现有解法python超时，通过减少递归次数优化

problems/0037.解数独.md

@@ -366,40 +366,56 @@ class Solution:
         """
         Do not return anything, modify board in-place instead.
         """
-        self.backtracking(board)
+        row_used = [set() for _ in range(9)]
+        col_used = [set() for _ in range(9)]
+        box_used = [set() for _ in range(9)]
+        for row in range(9):
+            for col in range(9):
+                num = board[row][col]
+                if num == ".":
+                    continue
+                row_used[row].add(num)
+                col_used[col].add(num)
+                box_used[(row // 3) * 3 + col // 3].add(num)
+        self.backtracking(0, 0, board, row_used, col_used, box_used)
 
-    def backtracking(self, board: List[List[str]]) -> bool:
-        # 若有解，返回True；若无解，返回False
-        for i in range(len(board)): # 遍历行
-            for j in range(len(board[0])):  # 遍历列
-                # 若空格内已有数字，跳过
-                if board[i][j] != '.': continue
-                for k in range(1, 10):
-                    if self.is_valid(i, j, k, board):
-                        board[i][j] = str(k)
-                        if self.backtracking(board): return True
-                        board[i][j] = '.'
-                # 若数字1-9都不能成功填入空格，返回False无解
-                return False
-        return True # 有解
+    def backtracking(
+        self,
+        row: int,
+        col: int,
+        board: List[List[str]],
+        row_used: List[List[int]],
+        col_used: List[List[int]],
+        box_used: List[List[int]],
+    ) -> bool:
+        if row == 9:
+            return True
 
-    def is_valid(self, row: int, col: int, val: int, board: List[List[str]]) -> bool:
-        # 判断同一行是否冲突
-        for i in range(9):
-            if board[row][i] == str(val):
-                return False
-        # 判断同一列是否冲突
-        for j in range(9):
-            if board[j][col] == str(val):
-                return False
-        # 判断同一九宫格是否有冲突
-        start_row = (row // 3) * 3
-        start_col = (col // 3) * 3
-        for i in range(start_row, start_row + 3):
-            for j in range(start_col, start_col + 3):
-                if board[i][j] == str(val):
-                    return False
-        return True
+        next_row, next_col = (row, col + 1) if col < 8 else (row + 1, 0)
+        if board[row][col] != ".":
+            return self.backtracking(
+                next_row, next_col, board, row_used, col_used, box_used
+            )
+
+        for num in map(str, range(1, 10)):
+            if (
+                num not in row_used[row]
+                and num not in col_used[col]
+                and num not in box_used[(row // 3) * 3 + col // 3]
+            ):
+                board[row][col] = num
+                row_used[row].add(num)
+                col_used[col].add(num)
+                box_used[(row // 3) * 3 + col // 3].add(num)
+                if self.backtracking(
+                    next_row, next_col, board, row_used, col_used, box_used
+                ):
+                    return True
+                board[row][col] = "."
+                row_used[row].remove(num)
+                col_used[col].remove(num)
+                box_used[(row // 3) * 3 + col // 3].remove(num)
+        return False
 ```
 
 ### Go