Source code for networkx.algorithms.simple_paths

# -*- coding: utf-8 -*-
#    Copyright (C) 2012 by
#    Sergio Nery Simoes <sergionery@gmail.com>
#    All rights reserved.
#    BSD license.
import collections
from heapq import heappush, heappop
from itertools import count

import networkx as nx
from networkx.utils import not_implemented_for
from networkx.utils import pairwise

__author__ = """\n""".join(['Sérgio Nery Simões <sergionery@gmail.com>',
                            'Aric Hagberg <aric.hagberg@gmail.com>',
                            'Andrey Paramonov',
                            'Jordi Torrents <jordi.t21@gmail.com>'])

__all__ = [
    'all_simple_paths',
    'is_simple_path',
    'shortest_simple_paths',
]


[docs]def is_simple_path(G, nodes): """Returns True if and only if the given nodes form a simple path in `G`. A *simple path* in a graph is a nonempty sequence of nodes in which no node appears more than once in the sequence, and each adjacent pair of nodes in the sequence is adjacent in the graph. Parameters ---------- nodes : list A list of one or more nodes in the graph `G`. Returns ------- bool Whether the given list of nodes represents a simple path in `G`. Notes ----- A list of zero nodes is not a path and a list of one node is a path. Here's an explanation why. This function operates on *node paths*. One could also consider *edge paths*. There is a bijection between node paths and edge paths. The *length of a path* is the number of edges in the path, so a list of nodes of length *n* corresponds to a path of length *n* - 1. Thus the smallest edge path would be a list of zero edges, the empty path. This corresponds to a list of one node. To convert between a node path and an edge path, you can use code like the following:: >>> from networkx.utils import pairwise >>> nodes = [0, 1, 2, 3] >>> edges = list(pairwise(nodes)) >>> edges [(0, 1), (1, 2), (2, 3)] >>> nodes = [edges[0][0]] + [v for u, v in edges] >>> nodes [0, 1, 2, 3] Examples -------- >>> G = nx.cycle_graph(4) >>> nx.is_simple_path(G, [2, 3, 0]) True >>> nx.is_simple_path(G, [0, 2]) False """ # The empty list is not a valid path. Could also return # NetworkXPointlessConcept here. if len(nodes) == 0: return False # If the list is a single node, just check that the node is actually # in the graph. if len(nodes) == 1: return nodes[0] in G # Test that no node appears more than once, and that each # adjacent pair of nodes is adjacent. return (len(set(nodes)) == len(nodes) and all(v in G[u] for u, v in pairwise(nodes)))
[docs]def all_simple_paths(G, source, target, cutoff=None): """Generate all simple paths in the graph G from source to target. A simple path is a path with no repeated nodes. Parameters ---------- G : NetworkX graph source : node Starting node for path target : nodes Single node or iterable of nodes at which to end path cutoff : integer, optional Depth to stop the search. Only paths of length <= cutoff are returned. Returns ------- path_generator: generator A generator that produces lists of simple paths. If there are no paths between the source and target within the given cutoff the generator produces no output. Examples -------- This iterator generates lists of nodes:: >>> G = nx.complete_graph(4) >>> for path in nx.all_simple_paths(G, source=0, target=3): ... print(path) ... [0, 1, 2, 3] [0, 1, 3] [0, 2, 1, 3] [0, 2, 3] [0, 3] You can generate only those paths that are shorter than a certain length by using the `cutoff` keyword argument:: >>> paths = nx.all_simple_paths(G, source=0, target=3, cutoff=2) >>> print(list(paths)) [[0, 1, 3], [0, 2, 3], [0, 3]] To get each path as the corresponding list of edges, you can use the :func:`networkx.utils.pairwise` helper function:: >>> paths = nx.all_simple_paths(G, source=0, target=3) >>> for path in map(nx.utils.pairwise, paths): ... print(list(path)) [(0, 1), (1, 2), (2, 3)] [(0, 1), (1, 3)] [(0, 2), (2, 1), (1, 3)] [(0, 2), (2, 3)] [(0, 3)] Pass an iterable of nodes as target to generate all paths