




'''simple docstring'''



import torch
from torch import nn






class  UpperCamelCase__(     nn.Module ):




				def __init__(	self    :				List[str]   ,					lowerCAmelCase    :				Any   ,					lowerCAmelCase    :				List[str]   ,					lowerCAmelCase    :				Optional[Any]   ,					lowerCAmelCase    :				Dict   ,					lowerCAmelCase    :				Union[str, Any]=1   ,					lowerCAmelCase    :				Any=False   )->  Dict:

										"""simple docstring"""




										super().__init__()

										UpperCAmelCase         =			n_token
										UpperCAmelCase         =			d_embed
										UpperCAmelCase         =			d_proj

										UpperCAmelCase         =			cutoffs + [n_token]
										UpperCAmelCase         =			[0] + self.cutoffs
										UpperCAmelCase         =			div_val

										UpperCAmelCase         =			self.cutoffs[0]
										UpperCAmelCase         =			len(self.cutoffs   ) - 1
										UpperCAmelCase         =			self.shortlist_size + self.n_clusters

										if self.n_clusters > 0:
																UpperCAmelCase         =			nn.Parameter(torch.zeros(self.n_clusters   ,					self.d_embed   )   )
																UpperCAmelCase         =			nn.Parameter(torch.zeros(self.n_clusters   )   )

										UpperCAmelCase         =			nn.ModuleList()
										UpperCAmelCase         =			nn.ParameterList()

										if div_val == 1:
																for i in range(len(self.cutoffs   )   ):
																						if d_proj != d_embed:
																												self.out_projs.append(nn.Parameter(torch.FloatTensor(__UpperCamelCase   ,					__UpperCamelCase   )   )   )
																						else:
																												self.out_projs.append(__UpperCamelCase   )

																self.out_layers.append(nn.Linear(__UpperCamelCase   ,					__UpperCamelCase   )   )
										else:
																for i in range(len(self.cutoffs   )   ):
																						UpperCAmelCase       ,       UpperCAmelCase         =			self.cutoff_ends[i], self.cutoff_ends[i + 1]
																						UpperCAmelCase         =			d_embed // (div_val**i)

																						self.out_projs.append(nn.Parameter(torch.FloatTensor(__UpperCamelCase   ,					__UpperCamelCase   )   )   )

																						self.out_layers.append(nn.Linear(__UpperCamelCase   ,					r_idx - l_idx   )   )

										UpperCAmelCase         =			keep_order




				def   a__(	self    :				int   ,					lowerCAmelCase    :				List[str]   ,					lowerCAmelCase    :				str   ,					lowerCAmelCase    :				List[Any]   ,					lowerCAmelCase    :				List[Any]   )->  Optional[int]:

										"""simple docstring"""




										if proj is None:
																UpperCAmelCase         =			nn.functional.linear(__UpperCamelCase   ,					__UpperCamelCase   ,					bias=__UpperCamelCase   )
										else:
																# if CUDA_MAJOR <= 9 and CUDA_MINOR <= 1:
																UpperCAmelCase         =			nn.functional.linear(__UpperCamelCase   ,					proj.t().contiguous()   )
																UpperCAmelCase         =			nn.functional.linear(__UpperCamelCase   ,					__UpperCamelCase   ,					bias=__UpperCamelCase   )
																# else:
																#     logit = torch.einsum('bd,de,ev->bv', (hidden, proj, weight.t()))
																#     if bias is not None:
																#         logit = logit + bias

										return logit




				def   a__(	self    :				Optional[int]   ,					lowerCAmelCase    :				Optional[Any]   ,					lowerCAmelCase    :				List[str]=None   ,					lowerCAmelCase    :				Union[str, Any]=False   )->  Any:

										"""simple docstring"""




										if labels is not None:
																# Shift so that tokens < n predict n
																UpperCAmelCase         =			hidden[..., :-1, :].contiguous()
																UpperCAmelCase         =			labels[..., 1:].contiguous()
																UpperCAmelCase         =			hidden.view(-1   ,					hidden.size(-1   )   )
																UpperCAmelCase         =			labels.view(-1   )
																if hidden.size(0   ) != labels.size(0   ):
																						raise RuntimeError('''Input and labels should have the same size in the batch dimension.'''   )
										else:
																UpperCAmelCase         =			hidden.view(-1   ,					hidden.size(-1   )   )

