<p>You are given an <code>n x n</code> binary matrix <code>grid</code> where <code>1</code> represents land and <code>0</code> represents water.</p>

<p>An <strong>island</strong> is a 4-directionally connected group of <code>1</code>&#39;s not connected to any other <code>1</code>&#39;s. There are <strong>exactly two islands</strong> in <code>grid</code>.</p>

<p>You may change <code>0</code>&#39;s to <code>1</code>&#39;s to connect the two islands to form <strong>one island</strong>.</p>

<p>Return <em>the smallest number of </em><code>0</code><em>&#39;s you must flip to connect the two islands</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> grid = [[0,1],[1,0]]
<strong>Output:</strong> 1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [[0,1,0],[0,0,0],[0,0,1]]
<strong>Output:</strong> 2
</pre>

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

<pre>
<strong>Input:</strong> grid = [[1,1,1,1,1],[1,0,0,0,1],[1,0,1,0,1],[1,0,0,0,1],[1,1,1,1,1]]
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>n == grid.length == grid[i].length</code></li>
	<li><code>2 &lt;= n &lt;= 100</code></li>
	<li><code>grid[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
	<li>There are exactly two islands in <code>grid</code>.</li>
</ul>

constexpr int dx[4] = {-1, 1, 0, 0};
constexpr int dy[4] = {0, 0, -1, 1};
constexpr int INF = 1e9;

using Graph = std::vector<std::vector<int>>;

class Solution {
private:
    int m = 0, n = 0;
    Graph graph;

    void Dfs(int x, int y) {
        graph[x][y] = 2;

        for (int d = 0; d < 4; d++) {
            int nx = x + dx[d], ny = y + dy[d];
            if (nx >= 0 && nx < n && ny >= 0 && ny < m && graph[nx][ny] == 1) {
                Dfs(nx, ny);
            }
        }
    }

public:
    int shortestBridge(vector<vector<int>>& grid) {
        n = grid.size();
        m = grid[0].size();
        graph = grid;

        bool found = false;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                if (graph[i][j] == 1) {
                    Dfs(i, j);
                    found = true;
                    break;
                }
            }
            if (found) break;
        }

        std::queue<std::pair<int, int>> q;
        Graph dist(n, std::vector<int>(m, INF));
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                if (graph[i][j] == 2) {
                    dist[i][j] = 0;
                    q.push({i, j}); 
                }
            }
        }

        while (!q.empty()) {
            auto [x, y] = q.front();
            q.pop();
            for (int d = 0; d < 4; d++) {
                int nx = x + dx[d], ny = y + dy[d];
                if (nx >= 0 && nx < n && ny >= 0 && ny < m) {
                    if (dist[nx][ny] > dist[x][y] + 1) {
                        dist[nx][ny] = dist[x][y] + 1;
                        q.push({nx, ny});
                    }
                    if (graph[nx][ny] == 1) return dist[nx][ny] - 1;
                }
            }
        }

        return -1;
    }
};
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