def strongly_connected_components_path(vertices, edges):
"""
Find the strongly connected components of a directed graph.
Uses a recursive linear-time algorithm described by Gabow [1]_ to find all
strongly connected components of a directed graph.
Parameters
----------
vertices : iterable
A sequence or other iterable of vertices. Each vertex should be
hashable.
edges : mapping
Dictionary (or mapping) that maps each vertex v to an iterable of the
vertices w that are linked to v by a directed edge (v, w).
Returns
-------
components : iterator
An iterator that yields sets of vertices. Each set produced gives the
vertices of one strongly connected component.
Raises
------
RuntimeError
If the graph is deep enough that the algorithm exceeds Python's
recursion limit.
Notes
-----
The algorithm has running time proportional to the total number of vertices
and edges. It's practical to use this algorithm on graphs with hundreds of
thousands of vertices and edges.
The algorithm is recursive. Deep graphs may cause Python to exceed its
recursion limit.
`vertices` will be iterated over exactly once, and `edges[v]` will be
iterated over exactly once for each vertex `v`. `edges[v]` is permitted to
specify the same vertex multiple times, and it's permissible for `edges[v]`
to include `v` itself. (In graph-theoretic terms, loops and multiple edges
are permitted.)
References
----------
.. [1] Harold N. Gabow, "Path-based depth-first search for strong and
biconnected components," Inf. Process. Lett. 74 (2000) 107--114.
.. [2] Robert E. Tarjan, "Depth-first search and linear graph algorithms,"
SIAM J.Comput. 1 (2) (1972) 146--160.
Examples
--------
Example from Gabow's paper [1]_.
>>> vertices = [1, 2, 3, 4, 5, 6]
>>> edges = {1: [2, 3], 2: [3, 4], 3: [], 4: [3, 5], 5: [2, 6], 6: [3, 4]}
>>> for scc in strongly_connected_components_path(vertices, edges):
... print(scc)
...
set([3])
set([2, 4, 5, 6])
set([1])
Example from Tarjan's paper [2]_.
>>> vertices = [1, 2, 3, 4, 5, 6, 7, 8]
>>> edges = {1: [2], 2: [3, 8], 3: [4, 7], 4: [5],
... 5: [3, 6], 6: [], 7: [4, 6], 8: [1, 7]}
>>> for scc in strongly_connected_components_path(vertices, edges):
... print(scc)
...
set([6])
set([3, 4, 5, 7])
set([8, 1, 2])
"""
identified = set()
stack = []
index = {}
boundaries = []
def dfs(v):
index[v] = len(stack)
stack.append(v)
boundaries.append(index[v])
for w in edges[v]:
if w not in index:
# For Python >= 3.3, replace with "yield from dfs(w)"
for scc in dfs(w):
yield scc
elif w not in identified:
while index[w] < boundaries[-1]:
boundaries.pop()
if boundaries[-1] == index[v]:
boundaries.pop()
scc = set(stack[index[v]:])
del stack[index[v]:]
identified.update(scc)
yield scc
for v in vertices:
if v not in index:
# For Python >= 3.3, replace with "yield from dfs(v)"
for scc in dfs(v):
yield scc
def strongly_connected_components_tree(vertices, edges):
"""
Find the strongly connected components of a directed graph.
Uses a recursive linear-time algorithm described by Tarjan [2]_ to find all
strongly connected components of a directed graph.
Parameters
----------
vertices : iterable
A sequence or other iterable of vertices. Each vertex should be
hashable.
edges : mapping
Dictionary (or mapping) that maps each vertex v to an iterable of the
vertices w that are linked to v by a directed edge (v, w).
Returns
-------
components : iterator
An iterator that yields sets of vertices. Each set produced gives the
vertices of one strongly connected component.
Raises
------
RuntimeError
If the graph is deep enough that the algorithm exceeds Python's
recursion limit.
Notes
-----
The algorithm has running time proportional to the total number of vertices
and edges. It's practical to use this algorithm on graphs with hundreds of
thousands of vertices and edges.
The algorithm is recursive. Deep graphs may cause Python to exceed its
recursion limit.
