{"id":58575,"date":"2025-08-16T12:49:48","date_gmt":"2025-08-16T04:49:48","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/58575.html"},"modified":"2025-08-16T12:49:48","modified_gmt":"2025-08-16T04:49:48","slug":"20250812-%e5%8d%95%e6%ba%90%e6%9c%80%e7%9f%ad%e8%b7%af%e6%80%bb%e7%bb%93","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/58575.html","title":{"rendered":"20250812 \u5355\u6e90\u6700\u77ed\u8def\u603b\u7ed3"},"content":{"rendered":"

\u5f15\u5165<\/h3>\n

\u6700\u77ed\u8def\u95ee\u9898\u662f\u4e00\u7c7b\u7ecf\u5178\u7684\u56fe\u8bba\u95ee\u9898，\u4e3b\u8981\u662f\u6c42\u89e3\u6709\u6743\u56fe\u4e0a\u4e24\u70b9\u95f4\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n

\u6211\u4eec\u4eca\u5929\u4e3b\u8981\u7684\u5185\u5bb9\u662fdij\u548cbf\u4e24\u79cd\u5355\u6e90\u6700\u77ed\u8def\u7b97\u6cd5\u3002<\/p>\n

\u6811<\/h3>\n

\u6811\u6709\u65e0\u73af\u3001\u8054\u901a\u7684\u7279\u70b9，\u6240\u4ee5\u4efb\u610f\u4e24\u70b9\u95f4\u7b80\u5355\u8def\u5f84\u552f\u4e00，\u4ee5\u8d77\u70b9\u4f5c\u4e3a\u6839\u8282\u70b9DFS\/BFS\u6c42\u8ddd\u79bb\u5373\u53ef\u3002<\/p>\n

\u65e0（\u7b49）\u6743\u56fe<\/h3>\n

\u65e0\u6743\u56fe\u4e0a\u6bcf\u6761\u8fb9\u7684\u8fb9\u6743\u76f8\u540c，\u6240\u4ee5bfs\u5373\u53ef\u627e\u5230\u4ece\u8d77\u70b9\u51fa\u53d1\u5230\u4efb\u610f\u4e00\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n

\u975e\u8d1f\u6743\u56fe<\/h3>\n

\u5bf9\u4e8e\u975e\u8d1f\u6743\u56fe，\u6211\u4eec\u4e00\u822c\u4f7f\u7528dijkstra\u7b97\u6cd5\u6c42\u89e3\u5355\u6e90\u6700\u77ed\u8def\u3002<\/p>\n

\u8fc7\u7a0b<\/h4>\n

dij\u662f\u4e00\u79cd\u70b9\u6838\u5fc3\u7684\u5355\u6e90\u6700\u77ed\u8def\u7b97\u6cd5，\u6838\u5fc3\u601d\u60f3\u662f：<\/p>\n

1\u3001\u627e\u5230\u6ca1\u6709\u786e\u5b9a\u6700\u77ed\u8def\u7684\u70b9\u91cc\u9762，\u8ddd\u79bb\u8d77\u70b9\u6700\u8fd1\u7684\u70b9u\u3002<\/p>\n

2\u3001\u7528\u8d77\u70b9\u5230\u70b9u\u7684\u8ddd\u79bbdis[u]，\u5c1d\u8bd5\u66f4\u65b0u\u7684\u90bb\u5c45\u4eec\u7684\u6700\u77ed\u8ddd\u79bb\u3002<\/p>\n

\u91cd\u590d\u4ee5\u4e0a\u64cd\u4f5c\u76f4\u5230\u6240\u6709\u70b9\u7684\u6700\u77ed\u8ddd\u79bb\u90fd\u88ab\u786e\u5b9a\u3002<\/p>\n

