TheAlgorithms · cclauss · Sep 7, 2021 · Sep 5, 2021 · Sep 5, 2021
diff --git a/DIRECTORY.md b/DIRECTORY.md
@@ -545,7 +545,7 @@
 ## Other
   * [Activity Selection](https://github.com/TheAlgorithms/Python/blob/master/other/activity_selection.py)
   * [Date To Weekday](https://github.com/TheAlgorithms/Python/blob/master/other/date_to_weekday.py)
-  * [Davis–Putnam–Logemann–Loveland](https://github.com/TheAlgorithms/Python/blob/master/other/davis–putnam–logemann–loveland.py)
+  * [Davisb Putnamb Logemannb Loveland](https://github.com/TheAlgorithms/Python/blob/master/other/davisb_putnamb_logemannb_loveland.py)
   * [Dijkstra Bankers Algorithm](https://github.com/TheAlgorithms/Python/blob/master/other/dijkstra_bankers_algorithm.py)
   * [Doomsday](https://github.com/TheAlgorithms/Python/blob/master/other/doomsday.py)
   * [Fischer Yates Shuffle](https://github.com/TheAlgorithms/Python/blob/master/other/fischer_yates_shuffle.py)
@@ -860,6 +860,7 @@
   * [Counting Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/counting_sort.py)
   * [Cycle Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/cycle_sort.py)
   * [Double Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/double_sort.py)
+  * [Dutch National Flag Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/dutch_national_flag_sort.py)
   * [Exchange Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/exchange_sort.py)
   * [External Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/external_sort.py)
   * [Gnome Sort](https://github.com/TheAlgorithms/Python/blob/master/sorts/gnome_sort.py)

diff --git a/arithmetic_analysis/in_static_equilibrium.py b/arithmetic_analysis/in_static_equilibrium.py
@@ -1,14 +1,14 @@
 """
 Checks if a system of forces is in static equilibrium.
 """
-from typing import List
+from __future__ import annotations
 
 from numpy import array, cos, cross, ndarray, radians, sin
 
 
 def polar_force(
     magnitude: float, angle: float, radian_mode: bool = False
-) -> List[float]:
+) -> list[float]:
     """
     Resolves force along rectangular components.
     (force, angle) => (force_x, force_y)

diff --git a/arithmetic_analysis/lu_decomposition.py b/arithmetic_analysis/lu_decomposition.py
@@ -3,12 +3,12 @@
 Reference:
 - https://en.wikipedia.org/wiki/LU_decomposition
 """
-from typing import Tuple
+from __future__ import annotations
 
 import numpy as np
 
 
-def lower_upper_decomposition(table: np.ndarray) -> Tuple[np.ndarray, np.ndarray]:
+def lower_upper_decomposition(table: np.ndarray) -> tuple[np.ndarray, np.ndarray]:
     """Lower-Upper (LU) Decomposition
 
     Example:

diff --git a/arithmetic_analysis/newton_forward_interpolation.py b/arithmetic_analysis/newton_forward_interpolation.py
@@ -1,7 +1,7 @@
 # https://www.geeksforgeeks.org/newton-forward-backward-interpolation/
+from __future__ import annotations
 
 import math
-from typing import List
 
 
 # for calculating u value
@@ -22,7 +22,7 @@ def ucal(u: float, p: int) -> float:
 
 def main() -> None:
     n = int(input("enter the numbers of values: "))
-    y: List[List[float]] = []
+    y: list[list[float]] = []
     for i in range(n):
         y.append([])
     for i in range(n):

diff --git a/arithmetic_analysis/newton_raphson.py b/arithmetic_analysis/newton_raphson.py
@@ -2,15 +2,16 @@
 # Author: Syed Haseeb Shah (github.com/QuantumNovice)
 # The Newton-Raphson method (also known as Newton's method) is a way to
 # quickly find a good approximation for the root of a real-valued function
+from __future__ import annotations
+
 from decimal import Decimal
 from math import *  # noqa: F401, F403
-from typing import Union
 
