NetworkX系列教程(10)-算法之一:最短路径问题

小书匠Graph图论

重头戏部分来了,写到这里我感觉得仔细认真点了,可能在NetworkX中,实现某些算法就一句话的事,但是这个算法是做什么的,用在什么地方,原理是怎么样的,不清除,所以,我决定先把图论中常用算法弄个明白在写这部分.

图论常用算法看我的博客:

下面我将使用NetworkX实现上面的算法,建议不清楚的部分打开两篇博客对照理解.

我将图论的经典问题及常用算法的总结写在下面两篇博客中:

图论---问题篇

图论---算法篇

目录:

• 11.Graph相关算法
  • 11.1最短路径
    • 11.1.1无向图和有向图
    • 11.1.2无权图
    • 11.1.3有权图(迪杰斯特拉)
    • 11.1.4贝尔曼-福特(Bellman-Ford)算法
    • 11.1.5检测负权重边
    • 11.1.6使用约翰逊(Johnson)的算法
    • 11.1.7弗洛伊德算法(Floyd-Warshall)
    • 11.1.8A*算法

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注意:如果代码出现找不库,请返回第一个教程,把库文件导入.

11.Graph相关算法

11.1最短路径

11.1.1无向图和有向图

1. #定义并画出该图 
   

   ---

2. G = nx.path_graph(5) 
   

   ---

3. nx.add_path(G,[0,5,2]) 
   

   ---

4. nx.add_path(G,[0,6,4]) 
   

   ---

5. nx.draw(G,with_labels=True) 
   

   ---

6. plt.title('无向图',fontproperties=myfont) 
   

   ---

7. plt.axis('on') 
   

   ---

8. plt.xticks([]) 
   

   ---

9. plt.yticks([]) 
   

   ---

10. plt.show() 
    

    ---

11. ---

12. #计算最短路径 
    

    ---

13. print('0节点到4节点最短路径: ',nx.shortest_path(G, source=0, target=4)) 
    

    ---

14. p1 = nx.shortest_path(G, source=0) 
    

    ---

15. print('0节点到所有节点最短路径: ',p1) 
    

    ---

16. ---

17. #计算图中所有的最短路径 
    

    ---

18. print('计算图中节点0到节点2的所有最短路径: ',[p for p in nx.all_shortest_paths(G, source=0, target=2)]) 
    

    ---

19. ---

20. #计算最短路径长度 
    

    ---

21. p2=nx.shortest_path_length(G, source=0, target=2) #最短路径长度 
    

    ---

22. p3=nx.average_shortest_path_length(G) #计算平均最短路径长度 
    

    ---

23. print('节点0到节点2的最短路径长度:',p2,' 平均最短路径长度: ',p3) 
    

    ---

24. ---

25. #检测是否有路径 
    

    ---

26. print('检测节点0到节点2是否有路径',nx.has_path(G,0,2)) 
    

    ---

无向图和有向图最短路径示例

输出:

1. 0节点到4节点最短路径: [0, 6, 4] 
   

   ---

2. 0节点到所有节点最短路径: {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [0, 6, 4], 5: [0, 5], 6: [0, 6]} 
   

   ---

3. 计算图中节点0到节点2的所有最短路径: [[0, 1, 2], [0, 5, 2]] 
   

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4. 节点0到节点2的最短路径长度: 2 平均最短路径长度: 1.8095238095238095 
   

   ---

5. 检测节点0到节点2是否有路径 True 
   

   ---

---

11.1.2无权图

1. G = nx.path_graph(3) 
   

   ---

2. nx.draw(G,with_labels=True) 
   

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3. plt.title('无权图',fontproperties=myfont) 
   

   ---

4. plt.axis('on') 
   

   ---

5. plt.xticks([]) 
   

   ---

6. plt.yticks([]) 
   

   ---

7. plt.show() 
   

