618 lines
21 KiB
Python
618 lines
21 KiB
Python
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"""
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========================
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Cycle finding algorithms
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========================
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"""
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from collections import defaultdict
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import networkx as nx
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from networkx.utils import not_implemented_for, pairwise
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__all__ = [
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"cycle_basis",
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"simple_cycles",
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"recursive_simple_cycles",
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"find_cycle",
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"minimum_cycle_basis",
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]
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def cycle_basis(G, root=None):
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"""Returns a list of cycles which form a basis for cycles of G.
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A basis for cycles of a network is a minimal collection of
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cycles such that any cycle in the network can be written
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as a sum of cycles in the basis. Here summation of cycles
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is defined as "exclusive or" of the edges. Cycle bases are
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useful, e.g. when deriving equations for electric circuits
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using Kirchhoff's Laws.
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Parameters
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----------
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G : NetworkX Graph
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root : node, optional
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Specify starting node for basis.
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Returns
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-------
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A list of cycle lists. Each cycle list is a list of nodes
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which forms a cycle (loop) in G.
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Examples
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--------
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>>> G = nx.Graph()
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>>> nx.add_cycle(G, [0, 1, 2, 3])
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>>> nx.add_cycle(G, [0, 3, 4, 5])
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>>> print(nx.cycle_basis(G, 0))
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[[3, 4, 5, 0], [1, 2, 3, 0]]
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Notes
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-----
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This is adapted from algorithm CACM 491 [1]_.
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References
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----------
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.. [1] Paton, K. An algorithm for finding a fundamental set of
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cycles of a graph. Comm. ACM 12, 9 (Sept 1969), 514-518.
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See Also
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--------
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simple_cycles
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"""
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gnodes = set(G.nodes())
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cycles = []
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while gnodes: # loop over connected components
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if root is None:
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root = gnodes.pop()
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stack = [root]
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pred = {root: root}
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used = {root: set()}
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while stack: # walk the spanning tree finding cycles
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z = stack.pop() # use last-in so cycles easier to find
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zused = used[z]
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for nbr in G[z]:
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if nbr not in used: # new node
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pred[nbr] = z
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stack.append(nbr)
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used[nbr] = {z}
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elif nbr == z: # self loops
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cycles.append([z])
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elif nbr not in zused: # found a cycle
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pn = used[nbr]
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cycle = [nbr, z]
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p = pred[z]
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while p not in pn:
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cycle.append(p)
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p = pred[p]
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cycle.append(p)
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cycles.append(cycle)
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used[nbr].add(z)
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gnodes -= set(pred)
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root = None
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return cycles
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@not_implemented_for("undirected")
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def simple_cycles(G):
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"""Find simple cycles (elementary circuits) of a directed graph.
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A `simple cycle`, or `elementary circuit`, is a closed path where
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no node appears twice. Two elementary circuits are distinct if they
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are not cyclic permutations of each other.
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This is a nonrecursive, iterator/generator version of Johnson's
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algorithm [1]_. There may be better algorithms for some cases [2]_ [3]_.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed graph
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Yields
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------
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list of nodes
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Each cycle is represented by a list of nodes along the cycle.
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Examples
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--------
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>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
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>>> G = nx.DiGraph(edges)
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>>> sorted(nx.simple_cycles(G))
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[[0], [0, 1, 2], [0, 2], [1, 2], [2]]
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To filter the cycles so that they don't include certain nodes or edges,
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copy your graph and eliminate those nodes or edges before calling.
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For example, to exclude self-loops from the above example:
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>>> H = G.copy()
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>>> H.remove_edges_from(nx.selfloop_edges(G))
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>>> sorted(nx.simple_cycles(H))
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[[0, 1, 2], [0, 2], [1, 2]]
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Notes
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-----
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The implementation follows pp. 79-80 in [1]_.
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The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
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elementary circuits.
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References
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----------
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.. [1] Finding all the elementary circuits of a directed graph.
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D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
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https://doi.org/10.1137/0204007
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.. [2] Enumerating the cycles of a digraph: a new preprocessing strategy.
