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Construct the Rectangle

Difficulty: Easy

Problem Description

Given a specific rectangular web page’s area, design a rectangular web page whose length L and width W satisfy the following requirements:

  1. The area of the rectangular web page must equal the given target area.
  2. The width W should not be larger than the length L (i.e., L >= W).
  3. The difference between length L and width W should be as small as possible.

Return an array [L, W] where L and W are the length and width of the web page designed in sequence.

Key Insights

  • The area of the rectangle is given, and we need to find pairs of (L, W) such that L * W = area.
  • To minimize the difference |L - W|, L should be as close to W as possible, leading to the idea of starting the search from the square root of the area.
  • We can iterate downwards from the integer value of the square root of the area to find the largest W such that L is also an integer.

Space and Time Complexity

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

Solution

To solve the problem, we use the following steps:

  1. Calculate the square root of the given area to find a starting point for W.
  2. Iterate downwards from this point until we find a width W that divides the area evenly (i.e., area % W == 0).
  3. Once a valid W is found, calculate L as area / W.
  4. Return the pair [L, W] ensuring L >= W.

The main data structure involved is simply an array to return the final dimensions.

Code Solutions

import math

def constructRectangle(area):
    # Start from the square root of the area
    W = int(math.sqrt(area))
    
    # Iterate downwards until we find a valid W
    while area % W != 0:
        W -= 1
        
    L = area // W  # Calculate L
    return [L, W]  # Return the dimensions
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