{"id":2673,"date":"2020-01-05T14:16:02","date_gmt":"2020-01-05T06:16:02","guid":{"rendered":"http:\/\/www.sniper97.cn\/?p=2673"},"modified":"2020-01-05T14:16:02","modified_gmt":"2020-01-05T06:16:02","slug":"%e3%80%90%e7%ae%97%e6%b3%95%e5%88%86%e6%9e%90%e4%b8%8e%e8%ae%be%e8%ae%a1%e3%80%91%e6%9c%80%e5%a4%a7%e6%b5%81","status":"publish","type":"post","link":"http:\/\/www.sniper97.cn\/index.php\/note\/introduction_to_algorithms\/2673\/","title":{"rendered":"\u3010\u7b97\u6cd5\u5bfc\u8bba\u3011\u6700\u5927\u6d41"},"content":{"rendered":"\n

\u6b63\u5982\u53ef\u4ee5\u901a\u8fc7\u5c06\u9053\u8def\u4ea4\u901a\u56fe\u6a21\u578b\u5316\u4e3a\u6709\u5411\u56fe\u6765\u627e\u5230\u4ece\u4e00\u4e2a\u57ce\u5e02\u5230\u53e6\u4e00\u4e2a\u57ce\u5e02\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5c06\u4e00\u4e2a\u6709\u5411\u56fe\u770b\u4f5c\u662f\u4e00\u4e2a\u201c\u6d41\u7f51\u7edc\u201d\u5e76\u4f7f\u7528\u5b83\u6765\u89e3\u51b3\u5173\u4e8e\u7269\u6599\u6d41\u52a8\u65b9\u9762\u7684\u95ee\u9898\u3002<\/p>\n\n\n

\u6211\u4eec\u628a\u6d41\u7f51\u7edc\u4e2d\u6bcf\u6761\u6709\u5411\u8fb9\u770b\u4f5c\u662f\u7269\u6599\u6d41\u52a8\u7684\u901a\u9053\uff0c\u662f\u7269\u6599\u6d41\u7ecf\u8be5\u901a\u9053\u7684\u6700\u5927\u901f\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u8ba1\u7b97\u4ece\u6e90\u8282\u70b9\u8fd0\u884c\u7269\u6599\u5230\u67d0\u4e2a\u70b9\u7684\u6700\u5927\u901f\u7387\uff0c\u8fd9\u662f\u6d41\u7f51\u7edc\u6709\u5173\u7684\u6240\u6709\u95ee\u9898\u4e2d\u6700\u7b80\u5355\u7684\u95ee\u9898\u4e4b\u4e00\u3002<\/p>\n\n\n

1.\u6d41\u7f51\u7edc<\/h2>\n\n\n

\u5982\u4e0b\u56fe\u5373\u4e3as->t\u7684\u6700\u5927\u6d41\u7f51\u7edc\uff08\u9700\u8981\u6ce8\u610f\uff0cb\u56fe\u8868\u793a \u5f53\u524d\u6d41\u91cf\/\u6700\u5927\u6d41\u91cf\uff0c\u5e76\u4e0d\u4ee3\u8868\u9664\u6cd5\uff09\u3002<\/p>\n\n\n

\"\"<\/figure><\/div>\n\n\n

\u4f7f\u7528\u53cd\u5e73\u884c\u8fb9\u6765\u6a21\u62df\u95ee\u9898<\/h3>\n\n\n

\u5bf9\u4e8e\u6709\u53cc\u5411\u8fb9\u7684\u60c5\u51b5\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u865a\u62df\u7ed3\u70b9\u6765\u6a21\u62df\u8be5\u95ee\u9898<\/p>\n\n\n

\"\"<\/figure><\/div>\n\n\n

\u5177\u6709\u591a\u4e2a\u6e90\u7ed3\u70b9\u548c\u591a\u4e2a\u6c47\u70b9\u7684\u7f51\u7edc<\/h3>\n\n\n

\u4e00\u4e2a\u6700\u5927\u6d41\u95ee\u9898\u53ef\u80fd\u6709\u51e0\u4e2a\u6e90\u7ed3\u70b9\u548c\u51e0\u4e2a\u6c47\u70b9\uff08\u7ec8\u70b9\uff09\u3002<\/p>\n\n\n

