Number of Islands — C++

// Time:  O(m * n * α(n)) ~= O(m * n)
// Space: O(m * n)

class Solution {
public:
    int numIslands(vector<vector<char>>& grid) {
        if (grid.empty()) {
            return 0;
        }
        int zero_count = 0;
        UnionFind union_find(grid.size() * grid[0].size());
        for (int i = 0; i < grid.size(); ++i) {
            for (int j = 0; j < grid[0].size(); ++j) {
                if (grid[i][j] == '1') {
                    if (i && grid[i][j] == grid[i - 1][j]) {
                        union_find.union_set(index(grid[0].size(), i - 1, j),
                                             index(grid[0].size(), i, j));
                    }
                    if (j && grid[i][j] == grid[i][j - 1]) {
                        union_find.union_set(index(grid[0].size(), i, j - 1),
                                             index(grid[0].size(), i, j));
                    }
                } else {
                    ++zero_count;
                }
            }
        }
        return union_find.size() - zero_count;
    }

private:
    class UnionFind {
    public:
        UnionFind(const int n) : set_(n), count_(n) {
            iota(set_.begin(), set_.end(), 0);
        }

        int find_set(const int x) {
           if (set_[x] != x) {
               set_[x] = find_set(set_[x]);  // Path compression.
           }
           return set_[x];
        }

        void union_set(const int x, const int y) {
            int x_root = find_set(x), y_root = find_set(y);
            if (x_root != y_root) {
                set_[min(x_root, y_root)] = max(x_root, y_root);
                --count_;
            }
        }

        int size() const {
            return count_;
        }

    private:
        vector<int> set_;
        int count_;
    };
    
    int index(int n, int i, int j) {
        return i * n + j;
    }
};

// Time:  O(m * n)
// Space: O(m * n)
// dfs solution
class Solution2 {
public:
    int numIslands(vector<vector<char>>& grid) {
        int count = 0;
        for (int i = 0; i < grid.size(); ++i) {
            for (int j = 0; j < grid[i].size(); ++j) {
                if (dfs(&grid, i, j)) {
                    ++count;
                }
            }
        }
        return count;
    }

private:
    bool dfs(vector<vector<char>> *grid, int i, int j) {
        static const vector<pair<int, int>> directions{{0, 1}, {1, 0}, {0, -1}, {-1, 0}};

        if ((*grid)[i][j] == '0') {
            return false;
        }
        (*grid)[i][j] = '0';
        vector<pair<int, int>> stk = {{i, j}};
        while (!stk.empty()) {
            const auto [r, c] = stk.back(); stk.pop_back();
            for (const auto& d : directions) {
                const auto nr = r + d.first;
                const auto nc = c + d.second;
                if (!(0 <= nr && nr < grid->size() &&
                      0 <= nc && nc < (*grid)[0].size() &&
                      (*grid)[nr][nc] == '1')) {
                    continue;
                }
                (*grid)[nr][nc] = '0';
                stk.emplace_back(nr, nc);
            }
        }
        return true;
    }
};

// Time:  O(m * n)
// Space: O(m * n)
// bfs solution
class Solution3 {
public:
    int numIslands(vector<vector<char>>& grid) {
        int count = 0;
        for (int i = 0; i < grid.size(); ++i) {
            for (int j = 0; j < grid[i].size(); ++j) {
                if (bfs(&grid, i, j)) {
                    ++count;
                }
            }
        }
        return count;
    }

private:
    bool bfs(vector<vector<char>> *grid, int i, int j) {
        static const vector<pair<int, int>> directions{{0, 1}, {1, 0}, {0, -1}, {-1, 0}};
        
        if ((*grid)[i][j] == '0') {
            return false;
        }
        (*grid)[i][j] = '0';
        queue<pair<int, int>> q({{i, j}});
        while (!q.empty()) {
            const auto [r, c] = q.front(); q.pop();
            for (const auto& d : directions) {
                const auto nr = r + d.first;
                const auto nc = c + d.second;
                if (!(0 <= nr && nr < grid->size() &&
                      0 <= nc && nc < (*grid)[0].size() &&
                      (*grid)[nr][nc] == '1')) {
                    continue;
                }
                (*grid)[nr][nc] = '0';
                q.emplace(nr, nc);
            }
        }
        return true;
    }
};