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from typing import List
from collections import defaultdict
from math import inf
from enum import Enum, auto


class EdgeTypes(Enum):
    CROSS_EDGE = auto()
    BACK_EDGE = auto()
    TREE_EDGE = auto()

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# Tarjan's Algo for Strongly Connected Components (SCC)
# Find Articulation Points using tarjan's algo
# Python implementation

from collections import defaultdict
from math import inf
from enum import Enum, auto
from typing import Tuple, List, Set

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# Tarjan's Algo for Strongly Connected Components (SCC)
# Python implementation

from collections import defaultdict
from math import inf
from enum import Enum, auto
from typing import Tuple, List, Set


# Rules:

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# Given a starting vertex visit all vertex

from typing import Dict, Set


class Graph:
    def __init__(self):
        self.edges = list()
        self.graph: Dict[int, Set] = dict()
        self.visited = dict()

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# Get all hamiltonian paths in a graph from a given start vertex
# Python implementation

from typing import Dict, Set, List
from collections import defaultdict


class Graph:
    def __init__(self, edges):
        self.edges = edges

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"""
----------------------------
N - Queens Problem
----------------------------
--- Python Implementation
--- Backtracking approach
--- Description: Given a chessboard of 4x4 size, place n queens on it so that they done attack each other.
"""
from typing import Generator, List, Set, Tuple, Dict

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"""
----------------------------
N - Queens Problem
----------------------------
--- Python Implementation
--- Backtracking approach
--- Description: Given a chessboard of 4x4 size, place n queens on it so that they done attack each other.
"""
from typing import Generator, List, Set, Tuple, Dict
from copy import deepcopy

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class UnionFind:
    def __init__(self):
        self.parent = dict()
        self.rank = dict()

    def find(self, item) -> int:
        origional_item = item

        parent = self.parent[item]
        while parent is not None:

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from typing import List


# This is an undirected graph.
# _graph[u] = list of all elements parallel to u (e.g. _graph[u] = [v])
# Also _graph[v] = [u] Since undirected graph the opposite edge is also present

def isBipartite(_graph: List[List[int]]) -> bool:
    # we are already given the adjacency matrix, so we can start BFS

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# Dijkstra Algo with no destination.
# Calculate -> Shortest path from start to all nodes

import math
import heapq
from typing import Dict, List, Tuple

# Maintains neighbours and weights
graph: Dict[int, List[Tuple[int, int]]] = dict()