






"""simple docstring"""





from __future__ import annotations

from collections.abc import Iterable







class 		_a       :





   '''simple docstring'''






   def __init__(   self			,				UpperCAmelCase_			,				UpperCAmelCase_)	-> Optional[Any]:




          '''simple docstring'''




          lowercase__  ,     lowercase__:    str        =      float(UpperCAmelCase_), float(UpperCAmelCase_)





   def __eq__(   self			,				UpperCAmelCase_)	-> Dict:




          '''simple docstring'''




          return self.x == other.x and self.y == other.y





   def __ne__(   self			,				UpperCAmelCase_)	-> Union[str, Any]:




          '''simple docstring'''




          return not self == other





   def __gt__(   self			,				UpperCAmelCase_)	-> Dict:




          '''simple docstring'''




          if self.x > other.x:
                 return True
          elif self.x == other.x:
                 return self.y > other.y
          return False





   def __lt__(   self			,				UpperCAmelCase_)	-> Optional[int]:




          '''simple docstring'''




          return not self > other





   def __ge__(   self			,				UpperCAmelCase_)	-> Union[str, Any]:




          '''simple docstring'''




          if self.x > other.x:
                 return True
          elif self.x == other.x:
                 return self.y >= other.y
          return False





   def __le__(   self			,				UpperCAmelCase_)	-> List[Any]:




          '''simple docstring'''




          if self.x < other.x:
                 return True
          elif self.x == other.x:
                 return self.y <= other.y
          return False





   def __repr__(   self)	-> List[Any]:




          '''simple docstring'''




          return F"""({self.x}, {self.y})"""





   def __hash__(   self)	-> Any:




          '''simple docstring'''




          return hash(self.x)







def     A(  snake_case_						,       ):
       """simple docstring"""





       lowercase__:    list[Point]        =      []
       if list_of_tuples:
              for p in list_of_tuples:
                     if isinstance(snake_case_						,       snake_case_		):
                            points.append(snake_case_		)
                     else:
                            try:
                                   points.append(Point(p[0]						,       p[1]		)		)
                            except (IndexError, TypeError):
                                   print(
                                       F"""Ignoring deformed point {p}. All points"""
                                       " must have at least 2 coordinates."		)
       return points






def     A(  snake_case_		):
       """simple docstring"""





       if not hasattr(snake_case_						,       "__iter__"		):
              lowercase__:    Tuple        =      F"""Expecting an iterable object but got an non-iterable type {points}"""
              raise ValueError(snake_case_		)

       if not points:
              lowercase__:    Dict        =      F"""Expecting a list of points but got {points}"""
              raise ValueError(snake_case_		)

       return _construct_points(snake_case_		)






def     A(  snake_case_						,       snake_case_						,       snake_case_		):
       """simple docstring"""





       lowercase__:    Any        =      (a.x * b.y + b.x * c.y + c.x * a.y) - (a.y * b.x + b.y * c.x + c.y * a.x)
       return det






def     A(  snake_case_		):
       """simple docstring"""





       lowercase__:    Optional[int]        =      sorted(_validate_input(snake_case_		)		)
       lowercase__:    int        =      len(snake_case_		)
       lowercase__:    Optional[int]        =      set()

       for i in range(n - 1		):
              for j in range(i + 1						,       snake_case_		):
                     lowercase__:    Dict        =      False
                     lowercase__:    List[str]        =      True
                     for k in range(snake_case_		):
                            if k != i and k != j:
                                   lowercase__:    List[Any]        =      _det(points[i]						,       points[j]						,       points[k]		)

                                   if det_k > 0:
                                          lowercase__:    List[Any]        =      True
                                   elif det_k < 0:
                                          lowercase__:    Any        =      True
                                   else:
                                          # point[i], point[j], point[k] all lie on a straight line
                                          # if point[k] is to the left of point[i] or it's to the
                                          # right of point[j], then point[i], point[j] cannot be
                                          # part of the convex hull of A
                                          if points[k] < points[i] or points[k] > points[j]:
                                                 lowercase__:    Any        =      False
                                                 break

                            if points_left_of_ij and points_right_of_ij:
                                   lowercase__:    Dict        =      False
                                   break

                     if ij_part_of_convex_hull:
                            convex_set.update([points[i], points[j]]		)

       return sorted(snake_case_		)






def     A(  snake_case_		):
       """simple docstring"""





       lowercase__:    Any        =      sorted(_validate_input(snake_case_		)		)
       lowercase__:    str        =      len(snake_case_		)

       # divide all the points into an upper hull and a lower hull
       # the left most point and the right most point are definitely
       # members of the convex hull by definition.
       # use these two anchors to divide all the points into two hulls,
       # an upper hull and a lower hull.

