Painting a Grid With Three Different Colors
\n# Approach\n Describe your approach to solving the problem. \nGenerate Valid Row Configurations:\n\nEnumerate all possible ways to color a single row of lengt
\n# Approach\n<!-- Describe your approach to solving the problem. -->\nGenerate Valid Row Configurations:\n\nEnumerate all possible ways to color a single row of length \n\uD835\uDC5A\nm with three colors while ensuring that no two adjacent cells share the same color.\nDetermine Valid Transitions Between Rows:\n\nPrecompute which row configurations can follow another row configuration without violating the adjacent coloring constraints.\nDynamic Programming Table Setup:\n\nLet dp[j][config] represent the number of ways to color the grid up to column j, ending with row configuration config.\nTransition from column j to j+1 based on valid row configurations.\nModulo Arithmetic:\n\nSince the result can be large, use modulo 10^9+7 for the calculations.\n\n# Complexity\n- Time complexity:\n<!-- Add your time complexity here, e.g. $$O(n)$$ -->\nGenerating Valid Rows: O(3^m), where 3^m is the number of potential configurations.\n\nTransition Checking: O(3^m . 3^m . m)\n\nDynamic Programming Filling: (n . 3^m . 3^m)).\n\n# Code\n```python []\nclass Solution(object):\n    def colorTheGrid(self, m, n):\n        """\n        :type m: int\n        :type n: int\n        :rtype: int\n        """\n        MOD = 10**9 + 7\n        \n        # Helper function to generate all valid row configurations\n        def generate_valid_rows(m):\n            colors = [0, 1, 2]  # 0: red, 1: green, 2: blue\n            valid_rows = []\n            \n            def backtrack(row):\n                if len(row) == m:\n                    valid_rows.append(tuple(row))\n                    return\n                for color in colors:\n                    if not row or row[-1] != color:  # Ensure no two adjacent cells are the same\n                        row.append(color)\n                        backtrack(row)\n                        row.pop()\n            \n            backtrack([])\n            return valid_rows\n        \n        # Generate all valid row configurations for m\n        valid_rows = generate_valid_rows(m)\n        num_configs = len(valid_rows)\n        \n        # Precompute transitions between row configurations\n        transitions = [[] for _ in range(num_configs)]\n        for i in range(num_configs):\n            for j in range(num_configs):\n                # Check if row i can transition to row j\n                if all(valid_rows[i][k] != valid_rows[j][k] for k in range(m)):\n                    transitions[i].append(j)\n        \n        # Initialize the DP table\n        dp = [[0] * num_configs for _ in range(n)]\n        \n        # Base case: first column can be any valid configuration\n        for i in range(num_configs):\n            dp[0][i] = 1\n        \n        # Fill the DP table\n        for j in range(1, n):\n            for i in range(num_configs):\n                for prev in transitions[i]:\n                    dp[j][i] = (dp[j][i] + dp[j-1][prev]) % MOD\n        \n        # Sum all ways to color the grid up to column n-1\n        result = sum(dp[n-1][i] for i in range(num_configs)) % MOD\n        \n        return result\n\n```
['Python']