid + 1;
    }
    
    // Return the minimum size of the
    // square required
    return j;
}

// Driver Code
public static void Main()
{
    int W = 2;
    int H = 3;
    int N = 10;

    // Function Call
    Console.WriteLine(FindSquare(N, W, H));
}
}

// This code is contributed by ukasp

<script>

// Javascript program for the above approach

// Function to check if side of square X
// can pack all the N rectangles or not
function bound(w, h, N, x)
{
    
    // Find the number of rectangle
    // it can pack
    let val = parseInt(x / w) * parseInt(x / h);

    // If val is atleast N,
    // then return true
    if (val >= N)
        return true;

    // Otherwise, return false
    else
        return false;
}

// Function to find the size of the
// smallest square that can contain
// N rectangles of dimensions W * H
function FindSquare(N, W, H)
{
    
    // Stores the lower bound
    let i = 1;

    // Stores the upper bound
    let j = W * H * N;

    // Iterate until i is less than j
    while (i < j) 
    {
        
        // Calculate the mid value
        let mid = i + parseInt((j - i) / 2);

        // If the current size of square
        // cam contain N rectangles
        if (bound(W, H, N, mid))
            j = mid;

        // Otherwise, update i
        else
            i = mid + 1;
    }

    // Return the minimum size of the
    // square required
    return j;
}

// Driver code
    let W = 2;
    let H = 3;
    let N = 10;

    // Function Call
    document.write(FindSquare(N, W, H));
    
</script>

9

Time Complexity: O(log(W*H))
Auxiliary Space: O(1)