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130. 被围绕的区域

题目描述

给你一个 m x n 的矩阵 board ，由若干字符 'X' 和 'O' 组成，捕获 所有 被围绕的区域：

  • 连接：一个单元格与水平或垂直方向上相邻的单元格连接。
  • 区域：连接所有 'O' 的单元格来形成一个区域。
  • 围绕：如果您可以用 'X' 单元格 连接这个区域，并且区域中没有任何单元格位于 board 边缘，则该区域被 'X' 单元格围绕。

通过将输入矩阵 board 中的所有 'O' 替换为 'X' 来 捕获被围绕的区域。

 

示例 1：

输入：board = [["X","X","X","X"],["X","O","O","X"],["X","X","O","X"],["X","O","X","X"]]

输出：[["X","X","X","X"],["X","X","X","X"],["X","X","X","X"],["X","O","X","X"]]

解释：

在上图中，底部的区域没有被捕获，因为它在 board 的边缘并且不能被围绕。

示例 2：

输入：board = [["X"]]

输出：[["X"]]

 

提示：

  • m == board.length
  • n == board[i].length
  • 1 <= m, n <= 200
  • board[i][j] 为 'X' 或 'O'

解法

方法一：DFS

我们可以从矩阵的边界开始，将矩阵边界上的每个 O 作为起始点，开始进行深度优先搜索。将搜索到的 O 全部替换成 .。

然后我们再遍历这个矩阵，对于每个位置：

  • 如果是 .，则替换成 O；
  • 否则如果是 O，则替换成 X。

时间复杂度 $O(m \times n)$，空间复杂度 $O(m \times n)$。其中 $m$ 和 $n$ 分别是矩阵的行数和列数。

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class Solution:
    def solve(self, board: List[List[str]]) -> None:
        def dfs(i: int, j: int):
            if not (0 <= i < m and 0 <= j < n and board[i][j] == "O"):
                return
            board[i][j] = "."
            for a, b in pairwise((-1, 0, 1, 0, -1)):
                dfs(i + a, j + b)

        m, n = len(board), len(board[0])
        for i in range(m):
            dfs(i, 0)
            dfs(i, n - 1)
        for j in range(n):
            dfs(0, j)
            dfs(m - 1, j)
        for i in range(m):
            for j in range(n):
                if board[i][j] == ".":
                    board[i][j] = "O"
                elif board[i][j] == "O":
                    board[i][j] = "X"

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class Solution {
    private final int[] dirs = {-1, 0, 1, 0, -1};
    private char[][] board;
    private int m;
    private int n;

    public void solve(char[][] board) {
        m = board.length;
        n = board[0].length;
        this.board = board;
        for (int i = 0; i < m; ++i) {
            dfs(i, 0);
            dfs(i, n - 1);
        }
        for (int j = 0; j < n; ++j) {
            dfs(0, j);
            dfs(m - 1, j);
        }
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                if (board[i][j] == '.') {
                    board[i][j] = 'O';
                } else if (board[i][j] == 'O') {
                    board[i][j] = 'X';
                }
            }
        }
    }

    private void dfs(int i, int j) {
        if (i < 0 || i >= m || j < 0 || j >= n || board[i][j] != 'O') {
            return;
        }
        board[i][j] = '.';
        for (int k = 0; k < 4; ++k) {
            dfs(i + dirs[k], j + dirs[k + 1]);
        }
    }
}

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class Solution {
public:
    void solve(vector<vector<char>>& board) {
        int m = board.size(), n = board[0].size();
        int dirs[5] = {-1, 0, 1, 0, -1};
        function<void(int, int)> dfs = [&](int i, int j) {
            if (i < 0 || i >= m || j < 0 || j >= n || board[i][j] != 'O') {
                return;
            }
            board[i][j] = '.';
            for (int k = 0; k < 4; ++k) {
                dfs(i + dirs[k], j + dirs[k + 1]);
            }
        };
        for (int i = 0; i < m; ++i) {
            dfs(i, 0);
            dfs(i, n - 1);
        }
        for (int j = 1; j < n - 1; ++j) {
            dfs(0, j);
            dfs(m - 1, j);
        }
        for (auto& row : board) {
            for (auto& c : row) {
                if (c == '.') {
                    c = 'O';
                } else if (c == 'O') {
                    c = 'X';
                }
            }
        }
    }
};