\u8bb0dis[i]dis[i]<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>i<\/span>]<\/span><\/span><\/span><\/span><\/span> \u8868\u793a\u5f53\u524d\u60c5\u51b5\u4e0b\u4ece\u8d77\u70b9\u51fa\u53d1\u5230ii<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u77ed\u8ddd\u79bb，\u5047\u8bbe\u6709\u4e00\u6761\u4eceuu<\/span><\/span>u<\/span><\/span><\/span><\/span><\/span>\u5230vv<\/span><\/span>v<\/span><\/span><\/span><\/span><\/span>，\u8fb9\u6743\u4e3aww<\/span><\/span>w<\/span><\/span><\/span><\/span><\/span>\u7684\u8fb9，\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u5c1d\u8bd5\u7528dis[u]+wdis[u] + w<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>u<\/span>]<\/span><\/span>+<\/span><\/span><\/span><\/span>w<\/span><\/span><\/span><\/span><\/span>\u6765\u66f4\u65b0dis[v]dis[v]<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>v<\/span>]<\/span><\/span><\/span><\/span><\/span>，\u8fd9\u4e2a\u64cd\u4f5c\u5c31\u53eb\u677e\u5f1b\u3002<\/p>\n

\u8bc1\u660e<\/h4>\n

\u8bb0\u4ece\u8d77\u70b9\u5230\u70b9u\u7684\u6700\u77ed\u8def\u4e3aD(u)D(u)<\/span><\/span>D<\/span>(<\/span>u<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5982\u679c\u6211\u4eec\u6bcf\u6b21\u786e\u5b9a\u7684\u70b9\u5f53\u65f6\u7684dis[u]=D(u)dis[u] = D(u)<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>u<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>u<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u90a3\u4e48\u6700\u7ec8\u6240\u6709\u7684dis[i]=D(i)dis[i] = D(i)<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>i<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>i<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002\u8bc1\u660e\u8fd9\u4e00\u70b9，\u5c31\u53ef\u4ee5\u8bc1\u660edij\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u3002<\/p>\n

\u4f7f\u7528\u53cd\u8bc1\u6cd5\u8bc1\u660e：<\/p>\n

\u5047\u8bbe\u5b58\u5728\u82e5\u5e72\u4e2a\u70b9\u88ab\u786e\u5b9a\u65f6disdis<\/span><\/span>d<\/span>i<\/span>s<\/span><\/span><\/span><\/span><\/span>\u5927\u4e8eDD<\/span><\/span>D<\/span><\/span><\/span><\/span><\/span>，\u8bb0\u5176\u4e2d\u7b2c\u4e00\u4e2a\u88ab\u786e\u5b9a\u7684\u70b9\u4e3axx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

xx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u663e\u7136\u4e0d\u4e3a\u8d77\u70b9，\u56e0\u4e3adis[st]=D(st)=0dis[st] = D(st) = 0<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>s<\/span>t<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>s<\/span>t<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5047\u8bbe\u5230xx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u77ed\u8def\u5f84\u662fst\u2192a\u2192b\u2192xst \\\\rightarrow a \\\\rightarrow b\\\\rightarrow x<\/span><\/span>s<\/span>t<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>a<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>b<\/span><\/span>\u2192<\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>，\u4e00\u5b9a\u5b58\u5728\u8fd9\u6761\u8def\u5f84，\u5426\u5219dis[x]=D(x)=INFdis[x] = D(x) = INF<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>x<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span>NF<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2dbb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u662fss<\/span><\/span>s<\/span><\/span><\/span><\/span><\/span>\u5230xx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u77ed\u8def\u5f84\u4e0a\u7b2c\u4e00\u4e2a\u672a\u88ab\u786e\u5b9a\u7684\u70b9<\/p>\n

\u5982\u679c\u8def\u5f84\u4e0a\u6240\u6709\u70b9\u90fd\u88ab\u786e\u5b9a，\u5c31\u6709dis[x]=D(x)dis[x] = D(x)<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>x<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u6240\u4ee5\u8def\u5f84\u4e0a\u4e00\u5b9a\u6709\u5c1a\u672a\u88ab\u786e\u5b9a\u7684\u70b9\u3002<\/p>\n

aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>\u662fstst<\/span><\/span>s<\/span>t<\/span><\/span><\/span><\/span><\/span>\u5230bb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u77ed\u8def\u5f84\u4e0a\u7684\u524d\u9a71\u8282\u70b9<\/p>\n

