I have some input like: [('A', 'B'),('C', 'D'),('D', 'C'),('C', 'D')]. I want to look for if the existence of a cycle in a directed graph represented by this edgeList.
I read a discussion: https://www.geeksforgeeks.org/detect-cycle-in-a-graph/, however it has some errors when the case is:
g = Graph(3)
g.addEdge('A', 'B')
g.addEdge('B', 'C')
g.addEdge('C', 'A')
Its result is 'Graph has no cycle'. This is clearly wrong. Can you help me to solve this problem?
Using the networkx library, we can use the simple_cycles function to find all simple cycles of a directed Graph.
Example Code:
import networkx as nx
edges = [('A', 'B'),('C', 'D'),('D', 'C'),('C', 'D')]
G = nx.DiGraph(edges)
for cycle in nx.simple_cycles(G):
print(cycle)
G = nx.DiGraph()
G.add_edge('A', 'B')
G.add_edge('B', 'C')
G.add_edge('C', 'A')
for cycle in nx.simple_cycles(G):
print(cycle)
Output:
['D', 'C']
['B', 'C', 'A']
The issue is the example given at [1]: https://www.geeksforgeeks.org/detect-cycle-in-a-graph/ works for integers only because they use the range() function to create a list of nodes,see the line
for node in range(self.V):
That makes the assumption that not only will all the nodes be integers but also that they will be a contiguous set i.e. [0,1,2,3] is okay but [0,3,10] is not.
You can fix the example if you like to work with any nodes by swapping the line given above with
for node in self.graph.keys():
which will loop through all the nodes instead of a range of numbers :)
My own implementation (non-recursive so without cycle length limit):
from collections import defaultdict
def has_cycle(graph):
try:
next(_iter_cycles(graph))
except StopIteration:
return False
return True
def _iter_cycles(edges):
"""Iterate over simple cycles in the directed graph."""
if isinstance(edges, dict):
graph = edges
else:
graph = defaultdict(set)
for x, y in edges:
graph[x].add(y)
SEP = object()
checked_nodes = set() # already checked nodes
for start_node in graph:
if start_node in checked_nodes:
continue
nodes_left = [start_node]
path = [] # current path from start_node
node_idx = {} # {node: path.index(node)}
while nodes_left:
node = nodes_left.pop()
if node is SEP:
checked_node = path.pop()
del node_idx[checked_node]
checked_nodes.add(checked_node)
continue
if node in checked_nodes:
continue
if node in node_idx:
cycle_path = path[node_idx[node]:]
cycle_path.append(node)
yield cycle_path
continue
next_nodes = graph.get(node)
if not next_nodes:
checked_nodes.add(node)
continue
node_idx[node] = len(path)
path.append(node)
nodes_left.append(SEP)
nodes_left.extend(next_nodes)
assert not has_cycle({0: [1, 2], 1: [3, 4], 5: [6, 7]})
assert has_cycle([(0, 1), (1, 0), (1, 2), (2, 1)])
def assert_cycles(graph, expected):
detected = sorted(_iter_cycles(graph))
if detected != expected:
raise Exception('expected cycles:\n{}\ndetected cycles:\n{}'.format(expected, detected))
assert_cycles([('A', 'B'),('C', 'D'),('D', 'C'),('C', 'D')], [['C', 'D', 'C']])
assert_cycles([('A', 'B'),('B', 'A'),('B', 'C'),('C', 'B')], [['A', 'B', 'A'], ['B', 'C', 'B']])
assert_cycles({1: [2, 3], 2: [3, 4]}, [])
assert_cycles([(1, 2), (1, 3), (2, 3), (2, 4)], [])
assert_cycles({1: [2, 4], 2: [3, 4], 3: [1]}, [[1, 2, 3, 1]])
assert_cycles([(1, 2), (1, 4), (2, 3), (2, 4), (3, 1)], [[1, 2, 3, 1]])
assert_cycles({0: [1, 2], 2: [3], 3: [4], 4: [2]}, [[2, 3, 4, 2]])
assert_cycles([(0, 1), (0, 2), (2, 3), (3, 4), (4, 2)], [[2, 3, 4, 2]])
assert_cycles({1: [2], 3: [4], 4: [5], 5: [3]}, [[3, 4, 5, 3]])
assert_cycles([(1, 2), (3, 4), (4, 5), (5, 3)], [[3, 4, 5, 3]])
assert_cycles({0: [], 1: []}, [])
assert_cycles([], [])
assert_cycles({0: [1, 2], 1: [3, 4], 5: [6, 7]}, [])
assert_cycles([(0, 1), (0, 2), (1, 3), (1, 4), (5, 6), (5, 7)], [])
assert_cycles({0: [1], 1: [0, 2], 2: [1]}, [[0, 1, 0], [1, 2, 1]])
assert_cycles([(0, 1), (1, 0), (1, 2), (2, 1)], [[0, 1, 0], [1, 2, 1]])
EDIT:
I found that while has_cycle seems to be correct,
the _iter_cycles does not iterate over all cycles!
Example in which _iter_cycles does not find all cycles:
assert_cycles([
(0, 1), (1, 2), (2, 0), # Cycle 0-1-2
(0, 2), (2, 0), # Cycle 0-2
(0, 1), (1, 4), (4, 0), # Cycle 0-1-4
],
[
[0, 1, 2, 0], # Not found (in Python 3.7)!
[0, 1, 4, 0],
[0, 2, 0],
]
)
I think the following function does the job and is 14 lines of code. The idea is to remove edges as we traverse them, starting at the leaves. I have not checked it exhaustivelly but it works with small graphs. Also mind that it is not optimized for speed in large graphs.
def get_cycle(DAG):
''' DAG is assumed a dict {node: [node,...],...}.
DAG keys are children and values (lists of) parents;
Roots are not DAG keys (since they have no parents).
The function returns the first cycle it will detect as a list
of nodes, or an empty list if the DAG has no cycles.
NB! the DAG is modified (eventually becomes totally disconnected)
'''
def get_leaves(DAG):
''' Return the leaves of the DAG.
A leaf is not a parent and not a root
'''
parents = set(i for v in DAG.values() for i in v)
return [c for c in DAG if len(DAG[c]) > 0 and c not in parents]
def ascend(node, DAG, visits):
''' Ascend a single path starting at node.
When a node is visited, remove it from the DAG.
If a node is visited multiple times we obviously have a cycle.
'''
if len(DAG.get(node, [])) > 0:
visits.append(parent := DAG[node].pop())
ascend(parent, DAG, visits)
return visits
def is_bag(lis: List):
''' Return True if lis contains multiplicities
'''
return len(set(lis)) < len(lis)
while len(leaves := get_leaves(DAG)) > 0:
if is_bag(visits := ascend(leaves[0], DAG=DAG, visits=[])):
return visits
return []
