feat: add solutions to lc problem: No.3565

solution/3500-3599/3565.Sequential Grid Path Cover/README.md

@@ -69,32 +69,294 @@ edit_url: https://github.com/doocs/leetcode/edit/main/solution/3500-3599/3565.Se
 
 <!-- solution:start -->
 
-### 方法一
+### 方法一：状态压缩 + DFS
+
+我们注意到，矩阵的大小不超过 $6 \times 6$，因此可以使用状态压缩来表示已经访问过的格子。我们可以使用一个整数 $\textit{st}$ 来表示已经访问过的格子，其中第 $i$ 位为 1 表示格子 $i$ 已经被访问过，0 表示未被访问过。
+
+接下来，我们遍历每一个格子作为起点，如果该格子是 0 或 1，则从该格子开始进行深度优先搜索（DFS）。在 DFS 中，我们将当前格子加入路径中，并将其标记为已访问。然后，我们检查当前格子的值，如果等于 $v$，则将 $v$ 加 1。接着，我们尝试向四个方向移动到相邻的格子，如果相邻格子未被访问且其值为 0 或 $v$，则继续进行 DFS。
+
+如果 DFS 成功找到了一条完整的路径，则返回该路径。如果无法找到完整路径，则回溯，撤销当前格子的访问标记，并尝试其他方向。
+
+时间复杂度 $O(m^2 \times n^2)$，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。
 
 <!-- tabs:start -->
 
 #### Python3
 
 ```python
-
+class Solution:
+    def findPath(self, grid: List[List[int]], k: int) -> List[List[int]]:
+        def f(i: int, j: int) -> int:
+            return i * n + j
+
+        def dfs(i: int, j: int, v: int):
+            nonlocal st
+            path.append([i, j])
+            if len(path) == m * n:
+                return True
+            st |= 1 << f(i, j)
+            if grid[i][j] == v:
+                v += 1
+            for a, b in pairwise(dirs):
+                x, y = i + a, j + b
+                if (
+                    0 <= x < m
+                    and 0 <= y < n
+                    and (st & 1 << f(x, y)) == 0
+                    and grid[x][y] in (0, v)
+                ):
+                    if dfs(x, y, v):
+                        return True
+            path.pop()
+            st ^= 1 << f(i, j)
+            return False
+
+        m, n = len(grid), len(grid[0])
+        st = 0
+        path = []
+        dirs = (-1, 0, 1, 0, -1)
+        for i in range(m):
+            for j in range(n):
+                if grid[i][j] in (0, 1):
+                    if dfs(i, j, 1):
+                        return path
+                    path.clear()
+                    st = 0
+        return []
 ```
 
 #### Java
 
 ```java
-
+class Solution {
+    private int m, n;
+    private long st = 0;
+    private List<List<Integer>> path = new ArrayList<>();
+    private final int[] dirs = {-1, 0, 1, 0, -1};
+
+    private int f(int i, int j) {
+        return i * n + j;
+    }
+
+    private boolean dfs(int i, int j, int v, int[][] grid) {
+        path.add(Arrays.asList(i, j));
+        if (path.size() == m * n) {
+            return true;
+        }
+        st |= 1L << f(i, j);
+        if (grid[i][j] == v) {
+            v += 1;
+        }
+        for (int t = 0; t < 4; t++) {
+            int a = dirs[t], b = dirs[t + 1];
+            int x = i + a, y = j + b;
+            if (0 <= x && x < m && 0 <= y && y < n && (st & (1L << f(x, y))) == 0
+                && (grid[x][y] == 0 || grid[x][y] == v)) {
+                if (dfs(x, y, v, grid)) {
+                    return true;
+                }
+            }
+        }
+        path.remove(path.size() - 1);
+        st ^= 1L << f(i, j);
+        return false;
+    }
+
+    public List<List<Integer>> findPath(int[][] grid, int k) {
+        m = grid.length;
+        n = grid[0].length;
+        for (int i = 0; i < m; i++) {
+            for (int j = 0; j < n; j++) {
+                if (grid[i][j] == 0 || grid[i][j] == 1) {
+                    if (dfs(i, j, 1, grid)) {
+                        return path;
+                    }
+                    path.clear();
+                    st = 0;
+                }
+            }
+        }
+        return List.of();
+    }
+}
 ```
 
