Painting a Grid With Three Different Colors

Problem Description

You are given two integers m and n. Consider an m x n grid where each cell is initially white. You can paint each cell red, green, or blue. All cells must be painted. Return the number of ways to color the grid with no two adjacent cells having the same color. Since the answer can be very large, return it modulo 10^9 + 7.

Key Insights

• Each cell in the grid can be painted in one of three colors.
• Adjacent cells (horizontally and vertically) cannot have the same color.
• The problem can be modeled using dynamic programming due to overlapping subproblems and optimal substructure properties.
• Constraints are relatively small for m (1 to 5) but large for n (1 to 1000), indicating a focus on efficient computation for the linear dimension.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we can use dynamic programming to maintain the number of ways to paint the grid. We only need to keep track of the previous state since the current state only depends on the previous row. The approach involves calculating the number of valid configurations for each row based on the configurations from the previous row, ensuring that no two adjacent cells have the same color.

1. Let dp[i] represent the number of ways to paint the first i columns of the grid.
2. For each cell in the current row, we calculate the possibilities based on the previous row's configurations.
3. Use the modulo operation to handle large numbers.

Code Solutions