#1931

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Painting a Grid With Three Different Colors

Count the number of ways to paint a grid using three colors while ensuring adjacent cells have different colors.

Problem Statement

You are given a grid of size m x n where each cell starts as white. You need to paint each cell with one of three colors: red, green, or blue. The objective is to count the number of ways to color the entire grid, ensuring no two adjacent cells (horizontally or vertically) share the same color. The final answer should be modulo 10^9 + 7.

The problem is constrained by the grid's dimensions where 1 <= m <= 5 and 1 <= n <= 1000. Due to the large number of potential color combinations, an efficient approach is necessary. The goal is to implement this using dynamic programming with state transitions to avoid redundant calculations.

Examples

Example 1

Input: m = 1, n = 1

Output: 3

The three possible colorings are shown in the image above.

Example 2

Input: m = 1, n = 2

Output: 6

The six possible colorings are shown in the image above.

Example 3

Input: m = 5, n = 5

Output: 580986

Example details omitted.

Constraints

1 <= m <= 5
1 <= n <= 1000

Solution Approach

State Transition Dynamic Programming

The solution involves using dynamic programming to track valid colorings. By representing each column's colors using bitmasks, we can efficiently calculate the number of valid ways to color subsequent columns. The state transition approach ensures that no adjacent cells share the same color, optimizing the calculation.

Bitmask Representation of Colors

Each column's color configuration is encoded as a bitmask where each bit represents the color of a cell in the column. This allows quick checking of adjacent cells' colors and facilitates transitions between states. The number of unique states is reduced by limiting the grid height (m ≤ 5), making the bitmask approach feasible.

Modulo Operation for Large Results

Given the problem's constraints, the number of colorings can be extremely large. To handle this, the result is calculated modulo 10^9 + 7, ensuring that intermediate calculations do not overflow and the final answer is manageable.

Complexity Analysis

Metric	Value
Time	O(3^{2m} \cdot n)
Space	O(3^{2m})

The time complexity is O(3^{2m} * n), as there are 3^{2m} possible color configurations for each column, and the solution must iterate through all columns. The space complexity is O(3^{2m}) due to the storage of the state transitions for each column configuration.

What Interviewers Usually Probe

Focus on the bitmask-based state transition approach for solving this problem.
Test if the candidate can handle the modulo operation correctly when dealing with large results.
Look for an understanding of dynamic programming and how to optimize it with state transitions.

Common Pitfalls or Variants

Common pitfalls

Forgetting to use modulo 10^9 + 7 for the result, leading to overflow errors.
Incorrectly representing column colorings as bitmasks, which can lead to invalid transitions.
Not efficiently handling state transitions, which could result in a time complexity that exceeds limits.

Follow-up variants

Adjust the grid size (m or n) to test scalability of the solution.
Change the number of colors to see how the algorithm adapts to more complex cases.
Explore a version where no adjacent cells can share the same color diagonally.

FAQ

How do I approach the "Painting a Grid With Three Different Colors" problem?

Use dynamic programming with a state transition approach. Represent each column's configuration as a bitmask and calculate the number of valid colorings efficiently.

Why use bitmasking for this problem?

Bitmasking allows you to represent each column's color choices compactly and transition between states efficiently without redundant calculations.

What is the time complexity of the solution?

The time complexity is O(3^{2m} * n), where m is the number of rows and n is the number of columns. This complexity is manageable due to the small size of m (≤ 5).

How does GhostInterview assist with the "Painting a Grid With Three Different Colors" problem?

GhostInterview helps by providing structured guidance on the dynamic programming approach, explaining bitmasking, and offering solutions to common pitfalls.

What is the significance of modulo 10^9 + 7 in this problem?

The modulo operation ensures that the result doesn't overflow and remains within the limits of typical integer types, which is crucial given the large number of possible configurations.

terminal

Solution

Solution 1: State Compression + Dynamic Programming

We notice that the number of rows in the grid does not exceed $5$, so there are at most $3^5=243$ different color schemes in a column.

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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

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