   ---

8. ---

9. path1 = nx.single_source_shortest_path(G, 0) #计算当前源与所有可达节点的最短路径 
   

   ---

10. length1 = nx.single_source_shortest_path_length(G, 0) #计算当前源与所有可达节点的最短路径的长度 
    

    ---

11. path2 = dict(nx.all_pairs_shortest_path(G)) #计算graph两两节点之间的最短路径 
    

    ---

12. length2 = dict(nx.all_pairs_shortest_path_length(G)) #计算graph两两节点之间的最短路径的长度 
    

    ---

13. prede1=nx.predecessor(G, 0) #返回G中从源到所有节点最短路径的前驱 
    

    ---

14. ---

15. print('当前源与所有可达节点的最短路径: ',path1,'\n当前源与所有可达节点的最短路径的长度: ',length1) 
    

    ---

16. print('\ngraph两两节点之间的最短路径: ',path2,'\ngraph两两节点之间的最短路径的长度: ',length2) 
    

    ---

17. print('\nG中从源到所有节点最短路径的前驱: ',prede1) 
    

    ---

无权图

输出:

1. 当前源与所有可达节点的最短路径: {0: [0], 1: [0, 1], 2: [0, 1, 2]}  
   

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2. 当前源与所有可达节点的最短路径的长度: {0: 0, 1: 1, 2: 2} 
   

   ---

3. graph两两节点之间的最短路径: {0: {0: [0], 1: [0, 1], 2: [0, 1, 2]}, 1: {0: [1, 0], 1: [1], 2: [1, 2]}, 2: {0: [2, 1, 0], 1: [2, 1], 2: [2]}}  
   

   ---

4. graph两两节点之间的最短路径的长度: {0: {0: 0, 1: 1, 2: 2}, 1: {0: 1, 1: 0, 2: 1}, 2: {0: 2, 1: 1, 2: 0}} 
   

   ---

5. G中从源到所有节点最短路径的前驱: {0: [], 1: [0], 2: [1]} 
   

   ---

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11.1.3有权图(迪杰斯特拉)

1. G = nx.path_graph(5, create_using = nx.DiGraph())  
   

   ---

2. nx.draw(G,with_labels=True) 
   

   ---

3. plt.title('有向图',fontproperties=myfont) 
   

   ---

4. plt.axis('on') 
   

   ---

5. plt.xticks([]) 
   

   ---

6. plt.yticks([]) 
   

   ---

7. plt.show() 
   

   ---

8. ---

9. #计算加权图最短路径长度和前驱 
   

   ---

10. pred, dist = nx.dijkstra_predecessor_and_distance(G, 0) 
    

    ---

11. print('\n加权图最短路径长度和前驱: ',pred, dist) 
    

    ---

12. ---

13. #返回G中从源到目标的最短加权路径,要求边权重必须为数值 
    

    ---

14. print('\nG中从源0到目标4的最短加权路径: ',nx.dijkstra_path(G,0,4)) 
    

    ---

15. print('\nG中从源0到目标4的最短加权路径的长度: ',nx.dijkstra_path_length(G,0,4)) #最短路径长度 
    

    ---

16. ---

17. #单源节点最短加权路径和长度。 
    

    ---

18. length1, path1 = nx.single_source_dijkstra(G, 0) 
    

    ---

19. print('\n单源节点最短加权路径和长度: ',length1, path1) 
    

    ---

20. #下面两条和是前面的分解 
    

    ---

21. # path2=nx.single_source_dijkstra_path(G,0) 
    

    ---

22. # length2 = nx.single_source_dijkstra_path_length(G, 0) 
    

    ---

23. #print(length1,'$', path1,'$',length2,'$',path2) 
    

    ---

24. ---

25. #多源节点最短加权路径和长度。 
    

    ---

26. path1 = nx.multi_source_dijkstra_path(G, {0, 4}) 
    

    ---

27. length1 = nx.multi_source_dijkstra_path_length(G, {0, 4}) 
    