ending in any of several nodes:: >>> G = nx.complete_graph(4) >>> for path in nx.all_simple_paths(G, source=0, target=[3, 2]): ... print(path) ... [0, 1, 2] [0, 1, 2, 3] [0, 1, 3] [0, 1, 3, 2] [0, 2] [0, 2, 1, 3] [0, 2, 3] [0, 3] [0, 3, 1, 2] [0, 3, 2] Iterate over each path from the root nodes to the leaf nodes in a directed acyclic graph using a functional programming approach:: >>> from itertools import chain >>> from itertools import product >>> from itertools import starmap >>> from functools import partial >>> >>> chaini = chain.from_iterable >>> >>> G = nx.DiGraph([(0, 1), (1, 2), (0, 3), (3, 2)]) >>> roots = (v for v, d in G.in_degree() if d == 0) >>> leaves = (v for v, d in G.out_degree() if d == 0) >>> all_paths = partial(nx.all_simple_paths, G) >>> list(chaini(starmap(all_paths, product(roots, leaves)))) [[0, 1, 2], [0, 3, 2]] The same list computed using an iterative approach:: >>> G = nx.DiGraph([(0, 1), (1, 2), (0, 3), (3, 2)]) >>> roots = (v for v, d in G.in_degree() if d == 0) >>> leaves = (v for v, d in G.out_degree() if d == 0) >>> all_paths = [] >>> for root in roots: ... for leaf in leaves: ... paths = nx.all_simple_paths(G, root, leaf) ... all_paths.extend(paths) >>> all_paths [[0, 1, 2], [0, 3, 2]] Iterate over each path from the root nodes to the leaf nodes in a directed acyclic graph passing all leaves together to avoid unnecessary compute:: >>> G = nx.DiGraph([(0, 1), (2, 1), (1, 3), (1, 4)]) >>> roots = (v for v, d in G.in_degree() if d == 0) >>> leaves = [v for v, d in G.out_degree() if d == 0] >>> all_paths = [] >>> for root in roots: ... paths = nx.all_simple_paths(G, root, leaves) ... all_paths.extend(paths) >>> all_paths [[0, 1, 3], [0, 1, 4], [2, 1, 3], [2, 1, 4]] Notes ----- This algorithm uses a modified depth-first search to generate the paths [1]_. A single path can be found in $O(V+E)$ time but the number of simple paths in a graph can be very large, e.g. $O(n!)$ in the complete graph of order $n$. References ---------- .. [1] R. Sedgewick, "Algorithms in C, Part 5: Graph Algorithms", Addison Wesley Professional, 3rd ed., 2001. See Also -------- all_shortest_paths, shortest_path """ if source not in G: raise nx.NodeNotFound('source node %s not in graph' % source) if target in G: targets = {target} else: try: targets = set(target) except TypeError: raise nx.NodeNotFound('target node %s not in graph' % target) if source in targets: return [] if cutoff is None: cutoff = len(G) - 1 if cutoff < 1: return [] if G.is_multigraph(): return _all_simple_paths_multigraph(G, source, targets, cutoff) else: return _all_simple_paths_graph(G, source, targets, cutoff)
def _all_simple_paths_graph(G, source, targets, cutoff): visited = collections.OrderedDict.fromkeys([source]) stack = [iter(G[source])] while stack: children = stack[-1] child = next(children, None) if child is None: stack.pop() visited.popitem() elif len(visited) < cutoff: if child in visited: continue if child in targets: yield list(visited) + [child] visited[child] = None if targets - set(visited.keys()): # expand stack until find all targets stack.append(iter(G[child])) else: visited.popitem() # maybe other ways to child else: # len(visited) == cutoff: for target in (targets & (set(children) | {child})) - set(visited.keys()): yield list(visited) + [target] stack.pop() visited.popitem() def _all_simple_paths_multigraph(G, source, targets, cutoff): visited = collections.OrderedDict.fromkeys([source]) stack = [(v for u, v in G.edges(source))] while stack: children = stack[-1] child = next(children, None) if child is None: stack.pop() visited.popitem() elif len(visited) < cutoff: if child in visited: continue if child in targets: yield list(visited) + [child] visited[child] = None if targets - set(visited.keys()): stack.append((v for u, v in G.edges(child))) else: visited.popitem() else: # len(visited) == cutoff: for target in targets - set(visited.keys()): count = ([child] + list(children)).count(target) for i in range(count): yield list(visited) + [target] stack.pop() visited.popitem()