\u5728\u5177\u6709\u591a\u4e2a\u6e90\u8282\u70b9\u548c\u591a\u4e2a\u6c47\u70b9\u7684\u7f51\u7edc\u4e2d\uff0c\u6211\u4eec\u6dfb\u52a0\u4e00\u4e2a\u8d85\u7ea7\u6e90\u7ed3\u70b9<\/strong>\u548c\u4e00\u4e2a\u8d85\u7ea7\u6c47\u70b9<\/strong>\uff0c\u5b83\u4eec\u5206\u522b\u8fde\u63a5\u7740\u5404\u4e2a\u6e90\u8282\u70b9\u548c\u5404\u4e2a\u6c47\u70b9\u3002<\/p>\n\n\n

\"\"<\/figure><\/div>\n\n\n

2.Ford-Fulkerson\u65b9\u6cd5<\/h2>\n\n\n

\u672c\u8282\u8ba8\u8bba\u7528\u6765\u89e3\u51b3\u6700\u5927\u6d41\u95ee\u9898\u7684Ford-Fullkerson\u65b9\u6cd5\u3002\u4e4b\u6240\u4ee5\u79f0\u5176\u4e3a\u65b9\u6cd5\u800c\u4e0d\u662f\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a Ford-Fullkerson \u5305\u542b\u4e86\u51e0\u79cd\u8fd0\u884c\u65f6\u95f4\u5404\u4e0d\u76f8\u540c\u7684\u5177\u4f53\u5b9e\u73b0\u3002<\/p>\n\n\n

\u5728\u8fd9\u4e4b\u524d\u5148\u89e3\u91ca\u51e0\u4e2a\u6982\u5ff5<\/p>\n\n\n

\u6b8b\u5b58\u7f51\u7edc<\/h3>\n\n\n

\u4e00\u4e2a\u663e\u793a\u7ba1\u9053\u8fd8\u6709\u591a\u5c11\u5269\u4f59\u80fd\u529b\u7684\u7f51\u7edc\uff0c\u540c\u65f6\u4e3a\u4e86\u65b9\u4fbf\u8ba1\u7b97\uff0c\u8fd8\u6709\u4e00\u6761\u76f8\u53cd\u7684\u8fb9\u7528\u6765\u8bb0\u5f55\u53cd\u5411\u5269\u4f59\u591a\u5c11\uff0c\u6bd4\u5982\u70b9i\u5230\u70b9j\u603b\u80fd\u529b20\u7684\u7ba1\u9053\uff0c\u4f7f\u7528\u4e8615\uff0c\u5219i->j\u4e3a15\uff0cj->i\u4e3a5\u3002<\/p>\n\n\n

\"\"<\/figure><\/div>\n\n\n

\u589e\u5e7f\u8def\u5f84<\/h3>\n\n\n

\u589e\u5e7f\u8def\u5f84\u4e3a\u6b8b\u5b58\u7f51\u7edc\u4e0a\u4e00\u6761\u53ef\u4ee5\u4ece\u7ed3\u70b9\u901a\u5f80\u6c47\u70b9\u7684\u4e00\u6761\u8def\u5f84\u3002<\/p>\n\n\n

\"\"<\/figure><\/div>\n\n\n

\u8fd9\u6761\u8def\u4e0a\u6700\u5c0f\u7684\u6b8b\u5b58\u5bb9\u91cf\u662f4\uff0c\u6211\u4eec\u79f0\u4e00\u6761\u589e\u5e7f\u8def\u5f84p\u4e0a\u80fd\u591f\u4e3a\u6bcf\u6761\u8fb9\u589e\u52a0\u7684\u6d41\u91cf\u7684\u6700\u5927\u503c\u4e3a\u8def\u5f84p\u7684\u6b8b\u5b58\u5bb9\u91cf<\/strong>\u3002<\/p>\n\n\n

\u57fa\u672c\u7684Ford-Fulkerson\u7b97\u6cd5<\/h3>\n\n\n

\u5728Ford-Fulkerson\u65b9\u6cd5\u7684\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0c\u5bfb\u627e\u67d0\u6761\u589e\u5e7f\u8def\u5f84p\uff0c\u7136\u540e\u4f7f\u7528p\u6765\u5bf9\u6d41f\u8fdb\u884c\u4fee\u6539\u3002<\/p>\n\n\n

