






from __future__ import annotations

from collections.abc import Callable

A__	:			Dict			   = list[list[float | int]]






def   a			(						lowerCamelCase_		,   lowerCamelCase_				):






				'''simple docstring'''
				lowercase__    =			len(SCREAMING_SNAKE_CASE_				)
				lowercase__    =			[[0 for _ in range(size + 1				)] for _ in range(SCREAMING_SNAKE_CASE_				)]
				lowercase__    =			42
				lowercase__    =			42
				lowercase__    =			42
				lowercase__    =			42
				lowercase__    =			42
				lowercase__    =			42

				for row in range(SCREAMING_SNAKE_CASE_				):
								for col in range(SCREAMING_SNAKE_CASE_				):
												lowercase__    =			matrix[row][col]

								lowercase__    =			vector[row][0]

				lowercase__    =			0
				lowercase__    =			0
				while row < size and col < size:
								# pivoting
								lowercase__    =			max((abs(augmented[rowa][col]				), rowa) for rowa in range(SCREAMING_SNAKE_CASE_		,   SCREAMING_SNAKE_CASE_				)				)[
								    1
								]
								if augmented[pivot_row][col] == 0:
												col += 1
												continue
								else:
												lowercase__ ,						lowercase__    =			augmented[pivot_row], augmented[row]

								for rowa in range(row + 1		,   SCREAMING_SNAKE_CASE_				):
												lowercase__    =			augmented[rowa][col] / augmented[row][col]
												lowercase__    =			0
												for cola in range(col + 1		,   size + 1				):
																augmented[rowa][cola] -= augmented[row][cola] * ratio

								row += 1
								col += 1

				# back substitution
				for col in range(1		,   SCREAMING_SNAKE_CASE_				):
								for row in range(SCREAMING_SNAKE_CASE_				):
												lowercase__    =			augmented[row][col] / augmented[col][col]
												for cola in range(SCREAMING_SNAKE_CASE_		,   size + 1				):
																augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
				return [
				    [round(augmented[row][size] / augmented[row][row]		,   10				)] for row in range(SCREAMING_SNAKE_CASE_				)
				]






def   a			(						lowerCamelCase_				):






				'''simple docstring'''
				lowercase__    =			len(SCREAMING_SNAKE_CASE_				)
				lowercase__    =			[[0 for _ in range(SCREAMING_SNAKE_CASE_				)] for _ in range(SCREAMING_SNAKE_CASE_				)]
				lowercase__    =			[[0] for _ in range(SCREAMING_SNAKE_CASE_				)]
				lowercase__    =			42
				lowercase__    =			42
				lowercase__    =			42
				lowercase__    =			42

				for x_val, y_val in enumerate(SCREAMING_SNAKE_CASE_				):
								for col in range(SCREAMING_SNAKE_CASE_				):
												lowercase__    =			(x_val + 1) ** (size - col - 1)
								lowercase__    =			y_val

				lowercase__    =			solve(SCREAMING_SNAKE_CASE_		,   SCREAMING_SNAKE_CASE_				)

				def interpolated_func(lowerCamelCase_				) -> int:
								return sum(
								    round(coeffs[x_val][0]				) * (var ** (size - x_val - 1))
								    for x_val in range(SCREAMING_SNAKE_CASE_				)				)

				return interpolated_func






def   a			(						lowerCamelCase_				):






				'''simple docstring'''
				return (
				    1
				    - variable
				    + variable**2
				    - variable**3
				    + variable**4
				    - variable**5
				    + variable**6
				    - variable**7
				    + variable**8
				    - variable**9
				    + variable**10
				)






def   a			(						lowerCamelCase_ = question_function		,   lowerCamelCase_ = 10				):