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G. Loizou and P. Thanish, Information Sciences, v. 27, 163-182, 1982.
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.. [3] A search strategy for the elementary cycles of a directed graph.
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J.L. Szwarcfiter and P.E. Lauer, BIT NUMERICAL MATHEMATICS,
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v. 16, no. 2, 192-204, 1976.
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See Also
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--------
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cycle_basis
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"""
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def _unblock(thisnode, blocked, B):
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stack = {thisnode}
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while stack:
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node = stack.pop()
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if node in blocked:
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blocked.remove(node)
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stack.update(B[node])
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B[node].clear()
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# Johnson's algorithm requires some ordering of the nodes.
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# We assign the arbitrary ordering given by the strongly connected comps
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# There is no need to track the ordering as each node removed as processed.
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# Also we save the actual graph so we can mutate it. We only take the
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# edges because we do not want to copy edge and node attributes here.
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subG = type(G)(G.edges())
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sccs = [scc for scc in nx.strongly_connected_components(subG) if len(scc) > 1]
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# Johnson's algorithm exclude self cycle edges like (v, v)
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# To be backward compatible, we record those cycles in advance
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# and then remove from subG
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for v in subG:
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if subG.has_edge(v, v):
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yield [v]
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subG.remove_edge(v, v)
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while sccs:
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scc = sccs.pop()
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sccG = subG.subgraph(scc)
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# order of scc determines ordering of nodes
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startnode = scc.pop()
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# Processing node runs "circuit" routine from recursive version
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path = [startnode]
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blocked = set() # vertex: blocked from search?
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closed = set() # nodes involved in a cycle
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blocked.add(startnode)
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B = defaultdict(set) # graph portions that yield no elementary circuit
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stack = [(startnode, list(sccG[startnode]))] # sccG gives comp nbrs
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while stack:
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thisnode, nbrs = stack[-1]
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if nbrs:
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nextnode = nbrs.pop()
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if nextnode == startnode:
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yield path[:]
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closed.update(path)
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# print "Found a cycle", path, closed
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elif nextnode not in blocked:
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path.append(nextnode)
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stack.append((nextnode, list(sccG[nextnode])))
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closed.discard(nextnode)
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blocked.add(nextnode)
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continue
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# done with nextnode... look for more neighbors
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if not nbrs: # no more nbrs
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if thisnode in closed:
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_unblock(thisnode, blocked, B)
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else:
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for nbr in sccG[thisnode]:
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if thisnode not in B[nbr]:
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B[nbr].add(thisnode)
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stack.pop()
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# assert path[-1] == thisnode
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path.pop()
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# done processing this node
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H = subG.subgraph(scc) # make smaller to avoid work in SCC routine
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sccs.extend(scc for scc in nx.strongly_connected_components(H) if len(scc) > 1)
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@not_implemented_for("undirected")
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def recursive_simple_cycles(G):
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"""Find simple cycles (elementary circuits) of a directed graph.
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A `simple cycle`, or `elementary circuit`, is a closed path where
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no node appears twice. Two elementary circuits are distinct if they
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are not cyclic permutations of each other.
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This version uses a recursive algorithm to build a list of cycles.
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You should probably use the iterator version called simple_cycles().
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Warning: This recursive version uses lots of RAM!
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It appears in NetworkX for pedagogical value.
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Parameters
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----------
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G : NetworkX DiGraph
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A directed graph
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Returns
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-------
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A list of cycles, where each cycle is represented by a list of nodes
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along the cycle.
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Example:
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>>> edges = [(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
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>>> G = nx.DiGraph(edges)
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>>> nx.recursive_simple_cycles(G)
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[[0], [2], [0, 1, 2], [0, 2], [1, 2]]
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Notes
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-----
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The implementation follows pp. 79-80 in [1]_.
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The time complexity is $O((n+e)(c+1))$ for $n$ nodes, $e$ edges and $c$
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elementary circuits.
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References
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----------
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.. [1] Finding all the elementary circuits of a directed graph.
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D. B. Johnson, SIAM Journal on Computing 4, no. 1, 77-84, 1975.