`vertices` will be iterated over exactly once, and `edges[v]` will be
iterated over exactly once for each vertex `v`. `edges[v]` is permitted to
specify the same vertex multiple times, and it's permissible for `edges[v]`
to include `v` itself. (In graph-theoretic terms, loops and multiple edges
are permitted.)
References
----------
.. [1] Harold N. Gabow, "Path-based depth-first search for strong and
biconnected components," Inf. Process. Lett. 74 (2000) 107--114.
.. [2] Robert E. Tarjan, "Depth-first search and linear graph algorithms,"
SIAM J.Comput. 1 (2) (1972) 146--160.
Examples
--------
Example from Gabow's paper [1]_.
>>> vertices = [1, 2, 3, 4, 5, 6]
>>> edges = {1: [2, 3], 2: [3, 4], 3: [], 4: [3, 5], 5: [2, 6], 6: [3, 4]}
>>> for scc in strongly_connected_components_tree(vertices, edges):
... print(scc)
...
set([3])
set([2, 4, 5, 6])
set([1])
Example from Tarjan's paper [2]_.
>>> vertices = [1, 2, 3, 4, 5, 6, 7, 8]
>>> edges = {1: [2], 2: [3, 8], 3: [4, 7], 4: [5],
... 5: [3, 6], 6: [], 7: [4, 6], 8: [1, 7]}
>>> for scc in strongly_connected_components_tree(vertices, edges):
... print(scc)
...
set([6])
set([3, 4, 5, 7])
set([8, 1, 2])
"""
identified = set()
stack = []
index = {}
lowlink = {}
def dfs(v):
index[v] = len(stack)
stack.append(v)
lowlink[v] = index[v]
for w in edges[v]:
if w not in index:
# For Python >= 3.3, replace with "yield from dfs(w)"
for scc in dfs(w):
yield scc
lowlink[v] = min(lowlink[v], lowlink[w])
elif w not in identified:
lowlink[v] = min(lowlink[v], lowlink[w])
if lowlink[v] == index[v]:
scc = set(stack[index[v]:])
del stack[index[v]:]
identified.update(scc)
yield scc
for v in vertices:
if v not in index:
# For Python >= 3.3, replace with "yield from dfs(v)"
for scc in dfs(v):
yield scc
def strongly_connected_components_iterative(vertices, edges):
"""
This is a non-recursive version of strongly_connected_components_path.
See the docstring of that function for more details.
Examples
--------
Example from Gabow's paper [1]_.
>>> vertices = [1, 2, 3, 4, 5, 6]
>>> edges = {1: [2, 3], 2: [3, 4], 3: [], 4: [3, 5], 5: [2, 6], 6: [3, 4]}
>>> for scc in strongly_connected_components_iterative(vertices, edges):
... print(scc)
...
set([3])
set([2, 4, 5, 6])
set([1])
Example from Tarjan's paper [2]_.
>>> vertices = [1, 2, 3, 4, 5, 6, 7, 8]
>>> edges = {1: [2], 2: [3, 8], 3: [4, 7], 4: [5],
... 5: [3, 6], 6: [], 7: [4, 6], 8: [1, 7]}
>>> for scc in strongly_connected_components_iterative(vertices, edges):
... print(scc)
...
set([6])
set([3, 4, 5, 7])
set([8, 1, 2])
"""
identified = set()
stack = []
index = {}
boundaries = []
for v in vertices:
if v not in index:
to_do = [('VISIT', v)]
while to_do:
operation_type, v = to_do.pop()
if operation_type == 'VISIT':
index[v] = len(stack)
stack.append(v)
boundaries.append(index[v])
to_do.append(('POSTVISIT', v))
# We reverse to keep the search order identical to that of
# the recursive code; the reversal is not necessary for
# correctness, and can be omitted.
to_do.extend(
reversed([('VISITEDGE', w) for w in edges[v]]))
elif operation_type == 'VISITEDGE':
if v not in index:
to_do.append(('VISIT', v))
elif v not in identified:
while index[v] < boundaries[-1]:
boundaries.pop()
else:
# operation_type == 'POSTVISIT'
if boundaries[-1] == index[v]:
boundaries.pop()
scc = set(stack[index[v]:])
del stack[index[v]:]
identified.update(scc)
yield scc