    ---

28. ---

29. print('\n多源节点最短加权路径和长度:', path1,length1) 
    

    ---

30. ---

31. #两两节点之间最短加权路径和长度。 
    

    ---

32. path1 = dict(nx.all_pairs_dijkstra_path(G)) 
    

    ---

33. length1 = dict(nx.all_pairs_dijkstra_path_length(G)) 
    

    ---

34. print('\n两两节点之间最短加权路径和长度: ',path1,length1) 
    

    ---

35. ---

36. #双向搜索的迪杰斯特拉 
    

    ---

37. length, path = nx.bidirectional_dijkstra(G, 0, 4) 
    

    ---

38. print('\n双向搜索的迪杰斯特拉:',length, path) 
    

    ---

迪杰斯特拉算法使用

输出:

1. 加权图最短路径长度和前驱: {0: [], 1: [0], 2: [1], 3: [2], 4: [3]} {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} 
   

   ---

2. G中从源0到目标4的最短加权路径: [0, 1, 2, 3, 4] 
   

   ---

3. G中从源0到目标4的最短加权路径的长度: 4 
   

   ---

4. 单源节点最短加权路径和长度: {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [0, 1, 2, 3, 4]} 
   

   ---

5. 多源节点最短加权路径和长度: {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [4]} {0: 0, 1: 1, 2: 2, 3: 3, 4: 0} 
   

   ---

6. 两两节点之间最短加权路径和长度: {0: {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [0, 1, 2, 3, 4]}, 1: {1: [1], 2: [1, 2], 3: [1, 2, 3], 4: [1, 2, 3, 4]}, 2: {2: [2], 3: [2, 3], 4: [2, 3, 4]}, 3: {3: [3], 4: [3, 4]}, 4: {4: [4]}} {0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 4}, 1: {1: 0, 2:1, 3: 2, 4: 3}, 2: {2: 0, 3: 1, 4: 2}, 3: {3: 0, 4: 1}, 4: {4: 0}} 
   

   ---

7. 双向搜索的迪杰斯特拉: 4 [0, 1, 2, 3, 4] 
   

   ---

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11.1.4贝尔曼-福特(Bellman-Ford)算法

1. G = nx.path_graph(5, create_using = nx.DiGraph())  
   

   ---

2. nx.draw(G,with_labels=True) 
   

   ---

3. plt.title('有权图',fontproperties=myfont) 
   

   ---

4. plt.axis('on') 
   

   ---

5. plt.xticks([]) 
   

   ---

6. plt.yticks([]) 
   

   ---

7. plt.show() 
   

   ---

8. ---

9. print('G中从源到目标的最短加权路径: ',nx.bellman_ford_path(G, 0, 4)) 
   

   ---

10. print('\nG中从源到目标的最短加权路径的长度:',nx.bellman_ford_path_length(G,0,4)) 
    

    ---

11. ---

12. path1=nx.single_source_bellman_ford_path(G,0) 
    

    ---

13. length1 = dict(nx.single_source_bellman_ford_path_length(G, 0)) 
    

    ---

14. print('\n单源节点最短加权路径和长度: ',path1,'\n单源节点最短加权路径和长度: ',length1) 
    

    ---

15. ---

16. path2 = dict(nx.all_pairs_bellman_ford_path(G)) 
    

    ---

17. length2 = dict(nx.all_pairs_bellman_ford_path_length(G)) 
    

    ---

18. print('\n两两节点之间最短加权路径和长度: ',path2,length2) 
    

    ---

19. ---

20. length, path = nx.single_source_bellman_ford(G, 0) 
    

    ---

21. pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0) 
    

    ---

22. print('\n加权图最短路径长度和前驱: ',pred,dist) 
    

    ---

贝尔曼-福特(Bellman-Ford)算法使用示例

输出:

