





__A    :   Any          =       float("inf")

class      A_	:
	def __init__(   self							, _A		):






				'''simple docstring'''







				UpperCAmelCase       		=    [0] * n
				UpperCAmelCase       		=    [0] * n
				UpperCAmelCase       		=    [0] * n
				UpperCAmelCase       		=    [[] for _ in range(_A		)]
	def 					_lowercase    (   self							, _A							, _A							, _A							, _A=0		):






				'''simple docstring'''







				self.adj[a].append([b, len(self.adj[b]		), c, 0]		)
				self.adj[b].append([a, len(self.adj[a]		) - 1, rcap, 0]		)
	def 					_lowercase    (   self							, _A							, _A							, _A		):






				'''simple docstring'''







				if vertex == sink or not flow:
							return flow

				for i in range(self.ptr[vertex]							, len(self.adj[vertex]		)		):
							UpperCAmelCase       		=    self.adj[vertex][i]
							if self.lvl[e[0]] == self.lvl[vertex] + 1:
										UpperCAmelCase       		=    self.depth_first_search(e[0]							, _A							, min(_A							, e[2] - e[3]		)		)
										if p:
													self.adj[vertex][i][3] += p
													self.adj[e[0]][e[1]][3] -= p
													return p
							UpperCAmelCase       		=    self.ptr[vertex] + 1
				return 0







	def 					_lowercase    (   self							, _A							, _A		):






				'''simple docstring'''







				UpperCAmelCase		,      UpperCAmelCase       		=    0, source
				for l in range(3_1		):  # noqa: E741  l = 30 maybe faster for random data
							while True:
										UpperCAmelCase		,      UpperCAmelCase       		=    [0] * len(self.q		), [0] * len(self.q		)
										UpperCAmelCase		,      UpperCAmelCase		,      UpperCAmelCase       		=    0, 1, 1
										while qi < qe and not self.lvl[sink]:
													UpperCAmelCase       		=    self.q[qi]
													qi += 1
													for e in self.adj[v]:
																if not self.lvl[e[0]] and (e[2] - e[3]) >> (3_0 - l):
																			UpperCAmelCase       		=    e[0]
																			qe += 1
																			UpperCAmelCase       		=    self.lvl[v] + 1

										UpperCAmelCase       		=    self.depth_first_search(_A							, _A							, _A		)
										while p:
													flow += p
													UpperCAmelCase       		=    self.depth_first_search(_A							, _A							, _A		)

										if not self.lvl[sink]:
													break

				return flow
__A    :   Tuple          =       Dinic(10)
__A    :   str          =       0
__A    :   int          =       9
for vertex in range(1, 5):
				graph.add_edge(source, vertex, 1)
for vertex in range(5, 9):
				graph.add_edge(vertex, sink, 1)
for vertex in range(1, 5):
				graph.add_edge(vertex, vertex + 4, 1)

# Now we can know that is the maximum flow(source -> sink)
print(graph.max_flow(source, sink))










from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve





def 						__SCREAMING_SNAKE_CASE  (    UpperCamelCase__      ,      UpperCamelCase__      ,      UpperCamelCase__			)  ->    np.ndarray:







			'''simple docstring'''

			UpperCAmelCase       		=    np.stack((horizontal_flow, vertical_flow)      ,      2			)

			# Create a grid of all pixel coordinates and subtract the flow to get the
			# target pixels coordinates
			UpperCAmelCase       		=    np.stack(
			    np.meshgrid(np.arange(0      ,      image.shape[1]			)      ,      np.arange(0      ,      image.shape[0]			)			)      ,      2			)
			UpperCAmelCase       		=    np.round(grid - flow			).astype(np.intaa			)

			# Find the locations outside of the original image
			UpperCAmelCase       		=    (grid < 0) | (grid >= np.array([image.shape[1], image.shape[0]]			))
			UpperCAmelCase       		=    0

			UpperCAmelCase       		=    image[grid[:, :, 1], grid[:, :, 0]]

			# Set pixels at invalid locations to 0
			UpperCAmelCase       		=    0

			return warped





def 						__SCREAMING_SNAKE_CASE  (    UpperCamelCase__      ,      UpperCamelCase__      ,      UpperCamelCase__      ,      UpperCamelCase__ = None      ,      )  ->    tuple[np.ndarray, np.ndarray]:







			'''simple docstring'''

			if alpha is None:
						UpperCAmelCase       		=    0.1

			# Initialize flow
			UpperCAmelCase       		=    np.zeros_like(UpperCamelCase__			)
			UpperCAmelCase       		=    np.zeros_like(UpperCamelCase__			)

			# Prepare kernels for the calculation of the derivatives and the average velocity
			UpperCAmelCase       		=    np.array([[-1, 1], [-1, 1]]			) * 0.25
			UpperCAmelCase       		=    np.array([[-1, -1], [1, 1]]			) * 0.25
			UpperCAmelCase       		=    np.array([[1, 1], [1, 1]]			) * 0.25
			UpperCAmelCase       		=    np.array(
			    [[1 / 12, 1 / 6, 1 / 12], [1 / 6, 0, 1 / 6], [1 / 12, 1 / 6, 1 / 12]]			)

			# Iteratively refine the flow
			for _ in range(UpperCamelCase__			):
						UpperCAmelCase       		=    warp(UpperCamelCase__      ,      UpperCamelCase__      ,      UpperCamelCase__			)
						UpperCAmelCase       		=    convolve(UpperCamelCase__      ,      UpperCamelCase__			) + convolve(UpperCamelCase__      ,      UpperCamelCase__			)
						UpperCAmelCase       		=    convolve(UpperCamelCase__      ,      UpperCamelCase__			) + convolve(UpperCamelCase__      ,      UpperCamelCase__			)
						UpperCAmelCase       		=    convolve(UpperCamelCase__      ,      UpperCamelCase__			) + convolve(UpperCamelCase__      ,      -kernel_t			)

						UpperCAmelCase       		=    convolve(UpperCamelCase__      ,      UpperCamelCase__			)
						UpperCAmelCase       		=    convolve(UpperCamelCase__      ,      UpperCamelCase__			)

						# This updates the flow as proposed in the paper (Step 12)
						UpperCAmelCase       		=    (
						    derivative_x * avg_horizontal_velocity
						    + derivative_y * avg_vertical_velocity
						    + derivative_t
						)
						UpperCAmelCase       		=    update / (alpha**2 + derivative_x**2 + derivative_y**2)

						UpperCAmelCase       		=    avg_horizontal_velocity - derivative_x * update
						UpperCAmelCase       		=    avg_vertical_velocity - derivative_y * update

			return horizontal_flow, vertical_flow


if __name__ == "__main__":
				import doctest

				doctest.testmod()