										if self.n_clusters == 0:
																UpperCAmelCase         =			self._compute_logit(__UpperCamelCase   ,					self.out_layers[0].weight   ,					self.out_layers[0].bias   ,					self.out_projs[0]   )
																if labels is not None:
																						UpperCAmelCase         =			labels != -100
																						UpperCAmelCase         =			torch.zeros_like(__UpperCamelCase   ,					dtype=hidden.dtype   ,					device=hidden.device   )
																						UpperCAmelCase         =			(
																						    -nn.functional.log_softmax(__UpperCamelCase   ,					dim=-1   )[mask].gather(1   ,					labels[mask].unsqueeze(1   )   ).squeeze(1   )
																						)
																else:
																						UpperCAmelCase         =			nn.functional.log_softmax(__UpperCamelCase   ,					dim=-1   )
										else:
																# construct weights and biases
																UpperCAmelCase       ,       UpperCAmelCase         =			[], []
																for i in range(len(self.cutoffs   )   ):
																						if self.div_val == 1:
																												UpperCAmelCase       ,       UpperCAmelCase         =			self.cutoff_ends[i], self.cutoff_ends[i + 1]
																												UpperCAmelCase         =			self.out_layers[0].weight[l_idx:r_idx]
																												UpperCAmelCase         =			self.out_layers[0].bias[l_idx:r_idx]
																						else:
																												UpperCAmelCase         =			self.out_layers[i].weight
																												UpperCAmelCase         =			self.out_layers[i].bias

																						if i == 0:
																												UpperCAmelCase         =			torch.cat([weight_i, self.cluster_weight]   ,					dim=0   )
																												UpperCAmelCase         =			torch.cat([bias_i, self.cluster_bias]   ,					dim=0   )

																						weights.append(__UpperCamelCase   )
																						biases.append(__UpperCamelCase   )

																UpperCAmelCase       ,       UpperCAmelCase       ,       UpperCAmelCase         =			weights[0], biases[0], self.out_projs[0]

																UpperCAmelCase         =			self._compute_logit(__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   )
																UpperCAmelCase         =			nn.functional.log_softmax(__UpperCamelCase   ,					dim=1   )

																if labels is None:
																						UpperCAmelCase         =			hidden.new_empty((head_logit.size(0   ), self.n_token)   )
																else:
																						UpperCAmelCase         =			torch.zeros_like(__UpperCamelCase   ,					dtype=hidden.dtype   ,					device=hidden.device   )

																UpperCAmelCase         =			0
																UpperCAmelCase         =			[0] + self.cutoffs
																for i in range(len(__UpperCamelCase   ) - 1   ):
																						UpperCAmelCase       ,       UpperCAmelCase         =			cutoff_values[i], cutoff_values[i + 1]

																						if labels is not None:
																												UpperCAmelCase         =			(labels >= l_idx) & (labels < r_idx)
																												UpperCAmelCase         =			mask_i.nonzero().squeeze()

																												if indices_i.numel() == 0:
																																		continue

																												UpperCAmelCase         =			labels.index_select(0   ,					__UpperCamelCase   ) - l_idx
																												UpperCAmelCase         =			head_logprob.index_select(0   ,					__UpperCamelCase   )
																												UpperCAmelCase         =			hidden.index_select(0   ,					__UpperCamelCase   )
																						else:
																												UpperCAmelCase         =			hidden

																						if i == 0:
																												if labels is not None:
																																		UpperCAmelCase         =			head_logprob_i.gather(1   ,					target_i[:, None]   ).squeeze(1   )
																												else:
																																		UpperCAmelCase         =			head_logprob[:, : self.cutoffs[0]]
																						else:
																												UpperCAmelCase       ,       UpperCAmelCase       ,       UpperCAmelCase         =			weights[i], biases[i], self.out_projs[i]

																												UpperCAmelCase         =			self._compute_logit(__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   )
																												UpperCAmelCase         =			nn.functional.log_softmax(__UpperCamelCase   ,					dim=1   )
																												UpperCAmelCase         =			self.cutoffs[0] + i - 1  # No probability for the head cluster
																												if labels is not None:
																																		UpperCAmelCase         =			head_logprob_i[:, cluster_prob_idx] + tail_logprob_i.gather(
																																		    1   ,					target_i[:, None]   ).squeeze(1   )
																												else:
																																		UpperCAmelCase         =			head_logprob[:, cluster_prob_idx, None] + tail_logprob_i
																																		UpperCAmelCase         =			logprob_i

																						if labels is not None:
																												if (hasattr(self   ,					'''keep_order'''   ) and self.keep_order) or keep_order:
																																		out.index_copy_(0   ,					__UpperCamelCase   ,					-logprob_i   )
																												else:
																																		out[offset : offset + logprob_i.size(0   )].copy_(-logprob_i   )
																												offset += logprob_i.size(0   )

										return out





				def   a__(	self    :				Optional[int]   ,					lowerCAmelCase    :				int   )->  Optional[int]:

