mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-24 05:21:09 +00:00
c824b90ead
* Remove redundent function After reviewing this code, I've noticed that the `is_completed` function is a redundant operation. Increasing the number of loops required for each step of the sudoku solver. This should remove n² operations where n is the width of the grid. * Update sudoku.py Remove additional newline
133 lines
3.9 KiB
Python
133 lines
3.9 KiB
Python
"""
|
||
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]]
|
||
|
||
# assigning initial values to the 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],
|
||
[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
|
||
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],
|
||
[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 is_safe(grid: Matrix, row: int, column: int, n: int) -> bool:
|
||
"""
|
||
This function checks the grid to see if each row,
|
||
column, and the 3x3 subgrids contain the digit 'n'.
|
||
It returns False if it is not 'safe' (a duplicate digit
|
||
is found) else returns True if it is 'safe'
|
||
"""
|
||
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 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.
|
||
"""
|
||
for i in range(9):
|
||
for j in range(9):
|
||
if grid[i][j] == 0:
|
||
return i, j
|
||
return None
|
||
|
||
|
||
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
|
||
for Sudoku solution (non-duplication across rows, columns, and boxes)
|
||
|
||
>>> sudoku(initial_grid) # doctest: +NORMALIZE_WHITESPACE
|
||
[[3, 1, 6, 5, 7, 8, 4, 9, 2],
|
||
[5, 2, 9, 1, 3, 4, 7, 6, 8],
|
||
[4, 8, 7, 6, 2, 9, 5, 3, 1],
|
||
[2, 6, 3, 4, 1, 5, 9, 8, 7],
|
||
[9, 7, 4, 8, 6, 3, 1, 2, 5],
|
||
[8, 5, 1, 7, 9, 2, 6, 4, 3],
|
||
[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) is None
|
||
True
|
||
"""
|
||
if location := find_empty_location(grid):
|
||
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) is not None:
|
||
return grid
|
||
|
||
grid[row][column] = 0
|
||
|
||
return None
|
||
|
||
|
||
def print_solution(grid: Matrix) -> None:
|
||
"""
|
||
A function to print the solution in the form
|
||
of a 9x9 grid
|
||
"""
|
||
for row in grid:
|
||
for cell in row:
|
||
print(cell, 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:")
|
||
solution = sudoku(example_grid)
|
||
if solution is not None:
|
||
print_solution(solution)
|
||
else:
|
||
print("Cannot find a solution.")
|