[docs]@not_implemented_for('multigraph') def shortest_simple_paths(G, source, target, weight=None): """Generate all simple paths in the graph G from source to target, starting from shortest ones. A simple path is a path with no repeated nodes. If a weighted shortest path search is to be used, no negative weights are allowed. Parameters ---------- G : NetworkX graph source : node Starting node for path target : node Ending node for path weight : string Name of the edge attribute to be used as a weight. If None all edges are considered to have unit weight. Default value None. Returns ------- path_generator: generator A generator that produces lists of simple paths, in order from shortest to longest. Raises ------ NetworkXNoPath If no path exists between source and target. NetworkXError If source or target nodes are not in the input graph. NetworkXNotImplemented If the input graph is a Multi[Di]Graph. Examples -------- >>> G = nx.cycle_graph(7) >>> paths = list(nx.shortest_simple_paths(G, 0, 3)) >>> print(paths) [[0, 1, 2, 3], [0, 6, 5, 4, 3]] You can use this function to efficiently compute the k shortest/best paths between two nodes. >>> from itertools import islice >>> def k_shortest_paths(G, source, target, k, weight=None): ... return list(islice(nx.shortest_simple_paths(G, source, target, weight=weight), k)) >>> for path in k_shortest_paths(G, 0, 3, 2): ... print(path) [0, 1, 2, 3] [0, 6, 5, 4, 3] Notes ----- This procedure is based on algorithm by Jin Y. Yen [1]_. Finding the first $K$ paths requires $O(KN^3)$ operations. See Also -------- all_shortest_paths shortest_path all_simple_paths References ---------- .. [1] Jin Y. Yen, "Finding the K Shortest Loopless Paths in a Network", Management Science, Vol. 17, No. 11, Theory Series (Jul., 1971), pp. 712-716. """ if source not in G: raise nx.NodeNotFound('source node %s not in graph' % source) if target not in G: raise nx.NodeNotFound('target node %s not in graph' % target) if weight is None: length_func = len shortest_path_func = _bidirectional_shortest_path else: def length_func(path): return sum(G.adj[u][v][weight] for (u, v) in zip(path, path[1:])) shortest_path_func = _bidirectional_dijkstra listA = list() listB = PathBuffer() prev_path = None while True: if not prev_path: length, path = shortest_path_func(G, source, target, weight=weight) listB.push(length, path) else: ignore_nodes = set() ignore_edges = set() for i in range(1, len(prev_path)): root = prev_path[:i] root_length = length_func(root) for path in listA: if path[:i] == root: ignore_edges.add((path[i - 1], path[i])) try: length, spur = shortest_path_func(G, root[-1], target, ignore_nodes=ignore_nodes, ignore_edges=ignore_edges, weight=weight) path = root[:-1] + spur listB.push(root_length + length, path) except nx.NetworkXNoPath: pass ignore_nodes.add(root[-1]) if listB: path = listB.pop() yield path listA.append(path) prev_path = path else: break
class PathBuffer(object): def __init__(self): self.paths = set() self.sortedpaths = list() self.counter = count() def __len__(self): return len(self.sortedpaths) def push(self, cost, path): hashable_path = tuple(path) if hashable_path not in self.paths: heappush(self.sortedpaths, (cost, next(self.counter), path)) self.paths.add(hashable_path) def pop(self): (cost, num, path) = heappop(self.sortedpaths) hashable_path = tuple(path) self.paths.remove(hashable_path) return path def _bidirectional_shortest_path(G, source, target, ignore_nodes=None, ignore_edges=None, weight=None): """Returns the shortest path between source and target ignoring nodes and edges in the containers ignore_nodes and ignore_edges. This is a custom modification of the standard bidirectional shortest path implementation at networkx.algorithms.unweighted Parameters ---------- G : NetworkX graph source : node starting node for path target : node ending node for path ignore_nodes : container of nodes nodes to ignore, optional ignore_edges : container of edges edges to ignore, optional