
from __future__ import annotations

from collections.abc import Iterable




class 	UpperCAmelCase_      :


  '''simple docstring'''







  def __init__(			self      :			Union[str, Any]      ,							UpperCamelCase__      :			Dict      ,							UpperCamelCase__      :			int   )      ->     Dict:







    """simple docstring"""






    __magic_name__					,   __magic_name__    					=	float(UpperCamelCase__   ), float(UpperCamelCase__   )
  def __eq__(			self      :			str      ,							UpperCamelCase__      :			Any   )      ->     Optional[int]:







    """simple docstring"""






    return self.x == other.x and self.y == other.y
  def __ne__(			self      :			int      ,							UpperCamelCase__      :			Any   )      ->     str:







    """simple docstring"""






    return not self == other
  def __gt__(			self      :			List[str]      ,							UpperCamelCase__      :			Tuple   )      ->     List[Any]:







    """simple docstring"""






    if self.x > other.x:
      return True
    elif self.x == other.x:
      return self.y > other.y
    return False
  def __lt__(			self      :			Any      ,							UpperCamelCase__      :			int   )      ->     Optional[Any]:







    """simple docstring"""






    return not self > other
  def __ge__(			self      :			Optional[Any]      ,							UpperCamelCase__      :			Union[str, Any]   )      ->     int:







    """simple docstring"""






    if self.x > other.x:
      return True
    elif self.x == other.x:
      return self.y >= other.y
    return False
  def __le__(			self      :			Any      ,							UpperCamelCase__      :			Tuple   )      ->     Tuple:







    """simple docstring"""






    if self.x < other.x:
      return True
    elif self.x == other.x:
      return self.y <= other.y
    return False
  def __repr__(			self      :			Union[str, Any]   )      ->     int:







    """simple docstring"""






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





  def __hash__(			self      :			List[Any]   )      ->     Dict:







    """simple docstring"""






    return hash(self.x   )







def a__   (	A_,					):






  '''simple docstring'''





  __magic_name__    					=	[]
  if list_of_tuples:
    for p in list_of_tuples:
      if isinstance(A_,					A_ ):
        points.append(A_ )
      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__   (	A_ ):






  '''simple docstring'''





  if not hasattr(A_,					"""__iter__""" ):
    __magic_name__    					=	f'''Expecting an iterable object but got an non-iterable type {points}'''
    raise ValueError(A_ )

  if not points:
    __magic_name__    					=	f'''Expecting a list of points but got {points}'''
    raise ValueError(A_ )

  return _construct_points(A_ )

def a__   (	A_,					A_,					A_ ):






  '''simple docstring'''





  __magic_name__    					=	(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__   (	A_ ):






  '''simple docstring'''





  __magic_name__    					=	sorted(_validate_input(A_ ) )
  __magic_name__    					=	len(A_ )
  __magic_name__    					=	set()

  for i in range(n - 1 ):
    for j in range(i + 1,					A_ ):
      __magic_name__    					=	__magic_name__    					=	False
      __magic_name__    					=	True
      for k in range(A_ ):
        if k != i and k != j:
          __magic_name__    					=	_det(points[i],					points[j],					points[k] )

          if det_k > 0:
            __magic_name__    					=	True
          elif det_k < 0:
            __magic_name__    					=	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]:
              __magic_name__    					=	False
              break

        if points_left_of_ij and points_right_of_ij:
          __magic_name__    					=	False
          break

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

  return sorted(A_ )

def a__   (	A_ ):






  '''simple docstring'''





  __magic_name__    					=	sorted(_validate_input(A_ ) )
  __magic_name__    					=	len(A_ )

  # 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

  __magic_name__    					=	points[0]
  __magic_name__    					=	points[n - 1]

  __magic_name__    					=	{left_most_point, right_most_point}
  __magic_name__    					=	[]
  __magic_name__    					=	[]

  for i in range(1,					n - 1 ):
    __magic_name__    					=	_det(A_,					A_,					points[i] )

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

  _construct_hull(A_,					A_,					A_,					A_ )
  _construct_hull(A_,					A_,					A_,					A_ )

  return sorted(A_ )

def a__   (	A_,					A_,					A_,					A_ ):






  '''simple docstring'''





  if points:
    __magic_name__    					=	None
    __magic_name__    					=	float("""-inf""" )
    __magic_name__    					=	[]

    for p in points:
      __magic_name__    					=	_det(A_,					A_,					A_ )

      if det > 0:
        candidate_points.append(A_ )

        if det > extreme_point_distance:
          __magic_name__    					=	det
          __magic_name__    					=	p

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

def a__   (	A_ ):






  '''simple docstring'''





  __magic_name__    					=	sorted(_validate_input(A_ ) )
  __magic_name__    					=	len(A_ )

  __magic_name__    					=	points[:2]
  for i in range(2,					A_ ):
    __magic_name__    					=	_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:
      __magic_name__    					=	points[i]
  i += 1

  for j in range(A_,					A_ ):
    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(A_ ) > 3 else convex_hull )

def a__   (	):






  '''simple docstring'''





  __magic_name__    					=	[
      (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)]
  __magic_name__    					=	convex_hull_bf(A_ )

  __magic_name__    					=	convex_hull_recursive(A_ )
  assert results_bf == results_recursive

  __magic_name__    					=	convex_hull_melkman(A_ )
  assert results_bf == results_melkman

  print(A_ )


if __name__ == "__main__":
    main()



def a__   (	A_ ):






  '''simple docstring'''





  __magic_name__    					=	str(A_ ).strip()
  if not bin_string:
    raise ValueError("""Empty string was passed to the function""" )
  __magic_name__    					=	bin_string[0] == """-"""
  if is_negative:
    __magic_name__    					=	bin_string[1:]
  if not all(char in """01""" for char in bin_string ):
    raise ValueError("""Non-binary value was passed to the function""" )
  __magic_name__    					=	0
  for char in bin_string:
    __magic_name__    					=	2 * decimal_number + int(A_ )
  return -decimal_number if is_negative else decimal_number


if __name__ == "__main__":
    from doctest import testmod

    testmod()

