


'''simple docstring'''
from __future__ import annotations

__snake_case											=					[
    [-1, 0],  # left
    [0, -1],  # down
    [1, 0],  # right
    [0, 1],  # up
]


def       a				(      __a			, __a			, __a			, __a			, __a			, )  ->     tuple[list[list[int]], list[list[int]]]:



					'''simple docstring'''






					UpperCamelCase__       :Any							 =			[
					    [0 for col in range(len(grid[0]       )       )] for row in range(len(__a       )       )
					]  # the reference grid
					UpperCamelCase__       :Any							 =			1
					UpperCamelCase__       :Optional[Any]							 =			[
					    [0 for col in range(len(grid[0]       )       )] for row in range(len(__a       )       )
					]  # the action grid

					UpperCamelCase__       :str							 =			init[0]
					UpperCamelCase__       :Tuple							 =			init[1]
					UpperCamelCase__       :List[str]							 =			0
					UpperCamelCase__       :Tuple							 =			g + heuristic[x][y]  # cost from starting cell to destination cell
					UpperCamelCase__       :int							 =			[[f, g, x, y]]

					UpperCamelCase__       :Optional[int]							 =			False  # flag that is set when search is complete
					UpperCamelCase__       :Optional[Any]							 =			False  # flag set if we can't find expand

					while not found and not resign:
										if len(__a       ) == 0:
															raise ValueError('''Algorithm is unable to find solution'''       )
										else:  # to choose the least costliest action so as to move closer to the goal
															cell.sort()
															cell.reverse()
															UpperCamelCase__       :List[Any]							 =			cell.pop()
															UpperCamelCase__       :Dict							 =			next_cell[2]
															UpperCamelCase__       :int							 =			next_cell[3]
															UpperCamelCase__       :List[Any]							 =			next_cell[1]

															if x == goal[0] and y == goal[1]:
																				UpperCamelCase__       :Tuple							 =			True
															else:
																				for i in range(len(__a       )       ):  # to try out different valid actions
																									UpperCamelCase__       :List[str]							 =			x + DIRECTIONS[i][0]
																									UpperCamelCase__       :Optional[int]							 =			y + DIRECTIONS[i][1]
																									if xa >= 0 and xa < len(__a       ) and ya >= 0 and ya < len(grid[0]       ):
																														if closed[xa][ya] == 0 and grid[xa][ya] == 0:
																																			UpperCamelCase__       :Any							 =			g + cost
																																			UpperCamelCase__       :Optional[int]							 =			ga + heuristic[xa][ya]
																																			cell.append([fa, ga, xa, ya]       )
																																			UpperCamelCase__       :List[Any]							 =			1
																																			UpperCamelCase__       :Any							 =			i
					UpperCamelCase__       :List[str]							 =			[]
					UpperCamelCase__       :List[Any]							 =			goal[0]
					UpperCamelCase__       :int							 =			goal[1]
					invpath.append([x, y]       )  # we get the reverse path from here
					while x != init[0] or y != init[1]:
										UpperCamelCase__       :Optional[int]							 =			x - DIRECTIONS[action[x][y]][0]
										UpperCamelCase__       :Dict							 =			y - DIRECTIONS[action[x][y]][1]
										UpperCamelCase__       :Optional[Any]							 =			xa
										UpperCamelCase__       :Tuple							 =			ya
										invpath.append([x, y]       )

					UpperCamelCase__       :Dict							 =			[]
					for i in range(len(__a       )       ):
										path.append(invpath[len(__a       ) - 1 - i]       )
					return path, action


if __name__ == "__main__":
		__snake_case											=					[
		    [0, 1, 0, 0, 0, 0],
		    [0, 1, 0, 0, 0, 0],  # 0 are free path whereas 1's are obstacles
		    [0, 1, 0, 0, 0, 0],
		    [0, 1, 0, 0, 1, 0],
		    [0, 0, 0, 0, 1, 0],
		]

		__snake_case											=					[0, 0]
		# all coordinates are given in format [y,x]
		__snake_case											=					[len(grid) - 1, len(grid[0]) - 1]
		__snake_case											=					1

		# the cost map which pushes the path closer to the goal
		__snake_case											=					[[0 for row in range(len(grid[0]))] for col in range(len(grid))]
		for i in range(len(grid)):
				for j in range(len(grid[0])):
						__snake_case											=					abs(i - goal[0]) + abs(j - goal[1])
						if grid[i][j] == 1:
								# added extra penalty in the heuristic map
								__snake_case											=					99

		__snake_case				, __snake_case											=					search(grid, init, goal, cost, heuristic)

		print('''ACTION MAP''')
		for i in range(len(action)):
				print(action[i])

		for i in range(len(path)):
				print(path[i])



'''simple docstring'''
from __future__ import annotations

__snake_case											=					list[list[int]]

# assigning initial values to the grid
__snake_case											=					[
    [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],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
__snake_case											=					[
    [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],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]


def       a				(      __a			, __a			, __a			, __a       )  ->     bool:



					'''simple docstring'''






					for i in range(9       ):
										if grid[row][i] == n or grid[i][column] == n:
															return False

					for i in range(3       ):
										for j in range(3       ):
															if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
																				return False

					return True


def       a				(      __a       )  ->     tuple[int, int] | None:



					'''simple docstring'''






					for i in range(9       ):
										for j in range(9       ):
															if grid[i][j] == 0:
																				return i, j
					return None


def       a				(      __a       )  ->     Matrix | None:



					'''simple docstring'''






					if location := find_empty_location(__a       ):
										UpperCamelCase__ ,    UpperCamelCase__       :Optional[int]							 =			location
					else:
										# If the location is ``None``, then the grid is solved.
										return grid

					for digit in range(1			, 10       ):
										if is_safe(__a			, __a			, __a			, __a       ):
															UpperCamelCase__       :Tuple							 =			digit

															if sudoku(__a       ) is not None:
																				return grid

															UpperCamelCase__       :Union[str, Any]							 =			0

					return None


def       a				(      __a       )  ->     None:



					'''simple docstring'''






					for row in grid:
										for cell in row:
															print(__a			, end=''' '''       )
										print()


if __name__ == "__main__":
		# make a copy of grid so that you can compare with the unmodified grid
		for example_grid in (initial_grid, no_solution):
				print('''\nExample grid:\n''' + '''=''' * 20)
				print_solution(example_grid)
				print('''\nExample grid solution:''')
				__snake_case											=					sudoku(example_grid)
				if solution is not None:
						print_solution(solution)
				else:
						print('''Cannot find a solution.''')