\u56e0\u4e3abb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u662f\u7b2c\u4e00\u4e2a\u672a\u88ab\u786e\u5b9a\u7684\u70b9，\u6240\u4ee5aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>\u4e00\u5b9a\u88ab\u786e\u5b9a\u3002（aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>\u53ef\u80fd\u7b49\u4e8estst<\/span><\/span>s<\/span>t<\/span><\/span><\/span><\/span><\/span>）<\/p>\n

\u56e0\u4e3axx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u662f\u7b2c\u4e00\u4e2adisdis<\/span><\/span>d<\/span>i<\/span>s<\/span><\/span><\/span><\/span><\/span>\u5927\u4e8eDD<\/span><\/span>D<\/span><\/span><\/span><\/span><\/span>\u7684\u70b9，\u6240\u4ee5\u5df2\u7ecf\u88ab\u786e\u5b9a\u7684aa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>\u6ee1\u8db3dis[a]=D(a)dis[a] = D(a)<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>a<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>a<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u4e3aaa<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span>\u662f\u5230bb<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u7684\u6700\u77ed\u8def\u5f84\u4e0a\u7684\u524d\u9a71\u8282\u70b9，\u6240\u4ee5D(b)=dis[b]=D(a)+w(a,b)D(b) = dis[b] = D(a) + w(a, b)<\/span><\/span>D<\/span>(<\/span>b<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>b<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>a<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>w<\/span>(<\/span>a<\/span>,<\/span><\/span>b<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u4e3axx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u662f\u88ab\u786e\u5b9a\u7684\u70b9，\u6240\u4ee5\u6b64\u65f6dis[x]dis[x]<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>x<\/span>]<\/span><\/span><\/span><\/span><\/span>\u5e94\u5f53\u5c0f\u4e8e\u7b49\u4e8e\u6240\u6709\u672a\u88ab\u786e\u5b9a\u7684\u70b9\u7684disdis<\/span><\/span>d<\/span>i<\/span>s<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u53c8\u6709D(x)=D(b)+w(b,x)D(x) = D(b) + w(b, x)<\/span><\/span>D<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>b<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>w<\/span>(<\/span>b<\/span>,<\/span><\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2dw(b,x)>=0w(b, x) >= 0<\/span><\/span>w<\/span>(<\/span>b<\/span>,<\/span><\/span>x<\/span>)<\/span><\/span>>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，<\/p>\n

\u7efc\u4e0a，dis[x]>D(x)=D(b)+w(b,x)>=D(b)=dis[b]dis[x] > D(x) = D(b) + w(b, x) >= D(b) = dis[b]<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>x<\/span>]<\/span><\/span>><\/span><\/span><\/span><\/span>D<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>b<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>w<\/span>(<\/span>b<\/span>,<\/span><\/span>x<\/span>)<\/span><\/span>>=<\/span><\/span><\/span><\/span>D<\/span>(<\/span>b<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>b<\/span>]<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u63a8\u51fa\u77db\u76fe，\u6240\u4ee5\u4e0d\u5b58\u5728\u70b9xx<\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u88ab\u786e\u5b9a\u65f6dis[x]>D[x]dis[x] > D[x]<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>x<\/span>]<\/span><\/span>><\/span><\/span><\/span><\/span>D<\/span>[<\/span>x<\/span>]<\/span><\/span><\/span><\/span><\/span>，\u539f\u7ed3\u8bba\u6210\u7acb\u3002<\/p>\n

