





from __future__ import annotations

from scipy.special import comb  # type: ignore







class 				__lowerCAmelCase       :







      def __init__(    self		,      lowerCAmelCase__							)   ->						List[Any]:



        '''simple docstring'''




        a__	:					Any										=list_of_points
        # Degree determines the flexibility of the curve.
        # Degree = 1 will produce a straight line.
        a__	:					List[Any]										=len(lowerCAmelCase__							) - 1







      def    _lowercase  (    self		,      lowerCAmelCase__							)   ->						list[float]:



        '''simple docstring'''




        assert 0 <= t <= 1, "Time t must be between 0 and 1."
        a__	:					list[float]										=[]
        for i in range(len(self.list_of_points							)							):
          # basis function for each i
          output_values.append(
              comb(self.degree		,      lowerCAmelCase__							) * ((1 - t) ** (self.degree - i)) * (t**i)							)
        # the basis must sum up to 1 for it to produce a valid Bezier curve.
        assert round(sum(lowerCAmelCase__							)		,      5							) == 1
        return output_values







      def    _lowercase  (    self		,      lowerCAmelCase__							)   ->						tuple[float, float]:



        '''simple docstring'''




        assert 0 <= t <= 1, "Time t must be between 0 and 1."

        a__	:					Optional[Any]										=self.basis_function(lowerCAmelCase__							)
        a__	:					Dict										=0.0
        a__	:					Dict										=0.0
        for i in range(len(self.list_of_points							)							):
          # For all points, sum up the product of i-th basis function and i-th point.
          x += basis_function[i] * self.list_of_points[i][0]
          y += basis_function[i] * self.list_of_points[i][1]
        return (x, y)







      def    _lowercase  (    self		,      lowerCAmelCase__ = 0.01							)   ->						Any:



        '''simple docstring'''




        from matplotlib import pyplot as plt  # type: ignore

        a__	:					list[float]										=[]  # x coordinates of points to plot
        a__	:					list[float]										=[]  # y coordinates of points to plot

        a__	:					Union[str, Any]										=0.0
        while t <= 1:
          a__	:					Optional[int]										=self.bezier_curve_function(lowerCAmelCase__							)
          to_plot_x.append(value[0]							)
          to_plot_y.append(value[1]							)
          t += step_size

        a__	:					List[Any]										=[i[0] for i in self.list_of_points]
        a__	:					Dict										=[i[1] for i in self.list_of_points]

        plt.plot(
            lowerCAmelCase__		,      lowerCAmelCase__		,      color="blue"		,      label="Curve of Degree " + str(self.degree							)		,      )
        plt.scatter(lowerCAmelCase__		,      lowerCAmelCase__		,      color="red"		,      label="Control Points"							)
        plt.legend()
        plt.show()


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     BezierCurve([(1, 2), (3, 5)]).plot_curve()  # degree 1
     BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve()  # degree 2
     BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve()  # degree 3









# A Bipartite Graph is a graph whose vertices can be divided into two independent sets,
# U and V such that every edge (u, v) either connects a vertex from U to V or a vertex
# from V to U. In other words, for every edge (u, v), either u belongs to U and v to V,
# or u belongs to V and v to U. We can also say that there is no edge that connects
# vertices of same set.






def        UpperCamelCase_(							_snake_case   :  Tuple	):




			"""simple docstring"""






			__a        =[False] * len(_snake_case	)
			__a        =[-1] * len(_snake_case	)

			def dfs(_snake_case   :  Dict       ,     _snake_case   :  Any	):
						__a        =True
						__a        =c
						for u in graph[v]:
									if not visited[u]:
												dfs(_snake_case       ,     1 - c	)

			for i in range(len(_snake_case	)	):
						if not visited[i]:
									dfs(_snake_case       ,     0	)

			for i in range(len(_snake_case	)	):
						for j in graph[i]:
									if color[i] == color[j]:
												return False

			return True


# Adjacency list of graph
_lowerCAmelCase					:			int  				=			{0: [1, 3], 1: [0, 2], 2: [1, 3], 3: [0, 2], 4: []}


print(check_bipartite_dfs(graph))