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https://doi.org/10.1137/0204007
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See Also
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--------
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simple_cycles, cycle_basis
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"""
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# Jon Olav Vik, 2010-08-09
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def _unblock(thisnode):
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"""Recursively unblock and remove nodes from B[thisnode]."""
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if blocked[thisnode]:
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blocked[thisnode] = False
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while B[thisnode]:
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_unblock(B[thisnode].pop())
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def circuit(thisnode, startnode, component):
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closed = False # set to True if elementary path is closed
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path.append(thisnode)
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blocked[thisnode] = True
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for nextnode in component[thisnode]: # direct successors of thisnode
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if nextnode == startnode:
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result.append(path[:])
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closed = True
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elif not blocked[nextnode]:
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if circuit(nextnode, startnode, component):
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closed = True
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if closed:
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_unblock(thisnode)
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else:
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for nextnode in component[thisnode]:
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if thisnode not in B[nextnode]: # TODO: use set for speedup?
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B[nextnode].append(thisnode)
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path.pop() # remove thisnode from path
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return closed
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path = [] # stack of nodes in current path
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blocked = defaultdict(bool) # vertex: blocked from search?
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B = defaultdict(list) # graph portions that yield no elementary circuit
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result = [] # list to accumulate the circuits found
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# Johnson's algorithm exclude self cycle edges like (v, v)
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# To be backward compatible, we record those cycles in advance
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# and then remove from subG
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for v in G:
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if G.has_edge(v, v):
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result.append([v])
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G.remove_edge(v, v)
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# Johnson's algorithm requires some ordering of the nodes.
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# They might not be sortable so we assign an arbitrary ordering.
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ordering = dict(zip(G, range(len(G))))
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for s in ordering:
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# Build the subgraph induced by s and following nodes in the ordering
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subgraph = G.subgraph(node for node in G if ordering[node] >= ordering[s])
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# Find the strongly connected component in the subgraph
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# that contains the least node according to the ordering
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strongcomp = nx.strongly_connected_components(subgraph)
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mincomp = min(strongcomp, key=lambda ns: min(ordering[n] for n in ns))
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component = G.subgraph(mincomp)
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if len(component) > 1:
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# smallest node in the component according to the ordering
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startnode = min(component, key=ordering.__getitem__)
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for node in component:
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blocked[node] = False
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B[node][:] = []
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dummy = circuit(startnode, startnode, component)
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return result
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def find_cycle(G, source=None, orientation=None):
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"""Returns a cycle found via depth-first traversal.
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The cycle is a list of edges indicating the cyclic path.
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Orientation of directed edges is controlled by `orientation`.
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Parameters
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----------
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G : graph
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A directed/undirected graph/multigraph.
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source : node, list of nodes
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The node from which the traversal begins. If None, then a source
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is chosen arbitrarily and repeatedly until all edges from each node in
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the graph are searched.
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orientation : None | 'original' | 'reverse' | 'ignore' (default: None)
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For directed graphs and directed multigraphs, edge traversals need not
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respect the original orientation of the edges.
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When set to 'reverse' every edge is traversed in the reverse direction.
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When set to 'ignore', every edge is treated as undirected.
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When set to 'original', every edge is treated as directed.
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In all three cases, the yielded edge tuples add a last entry to
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indicate the direction in which that edge was traversed.
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If orientation is None, the yielded edge has no direction indicated.
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The direction is respected, but not reported.
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Returns
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-------
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edges : directed edges
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A list of directed edges indicating the path taken for the loop.
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If no cycle is found, then an exception is raised.
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For graphs, an edge is of the form `(u, v)` where `u` and `v`
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are the tail and head of the edge as determined by the traversal.
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For multigraphs, an edge is of the form `(u, v, key)`, where `key` is
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the key of the edge. When the graph is directed, then `u` and `v`
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are always in the order of the actual directed edge.
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If orientation is not None then the edge tuple is extended to include
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the direction of traversal ('forward' or 'reverse') on that edge.
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Raises
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------
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NetworkXNoCycle
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If no cycle was found.
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Examples
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--------
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In this example, we construct a DAG and find, in the first call, that there
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are no directed cycles, and so an exception is raised. In the second call,
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we ignore edge orientations and find that there is an undirected cycle.
