






"""simple docstring"""

from __future__ import annotations

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


class 	__lowerCamelCase       (				A__    ):





		'''simple docstring'''






		a_			:       List[str]      =							1
		a_			:       Optional[Any]      =							2
		a_			:       Dict      =							3





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







def 	__lowerCamelCase			(						__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase )		->					float:

		"""simple docstring"""







		lowerCAmelCase_       ,			lowerCAmelCase_     :							Union[str, Any]             =    point
		return degrees(atana(y - miny   , x - minx ) )

def 	__lowerCamelCase			(						__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase )		->					Direction:

		"""simple docstring"""







		lowerCAmelCase_       ,			lowerCAmelCase_     :							Tuple             =    starting
		lowerCAmelCase_       ,			lowerCAmelCase_     :							Any             =    via
		lowerCAmelCase_       ,			lowerCAmelCase_     :							str             =    target
		lowerCAmelCase_     :							List[Any]             =    degrees(atana(ya - ya   , xa - xa ) )
		via_angle %= 360
		lowerCAmelCase_     :							Optional[Any]             =    degrees(atana(ya - ya   , xa - xa ) )
		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 	__lowerCamelCase			(						__UpperCamelCase )		->					list[tuple[int, int]]:

		"""simple docstring"""







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

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

		lowerCAmelCase_     :							Optional[int]             =    sorted(__UpperCamelCase   , key=lambda __UpperCamelCase : angle_comparer(__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase ) )
		# 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) )

		lowerCAmelCase_     :							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.
		lowerCAmelCase_     :							Dict             =    Direction.left

		for i in range(3   , len(__UpperCamelCase ) ):
				while True:
						lowerCAmelCase_     :							Any             =    stack[-2]
						lowerCAmelCase_     :							int             =    stack[-1]
						lowerCAmelCase_     :							Dict             =    sorted_points[i]
						lowerCAmelCase_     :							List[Any]             =    check_direction(__UpperCamelCase   , __UpperCamelCase   , __UpperCamelCase )

						if next_direction == Direction.left:
								lowerCAmelCase_     :							List[str]             =    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(__UpperCamelCase )








"""simple docstring"""

from __future__ import annotations

import random

# Adjacency list representation of this graph:
# https://en.wikipedia.org/wiki/File:Single_run_of_Karger%E2%80%99s_Mincut_algorithm.svg
lowercase__      =				{
    """1""": ["""2""", """3""", """4""", """5"""],
    """2""": ["""1""", """3""", """4""", """5"""],
    """3""": ["""1""", """2""", """4""", """5""", """10"""],
    """4""": ["""1""", """2""", """3""", """5""", """6"""],
    """5""": ["""1""", """2""", """3""", """4""", """7"""],
    """6""": ["""7""", """8""", """9""", """10""", """4"""],
    """7""": ["""6""", """8""", """9""", """10""", """5"""],
    """8""": ["""6""", """7""", """9""", """10"""],
    """9""": ["""6""", """7""", """8""", """10"""],
    """10""": ["""6""", """7""", """8""", """9""", """3"""],
}

def 	__lowerCamelCase			(						__UpperCamelCase )		->					set[tuple[str, str]]:

		"""simple docstring"""







		lowerCAmelCase_     :							Union[str, Any]             =    {node: {node} for node in graph}

		lowerCAmelCase_     :							List[str]             =    {node: graph[node][:] for node in graph}

		while len(__UpperCamelCase ) > 2:
				# Choose a random edge.
				lowerCAmelCase_     :							int             =    random.choice(list(graph_copy.keys() ) )
				lowerCAmelCase_     :							Any             =    random.choice(graph_copy[u] )

				# Contract edge (u, v) to new node uv
				lowerCAmelCase_     :							Union[str, Any]             =    u + v
				lowerCAmelCase_     :							List[Any]             =    list(set(graph_copy[u] + graph_copy[v] ) )
				uv_neighbors.remove(__UpperCamelCase )
				uv_neighbors.remove(__UpperCamelCase )
				lowerCAmelCase_     :							str             =    uv_neighbors
				for neighbor in uv_neighbors:
						graph_copy[neighbor].append(__UpperCamelCase )

				lowerCAmelCase_     :							Optional[int]             =    set(contracted_nodes[u].union(contracted_nodes[v] ) )

				# Remove nodes u and v.
				del graph_copy[u]
				del graph_copy[v]
				for neighbor in uv_neighbors:
						if u in graph_copy[neighbor]:
								graph_copy[neighbor].remove(__UpperCamelCase )
						if v in graph_copy[neighbor]:
								graph_copy[neighbor].remove(__UpperCamelCase )

    # Find cutset.
		lowerCAmelCase_     :							Union[str, Any]             =    [contracted_nodes[node] for node in graph_copy]
		return {
		    (node, neighbor)
		    for node in groups[0]
		    for neighbor in graph[node]
		    if neighbor in groups[1]
		}


if __name__ == "__main__":
					print(partition_graph(TEST_GRAPH))