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func solve(board [][]byte) {
    m, n := len(board), len(board[0])
    dirs := [5]int{-1, 0, 1, 0, -1}
    var dfs func(i, j int)
    dfs = func(i, j int) {
        if i < 0 || i >= m || j < 0 || j >= n || board[i][j] != 'O' {
            return
        }
        board[i][j] = '.'
        for k := 0; k < 4; k++ {
            dfs(i+dirs[k], j+dirs[k+1])
        }
    }
    for i := 0; i < m; i++ {
        dfs(i, 0)
        dfs(i, n-1)
    }
    for j := 0; j < n; j++ {
        dfs(0, j)
        dfs(m-1, j)
    }
    for i, row := range board {
        for j, c := range row {
            if c == '.' {
                board[i][j] = 'O'
            } else if c == 'O' {
                board[i][j] = 'X'
            }
        }
    }
}

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function solve(board: string[][]): void {
    const m = board.length;
    const n = board[0].length;
    const dirs: number[] = [-1, 0, 1, 0, -1];
    const dfs = (i: number, j: number): void => {
        if (i < 0 || i >= m || j < 0 || j >= n || board[i][j] !== 'O') {
            return;
        }
        board[i][j] = '.';
        for (let k = 0; k < 4; ++k) {
            dfs(i + dirs[k], j + dirs[k + 1]);
        }
    };
    for (let i = 0; i < m; ++i) {
        dfs(i, 0);
        dfs(i, n - 1);
    }
    for (let j = 0; j < n; ++j) {
        dfs(0, j);
        dfs(m - 1, j);
    }
    for (let i = 0; i < m; ++i) {
        for (let j = 0; j < n; ++j) {
            if (board[i][j] === '.') {
                board[i][j] = 'O';
            } else if (board[i][j] === 'O') {
                board[i][j] = 'X';
            }
        }
    }
}

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impl Solution {
    pub fn solve(board: &mut Vec<Vec<char>>) {
        let m = board.len();
        let n = board[0].len();
        let dirs = vec![-1, 0, 1, 0, -1];

        fn dfs(
            board: &mut Vec<Vec<char>>,
            i: usize,
            j: usize,
            dirs: &Vec<i32>,
            m: usize,
            n: usize,
        ) {
            if i >= 0 && i < m && j >= 0 && j < n && board[i][j] == 'O' {
                board[i][j] = '.';
                for k in 0..4 {
                    dfs(
                        board,
                        ((i as i32) + dirs[k]) as usize,
                        ((j as i32) + dirs[k + 1]) as usize,
                        dirs,
                        m,
                        n,
                    );
                }
            }
        }

        for i in 0..m {
            dfs(board, i, 0, &dirs, m, n);
            dfs(board, i, n - 1, &dirs, m, n);
        }
        for j in 0..n {
            dfs(board, 0, j, &dirs, m, n);
            dfs(board, m - 1, j, &dirs, m, n);
        }

        for i in 0..m {
            for j in 0..n {
                if board[i][j] == '.' {
                    board[i][j] = 'O';
                } else if board[i][j] == 'O' {
                    board[i][j] = 'X';
                }
            }
        }
    }
}

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public class Solution {
    private readonly int[] dirs = {-1, 0, 1, 0, -1};
    private char[][] board;
    private int m;
    private int n;

    public void Solve(char[][] board) {
        m = board.Length;
        n = board[0].Length;
        this.board = board;

        for (int i = 0; i < m; ++i) {
            Dfs(i, 0);
            Dfs(i, n - 1);
        }
        for (int j = 0; j < n; ++j) {
            Dfs(0, j);
            Dfs(m - 1, j);
        }

        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                if (board[i][j] == '.') {
                    board[i][j] = 'O';
                } else if (board[i][j] == 'O') {
                    board[i][j] = 'X';
                }
            }
        }
    }

    private void Dfs(int i, int j) {
        if (i < 0 || i >= m || j < 0 || j >= n || board[i][j] != 'O') {
            return;
        }
        board[i][j] = '.';
        for (int k = 0; k < 4; ++k) {
            Dfs(i + dirs[k], j + dirs[k + 1]);
        }
    }
}

方法二：并查集

我们也可以使用并查集，将矩阵边界上的每个 O 与一个超级节点 $m \times n$ 相连，将矩阵中的每个 O 与其上下左右的 O 相连。

然后我们遍历这个矩阵，对于每个位置，如果是 O，并且其与超级节点不相连，则将其替换成 X。

时间复杂度 $O(m \times n \times \alpha(m \times n))$，空间复杂度 $O(m \times n)$。其中 $m$ 和 $n$ 分别是矩阵的行数和列数，$\alpha$ 是阿克曼函数的反函数。

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class Solution:
    def solve(self, board: List[List[str]]) -> None:
        def find(x: int) -> int:
            if p[x] != x:
                p[x] = find(p[x])
            return p[x]

        m, n = len(board), len(board[0])
        p = list(range(m * n + 1))
        for i in range(m):
            for j in range(n):
                if board[i][j] == "O":
                    if i in (0, m - 1) or j in (0, n - 1):
                        p[find(i * n + j)] = find(m * n)
                    else:
                        for a, b in pairwise((-1, 0, 1, 0, -1)):
                            x, y = i + a, j + b
                            if board[x][y] == "O":
                                p[find(x * n + y)] = find(i * n + j)
        for i in range(m):
            for j in range(n):
                if board[i][j] == "O" and find(i * n + j) != find(m * n):
                    board[i][j] = "X"