				'''simple docstring'''
				lowercase__    =			[func(SCREAMING_SNAKE_CASE_				) for x_val in range(1		,   order + 1				)]

				lowercase__    =			[
				    interpolate(data_points[:max_coeff]				) for max_coeff in range(1		,   order + 1				)
				]

				lowercase__    =			0
				lowercase__    =			42
				lowercase__    =			42

				for poly in polynomials:
								lowercase__    =			1
								while func(SCREAMING_SNAKE_CASE_				) == poly(SCREAMING_SNAKE_CASE_				):
												x_val += 1

								ret += poly(SCREAMING_SNAKE_CASE_				)

				return ret


if __name__ == "__main__":
							print(F"{solution() = }")










from queue import PriorityQueue
from typing import Any

import numpy as np







def 	_a      (						SCREAMING_SNAKE_CASE_							:				dict      ,      SCREAMING_SNAKE_CASE_							:				str      ,      SCREAMING_SNAKE_CASE_							:				set      ,      SCREAMING_SNAKE_CASE_							:				set      ,      SCREAMING_SNAKE_CASE_							:				dict      ,      SCREAMING_SNAKE_CASE_							:				dict      ,      SCREAMING_SNAKE_CASE_							:				PriorityQueue      ,      SCREAMING_SNAKE_CASE_							:				dict      ,      SCREAMING_SNAKE_CASE_							:				float | int      ,      ):
					for nxt, d in graph[v]:
										if nxt in visited_forward:
															continue
										__lowerCAmelCase				   =       cst_fwd.get(SCREAMING_SNAKE_CASE_      ,      np.inf     )
										__lowerCAmelCase				   =       cst_fwd[v] + d
										if new_cost_f < old_cost_f:
															queue.put((new_cost_f, nxt)     )
															__lowerCAmelCase				   =       new_cost_f
															__lowerCAmelCase				   =       v
										if nxt in visited_backward:
															if cst_fwd[v] + d + cst_bwd[nxt] < shortest_distance:
																				__lowerCAmelCase				   =       cst_fwd[v] + d + cst_bwd[nxt]
					return shortest_distance







def 	_a      (						SCREAMING_SNAKE_CASE_							:				str      ,      SCREAMING_SNAKE_CASE_							:				str      ,      SCREAMING_SNAKE_CASE_							:				dict      ,      SCREAMING_SNAKE_CASE_							:				dict     ):
					__lowerCAmelCase				   =       -1

					__lowerCAmelCase				   =       set()
					__lowerCAmelCase				   =       set()
					__lowerCAmelCase				   =       {source: 0}
					__lowerCAmelCase				   =       {destination: 0}
					__lowerCAmelCase				   =       {source: None}
					__lowerCAmelCase				   =       {destination: None}
					__lowerCAmelCase				   =       PriorityQueue()
					__lowerCAmelCase				   =       PriorityQueue()

					__lowerCAmelCase				   =       np.inf

					queue_forward.put((0, source)     )
					queue_backward.put((0, destination)     )

					if source == destination:
										return 0

					while not queue_forward.empty() and not queue_backward.empty():
										__lowerCAmelCase   ,							__lowerCAmelCase				   =       queue_forward.get()
										visited_forward.add(SCREAMING_SNAKE_CASE_     )

										__lowerCAmelCase   ,							__lowerCAmelCase				   =       queue_backward.get()
										visited_backward.add(SCREAMING_SNAKE_CASE_     )

										__lowerCAmelCase				   =       pass_and_relaxation(
										    SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      )

										__lowerCAmelCase				   =       pass_and_relaxation(
										    SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      SCREAMING_SNAKE_CASE_      ,      )

										if cst_fwd[v_fwd] + cst_bwd[v_bwd] >= shortest_distance:
															break

					if shortest_distance != np.inf:
										__lowerCAmelCase				   =       shortest_distance
					return shortest_path_distance


UpperCamelCase__    			=			{
    """B""": [["""C""", 1]],
    """C""": [["""D""", 1]],
    """D""": [["""F""", 1]],
    """E""": [["""B""", 1], ["""G""", 2]],
    """F""": [],
    """G""": [["""F""", 1]],
}
UpperCamelCase__    			=			{
    """B""": [["""E""", 1]],
    """C""": [["""B""", 1]],
    """D""": [["""C""", 1]],
    """F""": [["""D""", 1], ["""G""", 1]],
    """E""": [[None, np.inf]],
    """G""": [["""E""", 2]],
}

if __name__ == "__main__":
					import doctest

					doctest.testmod()