       # all points to the left (above) the line joining the extreme points belong to the
       # upper hull
       # all points to the right (below) the line joining the extreme points below to the
       # lower hull
       # ignore all points on the line joining the extreme points since they cannot be
       # part of the convex hull

       lowercase__:    int        =      points[0]
       lowercase__:    int        =      points[n - 1]

       lowercase__:    Tuple        =      {left_most_point, right_most_point}
       lowercase__:    List[str]        =      []
       lowercase__:    List[Any]        =      []

       for i in range(1						,       n - 1		):
              lowercase__:    str        =      _det(snake_case_						,       snake_case_						,       points[i]		)

              if det > 0:
                     upper_hull.append(points[i]		)
              elif det < 0:
                     lower_hull.append(points[i]		)

       _construct_hull(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)
       _construct_hull(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)

       return sorted(snake_case_		)






def     A(  snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		):
       """simple docstring"""





       if points:
              lowercase__:    str        =      None
              lowercase__:    int        =      float("-inf"		)
              lowercase__:    Union[str, Any]        =      []

              for p in points:
                     lowercase__:    Any        =      _det(snake_case_						,       snake_case_						,       snake_case_		)

                     if det > 0:
                            candidate_points.append(snake_case_		)

                            if det > extreme_point_distance:
                                   lowercase__:    Dict        =      det
                                   lowercase__:    List[str]        =      p

              if extreme_point:
                     _construct_hull(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)
                     convex_set.add(snake_case_		)
                     _construct_hull(snake_case_						,       snake_case_						,       snake_case_						,       snake_case_		)






def     A(  snake_case_		):
       """simple docstring"""





       lowercase__:    Optional[Any]        =      sorted(_validate_input(snake_case_		)		)
       lowercase__:    List[str]        =      len(snake_case_		)

       lowercase__:    Union[str, Any]        =      points[:2]
       for i in range(2						,       snake_case_		):
              lowercase__:    List[Any]        =      _det(convex_hull[1]						,       convex_hull[0]						,       points[i]		)
              if det > 0:
                     convex_hull.insert(0						,       points[i]		)
                     break
              elif det < 0:
                     convex_hull.append(points[i]		)
                     break
              else:
                     lowercase__:    List[Any]        =      points[i]
       i += 1

       for j in range(snake_case_						,       snake_case_		):
              if (
                  _det(convex_hull[0]						,       convex_hull[-1]						,       points[j]		) > 0
                  and _det(convex_hull[-1]						,       convex_hull[0]						,       points[1]		) < 0
              ):
                     # The point lies within the convex hull
                     continue

              convex_hull.insert(0						,       points[j]		)
              convex_hull.append(points[j]		)
              while _det(convex_hull[0]						,       convex_hull[1]						,       convex_hull[2]		) >= 0:
                     del convex_hull[1]
              while _det(convex_hull[-1]						,       convex_hull[-2]						,       convex_hull[-3]		) <= 0:
                     del convex_hull[-2]

    # `convex_hull` is contains the convex hull in circular order
       return sorted(convex_hull[1:] if len(snake_case_		) > 3 else convex_hull		)






def     A(  ):
       """simple docstring"""





       lowercase__:    List[str]        =      [
           (0, 3),
           (2, 2),
           (1, 1),
           (2, 1),
           (3, 0),
           (0, 0),
           (3, 3),
           (2, -1),
           (2, -4),
           (1, -3),
       ]
       # the convex set of points is
       # [(0, 0), (0, 3), (1, -3), (2, -4), (3, 0), (3, 3)]
       lowercase__:    List[str]        =      convex_hull_bf(snake_case_		)

       lowercase__:    Union[str, Any]        =      convex_hull_recursive(snake_case_		)
       assert results_bf == results_recursive

       lowercase__:    Optional[Any]        =      convex_hull_melkman(snake_case_		)
       assert results_bf == results_melkman

       print(snake_case_		)


if __name__ == "__main__":
  main()














"""simple docstring"""





class 		_a       :





   '''simple docstring'''






   def __init__(   self			,				UpperCAmelCase_=None)	-> List[Any]:




          '''simple docstring'''




          lowercase__:    str        =      data
          lowercase__:    Any        =      None





   def __repr__(   self)	-> Optional[Any]:




          '''simple docstring'''




          lowercase__:    Any        =      ""
          lowercase__:    Dict        =      self
          while temp:
                 string_rep += F"""<{temp.data}> ---> """
                 lowercase__:    Tuple        =      temp.next
          string_rep += "<END>"
          return string_rep







def     A(  snake_case_		):
       """simple docstring"""





       if not elements_list:
              raise Exception("The Elements List is empty"		)

       # Set first element as Head
       lowercase__:    Any        =      Node(elements_list[0]		)
       lowercase__:    Optional[int]        =      head
       # Loop through elements from position 1
       for data in elements_list[1:]:
              lowercase__:    Dict        =      Node(snake_case_		)
              lowercase__:    Dict        =      current.next
       return head


UpperCamelCase			=      [1, 3, 5, 32, 44, 12, 43]
print(F"List: {list_data}")
print("""Creating Linked List from List.""")
UpperCamelCase			=      make_linked_list(list_data)
print("""Linked List:""")







print(linked_list)