weight : None This function accepts a weight argument for convenience of shortest_simple_paths function. It will be ignored. Returns ------- path: list List of nodes in a path from source to target. Raises ------ NetworkXNoPath If no path exists between source and target. See Also -------- shortest_path """ # call helper to do the real work results = _bidirectional_pred_succ(G, source, target, ignore_nodes, ignore_edges) pred, succ, w = results # build path from pred+w+succ path = [] # from w to target while w is not None: path.append(w) w = succ[w] # from source to w w = pred[path[0]] while w is not None: path.insert(0, w) w = pred[w] return len(path), path def _bidirectional_pred_succ(G, source, target, ignore_nodes=None, ignore_edges=None): """Bidirectional shortest path helper. Returns (pred,succ,w) where pred is a dictionary of predecessors from w to the source, and succ is a dictionary of successors from w to the target. """ # does BFS from both source and target and meets in the middle if ignore_nodes and (source in ignore_nodes or target in ignore_nodes): raise nx.NetworkXNoPath("No path between %s and %s." % (source, target)) if target == source: return ({target: None}, {source: None}, source) # handle either directed or undirected if G.is_directed(): Gpred = G.predecessors Gsucc = G.successors else: Gpred = G.neighbors Gsucc = G.neighbors # support optional nodes filter if ignore_nodes: def filter_iter(nodes): def iterate(v): for w in nodes(v): if w not in ignore_nodes: yield w return iterate Gpred = filter_iter(Gpred) Gsucc = filter_iter(Gsucc) # support optional edges filter if ignore_edges: if G.is_directed(): def filter_pred_iter(pred_iter): def iterate(v): for w in pred_iter(v): if (w, v) not in ignore_edges: yield w return iterate def filter_succ_iter(succ_iter): def iterate(v): for w in succ_iter(v): if (v, w) not in ignore_edges: yield w return iterate Gpred = filter_pred_iter(Gpred) Gsucc = filter_succ_iter(Gsucc) else: def filter_iter(nodes): def iterate(v): for w in nodes(v): if (v, w) not in ignore_edges \ and (w, v) not in ignore_edges: yield w return iterate Gpred = filter_iter(Gpred) Gsucc = filter_iter(Gsucc) # predecesssor and successors in search pred = {source: None} succ = {target: None} # initialize fringes, start with forward forward_fringe = [source] reverse_fringe = [target] while forward_fringe and reverse_fringe: if len(forward_fringe) <= len(reverse_fringe): this_level = forward_fringe forward_fringe = [] for v in this_level: for w in Gsucc(v): if w not in pred: forward_fringe.append(w) pred[w] = v if w in succ: # found path return pred, succ, w else: this_level = reverse_fringe reverse_fringe = [] for v in this_level: for w in Gpred(v): if w not in succ: succ[w] = v reverse_fringe.append(w) if w in pred: # found path return pred, succ, w raise nx.NetworkXNoPath("No path between %s and %s." % (source, target)) def _bidirectional_dijkstra(G, source, target, weight='weight', ignore_nodes=None, ignore_edges=None): """Dijkstra's algorithm for shortest paths using bidirectional search. This function returns the shortest path between source and target ignoring nodes and edges in the containers ignore_nodes and ignore_edges. This is a custom modification of the standard Dijkstra bidirectional shortest path implementation at networkx.algorithms.weighted Parameters ---------- G : NetworkX graph source : node Starting node. target : node Ending node. weight: string, optional (default='weight') Edge data key corresponding to the edge weight ignore_nodes : container of nodes nodes to ignore, optional ignore_edges : container of edges edges to ignore, optional Returns ------- length : number Shortest path length. Returns a tuple of two dictionaries keyed by node. The first dictionary stores distance from the source. The second stores the path from the source to that node. Raises ------ NetworkXNoPath If no path exists between source and target. Notes ----- Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed. In