[mypy] Add/fix type annotations for backtracking algorithms (#4055)

* Fix mypy errors for backtracking algorithms

* Fix CI failure

backtracking/sudoku.py

@@ -1,20 +1,20 @@
-from typing import List, Tuple, Union
+"""
+Given a partially filled 9×9 2D array, the objective is to fill a 9×9
+square grid with digits numbered 1 to 9, so that every row, column, and
+and each of the nine 3×3 sub-grids contains all of the digits.
+
+This can be solved using Backtracking and is similar to n-queens.
+We check to see if a cell is safe or not and recursively call the
+function on the next column to see if it returns True. if yes, we
+have solved the puzzle. else, we backtrack and place another number
+in that cell and repeat this process.
+"""
+from typing import List, Optional, Tuple
 
 Matrix = List[List[int]]
 
-"""
-    Given a partially filled 9×9 2D array, the objective is to fill a 9×9
-    square grid with digits numbered 1 to 9, so that every row, column, and
-    and each of the nine 3×3 sub-grids contains all of the digits.
-
-    This can be solved using Backtracking and is similar to n-queens.
-    We check to see if a cell is safe or not and recursively call the
-    function on the next column to see if it returns True. if yes, we
-    have solved the puzzle. else, we backtrack and place another number
-    in that cell and repeat this process.
-"""
 # assigning initial values to the grid
-initial_grid = [
+initial_grid: Matrix = [
     [3, 0, 6, 5, 0, 8, 4, 0, 0],
     [5, 2, 0, 0, 0, 0, 0, 0, 0],
     [0, 8, 7, 0, 0, 0, 0, 3, 1],
@@ -27,7 +27,7 @@ initial_grid = [
 ]
 
 # a grid with no solution
-no_solution = [
+no_solution: Matrix = [
     [5, 0, 6, 5, 0, 8, 4, 0, 3],
     [5, 2, 0, 0, 0, 0, 0, 0, 2],
     [1, 8, 7, 0, 0, 0, 0, 3, 1],
@@ -80,7 +80,7 @@ def is_completed(grid: Matrix) -> bool:
     return all(all(cell != 0 for cell in row) for row in grid)
 
 
-def find_empty_location(grid: Matrix) -> Tuple[int, int]:
+def find_empty_location(grid: Matrix) -> Optional[Tuple[int, int]]:
     """
     This function finds an empty location so that we can assign a number
     for that particular row and column.
@@ -89,9 +89,10 @@ def find_empty_location(grid: Matrix) -> Tuple[int, int]:
         for j in range(9):
             if grid[i][j] == 0:
                 return i, j
+    return None
 
 
-def sudoku(grid: Matrix) -> Union[Matrix, bool]:
+def sudoku(grid: Matrix) -> Optional[Matrix]:
     """
     Takes a partially filled-in grid and attempts to assign values to
     all unassigned locations in such a way to meet the requirements
@@ -107,25 +108,30 @@ def sudoku(grid: Matrix) -> Union[Matrix, bool]:
      [1, 3, 8, 9, 4, 7, 2, 5, 6],
      [6, 9, 2, 3, 5, 1, 8, 7, 4],
      [7, 4, 5, 2, 8, 6, 3, 1, 9]]
-     >>> sudoku(no_solution)
-     False
+     >>> sudoku(no_solution) is None
+     True
     """
 
     if is_completed(grid):
         return grid
 
-    row, column = find_empty_location(grid)
+    location = find_empty_location(grid)
+    if location is not None:
+        row, column = location
+    else:
+        # If the location is ``None``, then the grid is solved.
+        return grid
 
     for digit in range(1, 10):
         if is_safe(grid, row, column, digit):
             grid[row][column] = digit
 
-            if sudoku(grid):
+            if sudoku(grid) is not None:
                 return grid
 
             grid[row][column] = 0
 
-    return False
+    return None
 
 
 def print_solution(grid: Matrix) -> None:
@@ -141,11 +147,12 @@ def print_solution(grid: Matrix) -> None:
 
 if __name__ == "__main__":
     # make a copy of grid so that you can compare with the unmodified grid
-    for grid in (initial_grid, no_solution):
-        grid = list(map(list, grid))
-        solution = sudoku(grid)
-        if solution:
-            print("grid after solving:")
+    for example_grid in (initial_grid, no_solution):
+        print("\nExample grid:\n" + "=" * 20)
+        print_solution(example_grid)
+        print("\nExample grid solution:")
+        solution = sudoku(example_grid)
+        if solution is not None:
             print_solution(solution)
         else:
             print("Cannot find a solution.")