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Note that the second call finds a directed cycle while effectively
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traversing an undirected graph, and so, we found an "undirected cycle".
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This means that this DAG structure does not form a directed tree (which
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is also known as a polytree).
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>>> G = nx.DiGraph([(0, 1), (0, 2), (1, 2)])
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>>> nx.find_cycle(G, orientation="original")
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Traceback (most recent call last):
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...
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networkx.exception.NetworkXNoCycle: No cycle found.
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>>> list(nx.find_cycle(G, orientation="ignore"))
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[(0, 1, 'forward'), (1, 2, 'forward'), (0, 2, 'reverse')]
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See Also
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--------
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simple_cycles
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"""
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if not G.is_directed() or orientation in (None, "original"):
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def tailhead(edge):
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return edge[:2]
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elif orientation == "reverse":
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def tailhead(edge):
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return edge[1], edge[0]
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elif orientation == "ignore":
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def tailhead(edge):
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if edge[-1] == "reverse":
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return edge[1], edge[0]
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return edge[:2]
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explored = set()
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cycle = []
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final_node = None
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for start_node in G.nbunch_iter(source):
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if start_node in explored:
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# No loop is possible.
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continue
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edges = []
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# All nodes seen in this iteration of edge_dfs
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seen = {start_node}
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# Nodes in active path.
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active_nodes = {start_node}
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previous_head = None
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for edge in nx.edge_dfs(G, start_node, orientation):
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# Determine if this edge is a continuation of the active path.
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tail, head = tailhead(edge)
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if head in explored:
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# Then we've already explored it. No loop is possible.
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continue
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if previous_head is not None and tail != previous_head:
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# This edge results from backtracking.
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# Pop until we get a node whose head equals the current tail.
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# So for example, we might have:
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# (0, 1), (1, 2), (2, 3), (1, 4)
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# which must become:
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# (0, 1), (1, 4)
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while True:
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try:
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popped_edge = edges.pop()
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except IndexError:
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edges = []
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active_nodes = {tail}
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break
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else:
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popped_head = tailhead(popped_edge)[1]
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active_nodes.remove(popped_head)
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if edges:
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last_head = tailhead(edges[-1])[1]
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if tail == last_head:
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break
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edges.append(edge)
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if head in active_nodes:
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# We have a loop!
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cycle.extend(edges)
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final_node = head
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break
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else:
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seen.add(head)
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active_nodes.add(head)
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previous_head = head
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if cycle:
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break
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else:
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explored.update(seen)
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else:
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assert len(cycle) == 0
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raise nx.exception.NetworkXNoCycle("No cycle found.")
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# We now have a list of edges which ends on a cycle.
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# So we need to remove from the beginning edges that are not relevant.
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for i, edge in enumerate(cycle):
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tail, head = tailhead(edge)
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if tail == final_node:
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break
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return cycle[i:]
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def minimum_cycle_basis(G, weight=None):
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"""Returns a minimum weight cycle basis for G
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Minimum weight means a cycle basis for which the total weight
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(length for unweighted graphs) of all the cycles is minimum.
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Parameters
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----------
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G : NetworkX Graph
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weight: string
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name of the edge attribute to use for edge weights
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Returns
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-------
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A list of cycle lists. Each cycle list is a list of nodes
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which forms a cycle (loop) in G. Note that the nodes are not
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necessarily returned in a order by which they appear in the cycle
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Examples
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--------
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>>> G = nx.Graph()
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>>> nx.add_cycle(G, [0, 1, 2, 3])
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>>> nx.add_cycle(G, [0, 3, 4, 5])
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>>> print([sorted(c) for c in nx.minimum_cycle_basis(G)])
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[[0, 1, 2, 3], [0, 3, 4, 5]]
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References:
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[1] Kavitha, Telikepalli, et al. "An O(m^2n) Algorithm for
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Minimum Cycle Basis of Graphs."
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http://link.springer.com/article/10.1007/s00453-007-9064-z
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[2] de Pina, J. 1995. Applications of shortest path methods.