practice bidirectional Dijkstra is much more than twice as fast as ordinary Dijkstra. Ordinary Dijkstra expands nodes in a sphere-like manner from the source. The radius of this sphere will eventually be the length of the shortest path. Bidirectional Dijkstra will expand nodes from both the source and the target, making two spheres of half this radius. Volume of the first sphere is pi*r*r while the others are 2*pi*r/2*r/2, making up half the volume. This algorithm is not guaranteed to work if edge weights are negative or are floating point numbers (overflows and roundoff errors can cause problems). See Also -------- shortest_path shortest_path_length """ if ignore_nodes and (source in ignore_nodes or target in ignore_nodes): raise nx.NetworkXNoPath("No path between %s and %s." % (source, target)) if source == target: return (0, [source]) # handle either directed or undirected if G.is_directed(): Gpred = G.predecessors Gsucc = G.successors else: Gpred = G.neighbors Gsucc = G.neighbors # support optional nodes filter if ignore_nodes: def filter_iter(nodes): def iterate(v): for w in nodes(v): if w not in ignore_nodes: yield w return iterate Gpred = filter_iter(Gpred) Gsucc = filter_iter(Gsucc) # support optional edges filter if ignore_edges: if G.is_directed(): def filter_pred_iter(pred_iter): def iterate(v): for w in pred_iter(v): if (w, v) not in ignore_edges: yield w return iterate def filter_succ_iter(succ_iter): def iterate(v): for w in succ_iter(v): if (v, w) not in ignore_edges: yield w return iterate Gpred = filter_pred_iter(Gpred) Gsucc = filter_succ_iter(Gsucc) else: def filter_iter(nodes): def iterate(v): for w in nodes(v): if (v, w) not in ignore_edges \ and (w, v) not in ignore_edges: yield w return iterate Gpred = filter_iter(Gpred) Gsucc = filter_iter(Gsucc) push = heappush pop = heappop # Init: Forward Backward dists = [{}, {}] # dictionary of final distances paths = [{source: [source]}, {target: [target]}] # dictionary of paths fringe = [[], []] # heap of (distance, node) tuples for # extracting next node to expand seen = [{source: 0}, {target: 0}] # dictionary of distances to # nodes seen c = count() # initialize fringe heap push(fringe[0], (0, next(c), source)) push(fringe[1], (0, next(c), target)) # neighs for extracting correct neighbor information neighs = [Gsucc, Gpred] # variables to hold shortest discovered path #finaldist = 1e30000 finalpath = [] dir = 1 while fringe[0] and fringe[1]: # choose direction # dir == 0 is forward direction and dir == 1 is back dir = 1 - dir # extract closest to expand (dist, _, v) = pop(fringe[dir]) if v in dists[dir]: # Shortest path to v has already been found continue # update distance dists[dir][v] = dist # equal to seen[dir][v] if v in dists[1 - dir]: # if we have scanned v in both directions we are done # we have now discovered the shortest path return (finaldist, finalpath) for w in neighs[dir](v): if(dir == 0): # forward if G.is_multigraph(): minweight = min((dd.get(weight, 1) for k, dd in G[v][w].items())) else: minweight = G[v][w].get(weight, 1) vwLength = dists[dir][v] + minweight # G[v][w].get(weight,1) else: # back, must remember to change v,w->w,v if G.is_multigraph(): minweight = min((dd.get(weight, 1) for k, dd in G[w][v].items())) else: minweight = G[w][v].get(weight, 1) vwLength = dists[dir][v] + minweight # G[w][v].get(weight,1) if w in dists[dir]: if vwLength < dists[dir][w]: raise ValueError( "Contradictory paths found: negative weights?") elif w not in seen[dir] or vwLength < seen[dir][w]: # relaxing seen[dir][w] = vwLength push(fringe[dir], (vwLength, next(c), w)) paths[dir][w] = paths[dir][v] + [w] if w in seen[0] and w in seen[1]: # see if this path is better than than the already # discovered shortest path totaldist = seen[0][w] + seen[1][w] if finalpath == [] or finaldist > totaldist: finaldist = totaldist revpath = paths[1][w][:] revpath.reverse() finalpath = paths[0][w] + revpath[1:] raise nx.NetworkXNoPath("No path between %s and %s." % (source, target))