





import math
import sys

class      A_	:
 def __init__(   self		):






    '''simple docstring'''







    UpperCAmelCase       		=    0
    UpperCAmelCase       		=    []
    UpperCAmelCase       		=    {}  # To store the pos of node in array
 def 					_lowercase    (   self		):






    '''simple docstring'''







    return self.cur_size == 0
 def 					_lowercase    (   self							, _A		):






    '''simple docstring'''







    UpperCAmelCase       		=    self.left(_A		)
    UpperCAmelCase       		=    self.right(_A		)
    if lc < self.cur_size and self.array(_A		)[0] < self.array(_A		)[0]:
       UpperCAmelCase       		=    lc
    else:
       UpperCAmelCase       		=    idx
    if rc < self.cur_size and self.array(_A		)[0] < self.array(_A		)[0]:
       UpperCAmelCase       		=    rc
    if smallest != idx:
       self.swap(_A							, _A		)
       self.min_heapify(_A		)
 def 					_lowercase    (   self							, _A		):






    '''simple docstring'''







    UpperCAmelCase       		=    self.cur_size
    self.cur_size += 1
    self.array.append((sys.maxsize, tup[1])		)
    self.decrease_key((sys.maxsize, tup[1])							, tup[0]		)
 def 					_lowercase    (   self		):






    '''simple docstring'''







    UpperCAmelCase       		=    self.array[0][1]
    UpperCAmelCase       		=    self.array[self.cur_size - 1]
    self.cur_size -= 1
    self.min_heapify(1		)
    del self.pos[min_node]
    return min_node
 def 					_lowercase    (   self							, _A		):






    '''simple docstring'''







    return 2 * i + 1
 def 					_lowercase    (   self							, _A		):






    '''simple docstring'''







    return 2 * i + 2
 def 					_lowercase    (   self							, _A		):






    '''simple docstring'''







    return math.floor(i / 2		)
 def 					_lowercase    (   self							, _A							, _A		):






    '''simple docstring'''







    UpperCAmelCase       		=    j
    UpperCAmelCase       		=    i
    UpperCAmelCase       		=    self.array[i]
    UpperCAmelCase       		=    self.array[j]
    UpperCAmelCase       		=    temp







 def 					_lowercase    (   self							, _A							, _A		):






    '''simple docstring'''







    UpperCAmelCase       		=    self.pos[tup[1]]
    # assuming the new_d is atmost old_d
    UpperCAmelCase       		=    (new_d, tup[1])
    while idx > 0 and self.array[self.par(_A		)][0] > self.array[idx][0]:
       self.swap(_A							, self.par(_A		)		)
       UpperCAmelCase       		=    self.par(_A		)

class      A_	:
 def __init__(   self							, _A		):






    '''simple docstring'''







    UpperCAmelCase       		=    {}  # To store graph: u -> (v,w)
    UpperCAmelCase       		=    num  # Number of nodes in graph
    # To store the distance from source vertex
    UpperCAmelCase       		=    [0] * self.num_nodes
    UpperCAmelCase       		=    [-1] * self.num_nodes  # To store the path
 def 					_lowercase    (   self							, _A							, _A							, _A		):






    '''simple docstring'''







    if u in self.adjList:
       self.adjList[u].append((v, w)		)
    else:
       UpperCAmelCase       		=    [(v, w)]

    # Assuming undirected graph
    if v in self.adjList:
       self.adjList[v].append((u, w)		)
    else:
       UpperCAmelCase       		=    [(u, w)]
 def 					_lowercase    (   self		):






    '''simple docstring'''







    for u in self.adjList:
       print(_A							, '''->'''							, ''' -> '''.join(str(F"""{v}({w})"""		) for v, w in self.adjList[u]		)		)
 def 					_lowercase    (   self							, _A		):






    '''simple docstring'''







    UpperCAmelCase       		=    [-1] * self.num_nodes
    # src is the source node
    UpperCAmelCase       		=    0
    UpperCAmelCase       		=    PriorityQueue()
    q.insert((0, src)		)  # (dist from src, node)
    for u in self.adjList:
       if u != src:
          UpperCAmelCase       		=    sys.maxsize  # Infinity
          UpperCAmelCase       		=    -1

    while not q.is_empty():
       UpperCAmelCase       		=    q.extract_min()  # Returns node with the min dist from source
       # Update the distance of all the neighbours of u and
       # if their prev dist was INFINITY then push them in Q
       for v, w in self.adjList[u]:
          UpperCAmelCase       		=    self.dist[u] + w
          if self.dist[v] > new_dist:
             if self.dist[v] == sys.maxsize:
                q.insert((new_dist, v)		)
             else:
                q.decrease_key((self.dist[v], v)							, _A		)
             UpperCAmelCase       		=    new_dist
             UpperCAmelCase       		=    u

