Leetcode #1931: Painting a Grid With Three Different Colors
In this guide, we solve Leetcode #1931 Painting a Grid With Three Different Colors 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 two integers m and n. Consider an m x n grid where each cell is initially white.
Quick Facts
- Difficulty: Hard
- Premium: No
- Tags: Dynamic Programming
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: m = 1, n = 1
Output: 3
Explanation: The three possible colorings are shown in the image above.
Python Solution
class Solution:
def colorTheGrid(self, m: int, n: int) -> int:
def f1(x: int) -> bool:
last = -1
for _ in range(m):
if x % 3 == last:
return False
last = x % 3
x //= 3
return True
def f2(x: int, y: int) -> bool:
for _ in range(m):
if x % 3 == y % 3:
return False
x, y = x // 3, y // 3
return True
mod = 10**9 + 7
mx = 3**m
valid = {i for i in range(mx) if f1(i)}
d = defaultdict(list)
for x in valid:
for y in valid:
if f2(x, y):
d[x].append(y)
f = [int(i in valid) for i in range(mx)]
for _ in range(n - 1):
g = [0] * mx
for i in valid:
for j in d[i]:
g[i] = (g[i] + f[j]) % mod
f = g
return sum(f) % mod
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.