1. G中从源到目标的最短加权路径: [0, 1, 2, 3, 4] 
   

   ---

2. G中从源到目标的最短加权路径的长度: 4 
   

   ---

3. 单源节点最短加权路径和长度: {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [0, 1, 2, 3, 4]}  
   

   ---

4. 单源节点最短加权路径和长度: {0: 0, 1: 1, 2: 2, 3: 3, 4: 4} 
   

   ---

5. 两两节点之间最短加权路径和长度: {0: {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [0, 1, 2, 3, 4]}, 1: {1: [1], 2: [1, 2], 3: [1, 2, 3], 4: 
   

   ---

[1, 2, 3, 4]}, 2: {2: [2], 3: [2, 3], 4: [2, 3, 4]}, 3: {3: [3], 4:

[3, 4]}, 4: {4: [4]}} {0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 4}, 1: {1: 0, 2:

1, 3: 2, 4: 3}, 2: {2: 0, 3: 1, 4: 2}, 3: {3: 0, 4: 1}, 4: {4: 0}}

加权图最短路径长度和前驱: {0: [None], 1: [0], 2: [1], 3: [2], 4: [3]} {0: 0, 1: 1, 2: 2, 3: 3, 4: 4}

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11.1.5检测负权重边

1. #定义并画出该图 
   

   ---

2. G = nx.cycle_graph(5, create_using = nx.DiGraph()) 
   

   ---

3. ---

4. #添加负权重边前后 
   

   ---

5. print(nx.negative_edge_cycle(G)) 
   

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6. G[1][2]['weight'] = -7 
   

   ---

7. print(nx.negative_edge_cycle(G)) 
   

   ---

输出:

1. False 
   

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2. True 
   

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11.1.6使用约翰逊(Johnson)的算法

1. #生成graph 
   

   ---

2. G = nx.DiGraph() 
   

   ---

3. G.add_weighted_edges_from([('0', '3', 3), ('0', '1', -5),('0', '2', 2), ('1', '2', 4), ('2', '3', 1)]) 
   

   ---

4. ---

5. #边和节点信息 
   

   ---

6. edge_labels = nx.get_edge_attributes(G,'weight')  
   

   ---

7. labels={'0':'0','1':'1','2':'2','3':'3'} 
   

   ---

8. ---

9. #生成节点位置  
   

   ---

10. pos=nx.spring_layout(G)  
    

    ---

11. ---

12. #把节点画出来  
    

    ---

13. nx.draw_networkx_nodes(G,pos,node_color='g',node_size=500,alpha=0.8)  
    

    ---

14. ---

15. #把边画出来  
    

    ---

16. nx.draw_networkx_edges(G,pos,width=1.0,alpha=0.5,edge_color='b')  
    

    ---

17. ---

18. #把节点的标签画出来  
    

    ---

19. nx.draw_networkx_labels(G,pos,labels,font_size=16)  
    

    ---

20. ---

21. #把边权重画出来  
    

    ---

22. nx.draw_networkx_edge_labels(G, pos, edge_labels)  
    

    ---

23. ---

24. #显示graph 
    

    ---

25. plt.title('有权图',fontproperties=myfont) 
    

    ---

26. plt.axis('on') 
    

    ---

27. plt.xticks([]) 
    

    ---

28. plt.yticks([]) 
    

    ---

29. plt.show() 
    

    ---

30. ---

31. #使用johnson算法计算最短路径 
    

    ---

32. paths = nx.johnson(G, weight='weight') 
    

    ---

33. ---

34. print(paths) 
    

    ---

约翰逊(Johnson)的算法使用示例

输出:

1. {'2': {'2': ['2'], '3': ['2', '3']}, '3': {'3': ['3']}, '0': {'2': ['0', '1', '2'], '3': ['0', '1', '2', '3'], '0': ['0'], '1': ['0','1']}, '1': {'2': ['1', '2'], '3': ['1', '2', '3'], '1': ['1']}} 
   

   ---

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11.1.7弗洛伊德算法(Floyd-Warshall)

