"""
This is a pure Python implementation of the merge-insertion sort algorithm
Source: https://en.wikipedia.org/wiki/Graham_scan

For doctests run following command:
python3 -m doctest -v graham_scan.py
"""

from __future__ import annotations

from collections import deque
from enum import Enum
from math import atan2, degrees
from sys import maxsize


# traversal from the lowest and the most left point in anti-clockwise direction
# if direction gets right, the previous point is not the convex hull.
class Direction(Enum):
    left = 1
    straight = 2
    right = 3

    def __repr__(self):
        return f"{self.__class__.__name__}.{self.name}"


def angle_comparer(point: tuple[int, int], minx: int, miny: int) -> float:
    """Return the angle toward to point from (minx, miny)

    :param point: The target point
           minx: The starting point's x
           miny: The starting point's y
    :return: the angle

    Examples:
    >>> angle_comparer((1,1), 0, 0)
    45.0

    >>> angle_comparer((100,1), 10, 10)
    -5.710593137499642

    >>> angle_comparer((5,5), 2, 3)
    33.690067525979785
    """
    # sort the points accorgind to the angle from the lowest and the most left point
    x, y = point
    return degrees(atan2(y - miny, x - minx))


def check_direction(
    starting: tuple[int, int], via: tuple[int, int], target: tuple[int, int]
) -> Direction:
    """Return the direction toward to the line from via to target from starting

    :param starting: The starting point
           via: The via point
           target: The target point
    :return: the Direction

    Examples:
    >>> check_direction((1,1), (2,2), (3,3))
    Direction.straight

    >>> check_direction((60,1), (-50,199), (30,2))
    Direction.left

    >>> check_direction((0,0), (5,5), (10,0))
    Direction.right
    """
    x0, y0 = starting
    x1, y1 = via
    x2, y2 = target
    via_angle = degrees(atan2(y1 - y0, x1 - x0))
    via_angle %= 360
    target_angle = degrees(atan2(y2 - y0, x2 - x0))
    target_angle %= 360
    # t-
    #  \ \
    #   \ v
    #    \|
    #     s
    # via_angle is always lower than target_angle, if direction is left.
    # If they are same, it means they are on a same line of convex hull.
    if target_angle > via_angle:
        return Direction.left
    elif target_angle == via_angle:
        return Direction.straight
    else:
        return Direction.right


def graham_scan(points: list[tuple[int, int]]) -> list[tuple[int, int]]:
    """Pure implementation of graham scan algorithm in Python

    :param points: The unique points on coordinates.
    :return: The points on convex hell.

    Examples:
    >>> graham_scan([(9, 6), (3, 1), (0, 0), (5, 5), (5, 2), (7, 0), (3, 3), (1, 4)])
    [(0, 0), (7, 0), (9, 6), (5, 5), (1, 4)]

    >>> graham_scan([(0, 0), (1, 0), (1, 1), (0, 1)])
    [(0, 0), (1, 0), (1, 1), (0, 1)]

    >>> graham_scan([(0, 0), (1, 1), (2, 2), (3, 3), (-1, 2)])
    [(0, 0), (1, 1), (2, 2), (3, 3), (-1, 2)]

    >>> graham_scan([(-100, 20), (99, 3), (1, 10000001), (5133186, -25), (-66, -4)])
    [(5133186, -25), (1, 10000001), (-100, 20), (-66, -4)]
    """

    if len(points) <= 2:
        # There is no convex hull
        raise ValueError("graham_scan: argument must contain more than 3 points.")
    if len(points) == 3:
        return points
    # find the lowest and the most left point
    minidx = 0
    miny, minx = maxsize, maxsize
    for i, point in enumerate(points):
        x = point[0]
        y = point[1]
        if y < miny:
            miny = y
            minx = x
            minidx = i
        if y == miny:
            if x < minx:
                minx = x
                minidx = i

    # remove the lowest and the most left point from points for preparing for sort
    points.pop(minidx)

    sorted_points = sorted(points, key=lambda point: angle_comparer(point, minx, miny))
    # This insert actually costs complexity,
    # and you should instead add (minx, miny) into stack later.
    # I'm using insert just for easy understanding.
    sorted_points.insert(0, (minx, miny))

    stack: deque[tuple[int, int]] = deque()
    stack.append(sorted_points[0])
    stack.append(sorted_points[1])
    stack.append(sorted_points[2])
    # In any ways, the first 3 points line are towards left.
    # Because we sort them the angle from minx, miny.
    current_direction = Direction.left

    for i in range(3, len(sorted_points)):
        while True:
            starting = stack[-2]
            via = stack[-1]
            target = sorted_points[i]
            next_direction = check_direction(starting, via, target)

            if next_direction == Direction.left:
                current_direction = Direction.left
                break
            if next_direction == Direction.straight:
                if current_direction == Direction.left:
                    # We keep current_direction as left.
                    # Because if the straight line keeps as straight,
                    # we want to know if this straight line is towards left.
                    break
                elif current_direction == Direction.right:
                    # If the straight line is towards right,
                    # every previous points on those straigh line is not convex hull.
                    stack.pop()
            if next_direction == Direction.right:
                stack.pop()
        stack.append(sorted_points[i])
    return list(stack)