 from sympy import diff
 
 
 def newton_raphson(
-    func: str, a: Union[float, Decimal], precision: float = 10 ** -10
+    func: str, a: float | Decimal, precision: float = 10 ** -10
 ) -> float:
     """Finds root from the point 'a' onwards by Newton-Raphson method
     >>> newton_raphson("sin(x)", 2)

diff --git a/backtracking/all_combinations.py b/backtracking/all_combinations.py
@@ -3,16 +3,16 @@
         numbers out of 1 ... n. We use backtracking to solve this problem.
         Time complexity: O(C(n,k)) which is O(n choose k) = O((n!/(k! * (n - k)!)))
 """
-from typing import List
+from __future__ import annotations
 
 
-def generate_all_combinations(n: int, k: int) -> List[List[int]]:
+def generate_all_combinations(n: int, k: int) -> list[list[int]]:
     """
     >>> generate_all_combinations(n=4, k=2)
     [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]
     """
 
-    result: List[List[int]] = []
+    result: list[list[int]] = []
     create_all_state(1, n, k, [], result)
     return result
 
@@ -21,8 +21,8 @@ def create_all_state(
     increment: int,
     total_number: int,
     level: int,
-    current_list: List[int],
-    total_list: List[List[int]],
+    current_list: list[int],
+    total_list: list[list[int]],
 ) -> None:
     if level == 0:
         total_list.append(current_list[:])
@@ -34,7 +34,7 @@ def create_all_state(
         current_list.pop()
 
 
-def print_all_state(total_list: List[List[int]]) -> None:
+def print_all_state(total_list: list[list[int]]) -> None:
     for i in total_list:
         print(*i)
 

diff --git a/backtracking/all_permutations.py b/backtracking/all_permutations.py
@@ -5,18 +5,18 @@
         Time complexity: O(n! * n),
         where n denotes the length of the given sequence.
 """
-from typing import List, Union
+from __future__ import annotations
 
 
-def generate_all_permutations(sequence: List[Union[int, str]]) -> None:
+def generate_all_permutations(sequence: list[int | str]) -> None:
     create_state_space_tree(sequence, [], 0, [0 for i in range(len(sequence))])
 
 
 def create_state_space_tree(
-    sequence: List[Union[int, str]],
-    current_sequence: List[Union[int, str]],
+    sequence: list[int | str],
+    current_sequence: list[int | str],
     index: int,
-    index_used: List[int],
+    index_used: list[int],
 ) -> None:
     """
     Creates a state space tree to iterate through each branch using DFS.
@@ -44,8 +44,8 @@ def create_state_space_tree(
 sequence = list(map(int, input().split()))
 """
 
-sequence: List[Union[int, str]] = [3, 1, 2, 4]
+sequence: list[int | str] = [3, 1, 2, 4]
 generate_all_permutations(sequence)
 
-sequence_2: List[Union[int, str]] = ["A", "B", "C"]
+sequence_2: list[int | str] = ["A", "B", "C"]
 generate_all_permutations(sequence_2)
diff --git a/backtracking/all_subsequences.py b/backtracking/all_subsequences.py
@@ -5,15 +5,17 @@
 Time complexity: O(2^n),
 where n denotes the length of the given sequence.
 """
-from typing import Any, List
+from __future__ import annotations
 
+from typing import Any
 
-def generate_all_subsequences(sequence: List[Any]) -> None:
+
+def generate_all_subsequences(sequence: list[Any]) -> None:
     create_state_space_tree(sequence, [], 0)
 
 
 def create_state_space_tree(
-    sequence: List[Any], current_subsequence: List[Any], index: int
+    sequence: list[Any], current_subsequence: list[Any], index: int
 ) -> None:
     """
     Creates a state space tree to iterate through each branch using DFS.
@@ -32,7 +34,7 @@ def create_state_space_tree(
 
 
 if __name__ == "__main__":
-    seq: List[Any] = [3, 1, 2, 4]
+    seq: list[Any] = [3, 1, 2, 4]
     generate_all_subsequences(seq)
 
     seq.clear()

diff --git a/backtracking/coloring.py b/backtracking/coloring.py
@@ -5,11 +5,10 @@
 