1. #使用Floyd算法找到所有对最短路径长度。 
   

   ---

2. G = nx.DiGraph() 
   

   ---

3. G.add_weighted_edges_from([('0', '3', 3), ('0', '1', -5),('0', '2', 2), ('1', '2', 4), ('2', '3', 1)]) 
   

   ---

4. ---

5. #边和节点信息 
   

   ---

6. edge_labels = nx.get_edge_attributes(G,'weight')  
   

   ---

7. labels={'0':'0','1':'1','2':'2','3':'3'} 
   

   ---

8. ---

9. #生成节点位置  
   

   ---

10. pos=nx.spring_layout(G)  
    

    ---

11. ---

12. #把节点画出来  
    

    ---

13. nx.draw_networkx_nodes(G,pos,node_color='g',node_size=500,alpha=0.8)  
    

    ---

14. ---

15. #把边画出来  
    

    ---

16. nx.draw_networkx_edges(G,pos,width=1.0,alpha=0.5,edge_color='b')  
    

    ---

17. ---

18. #把节点的标签画出来  
    

    ---

19. nx.draw_networkx_labels(G,pos,labels,font_size=16)  
    

    ---

20. ---

21. #把边权重画出来  
    

    ---

22. nx.draw_networkx_edge_labels(G, pos, edge_labels)  
    

    ---

23. ---

24. #显示graph 
    

    ---

25. plt.title('有权图',fontproperties=myfont) 
    

    ---

26. plt.axis('on') 
    

    ---

27. plt.xticks([]) 
    

    ---

28. plt.yticks([]) 
    

    ---

29. plt.show() 
    

    ---

30. ---

31. #计算最短路径长度 
    

    ---

32. lenght=nx.floyd_warshall(G, weight='weight') 
    

    ---

33. ---

34. #计算最短路径上的前驱与路径长度 
    

    ---

35. predecessor,distance1=nx.floyd_warshall_predecessor_and_distance(G, weight='weight') 
    

    ---

36. ---

37. #计算两两节点之间的最短距离,并以numpy矩阵形式返回 
    

    ---

38. distance2=nx.floyd_warshall_numpy(G, weight='weight') 
    

    ---

39. ---

40. print(list(lenght)) 
    

    ---

41. print(predecessor) 
    

    ---

42. print(list(distance1)) 
    

    ---

43. print(distance2) 
    

    ---

弗洛伊德算法(Floyd-Warshall)使用示例

输出:

1. ['2', '3', '0', '1'] 
   

   ---

2. {'2': {'3': '2'}, '0': {'2': '1', '3': '2', '1': '0'}, '1': {'2': '1', '3': '2'}} 
   

   ---

3. ['2', '3', '0', '1'] 
   

   ---

4. [[ 0. 1. inf inf] 
   

   ---

5. [inf 0. inf inf] 
   

   ---

6. [-1. 0. 0. -5.] 
   

   ---

7. [ 4. 5. inf 0.]] 
   

   ---

注:输出中的矩阵不是按照节点0,1,2,3排序,而是2,1,3,0,即如图:

两点之间的最短距离

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11.1.8A*算法

1. G = nx.path_graph(5) 
   

   ---

2. ---

3. #显示graph 
   

   ---

4. nx.draw(G,with_labels=True) 
   

   ---

5. plt.title('有x向图',fontproperties=myfont) 
   

   ---

6. plt.axis('on') 
   

   ---

7. plt.xticks([]) 
   

   ---

8. plt.yticks([]) 
   

   ---

9. plt.show() 
   

   ---

10. ---

11. #直接输出路径和长度 
    

    ---

12. print(nx.astar_path(G, 0, 4)) 
    

    ---

13. print(nx.astar_path_length(G, 0, 4)) 
    

    ---

A*算法

输出:

1. [0, 1, 2, 3, 4] 
   

   ---

2. 4 
   

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