
"""simple docstring"""

from __future__ import annotations

from collections.abc import Iterable
class        _UpperCAmelCase							:




      def __init__(							self				,    lowercase_				,    lowercase_							)  ->   Union[str, Any]:
          UpperCAmelCase       =   float(lowercase_							), float(lowercase_							)




      def __eq__(							self				,    lowercase_							)  ->   Optional[Any]:
          return self.x == other.x and self.y == other.y




      def __ne__(							self				,    lowercase_							)  ->   Tuple:
          return not self == other




      def __gt__(							self				,    lowercase_							)  ->   Dict:




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




      def __lt__(							self				,    lowercase_							)  ->   Optional[Any]:
          return not self > other




      def __ge__(							self				,    lowercase_							)  ->   Optional[int]:




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




      def __le__(							self				,    lowercase_							)  ->   Dict:




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




      def __repr__(							self							)  ->   str:
          return F"({self.x}, {self.y})"




      def __hash__(							self							)  ->   List[str]:
          return hash(self.x							)




def       lowercase__  (   lowerCAmelCase			:   list[Point] | list[list[float]] | Iterable[list[float]]	,			)	-> Optional[Any]:






    """simple docstring"""




    UpperCAmelCase       =   []
    if list_of_tuples:
        for p in list_of_tuples:
            if isinstance(SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	):
                points.append(SCREAMING_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       lowercase__  (   lowerCAmelCase			:   list[Point] | list[list[float]]	)	-> Tuple:






    """simple docstring"""




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

    if not points:
        UpperCAmelCase       =   F"Expecting a list of points but got {points}"
        raise ValueError(SCREAMING_SNAKE_CASE__	)

    return _construct_points(SCREAMING_SNAKE_CASE__	)




def       lowercase__  (   lowerCAmelCase			:   Point	,			lowerCAmelCase			:   Point	,			lowerCAmelCase			:   Point	)	-> int:






    """simple docstring"""




    UpperCAmelCase       =   (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       lowercase__  (   lowerCAmelCase			:   list[Point]	)	-> List[Any]:






    """simple docstring"""




    UpperCAmelCase       =   sorted(_validate_input(SCREAMING_SNAKE_CASE__	)	)
    UpperCAmelCase       =   len(SCREAMING_SNAKE_CASE__	)
    UpperCAmelCase       =   set()

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

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

                if points_left_of_ij and points_right_of_ij:
                    UpperCAmelCase       =   False
                    break

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

    return sorted(SCREAMING_SNAKE_CASE__	)




def       lowercase__  (   lowerCAmelCase			:   list[Point]	)	-> Tuple:






    """simple docstring"""




    UpperCAmelCase       =   sorted(_validate_input(SCREAMING_SNAKE_CASE__	)	)
    UpperCAmelCase       =   len(SCREAMING_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

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

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

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

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

    _construct_hull(SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	)
    _construct_hull(SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	)

    return sorted(SCREAMING_SNAKE_CASE__	)




def       lowercase__  (   lowerCAmelCase			:   list[Point]	,			lowerCAmelCase			:   Point	,			lowerCAmelCase			:   Point	,			lowerCAmelCase			:   set[Point]	)	-> Optional[int]:






    """simple docstring"""




    if points:
        UpperCAmelCase       =   None
        UpperCAmelCase       =   float('-inf'	)
        UpperCAmelCase       =   []

        for p in points:
            UpperCAmelCase       =   _det(SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	,			SCREAMING_SNAKE_CASE__	)

            if det > 0:
                candidate_points.append(SCREAMING_SNAKE_CASE__	)

                if det > extreme_point_distance:
                    UpperCAmelCase       =   det
                    UpperCAmelCase       =   p

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




def       lowercase__  (   lowerCAmelCase			:   list[Point]	)	-> Any:






    """simple docstring"""




    UpperCAmelCase       =   sorted(_validate_input(SCREAMING_SNAKE_CASE__	)	)
    UpperCAmelCase       =   len(SCREAMING_SNAKE_CASE__	)

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

    for j in range(SCREAMING_SNAKE_CASE__	,			SCREAMING_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(SCREAMING_SNAKE_CASE__	) > 3 else convex_hull	)




def       lowercase__  (   )	-> Optional[Any]:






    """simple docstring"""




    UpperCAmelCase       =   [
        (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)]
    UpperCAmelCase       =   convex_hull_bf(SCREAMING_SNAKE_CASE__	)

    UpperCAmelCase       =   convex_hull_recursive(SCREAMING_SNAKE_CASE__	)
    assert results_bf == results_recursive

    UpperCAmelCase       =   convex_hull_melkman(SCREAMING_SNAKE_CASE__	)
    assert results_bf == results_melkman

    print(SCREAMING_SNAKE_CASE__	)


if __name__ == "__main__":
   main()





"""simple docstring"""

from __future__ import annotations

SCREAMING_SNAKE_CASE_         =  '''Alexander Joslin'''




def       lowercase__  (   lowerCAmelCase			:   int	)	-> int:






    """simple docstring"""




    if number <= 0:
        UpperCAmelCase       =   F"n must be greater than 0. Got n = {number}"
        raise ValueError(lowerCAmelCase	)

    UpperCAmelCase       =   [number + 1] * (number + 1)

    # starting position
    UpperCAmelCase       =   0
    for i in range(1	,			lowerCAmelCase	):
        UpperCAmelCase       =   min(table[i + 1]	,			table[i] + 1	)
        # check if out of bounds
        if i * 2 <= number:
            UpperCAmelCase       =   min(table[i * 2]	,			table[i] + 1	)
        # check if out of bounds
        if i * 3 <= number:
            UpperCAmelCase       =   min(table[i * 3]	,			table[i] + 1	)
    return table[number]


if __name__ == "__main__":
   import doctest

   doctest.testmod()



