Minimum White Tiles After Covering With Carpets

Problem Description

Ah, the age-old dilemma of home improvement: how to cover those pesky white tiles that seem to mock your every attempt at interior design. Imagine you’ve just moved into a new place, and the floor is a patchwork of white tiles that look like they were designed by a toddler with a paintbrush. You have a limited number of carpets and a specific length for each carpet. Your mission, should you choose to accept it, is to minimize the number of visible white tiles after strategically placing your carpets.

In simpler terms, you want to cover as many white tiles as possible with your carpets, but you can only use a certain number of them. It’s like trying to cover your embarrassing dance moves at a wedding with a limited number of friends—how do you make the most of what you have?

Code Solution


class Solution:
    def minimumWhiteTiles(
        self,
        floor: str,
        numCarpets: int,
        carpetLen: int,
    ) -> int:
        kMax = 1000

        @functools.lru_cache(None)
        def dp(i: int, j: int) -> int:
            if j < 0:
                return kMax
            if i >= len(floor):
                return 0
            return min(dp(i + carpetLen, j - 1),
                       dp(i + 1, j) + int(floor[i]))

        return dp(0, numCarpets)

Approach Explanation

The code uses a dynamic programming approach to solve the problem. The function dp(i, j) calculates the minimum number of visible white tiles starting from index i with j carpets left to use. The base cases handle scenarios where you run out of carpets or reach the end of the floor. The recursive calls explore two options: either placing a carpet or leaving the current tile as is. The minimum of these two options is returned, ensuring that you cover as many tiles as possible.

Time and Space Complexity

Complexity Type	Complexity
Time Complexity	O(n * m)
Space Complexity	O(n * m)

Real-World Example

Imagine you’re hosting a party, and your friends are coming over. You have a limited number of rugs to cover the ugly white tiles in your living room. You want to make sure that the most visible areas are covered, but you can only use a few rugs. This scenario mirrors the problem perfectly—how do you maximize the coverage with limited resources?