<p>An <code>n x n</code> grid is composed of <code>1 x 1</code> squares where each <code>1 x 1</code> square consists of a <code>&#39;/&#39;</code>, <code>&#39;\&#39;</code>, or blank space <code>&#39; &#39;</code>. These characters divide the square into contiguous regions.</p>

<p>Given the grid <code>grid</code> represented as a string array, return <em>the number of regions</em>.</p>

<p>Note that backslash characters are escaped, so a <code>&#39;\&#39;</code> is represented as <code>&#39;\\&#39;</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/15/1.png" style="width: 200px; height: 200px;" />
<pre>
<strong>Input:</strong> grid = [&quot; /&quot;,&quot;/ &quot;]
<strong>Output:</strong> 2
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/15/2.png" style="width: 200px; height: 198px;" />
<pre>
<strong>Input:</strong> grid = [&quot; /&quot;,&quot;  &quot;]
<strong>Output:</strong> 1
</pre>

<p><strong class="example">Example 3:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2018/12/15/4.png" style="width: 200px; height: 200px;" />
<pre>
<strong>Input:</strong> grid = [&quot;/\\&quot;,&quot;\\/&quot;]
<strong>Output:</strong> 5
<strong>Explanation: </strong>Recall that because \ characters are escaped, &quot;\\/&quot; refers to \/, and &quot;/\\&quot; refers to /\.
</pre>

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

<ul>
	<li><code>n == grid.length == grid[i].length</code></li>
	<li><code>1 &lt;= n &lt;= 30</code></li>
	<li><code>grid[i][j]</code> is either <code>&#39;/&#39;</code>, <code>&#39;\&#39;</code>, or <code>&#39; &#39;</code>.</li>
</ul>

class Solution {
public:
    static void dfs(int v, vector<vector<int>>& adj, bitset<962>& viz) {
        viz[v] = 1;
        for (int w : adj[v]) {
            if (viz[w]) continue;
            dfs(w, adj, viz);
        }
    }
    static int regionsBySlashes(vector<string>& grid) {
        const int n = grid.size();
        const int side = n+1; // side length of the extended grid
        const int N = side*side; // total vertices in the extended grid

        bitset<962> viz = 0; // Supports up to a 31x31 grid
        vector<vector<int>> adj(N);

        int N0 = N-side;
        for (int i = 0; i < n; i++) {
            adj[i].push_back(i+1);             // Top border
            adj[i+1].push_back(i);

            adj[i+N0].push_back(i+N0+1);   // Bottom border
            adj[i+N0+1].push_back(i+N0);

            adj[i*side].push_back((i+1)*side);    // Left border
            adj[(i+1)*side].push_back(i*side);

            adj[n+i*side].push_back(n+(i+1)*side); // Right border
            adj[n+(i+1)*side].push_back(n+i*side);
        }

        int E = 4*n; // Initial edges from the borders

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == '\\') {
                    int u = i*side+j;
                    int v = (i+1)*side+j+1;
                    adj[u].push_back(v);
                    adj[v].push_back(u);
                    E++;
                } 
                else if (grid[i][j] == '/') {
                    int u = i*side+j+1;
                    int v = (i+1)*side+j;
                    adj[u].push_back(v);
                    adj[v].push_back(u);
                    E++;
                }
            }
        }

        int component = 0;
        for (int i=0; i < N; i++)
            // don't count isolated 1-vertex components
            if (!adj[i].empty() && !viz[i]) {
            //    cout << "\nStarting DFS for component at vertex: "<<i<<endl;
                component++;
                dfs(i, adj, viz);
            }
        
        int V = viz.count();
        int F = E-V+1+component;
        
        // output
       // cout<<"V="<<V<<", E="<<E<<", component="<<component<<", F="<<F<< endl;
        //print(grid);
        return F - 1; // Final face count
    }
};
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