# python – 创建“最小连接”有向无环图

> NetworkX有内置功能吗？

> [可选]仅在1的答案为“否”的情况下,那么实现此目的的最有效算法是什么(对于相当密集的图形)？

>节点是：

``````['termsequence', 'maximumdegree', 'emptymultigraph', 'minimum', 'multiset', 'walk', 'nonemptymultigraph', 'euleriantrail', 'nonnullmultigraph', 'cycle', 'loop', 'abwalk', 'endvertices', 'simplegraph', 'vertex', 'multipletrails', 'edge', 'set', 'stroll', 'union', 'trailcondition', 'nullmultigraph', 'trivialmultigraph', 'sequence', 'multiplepaths', 'path', 'degreevertex', 'onedgesonvertices', 'nontrivialmultigraph', 'adjacentedges', 'adjacentvertices', 'simpleedge', 'maximum', 'multipleloops', 'length', 'circuit', 'class', 'euleriangraph', 'incident', 'minimumdegree', 'orderedpair', 'unique', 'closedwalk', 'multipleedges', 'pathcondition', 'multigraph', 'trail']
``````

>边缘是：

``````[('termsequence', 'endvertices'), ('emptymultigraph', 'nonemptymultigraph'), ('minimum', 'minimumdegree'), ('multiset', 'trailcondition'), ('multiset', 'pathcondition'), ('multiset', 'multigraph'), ('walk', 'length'), ('walk', 'closedwalk'), ('walk', 'abwalk'), ('walk', 'trail'), ('walk', 'endvertices'), ('euleriantrail', 'euleriangraph'), ('loop', 'simplegraph'), ('loop', 'degreevertex'), ('loop', 'simpleedge'), ('loop', 'multipleloops'), ('endvertices', 'abwalk'), ('vertex', 'adjacentvertices'), ('vertex', 'onedgesonvertices'), ('vertex', 'walk'), ('vertex', 'adjacentedges'), ('vertex', 'multipleedges'), ('vertex', 'edge'), ('vertex', 'multipleloops'), ('vertex', 'degreevertex'), ('vertex', 'incident'), ('edge', 'adjacentvertices'), ('edge', 'onedgesonvertices'), ('edge', 'multipleedges'), ('edge', 'simpleedge'), ('edge', 'adjacentedges'), ('edge', 'loop'), ('edge', 'trailcondition'), ('edge', 'pathcondition'), ('edge', 'walk'), ('edge', 'incident'), ('set', 'onedgesonvertices'), ('set', 'edge'), ('union', 'multiplepaths'), ('union', 'multipletrails'), ('trailcondition', 'trail'), ('nullmultigraph', 'nonnullmultigraph'), ('sequence', 'walk'), ('sequence', 'endvertices'), ('path', 'cycle'), ('path', 'multiplepaths'), ('degreevertex', 'maximumdegree'), ('degreevertex', 'minimumdegree'), ('onedgesonvertices', 'multigraph'), ('maximum', 'maximumdegree'), ('circuit', 'euleriangraph'), ('class', 'multiplepaths'), ('class', 'multipletrails'), ('incident', 'adjacentedges'), ('incident', 'degreevertex'), ('incident', 'onedgesonvertices'), ('orderedpair', 'multigraph'), ('closedwalk', 'circuit'), ('closedwalk', 'cycle'), ('closedwalk', 'stroll'), ('pathcondition', 'path'), ('multigraph', 'euleriangraph'), ('multigraph', 'nullmultigraph'), ('multigraph', 'trivialmultigraph'), ('multigraph', 'nontrivialmultigraph'), ('multigraph', 'emptymultigraph'), ('multigraph', 'euleriantrail'), ('multigraph', 'simplegraph'), ('trail', 'path'), ('trail', 'circuit'), ('trail', 'multipletrails')]
``````

``````import networkx as nx
from collections import defaultdict

def remove_redundant_edges(G):
processed_child_count = defaultdict(int)  #when all of a nodes children are processed, we'll add it to nodes_to_process
descendants = defaultdict(set)            #all descendants of a node (including children)
out_degree = {node:G.out_degree(node) for node in G.nodes_iter()}
nodes_to_process = [node for node in G.nodes_iter() if out_degree[node]==0] #initially it's all nodes without children
while nodes_to_process:
next_nodes = []
for node in nodes_to_process:
'''when we enter this loop, the descendants of a node are known, except for direct children.'''
for child in G.neighbors(node):
if child in descendants[node]:  #if the child is already an indirect descendant, delete the edge
G.remove_edge(node,child)
else:                                    #otherwise add it to the descendants
for predecessor in G.predecessors(node):             #update all parents' indirect descendants
