
'''simple docstring'''






# using dfs for finding eulerian path traversal







def 		A_							(						_lowerCAmelCase :     str					,						_lowerCAmelCase :     Union[str, Any]					,						_lowerCAmelCase :     Optional[Any]					,						_lowerCAmelCase :     Union[str, Any]=None				):



      """simple docstring"""




      _lowerCamelCase					: Optional[int]		    = (path or []) + [u]
      for v in graph[u]:
            if visited_edge[u][v] is False:
                  _lowerCamelCase    ,		_lowerCamelCase					: Optional[Any]		    = True, True
                  _lowerCamelCase					: str		    = dfs(_lowerCAmelCase					,						_lowerCAmelCase					,						_lowerCAmelCase					,						_lowerCAmelCase				)
      return path







def 		A_							(						_lowerCAmelCase :     Dict					,						_lowerCAmelCase :     List[str]				):



      """simple docstring"""




      _lowerCamelCase					: Tuple		    = 0
      _lowerCamelCase					: List[Any]		    = -1
      for i in range(_lowerCAmelCase				):
            if i not in graph.keys():
                  continue
            if len(graph[i]				) % 2 == 1:
                  odd_degree_nodes += 1
                  _lowerCamelCase					: Dict		    = i
      if odd_degree_nodes == 0:
            return 1, odd_node
      if odd_degree_nodes == 2:
            return 2, odd_node
      return 3, odd_node







def 		A_							(						_lowerCAmelCase :     List[Any]					,						_lowerCAmelCase :     Dict				):



      """simple docstring"""




      _lowerCamelCase					: List[Any]		    = [[False for _ in range(max_node + 1				)] for _ in range(max_node + 1				)]
      _lowerCamelCase    ,		_lowerCamelCase					: Dict		    = check_circuit_or_path(_lowerCAmelCase					,						_lowerCAmelCase				)
      if check == 3:
            print("graph is not Eulerian"				)
            print("no path"				)
            return
      _lowerCamelCase					: str		    = 1
      if check == 2:
            _lowerCamelCase					: int		    = odd_node
            print("graph has a Euler path"				)
      if check == 1:
            print("graph has a Euler cycle"				)
      _lowerCamelCase					: List[Any]		    = dfs(_lowerCAmelCase					,						_lowerCAmelCase					,						_lowerCAmelCase				)
      print(_lowerCAmelCase				)







def 		A_							(						):



      """simple docstring"""




      _lowerCamelCase					: Optional[Any]		    = {1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
      _lowerCamelCase					: Union[str, Any]		    = {1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
      _lowerCamelCase					: Tuple		    = {1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
      _lowerCamelCase					: Dict		    = {1: [2, 3], 2: [1, 3], 3: [1, 2]}
      _lowerCamelCase					: List[Any]		    = {
          1: [],
          2: []
          # all degree is zero
      }
      _lowerCamelCase					: Optional[Any]		    = 10
      check_euler(_lowerCAmelCase					,						_lowerCAmelCase				)
      check_euler(_lowerCAmelCase					,						_lowerCAmelCase				)
      check_euler(_lowerCAmelCase					,						_lowerCAmelCase				)
      check_euler(_lowerCAmelCase					,						_lowerCAmelCase				)
      check_euler(_lowerCAmelCase					,						_lowerCAmelCase				)


if __name__ == "__main__":
 main()





'''simple docstring'''

def 						__magic_name__       (      __UpperCAmelCase, __UpperCAmelCase, __UpperCAmelCase, __UpperCAmelCase      )						->   Optional[Any]:

					'''simple docstring'''

					snake_case_		     =							[False] * len(__UpperCAmelCase      )
					snake_case_		     =							[]
					queue.append(__UpperCAmelCase      )
					snake_case_		     =							True

					while queue:
										snake_case_		     =							queue.pop(0      )
										for ind in range(len(graph[u]      )      ):
															if visited[ind] is False and graph[u][ind] > 0:
																				queue.append(__UpperCAmelCase      )
																				snake_case_		     =							True
																				snake_case_		     =							u

					return visited[t]


def 						__magic_name__       (      __UpperCAmelCase, __UpperCAmelCase, __UpperCAmelCase      )						->   Tuple:

					'''simple docstring'''

					snake_case_		     =							[-1] * (len(__UpperCAmelCase      ))
					snake_case_		     =							0
					while bfs(__UpperCAmelCase, __UpperCAmelCase, __UpperCAmelCase, __UpperCAmelCase      ):
										snake_case_		     =							float('''Inf'''      )
										snake_case_		     =							sink

										while s != source:
															# Find the minimum value in select path
															snake_case_		     =							min(__UpperCAmelCase, graph[parent[s]][s]      )
															snake_case_		     =							parent[s]

										max_flow += path_flow
										snake_case_		     =							sink

										while v != source:
															snake_case_		     =							parent[v]
															graph[u][v] -= path_flow
															graph[v][u] += path_flow
															snake_case_		     =							parent[v]
					return max_flow


a    :							List[Any]										=    [
    [0, 16, 13, 0, 0, 0],
    [0, 0, 10, 12, 0, 0],
    [0, 4, 0, 0, 14, 0],
    [0, 0, 9, 0, 0, 20],
    [0, 0, 0, 7, 0, 4],
    [0, 0, 0, 0, 0, 0],
]

a						,a    :							int										=    0, 5

print(ford_fulkerson(graph, source, sink))

