

from collections.abc import Sequence
from queue import Queue


class 			A						:







    """simple docstring"""





    def __init__(				self			:	Union[str, Any],lowercase_			:	Optional[Any],lowercase_			:	Any,lowercase_			:	List[Any],lowercase_			:	str=None,lowercase_			:	List[str]=None		)->   Dict:





        '''simple docstring'''







        A__   				=       start
        A__   				=       end
        A__   				=       val
        A__   				=       (start + end) // 2
        A__   				=       left
        A__   				=       right


    def __repr__(				self			:	int		)->   int:





        '''simple docstring'''







        return F'SegmentTreeNode(start={self.start}, end={self.end}, val={self.val})'




class 			A						:







    """simple docstring"""





    def __init__(				self			:	Optional[int],lowercase_			:	Sequence,lowercase_			:	Optional[int]		)->   Any:





        '''simple docstring'''







        A__   				=       collection
        A__   				=       function
        if self.collection:
            A__   				=       self._build_tree(0,len(lowercase_		) - 1		)


    def 						snake_case__					(				self			:	Union[str, Any],lowercase_			:	Optional[int],lowercase_			:	int		)->   Any:





        '''simple docstring'''







        self._update_tree(self.root,lowercase_,lowercase_		)


    def 						snake_case__					(				self			:	List[str],lowercase_			:	Tuple,lowercase_			:	Tuple		)->   Any:





        '''simple docstring'''







        return self._query_range(self.root,lowercase_,lowercase_		)


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





        '''simple docstring'''







        if start == end:
            return SegmentTreeNode(lowercase_,lowercase_,self.collection[start]		)
        A__   				=       (start + end) // 2
        A__   				=       self._build_tree(lowercase_,lowercase_		)
        A__   				=       self._build_tree(mid + 1,lowercase_		)
        return SegmentTreeNode(lowercase_,lowercase_,self.fn(left.val,right.val		),lowercase_,lowercase_		)


    def 						snake_case__					(				self			:	Optional[Any],lowercase_			:	int,lowercase_			:	Union[str, Any],lowercase_			:	Optional[Any]		)->   int:





        '''simple docstring'''







        if node.start == i and node.end == i:
            A__   				=       val
            return
        if i <= node.mid:
            self._update_tree(node.left,lowercase_,lowercase_		)
        else:
            self._update_tree(node.right,lowercase_,lowercase_		)
        A__   				=       self.fn(node.left.val,node.right.val		)


    def 						snake_case__					(				self			:	str,lowercase_			:	Union[str, Any],lowercase_			:	Dict,lowercase_			:	Dict		)->   int:





        '''simple docstring'''







        if node.start == i and node.end == j:
            return node.val

        if i <= node.mid:
            if j <= node.mid:
                # range in left child tree
                return self._query_range(node.left,lowercase_,lowercase_		)
            else:
                # range in left child tree and right child tree
                return self.fn(
                    self._query_range(node.left,lowercase_,node.mid		),self._query_range(node.right,node.mid + 1,lowercase_		),)
        else:
            # range in right child tree
            return self._query_range(node.right,lowercase_,lowercase_		)


    def 						snake_case__					(				self			:	Optional[Any]		)->   Optional[int]:





        '''simple docstring'''







        if self.root is not None:
            A__   				=       Queue()
            queue.put(self.root		)
            while not queue.empty():
                A__   				=       queue.get()
                yield node

                if node.left is not None:
                    queue.put(node.left		)

                if node.right is not None:
                    queue.put(node.right		)


if __name__ == "__main__":
     import operator

     for fn in [operator.add, max, min]:
          print("*" * 50)
          lowercase_			=  SegmentTree([2, 1, 5, 3, 4], fn)
          for node in arr.traverse():
               print(node)
          print()

          arr.update(1, 5)
          for node in arr.traverse():
               print(node)
          print()

          print(arr.query_range(3, 4))  # 7
          print(arr.query_range(2, 2))  # 5
          print(arr.query_range(1, 3))  # 13
          print()




from __future__ import annotations

from scipy.special import comb  # type: ignore


class 			A						:







    """simple docstring"""





    def __init__(				self			:	Any,lowercase_			:	list[tuple[float, float]]		)->   Optional[int]:





        '''simple docstring'''







        A__   				=       list_of_points
        # Degree determines the flexibility of the curve.
        # Degree = 1 will produce a straight line.
        A__   				=       len(lowercase_		) - 1


    def 						snake_case__					(				self			:	List[Any],lowercase_			:	float		)->   list[float]:





        '''simple docstring'''







        assert 0 <= t <= 1, "Time t must be between 0 and 1."
        A__   				=       []
        for i in range(len(self.list_of_points		)		):
            # basis function for each i
            output_values.append(
                comb(self.degree,lowercase_		) * ((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(lowercase_		),5		) == 1
        return output_values


    def 						snake_case__					(				self			:	str,lowercase_			:	float		)->   tuple[float, float]:





        '''simple docstring'''







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

        A__   				=       self.basis_function(lowercase_		)
        A__   				=       0.0
        A__   				=       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 						snake_case__					(				self			:	str,lowercase_			:	float = 0.01		)->   str:





        '''simple docstring'''







        from matplotlib import pyplot as plt  # type: ignore

        A__   				=       []  # x coordinates of points to plot
        A__   				=       []  # y coordinates of points to plot

        A__   				=       0.0
        while t <= 1:
            A__   				=       self.bezier_curve_function(lowercase_		)
            to_plot_x.append(value[0]		)
            to_plot_y.append(value[1]		)
            t += step_size

        A__   				=       [i[0] for i in self.list_of_points]
        A__   				=       [i[1] for i in self.list_of_points]

        plt.plot(
            lowercase_,lowercase_,color='blue',label='Curve of Degree ' + str(self.degree		),)
        plt.scatter(lowercase_,lowercase_,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

