Leetcode #2209: Minimum White Tiles After Covering With Carpets
In this guide, we solve Leetcode #2209 Minimum White Tiles After Covering With Carpets in Python and focus on the core idea that makes the solution efficient.
You will see the intuition, the step-by-step method, and a clean Python implementation you can use in interviews.
Problem Statement
You are given a 0-indexed binary string floor, which represents the colors of tiles on a floor: floor[i] = '0' denotes that the ith tile of the floor is colored black. On the other hand, floor[i] = '1' denotes that the ith tile of the floor is colored white.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: String, Dynamic Programming, Prefix Sum
Intuition
The problem breaks into overlapping subproblems, so caching results prevents exponential repetition.
A carefully chosen DP state captures exactly what we need to build the final answer.
Approach
Define the DP state and recurrence, then compute states in the correct order.
Optionally compress space once the recurrence is clear.
Steps:
- Choose a DP state definition.
- Write the recurrence and base cases.
- Compute states in the correct order.
Example
Input: floor = "10110101", numCarpets = 2, carpetLen = 2
Output: 2
Explanation:
The figure above shows one way of covering the tiles with the carpets such that only 2 white tiles are visible.
No other way of covering the tiles with the carpets can leave less than 2 white tiles visible.
Python Solution
class Solution:
def minimumWhiteTiles(self, floor: str, numCarpets: int, carpetLen: int) -> int:
def dfs(i: int, j: int) -> int:
if i >= n:
return 0
if floor[i] == "0":
return dfs(i + 1, j)
if j == 0:
return s[-1] - s[i]
return min(1 + dfs(i + 1, j), dfs(i + carpetLen, j - 1))
n = len(floor)
s = [0] * (n + 1)
for i, c in enumerate(floor):
s[i + 1] = s[i] + int(c == "1")
ans = dfs(0, numCarpets)
dfs.cache_clear()
return ans
Complexity
The time complexity is , and the space complexity is . The space complexity is .
Edge Cases and Pitfalls
Watch for boundary values, empty inputs, and duplicate values where applicable. If the problem involves ordering or constraints, confirm the invariant is preserved at every step.
Summary
This Python solution focuses on the essential structure of the problem and keeps the implementation interview-friendly while meeting the constraints.