     Wikipedia: https://en.wikipedia.org/wiki/Graph_coloring
 """
-from typing import List
 
 
 def valid_coloring(
-    neighbours: List[int], colored_vertices: List[int], color: int
+    neighbours: list[int], colored_vertices: list[int], color: int
 ) -> bool:
     """
     For each neighbour check if coloring constraint is satisfied
@@ -35,7 +34,7 @@ def valid_coloring(
 
 
 def util_color(
-    graph: List[List[int]], max_colors: int, colored_vertices: List[int], index: int
+    graph: list[list[int]], max_colors: int, colored_vertices: list[int], index: int
 ) -> bool:
     """
     Pseudo-Code
@@ -86,7 +85,7 @@ def util_color(
     return False
 
 
-def color(graph: List[List[int]], max_colors: int) -> List[int]:
+def color(graph: list[list[int]], max_colors: int) -> list[int]:
     """
     Wrapper function to call subroutine called util_color
     which will either return True or False.

diff --git a/backtracking/hamiltonian_cycle.py b/backtracking/hamiltonian_cycle.py
@@ -6,11 +6,10 @@
 
     Wikipedia: https://en.wikipedia.org/wiki/Hamiltonian_path
 """
-from typing import List
 
 
 def valid_connection(
-    graph: List[List[int]], next_ver: int, curr_ind: int, path: List[int]
+    graph: list[list[int]], next_ver: int, curr_ind: int, path: list[int]
 ) -> bool:
     """
     Checks whether it is possible to add next into path by validating 2 statements
@@ -47,7 +46,7 @@ def valid_connection(
     return not any(vertex == next_ver for vertex in path)
 
 
-def util_hamilton_cycle(graph: List[List[int]], path: List[int], curr_ind: int) -> bool:
+def util_hamilton_cycle(graph: list[list[int]], path: list[int], curr_ind: int) -> bool:
     """
     Pseudo-Code
     Base Case:
@@ -108,7 +107,7 @@ def util_hamilton_cycle(graph: List[List[int]], path: List[int], curr_ind: int)
     return False
 
 
-def hamilton_cycle(graph: List[List[int]], start_index: int = 0) -> List[int]:
+def hamilton_cycle(graph: list[list[int]], start_index: int = 0) -> list[int]:
     r"""
     Wrapper function to call subroutine called util_hamilton_cycle,
     which will either return array of vertices indicating hamiltonian cycle

diff --git a/backtracking/knight_tour.py b/backtracking/knight_tour.py
@@ -1,9 +1,9 @@
 # Knight Tour Intro: https://www.youtube.com/watch?v=ab_dY3dZFHM
 
-from typing import List, Tuple
+from __future__ import annotations
 
 
-def get_valid_pos(position: Tuple[int, int], n: int) -> List[Tuple[int, int]]:
+def get_valid_pos(position: tuple[int, int], n: int) -> list[tuple[int, int]]:
     """
     Find all the valid positions a knight can move to from the current position.
 
@@ -32,7 +32,7 @@ def get_valid_pos(position: Tuple[int, int], n: int) -> List[Tuple[int, int]]:
     return permissible_positions
 
 
-def is_complete(board: List[List[int]]) -> bool:
+def is_complete(board: list[list[int]]) -> bool:
     """
     Check if the board (matrix) has been completely filled with non-zero values.
 