 #### C++
 
 ```cpp
-
+class Solution {
+    int m, n;
+    unsigned long long st = 0;
+    vector<vector<int>> path;
+    int dirs[5] = {-1, 0, 1, 0, -1};
+
+    int f(int i, int j) {
+        return i * n + j;
+    }
+
+    bool dfs(int i, int j, int v, vector<vector<int>>& grid) {
+        path.push_back({i, j});
+        if (path.size() == static_cast<size_t>(m * n)) {
+            return true;
+        }
+        st |= 1ULL << f(i, j);
+        if (grid[i][j] == v) {
+            v += 1;
+        }
+        for (int t = 0; t < 4; ++t) {
+            int a = dirs[t], b = dirs[t + 1];
+            int x = i + a, y = j + b;
+            if (0 <= x && x < m && 0 <= y && y < n && (st & (1ULL << f(x, y))) == 0
+                && (grid[x][y] == 0 || grid[x][y] == v)) {
+                if (dfs(x, y, v, grid)) {
+                    return true;
+                }
+            }
+        }
+        path.pop_back();
+        st ^= 1ULL << f(i, j);
+        return false;
+    }
+
+public:
+    vector<vector<int>> findPath(vector<vector<int>>& grid, int k) {
+        m = grid.size();
+        n = grid[0].size();
+        for (int i = 0; i < m; ++i) {
+            for (int j = 0; j < n; ++j) {
+                if (grid[i][j] == 0 || grid[i][j] == 1) {
+                    if (dfs(i, j, 1, grid)) {
+                        return path;
+                    }
+                    path.clear();
+                    st = 0;
+                }
+            }
+        }
+        return {};
+    }
+};
 ```
 
 #### Go
 
 ```go
+func findPath(grid [][]int, k int) [][]int {
+	_ = k
+	m := len(grid)
+	n := len(grid[0])
+	var st uint64
+	path := [][]int{}
+	dirs := []int{-1, 0, 1, 0, -1}
+
+	f := func(i, j int) int { return i*n + j }
+
+	var dfs func(int, int, int) bool
+	dfs = func(i, j, v int) bool {
+		path = append(path, []int{i, j})
+		if len(path) == m*n {
+			return true
+		}
+		idx := f(i, j)
+		st |= 1 << idx
+		if grid[i][j] == v {
+			v++
+		}
+		for t := 0; t < 4; t++ {
+			a, b := dirs[t], dirs[t+1]
+			x, y := i+a, j+b
+			if 0 <= x && x < m && 0 <= y && y < n {
+				idx2 := f(x, y)
+				if (st>>idx2)&1 == 0 && (grid[x][y] == 0 || grid[x][y] == v) {
+					if dfs(x, y, v) {
+						return true
+					}
+				}
+			}
+		}
+		path = path[:len(path)-1]
+		st ^= 1 << idx
+		return false
+	}
+
+	for i := 0; i < m; i++ {
+		for j := 0; j < n; j++ {
+			if grid[i][j] == 0 || grid[i][j] == 1 {
+				if dfs(i, j, 1) {
+					return path
+				}
+				path = path[:0]
+				st = 0
+			}
+		}
+	}
+	return [][]int{}
+}
+```
 
+#### TypeScript
+
+```ts
+function findPath(grid: number[][], k: number): number[][] {
+    const m = grid.length;
+    const n = grid[0].length;
+
+    const dirs = [-1, 0, 1, 0, -1];
+    const path: number[][] = [];
+    let st = 0;
+
+    function f(i: number, j: number): number {
+        return i * n + j;
+    }
+
+    function dfs(i: number, j: number, v: number): boolean {
+        path.push([i, j]);
+        if (path.length === m * n) {
+            return true;
+        }
+
+        st |= 1 << f(i, j);
+        if (grid[i][j] === v) {
+            v += 1;
+        }
+
+        for (let d = 0; d < 4; d++) {
+            const x = i + dirs[d];
+            const y = j + dirs[d + 1];
+            const pos = f(x, y);
+            if (
+                x >= 0 &&
+                x < m &&
+                y >= 0 &&
+                y < n &&
+                (st & (1 << pos)) === 0 &&
+                (grid[x][y] === 0 || grid[x][y] === v)
+            ) {
+                if (dfs(x, y, v)) {
+                    return true;
+                }
+            }
+        }
+
+        path.pop();
+        st ^= 1 << f(i, j);
+        return false;
+    }
+
+    for (let i = 0; i < m; i++) {
+        for (let j = 0; j < n; j++) {
+            if (grid[i][j] === 0 || grid[i][j] === 1) {
+                st = 0;
+                path.length = 0;
+                if (dfs(i, j, 1)) {
+                    return path;
+                }
+            }
+        }
+    }
+
+    return [];
+}
 ```
 
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