<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 UnionFind {
private:
    vector< int > root;
    vector< int > rank;

public:
    UnionFind( int size ) : root( size ), rank( size ) {
        for( int i = 0; i < size; i++ ) {
            root[ i ] = i;
            rank[ i ] = 1;
        }
    }

    int find( int x ) {
        if( root[ x ] == x ) {
            return x;
        }
        root[ x ] = find( root[ x ] );
        return root[ x ];
    }

    void doUnion( int x, int y ) {
        int rootX = find( x );
        int rootY = find( y );
        if( rootX != rootY ) {
            if( rank[ rootX ] > rank[ rootY ] ) {
                root[ rootY ] = rootX;
            } else if( rank[ rootY ] > rank[ rootX ] ) {
                root[ rootX ] = rootY;
            } else {
                root[ rootY ] = rootX;
                rank[ rootX ]++;
            }
        }
    }

    int numConnected() {
        int numVertices = root.size();
        unordered_set< int > uniqueRoots;
        for( int i = 0; i < numVertices; i++ ) {
            uniqueRoots.insert( find( i ) );
        }
        return uniqueRoots.size();
    }
};

class Solution {
public:
    int getTriangleIndex(int gridSize, int row, int col, int triangleNum) {
        return (gridSize * row + col) * 4 + triangleNum;
    }

    int regionsBySlashes(vector<string>& grid) {
        int n = grid.size();
        int numTriangles = n * n * 4;

        UnionFind uf( numTriangles );
        for( int i = 0; i < n; i++ ) {
            for( int j = 0; j < n; j++ ) {
                if( i > 0 ) {
                    uf.doUnion( getTriangleIndex( n, i-1, j, 2 ),
                                getTriangleIndex( n, i, j, 0 ) );
                }
                if( j > 0 ) {
                    uf.doUnion( getTriangleIndex( n, i, j-1, 1 ),
                                getTriangleIndex( n, i, j, 3 ) );
                }
                if( grid[ i ][ j ] != '/' ) {
                    uf.doUnion( getTriangleIndex( n, i, j, 0 ),
                                getTriangleIndex( n, i, j, 1 ) );
                    uf.doUnion( getTriangleIndex( n, i, j, 2 ),
                                getTriangleIndex( n, i, j, 3 ) );
                }
                if( grid[ i ][ j ] != '\\' ) {
                    uf.doUnion( getTriangleIndex( n, i, j, 0 ),
                                getTriangleIndex( n, i, j, 3 ) );
                    uf.doUnion( getTriangleIndex( n, i, j, 1 ),
                                getTriangleIndex( n, i, j, 2 ) );
                }
            }
        }

        return uf.numConnected();
    }
};
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