        # Show the shortest distances from src
    self.show_distances(_A		)
 def 					_lowercase    (   self							, _A		):






    '''simple docstring'''







    print(F"""Distance from node: {src}"""		)
    for u in range(self.num_nodes		):
       print(F"""Node {u} has distance: {self.dist[u]}"""		)







 def 					_lowercase    (   self							, _A							, _A		):






    '''simple docstring'''







    UpperCAmelCase       		=    []
    UpperCAmelCase       		=    0
    UpperCAmelCase       		=    dest
    # Backtracking from dest to src
    while self.par[temp] != -1:
       path.append(_A		)
       if temp != src:
          for v, w in self.adjList[temp]:
             if v == self.par[temp]:
                cost += w
                break
       UpperCAmelCase       		=    self.par[temp]
    path.append(_A		)
    path.reverse()

    print(F"""----Path to reach {dest} from {src}----"""		)
    for u in path:
       print(F"""{u}"""							, end=''' '''		)
       if u != dest:
          print('''-> '''							, end=''''''		)

    print('''\nTotal cost of path: '''							, _A		)


if __name__ == "__main__":
    __A    :   int          =       Graph(9)
    graph.add_edge(0, 1, 4)
    graph.add_edge(0, 7, 8)
    graph.add_edge(1, 2, 8)
    graph.add_edge(1, 7, 11)
    graph.add_edge(2, 3, 7)
    graph.add_edge(2, 8, 2)
    graph.add_edge(2, 5, 4)
    graph.add_edge(3, 4, 9)
    graph.add_edge(3, 5, 14)
    graph.add_edge(4, 5, 10)
    graph.add_edge(5, 6, 2)
    graph.add_edge(6, 7, 1)
    graph.add_edge(6, 8, 6)
    graph.add_edge(7, 8, 7)
    graph.show_graph()
    graph.dijkstra(0)
    graph.show_path(0, 4)

# OUTPUT
# 0 -> 1(4) -> 7(8)
# 1 -> 0(4) -> 2(8) -> 7(11)
# 7 -> 0(8) -> 1(11) -> 6(1) -> 8(7)
# 2 -> 1(8) -> 3(7) -> 8(2) -> 5(4)
# 3 -> 2(7) -> 4(9) -> 5(14)
# 8 -> 2(2) -> 6(6) -> 7(7)
# 5 -> 2(4) -> 3(14) -> 4(10) -> 6(2)
# 4 -> 3(9) -> 5(10)
# 6 -> 5(2) -> 7(1) -> 8(6)
# Distance from node: 0
# Node 0 has distance: 0
# Node 1 has distance: 4
# Node 2 has distance: 12
# Node 3 has distance: 19
# Node 4 has distance: 21
# Node 5 has distance: 11
# Node 6 has distance: 9
# Node 7 has distance: 8
# Node 8 has distance: 14
# ----Path to reach 4 from 0----
# 0 -> 7 -> 6 -> 5 -> 4
# Total cost of path:  21










import math





def 						__SCREAMING_SNAKE_CASE  (    UpperCamelCase__			)  ->    bool:







			'''simple docstring'''

			if 1 < number < 4:
						# 2 and 3 are primes
						return True
			elif number < 2 or number % 2 == 0 or number % 3 == 0:
						# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
						return False

			# All primes number are in format of 6k +/- 1
			for i in range(5      ,      int(math.sqrt(UpperCamelCase__			) + 1			)      ,      6			):
						if number % i == 0 or number % (i + 2) == 0:
									return False
			return True





def 						__SCREAMING_SNAKE_CASE  (    UpperCamelCase__ = 6008_5147_5143			)  ->    int:







			'''simple docstring'''

			try:
						UpperCAmelCase       		=    int(UpperCamelCase__			)
			except (TypeError, ValueError):
						raise TypeError('''Parameter n must be int or castable to int.'''			)
			if n <= 0:
						raise ValueError('''Parameter n must be greater than or equal to one.'''			)
			UpperCAmelCase       		=    0
			if is_prime(UpperCamelCase__			):
						return n
			while n % 2 == 0:
						n //= 2
			if is_prime(UpperCamelCase__			):
						return n
			for i in range(3      ,      int(math.sqrt(UpperCamelCase__			)			) + 1      ,      2			):
						if n % i == 0:
									if is_prime(n // i			):
												UpperCAmelCase       		=    n // i
												break
									elif is_prime(UpperCamelCase__			):
												UpperCAmelCase       		=    i
			return max_number


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