@@ -47,7 +47,7 @@ def is_complete(board: List[List[int]]) -> bool:
 
 
 def open_knight_tour_helper(
-    board: List[List[int]], pos: Tuple[int, int], curr: int
+    board: list[list[int]], pos: tuple[int, int], curr: int
 ) -> bool:
     """
     Helper function to solve knight tour problem.
@@ -68,7 +68,7 @@ def open_knight_tour_helper(
     return False
 
 
-def open_knight_tour(n: int) -> List[List[int]]:
+def open_knight_tour(n: int) -> list[list[int]]:
     """
     Find the solution for the knight tour problem for a board of size n. Raises
     ValueError if the tour cannot be performed for the given size.

diff --git a/backtracking/minimax.py b/backtracking/minimax.py
@@ -7,12 +7,13 @@
 leaves of game tree is stored in scores[]
 height is maximum height of Game tree
 """
+from __future__ import annotations
+
 import math
-from typing import List
 
 
 def minimax(
-    depth: int, node_index: int, is_max: bool, scores: List[int], height: float
+    depth: int, node_index: int, is_max: bool, scores: list[int], height: float
 ) -> int:
     """
     >>> import math

diff --git a/backtracking/n_queens.py b/backtracking/n_queens.py
@@ -7,12 +7,12 @@
  diagonal lines.
 
 """
-from typing import List
+from __future__ import annotations
 
 solution = []
 
 
-def isSafe(board: List[List[int]], row: int, column: int) -> bool:
+def isSafe(board: list[list[int]], row: int, column: int) -> bool:
     """
     This function returns a boolean value True if it is safe to place a queen there
     considering the current state of the board.
@@ -40,7 +40,7 @@ def isSafe(board: List[List[int]], row: int, column: int) -> bool:
     return True
 
 
-def solve(board: List[List[int]], row: int) -> bool:
+def solve(board: list[list[int]], row: int) -> bool:
     """
     It creates a state space tree and calls the safe function until it receives a
     False Boolean and terminates that branch and backtracks to the next
@@ -70,7 +70,7 @@ def solve(board: List[List[int]], row: int) -> bool:
     return False
 
 
-def printboard(board: List[List[int]]) -> None:
+def printboard(board: list[list[int]]) -> None:
     """
     Prints the boards that have a successful combination.
     """

diff --git a/backtracking/n_queens_math.py b/backtracking/n_queens_math.py
@@ -75,14 +75,14 @@
 for another one or vice versa.
 
 """
-from typing import List
+from __future__ import annotations
 
 
 def depth_first_search(
-    possible_board: List[int],
-    diagonal_right_collisions: List[int],
-    diagonal_left_collisions: List[int],
-    boards: List[List[str]],
+    possible_board: list[int],
+    diagonal_right_collisions: list[int],
+    diagonal_left_collisions: list[int],
+    boards: list[list[str]],
     n: int,
 ) -> None:
     """
@@ -139,7 +139,7 @@ def depth_first_search(
 
 
 def n_queens_solution(n: int) -> None:
-    boards: List[List[str]] = []
+    boards: list[list[str]] = []
     depth_first_search([], [], [], boards, n)
 
     # Print all the boards

diff --git a/backtracking/rat_in_maze.py b/backtracking/rat_in_maze.py
@@ -1,7 +1,7 @@
-from typing import List
+from __future__ import annotations
 
 
-def solve_maze(maze: List[List[int]]) -> bool:
+def solve_maze(maze: list[list[int]]) -> bool:
     """
     This method solves the "rat in maze" problem.
     In this problem we have some n by n matrix, a start point and an end point.
@@ -70,7 +70,7 @@ def solve_maze(maze: List[List[int]]) -> bool:
     return solved
 
 
-def run_maze(maze: List[List[int]], i: int, j: int, solutions: List[List[int]]) -> bool:
+def run_maze(maze: list[list[int]], i: int, j: int, solutions: list[list[int]]) -> bool:
     """
     This method is recursive starting from (i, j) and going in one of four directions:
     up, down, left, right.