<p>You are given an <code>n x n</code> binary matrix <code>grid</code>. You are allowed to change <strong>at most one</strong> <code>0</code> to be <code>1</code>.</p>

<p>Return <em>the size of the largest <strong>island</strong> in</em> <code>grid</code> <em>after applying this operation</em>.</p>

<p>An <strong>island</strong> is a 4-directionally connected group of <code>1</code>s.</p>

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

<pre>
<strong>Input:</strong> grid = [[1,0],[0,1]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> Change one 0 to 1 and connect two 1s, then we get an island with area = 3.
</pre>

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

<pre>
<strong>Input:</strong> grid = [[1,1],[1,0]]
<strong>Output:</strong> 4
<strong>Explanation: </strong>Change the 0 to 1 and make the island bigger, only one island with area = 4.</pre>

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

<pre>
<strong>Input:</strong> grid = [[1,1],[1,1]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> Can&#39;t change any 0 to 1, only one island with area = 4.
</pre>

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

<ul>
	<li><code>n == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= n &lt;= 500</code></li>
	<li><code>grid[i][j]</code> is either <code>0</code> or <code>1</code>.</li>
</ul>

class DisjointSet {
public:
    vector<int> rank, parent, size;
    DisjointSet(int n) {
        rank.resize(n + 1, 0);
        parent.resize(n + 1);
        size.resize(n + 1);
        for (int i = 0; i <= n; i++) {
            parent[i] = i;
            size[i] = 1;
        }
    }

    int findUParent(int node) {
        if (node == parent[node])
            return node;

        return parent[node] = findUParent(parent[node]);
    }

    void UnionByRank(int u, int v) {
        int ulp_u = findUParent(u);
        int ulp_v = findUParent(v);

        if (ulp_u == ulp_v)
            return;
        if (rank[ulp_u] < rank[ulp_v]) {
            parent[ulp_u] = ulp_v;
        } else if (rank[ulp_u] > rank[ulp_v]) {
            parent[ulp_v] = ulp_u;
        } else {
            parent[ulp_u] = ulp_v;
            rank[ulp_v]++;
        }
    }

    void UnionBySize(int u, int v) {
        int ulp_u = findUParent(u);
        int ulp_v = findUParent(v);

        if (ulp_u == ulp_v)
            return;
        if (size[ulp_u] < size[ulp_v]) {
            parent[ulp_u] = ulp_v;
            size[ulp_v] += size[ulp_u];
        } else {
            parent[ulp_v] = ulp_u;
            size[ulp_u] += size[ulp_v];
        }
    }
};
class Solution {
public:
    bool isValid(int i, int j, int n) {
        if (i >= 0 && j >= 0 && i < n && j < n)
            return 1;
        return 0;
    }
    // void BFs(int i, int j, vector<vector<int>>& grid, vector<vector<int>>& vis,
    //          DisjointSet& ds) {
    //     vis[i][j] = 1;
    //     int n = grid.size();
    //     queue<pair<int, int>> q;
    //     q.push({i, j});

    //     while (!q.empty()) {
    //         int row = q.front().first;
    //         int col = q.front().second;
    //         q.pop();

    //         int dx[] = {0, 0, -1, 1};
    //         int dy[] = {-1, 1, 0, 0};

    //         for (int l = 0; l < 4; l++) {
    //             int adjr = row + dx[l];
    //             int adjc = col + dy[l];

    //             if (isValid(adjr, adjc, n) && !vis[adjr][adjc] &&
    //                 grid[adjr][adjc] == 1) {
    //                 vis[adjr][adjc] = 1;
    //                 int node = row * n + col;
    //                 int adjnode = adjr * n + adjc;
    //                 q.push({adjr, adjc});

    //                 ds.UnionBySize(node, adjnode);
    //             }
    //         }
    //     }
    // }

    void solve(int i, int j, DisjointSet& ds, int& ans, int n, vector<vector<int>>& grid) {
        int dx[] = {-1, 1, 0, 0};
        int dy[] = {0, 0, -1, 1};
        int node = i * n + j;

        unordered_set<int> uniqueParents;

        for (int l = 0; l < 4; l++) {
            int adjr = i + dx[l];
            int adjc = j + dy[l];

            if (isValid(adjr, adjc, n) && grid[adjr][adjc] == 1) {
                int adjNode = adjr * n + adjc;
                uniqueParents.insert(ds.findUParent(adjNode));
            }
        }

        int newSize = 1;
        for (auto parent : uniqueParents) {
            newSize += ds.size[parent];
        }

        ans = max(ans, newSize);
    }

    int largestIsland(vector<vector<int>>& grid) {
        int n = grid.size();
        DisjointSet ds(n * n);

        //vector<vector<int>> vis(n, vector<int>(n, 0));
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 1) {
                    int dx[] = {0, 0, -1, 1};
                    int dy[] = {-1, 1, 0, 0};

                    for (int l = 0; l < 4; l++) {
                        int adjr = i + dx[l];
                        int adjc = j + dy[l];

                        if (isValid(adjr, adjc, n) && grid[adjr][adjc] == 1) {
                            int node = i * n + j;
                            int adjnode = adjr * n + adjc;
                     
                            ds.UnionBySize(node, adjnode);
                        }
                    }
                }
            }
        }

        int ans = 0;

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 0) {
                    solve(i, j, ds, ans, n, grid);
                }
            }
        }

        ans = max(ans, ds.size[ds.findUParent(0)]);
        return ans;
    }
};
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