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class Solution {
    private int[] p;

    public void solve(char[][] board) {
        int m = board.length;
        int n = board[0].length;
        p = new int[m * n + 1];
        for (int i = 0; i < p.length; ++i) {
            p[i] = i;
        }
        int[] dirs = {-1, 0, 1, 0, -1};
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                if (board[i][j] == 'O') {
                    if (i == 0 || i == m - 1 || j == 0 || j == n - 1) {
                        p[find(i * n + j)] = find(m * n);
                    } else {
                        for (int k = 0; k < 4; ++k) {
                            int x = i + dirs[k];
                            int y = j + dirs[k + 1];
                            if (board[x][y] == 'O') {
                                p[find(x * n + y)] = find(i * n + j);
                            }
                        }
                    }
                }
            }
        }
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                if (board[i][j] == 'O' && find(i * n + j) != find(m * n)) {
                    board[i][j] = 'X';
                }
            }
        }
    }

    private int find(int x) {
        if (p[x] != x) {
            p[x] = find(p[x]);
        }
        return p[x];
    }
}

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class Solution {
public:
    void solve(vector<vector<char>>& board) {
        int m = board.size(), n = board[0].size();
        vector<int> p(m * n + 1);
        iota(p.begin(), p.end(), 0);
        function<int(int)> find = [&](int x) {
            return p[x] == x ? x : p[x] = find(p[x]);
        };
        int dirs[5] = {-1, 0, 1, 0, -1};
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                if (board[i][j] == 'O') {
                    if (i == 0 || i == m - 1 || j == 0 || j == n - 1) {
                        p[find(i * n + j)] = find(m * n);
                    } else {
                        for (int k = 0; k < 4; ++k) {
                            int x = i + dirs[k];
                            int y = j + dirs[k + 1];
                            if (board[x][y] == 'O') {
                                p[find(x * n + y)] = find(i * n + j);
                            }
                        }
                    }
                }
            }
        }
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < n; ++j) {
                if (board[i][j] == 'O' && find(i * n + j) != find(m * n)) {
                    board[i][j] = 'X';
                }
            }
        }
    }
};

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func solve(board [][]byte) {
    m, n := len(board), len(board[0])
    p := make([]int, m*n+1)
    for i := range p {
        p[i] = i
    }
    var find func(x int) int
    find = func(x int) int {
        if p[x] != x {
            p[x] = find(p[x])
        }
        return p[x]
    }
    dirs := [5]int{-1, 0, 1, 0, -1}
    for i := 0; i < m; i++ {
        for j := 0; j < n; j++ {
            if board[i][j] == 'O' {
                if i == 0 || i == m-1 || j == 0 || j == n-1 {
                    p[find(i*n+j)] = find(m * n)
                } else {
                    for k := 0; k < 4; k++ {
                        x, y := i+dirs[k], j+dirs[k+1]
                        if board[x][y] == 'O' {
                            p[find(x*n+y)] = find(i*n + j)
                        }
                    }
                }
            }
        }
    }
    for i := 0; i < m; i++ {
        for j := 0; j < n; j++ {
            if board[i][j] == 'O' && find(i*n+j) != find(m*n) {
                board[i][j] = 'X'
            }
        }
    }
}

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function solve(board: string[][]): void {
    const m = board.length;
    const n = board[0].length;
    const p: number[] = Array(m * n + 1)
        .fill(0)
        .map((_, i) => i);
    const dirs: number[] = [-1, 0, 1, 0, -1];
    const find = (x: number): number => {
        if (p[x] !== x) {
            p[x] = find(p[x]);
        }
        return p[x];
    };
    for (let i = 0; i < m; ++i) {
        for (let j = 0; j < n; ++j) {
            if (board[i][j] === 'O') {
                if (i === 0 || i === m - 1 || j === 0 || j === n - 1) {
                    p[find(i * n + j)] = find(m * n);
                } else {
                    for (let k = 0; k < 4; ++k) {
                        const [x, y] = [i + dirs[k], j + dirs[k + 1]];
                        if (board[x][y] === 'O') {
                            p[find(x * n + y)] = find(i * n + j);
                        }
                    }
                }
            }
        }
    }
    for (let i = 0; i < m; ++i) {
        for (let j = 0; j < n; ++j) {
            if (board[i][j] === 'O' && find(i * n + j) !== find(m * n)) {
                board[i][j] = 'X';
            }
        }
    }
}

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