Painting a Grid With Three Different Colors
<p>You are given two integers <code>m</code> and <code>n</code>. Consider an <code>m x n</code> grid where each cell is initially white. You can paint each cell <strong>red</strong>, <strong>green</strong>, or <strong>blue</strong>. All cells <strong>must</strong> be painted.</p>

<p>Return<em> the number of ways to color the grid with <strong>no two adjacent cells having the same color</strong></em>. Since the answer can be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1900-1999/1931.Painting%20a%20Grid%20With%20Three%20Different%20Colors/images/colorthegrid.png" style="width: 200px; height: 50px;" />
<pre>
<strong>Input:</strong> m = 1, n = 1
<strong>Output:</strong> 3
<strong>Explanation:</strong> The three possible colorings are shown in the image above.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1900-1999/1931.Painting%20a%20Grid%20With%20Three%20Different%20Colors/images/copy-of-colorthegrid.png" style="width: 321px; height: 121px;" />
<pre>
<strong>Input:</strong> m = 1, n = 2
<strong>Output:</strong> 6
<strong>Explanation:</strong> The six possible colorings are shown in the image above.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> m = 5, n = 5
<strong>Output:</strong> 580986
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= m &lt;= 5</code></li>
	<li><code>1 &lt;= n &lt;= 1000</code></li>
</ul>
Dynamic Programming
class Solution {
public:
    int colorTheGrid(int m, int n) {
        auto f1 = [&](int x) {
            int last = -1;
            for (int i = 0; i < m; ++i) {
                if (x % 3 == last) {
                    return false;
                }
                last = x % 3;
                x /= 3;
            }
            return true;
        };
        auto f2 = [&](int x, int y) {
            for (int i = 0; i < m; ++i) {
                if (x % 3 == y % 3) {
                    return false;
                }
                x /= 3;
                y /= 3;
            }
            return true;
        };

        const int mod = 1e9 + 7;
        int mx = pow(3, m);
        unordered_set<int> valid;
        vector<int> f(mx);
        for (int i = 0; i < mx; ++i) {
            if (f1(i)) {
                valid.insert(i);
                f[i] = 1;
            }
        }
        unordered_map<int, vector<int>> d;
        for (int i : valid) {
            for (int j : valid) {
                if (f2(i, j)) {
                    d[i].push_back(j);
                }
            }
        }
        for (int k = 1; k < n; ++k) {
            vector<int> g(mx);
            for (int i : valid) {
                for (int j : d[i]) {
                    g[i] = (g[i] + f[j]) % mod;
                }
            }
            f = move(g);
        }
        int ans = 0;
        for (int x : f) {
            ans = (ans + x) % mod;
        }
        return ans;
    }
};