\u6211\u4eec\u4ece\u6b8b\u5b58\u7f51\u7edc\u4e2d\u5bfb\u627e\u4e00\u6761\u589e\u5e7f\u8def\u7ecf\uff0c\u7136\u540e\u5728\u6d41\u7f51\u7edc\u4e2d\u589e\u52a0\u6216\u51cf\u5c11\uff08\u589e\u52a0\u6216\u51cf\u5c11\u57fa\u4e8e\u589e\u5e7f\u8def\u5f84\u7684\u8fb9\u662f\u4e0d\u662f\u6e90\u7f51\u7edc\u4e2d\u7684\u8fb9\uff0c\u56e0\u4e3a\u6b8b\u5b58\u7f51\u7edc\u662f\u53cc\u5411\u8fb9\uff09\u589e\u5e7f\u77e9\u9635\u7684\u6b8b\u5b58\u5bb9\u91cf\uff0c\u76f4\u81f3\u6b8b\u5b58\u7f51\u7edc\u4e2d\u6ca1\u6709\u589e\u5e7f\u77e9\u9635\u4e3a\u6b62\uff0c\u5219\u6d41\u7f51\u7edc\u5373\u4e3a\u6700\u5927\u6d41\u7f51\u7edc\u3002<\/p>\n\n\n

\u4ee3\u7801\uff08\u7531\u4e8e\u8fd9\u6bb5\u4f7f\u7528\u4e86networkx\u6765\u8fdb\u884c\u56fe\u8fd0\u7b97\uff0c\u6240\u4ee5\u4e0a\u5b8c\u6574\u4ee3\u7801\uff09\uff1a<\/p>\n\n\n

# -*- coding:utf-8 -*-
import networkx as nx
import matplotlib.pyplot as plt


class FordFulkerson():
    def __init__(self):
        self.inf = float('inf')
        pass

    def ford_fulkerson_method(self, flowGraph, remainGraph, from_node, to_node):
        while nx.has_path(remainGraph, from_node, to_node):
            # \u5bfb\u627e\u8def\u5f84p\u7684\u6700\u5c0f\u6743\u91cd
            path = nx.shortest_path(remainGraph, from_node, to_node)
            min_weight = float('inf')
            min_weight_idx = 0
            for i in range(1, len(path)):
                if min_weight > remainGraph[path[i - 1]][path[i]]['weight']:
                    min_weight = remainGraph[path[i - 1]][path[i]]['weight']
                    min_weight_idx = i

            def judge_in_graph(G, from_node, to_node):
                for n, nbrs in G.adjacency():
                    for nbr, eattr in nbrs.items():
                        data = eattr['weight']
                        if n is from_node and nbr is to_node:
                            return True
                return False

            # \u6240\u6709\u6b8b\u5b58\u7f51\u7edc\u8def\u5f84\u51cf\uff08\u52a0\uff09\u53bb\u6700\u5c0f\u503c\uff0c\u6d41\u7f51\u7edc\u52a0\uff08\u51cf\uff09\u4e0a\u6700\u5c0f\u503c
            for i in range(1, len(path)):
                if judge_in_graph(flowGraph, path[i - 1], path[i]):
                    if remainGraph[path[i - 1]][path[i]]['weight'] - min_weight == 0:
                        remainGraph.remove_edge(path[i - 1], path[i])
                    else:
                        remainGraph[path[i - 1]][path[i]]['weight'] -= min_weight
                    flowGraph[path[i - 1]][path[i]]['weight'] += min_weight
                else:
                    if judge_in_graph(remainGraph, path[i - 1], path[i]):
                        remainGraph[path[i - 1]][path[i]]['weight'] -= min_weight
                    else:
                        remainGraph.add_edge(path[i - 1], path[i], weight=min_weight)
                    flowGraph[path[i - 1]][path[i]]['weight'] -= min_weight

        return flowGraph, remainGraph


if __name__ == '__main__':
    # \u6d41\u56fe
    flowGraph = nx.DiGraph()
    flowGraph.add_nodes_from(['s', 'a', 'b', 'c', 'd', 't'])
    flowGraph.add_edges_from(
        [('s', 'a', {'weight': 0}), ('s', 'd', {'weight': 0}), ('a', 'b', {'weight': 0}),
         ('b', 't', {'weight': 0}),
         ('d', 'c', {'weight': 0}), ('c', 'b', {'weight': 0}), ('b', 'd', {'weight': 0}),
         ('c', 't', {'weight': 0}),
         ('d', 'a', {'weight': 0})])