										"""simple docstring"""




										if self.n_clusters == 0:
																UpperCAmelCase         =			self._compute_logit(__UpperCamelCase   ,					self.out_layers[0].weight   ,					self.out_layers[0].bias   ,					self.out_projs[0]   )
																return nn.functional.log_softmax(__UpperCamelCase   ,					dim=-1   )
										else:
																# construct weights and biases
																UpperCAmelCase       ,       UpperCAmelCase         =			[], []
																for i in range(len(self.cutoffs   )   ):
																						if self.div_val == 1:
																												UpperCAmelCase       ,       UpperCAmelCase         =			self.cutoff_ends[i], self.cutoff_ends[i + 1]
																												UpperCAmelCase         =			self.out_layers[0].weight[l_idx:r_idx]
																												UpperCAmelCase         =			self.out_layers[0].bias[l_idx:r_idx]
																						else:
																												UpperCAmelCase         =			self.out_layers[i].weight
																												UpperCAmelCase         =			self.out_layers[i].bias

																						if i == 0:
																												UpperCAmelCase         =			torch.cat([weight_i, self.cluster_weight]   ,					dim=0   )
																												UpperCAmelCase         =			torch.cat([bias_i, self.cluster_bias]   ,					dim=0   )

																						weights.append(__UpperCamelCase   )
																						biases.append(__UpperCamelCase   )

																UpperCAmelCase       ,       UpperCAmelCase       ,       UpperCAmelCase         =			weights[0], biases[0], self.out_projs[0]
																UpperCAmelCase         =			self._compute_logit(__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   )

																UpperCAmelCase         =			hidden.new_empty((head_logit.size(0   ), self.n_token)   )
																UpperCAmelCase         =			nn.functional.log_softmax(__UpperCamelCase   ,					dim=1   )

																UpperCAmelCase         =			[0] + self.cutoffs
																for i in range(len(__UpperCamelCase   ) - 1   ):
																						UpperCAmelCase       ,       UpperCAmelCase         =			cutoff_values[i], cutoff_values[i + 1]

																						if i == 0:
																												UpperCAmelCase         =			head_logprob[:, : self.cutoffs[0]]
																						else:
																												UpperCAmelCase       ,       UpperCAmelCase       ,       UpperCAmelCase         =			weights[i], biases[i], self.out_projs[i]

																												UpperCAmelCase         =			self._compute_logit(__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   ,					__UpperCamelCase   )
																												UpperCAmelCase         =			nn.functional.log_softmax(__UpperCamelCase   ,					dim=1   )

																												UpperCAmelCase         =			head_logprob[:, -i] + tail_logprob_i
																												UpperCAmelCase         =			logprob_i

																return out









'''simple docstring'''



import heapq


def    lowerCamelCase__   ( A     :    dict  ):




						'''simple docstring'''






						UpperCAmelCase         =			[]

						# for each node and his adjacency list add them and the rank of the node to queue
						# using heapq module the queue will be filled like a Priority Queue
						# heapq works with a min priority queue, so I used -1*len(v) to build it
						for key, value in graph.items():
												# O(log(n))
												heapq.heappush(A   ,     [-1 * len(A  ), (key, value)]  )

						# chosen_vertices = set of chosen vertices
						UpperCAmelCase         =			set()

						# while queue isn't empty and there are still edges
						#   (queue[0][0] is the rank of the node with max rank)
						while queue and queue[0][0] != 0:
												# extract vertex with max rank from queue and add it to chosen_vertices
												UpperCAmelCase         =			heapq.heappop(A  )[1][0]
												chosen_vertices.add(A  )

												# Remove all arcs adjacent to argmax
												for elem in queue:
																		# if v haven't adjacent node, skip
																		if elem[0] == 0:
																								continue
																		# if argmax is reachable from elem
																		# remove argmax from elem's adjacent list and update his rank
																		if argmax in elem[1][1]:
																								UpperCAmelCase         =			elem[1][1].index(A  )
																								del elem[1][1][index]
																								elem[0] += 1
        # re-order the queue
												heapq.heapify(A  )
						return chosen_vertices


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	_lowercase    :    Optional[int]			     =  {0: [1, 3], 1: [0, 3], 2: [0, 3, 4], 3: [0, 1, 2], 4: [2, 3]}
	print(F"""Minimum vertex cover:\n{greedy_min_vertex_cover(graph)}""")