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Ph.D. thesis, University of Amsterdam, Netherlands
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See Also
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--------
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simple_cycles, cycle_basis
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"""
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# We first split the graph in commected subgraphs
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return sum(
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(_min_cycle_basis(G.subgraph(c), weight) for c in nx.connected_components(G)),
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[],
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)
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def _min_cycle_basis(comp, weight):
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cb = []
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# We extract the edges not in a spanning tree. We do not really need a
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# *minimum* spanning tree. That is why we call the next function with
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# weight=None. Depending on implementation, it may be faster as well
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spanning_tree_edges = list(nx.minimum_spanning_edges(comp, weight=None, data=False))
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edges_excl = [frozenset(e) for e in comp.edges() if e not in spanning_tree_edges]
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N = len(edges_excl)
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# We maintain a set of vectors orthogonal to sofar found cycles
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set_orth = [{edge} for edge in edges_excl]
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for k in range(N):
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# kth cycle is "parallel" to kth vector in set_orth
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new_cycle = _min_cycle(comp, set_orth[k], weight=weight)
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cb.append(list(set().union(*new_cycle)))
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# now update set_orth so that k+1,k+2... th elements are
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# orthogonal to the newly found cycle, as per [p. 336, 1]
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base = set_orth[k]
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set_orth[k + 1 :] = [
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orth ^ base if len(orth & new_cycle) % 2 else orth
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for orth in set_orth[k + 1 :]
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]
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return cb
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def _min_cycle(G, orth, weight=None):
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"""
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Computes the minimum weight cycle in G,
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orthogonal to the vector orth as per [p. 338, 1]
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"""
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T = nx.Graph()
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nodes_idx = {node: idx for idx, node in enumerate(G.nodes())}
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idx_nodes = {idx: node for node, idx in nodes_idx.items()}
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nnodes = len(nodes_idx)
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# Add 2 copies of each edge in G to T. If edge is in orth, add cross edge;
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|
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# otherwise in-plane edge
|
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for u, v, data in G.edges(data=True):
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uidx, vidx = nodes_idx[u], nodes_idx[v]
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|
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edge_w = data.get(weight, 1)
|
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|
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if frozenset((u, v)) in orth:
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T.add_edges_from(
|
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[(uidx, nnodes + vidx), (nnodes + uidx, vidx)], weight=edge_w
|
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)
|
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else:
|
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|
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T.add_edges_from(
|
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|
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[(uidx, vidx), (nnodes + uidx, nnodes + vidx)], weight=edge_w
|
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)
|
||
|
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|
||
|
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all_shortest_pathlens = dict(nx.shortest_path_length(T, weight=weight))
|
||
|
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cross_paths_w_lens = {
|
||
|
|
n: all_shortest_pathlens[n][nnodes + n] for n in range(nnodes)
|
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|
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}
|
||
|
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|
||
|
|
# Now compute shortest paths in T, which translates to cyles in G
|
||
|
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start = min(cross_paths_w_lens, key=cross_paths_w_lens.get)
|
||
|
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end = nnodes + start
|
||
|
|
min_path = nx.shortest_path(T, source=start, target=end, weight="weight")
|
||
|
|
|
||
|
|
# Now we obtain the actual path, re-map nodes in T to those in G
|
||
|
|
min_path_nodes = [node if node < nnodes else node - nnodes for node in min_path]
|
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|
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# Now remove the edges that occur two times
|
||
|
|
mcycle_pruned = _path_to_cycle(min_path_nodes)
|
||
|
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|
||
|
|
return {frozenset((idx_nodes[u], idx_nodes[v])) for u, v in mcycle_pruned}
|
||
|
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|
||
|
|
|
||
|
|
def _path_to_cycle(path):
|
||
|
|
"""
|
||
|
|
Removes the edges from path that occur even number of times.
|
||
|
|
Returns a set of edges
|
||
|
|
"""
|
||
|
|
edges = set()
|
||
|
|
for edge in pairwise(path):
|
||
|
|
# Toggle whether to keep the current edge.
|
||
|
|
edges ^= {edge}
|
||
|
|
return edges
|