    # \u6b8b\u5b58\u7f51\u7edc
    remainGraph = nx.DiGraph()
    remainGraph.add_nodes_from(['s', 'a', 'b', 'c', 'd', 't'])
    # remainGraph.add_edges_from(
    #     [('s', 'a', {'weight': 16}), ('a', 's', {'weight': 0}), ('s', 'd', {'weight': 13}),
    #      ('d', 's', {'weight': 0}),
    #      ('a', 'b', {'weight': 12}), ('b', 'a', {'weight': 0}), ('b', 't', {'weight': 20}),
    #      ('t', 'b', {'weight': 0}),
    #      ('d', 'c', {'weight': 14}), ('c', 'd', {'weight': 0}), ('c', 'b', {'weight': 7}),
    #      ('b', 'c', {'weight': 0}),
    #      ('b', 'd', {'weight': 9}), ('d', 'b', {'weight': 0}), ('c', 't', {'weight': 4}),
    #      ('t', 'c', {'weight': 0}),
    #      ('d', 'a', {'weight': 4}), ('a', 'd', {'weight': 0})])

    remainGraph.add_edges_from(
        [('s', 'a', {'weight': 16}), ('s', 'd', {'weight': 13}),
         ('a', 'b', {'weight': 12}), ('b', 't', {'weight': 20}),
         ('d', 'c', {'weight': 14}), ('c', 'b', {'weight': 7}),
         ('b', 'd', {'weight': 9}), ('c', 't', {'weight': 4}),
         ('d', 'a', {'weight': 4})])

    aaa = nx.shortest_path(flowGraph, 's', 't')
    print(aaa[1])
    print(type(aaa[1]))
    # print(nx.shortest_path(flowGraph, 's', 't'))
    print(flowGraph['s'][aaa[1]]['weight'])
    # remainGraph.add_edge('s', 'e', weight=10)
    print(FordFulkerson().ford_fulkerson_method(flowGraph, remainGraph, 's', 't'))
    # nx.shortest_path()
    # nx.draw(remainGraph)
    # plt.show()
<\/pre>\n\n\n
\u6700\u540e\u53ef\u4ee5\u7b97\u51fa\u7ed3\u679c\uff1a<\/p>\n\n\n
\"\"<\/figure><\/div>\n\n\n
\u56e0\u6b64\uff0c\u6700\u4f18\u7684\u8fd9\u6761\u7ba1\u9053\u4f53\u7cfb \u6700\u591a\u8f93\u9001s\u7684\u51fa\u5ea6\u6743\u91cd\u2014\u201412+11=23\u4e2a\u5355\u4f4d\u7684\u8d27\u7269\u3002<\/p>\n\n\n
3.\u6700\u5927\u4e8c\u5206\u5339\u914d<\/h2>\n\n\n
\u6700\u5927\u4e8c\u5206\u5339\u914d\u662f\u5728\u56fe\u4e2d\u627e\u5230\u6700\u5927\u7684\u4e8c\u5206\u5339\u914d\u9879\u3002<\/p>\n\n\n
\u5982\u56fe\uff0c\u4e3a\u57fa\u6570\u4e3a2\u7684\u6700\u5927\u5339\u914d<\/p>\n\n\n
\"\"<\/figure><\/div>\n\n\n
\u57fa\u6570\u4e3a3\u7684\u6700\u5927\u5339\u914d<\/p>\n\n\n
\"\"<\/figure><\/div>\n\n\n
\u56e0\u6b64\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8f6c\u6362\u6210\u6700\u5927\u6d41\u7b97\u6cd5\uff0c\u6dfb\u52a0\u4e00\u4e2a\u8d85\u7ea7\u6e90\u7ed3\u70b9\u548c\u8d85\u7ea7\u6c47\u70b9\u5e76\u8d4b\u4e88\u6bcf\u6761\u8fb9\u7684\u6743\u91cd\u4e3a1\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u95ee\u9898\u8f6c\u6362\u4e3a\u6700\u5927\u6d41\u95ee\u9898\u3002<\/p>\n\n\n
\"\"<\/figure><\/div>\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
\u6b63\u5982\u53ef\u4ee5\u901a\u8fc7\u5c06\u9053\u8def\u4ea4\u901a\u56fe\u6a21\u578b\u5316\u4e3a\u6709\u5411\u56fe\u6765\u627e\u5230\u4ece\u4e00\u4e2a\u57ce\u5e02\u5230\u53e6\u4e00\u4e2a\u57ce\u5e02\u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5c06\u4e00\u4e2a\u6709 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_mi_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[11],"tags":[],"views":9300,"_links":{"self":[{"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/posts\/2673"}],"collection":[{"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2673"}],"version-history":[{"count":0,"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/posts\/2673\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2673"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2673"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.sniper97.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2673"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}