descendants[predecessor].update(descendants[node])
processed_child_count[predecessor]+=1            #we have processed one more child of this parent
if processed_child_count[predecessor] == out_degree[predecessor]:  #if all children processed, add to list for next iteration.
next_nodes.append(predecessor)
nodes_to_process=next_nodes
``````

``````G=nx.DiGraph()
G.add_nodes_from(['termsequence', 'maximumdegree', 'emptymultigraph', 'minimum', 'multiset', 'walk', 'nonemptymultigraph', 'euleriantrail', 'nonnullmultigraph', 'cycle', 'loop', 'abwalk', 'endvertices', 'simplegraph', 'vertex', 'multipletrails', 'edge', 'set', 'stroll', 'union', 'trailcondition', 'nullmultigraph', 'trivialmultigraph', 'sequence', 'multiplepaths', 'path', 'degreevertex', 'onedgesonvertices', 'nontrivialmultigraph', 'adjacentedges', 'adjacentvertices', 'simpleedge', 'maximum', 'multipleloops', 'length', 'circuit', 'class', 'euleriangraph', 'incident', 'minimumdegree', 'orderedpair', 'unique', 'closedwalk', 'multipleedges', 'pathcondition', 'multigraph', 'trail'])
G.add_edges_from([('termsequence', 'endvertices'), ('emptymultigraph', 'nonemptymultigraph'), ('minimum', 'minimumdegree'), ('multiset', 'trailcondition'), ('multiset', 'pathcondition'), ('multiset', 'multigraph'), ('walk', 'length'), ('walk', 'closedwalk'), ('walk', 'abwalk'), ('walk', 'trail'), ('walk', 'endvertices'), ('euleriantrail', 'euleriangraph'), ('loop', 'simplegraph'), ('loop', 'degreevertex'), ('loop', 'simpleedge'), ('loop', 'multipleloops'), ('endvertices', 'abwalk'), ('vertex', 'adjacentvertices'), ('vertex', 'onedgesonvertices'), ('vertex', 'walk'), ('vertex', 'adjacentedges'), ('vertex', 'multipleedges'), ('vertex', 'edge'), ('vertex', 'multipleloops'), ('vertex', 'degreevertex'), ('vertex', 'incident'), ('edge', 'adjacentvertices'), ('edge', 'onedgesonvertices'), ('edge', 'multipleedges'), ('edge', 'simpleedge'), ('edge', 'adjacentedges'), ('edge', 'loop'), ('edge', 'trailcondition'), ('edge', 'pathcondition'), ('edge', 'walk'), ('edge', 'incident'), ('set', 'onedgesonvertices'), ('set', 'edge'), ('union', 'multiplepaths'), ('union', 'multipletrails'), ('trailcondition', 'trail'), ('nullmultigraph', 'nonnullmultigraph'), ('sequence', 'walk'), ('sequence', 'endvertices'), ('path', 'cycle'), ('path', 'multiplepaths'), ('degreevertex', 'maximumdegree'), ('degreevertex', 'minimumdegree'), ('onedgesonvertices', 'multigraph'), ('maximum', 'maximumdegree'), ('circuit', 'euleriangraph'), ('class', 'multiplepaths'), ('class', 'multipletrails'), ('incident', 'adjacentedges'), ('incident', 'degreevertex'), ('incident', 'onedgesonvertices'), ('orderedpair', 'multigraph'), ('closedwalk', 'circuit'), ('closedwalk', 'cycle'), ('closedwalk', 'stroll'), ('pathcondition', 'path'), ('multigraph', 'euleriangraph'), ('multigraph', 'nullmultigraph'), ('multigraph', 'trivialmultigraph'), ('multigraph', 'nontrivialmultigraph'), ('multigraph', 'emptymultigraph'), ('multigraph', 'euleriantrail'), ('multigraph', 'simplegraph'), ('trail', 'path'), ('trail', 'circuit'), ('trail', 'multipletrails')])

print G.size()
>71
print G.order()
>47
descendants = {}  #for testing below
for node in G.nodes():
descendants[node] = nx.descendants(G,node)

remove_redundant_edges(G)  #this removes the edges

print G.size()  #lots of edges gone
>56
print G.order() #no nodes changed.
>47
newdescendants = {}  #for comparison with above
for node in G.nodes():
newdescendants[node] = nx.descendants(G,node)

for node in G.nodes():
if descendants[node] != newdescendants[node]:
print 'descendants changed!!'   #all nodes have the same descendants
for child in G.neighbors(node):
if len(list(nx.all_simple_paths(G,node, child)))>1:
print 'bad edge'  #no alternate path exists from a node to its child.
``````

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