\u5b9e\u73b0<\/h4>\n

\u6734\u7d20dij<\/p>\n

vector <<\/span>int<\/span>><\/span> dis,<\/span> vis;<\/span>
\nvector <<\/span>pair<<\/span>int<\/span>,<\/span> int<\/span>><\/span> ><\/span> ve[<\/span>N]<\/span>;<\/span>
\nvoid<\/span> dij<\/span>(<\/span>int<\/span> st)<\/span> {<\/span>
\n dis =<\/span> vector<\/span><<\/span>int<\/span>><\/span><\/span><\/span>(<\/span>n +<\/span> 1<\/span>,<\/span> INF)<\/span>;<\/span>
\n vis =<\/span> vector<\/span><<\/span>int<\/span>><\/span><\/span><\/span>(<\/span>n +<\/span> 1<\/span>,<\/span> 0<\/span>)<\/span>;<\/span>
\n dis[<\/span>st]<\/span> =<\/span> 0<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> _i =<\/span> 0<\/span>;<\/span> _i <<\/span> n;<\/span> _i++<\/span>)<\/span> {<\/span>
\n int<\/span> id =<\/span> 0<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 1<\/span>;<\/span> i <=<\/span> n;<\/span> i++<\/span>)<\/span> {<\/span>
\n if<\/span> (<\/span>vis[<\/span>i]<\/span> ==<\/span> 0<\/span> &&<\/span> dis[<\/span>i]<\/span> <<\/span> dis[<\/span>id]<\/span>)<\/span> {<\/span>
\n id =<\/span> i;<\/span>
\n }<\/span>
\n }<\/span>
\n if<\/span> (<\/span>id ==<\/span> 0<\/span>)<\/span> return<\/span>;<\/span>
\n vis[<\/span>id]<\/span> =<\/span> 1<\/span>;<\/span>
\n for<\/span> (<\/span>auto<\/span> p :<\/span> ve[<\/span>id]<\/span>)<\/span> {<\/span>
\n int<\/span> nxt =<\/span> p.<\/span>first,<\/span> w =<\/span> p.<\/span>second;<\/span>
\n dis[<\/span>nxt]<\/span> =<\/span> min<\/span>(<\/span>dis[<\/span>nxt]<\/span>,<\/span> dis[<\/span>id]<\/span> +<\/span> w)<\/span>;<\/span>
\n }<\/span>
\n }<\/span>
\n}<\/span><\/p>\n

\u5806\u4f18\u5316dij<\/p>\n

vector <<\/span>int<\/span>><\/span> dis;<\/span>
\nvector <<\/span>pair<<\/span>int<\/span>,<\/span> int<\/span>><\/span> ><\/span> ve[<\/span>N]<\/span>;<\/span>
\nvoid<\/span> dij<\/span>(<\/span>int<\/span> st)<\/span> {<\/span>
\n dis =<\/span> vector<\/span><<\/span>int<\/span>><\/span><\/span><\/span>(<\/span>n +<\/span> 1<\/span>,<\/span> INF)<\/span>;<\/span>
\n dis[<\/span>st]<\/span> =<\/span> 0<\/span>;<\/span>
\n priority_queue<<\/span>pair<<\/span>int<\/span>,<\/span>int<\/span>><\/span>,<\/span>vector<<\/span>pair<<\/span>int<\/span>,<\/span>int<\/span>><\/span> ><\/span>,<\/span>greater<<\/span>pair<<\/span>int<\/span>,<\/span>int<\/span>><\/span> ><\/span> ><\/span> q;<\/span>
\n q.<\/span>push<\/span>(<\/span>{<\/span>0<\/span>,<\/span> st}<\/span>)<\/span>;<\/span>
\n while<\/span> (<\/span>q.<\/span>size<\/span>(<\/span>)<\/span>)<\/span> {<\/span>
\n int<\/span> d =<\/span> q.<\/span>top<\/span>(<\/span>)<\/span>.<\/span>first;<\/span>
\n int<\/span> now =<\/span> q.<\/span>top<\/span>(<\/span>)<\/span>.<\/span>second;<\/span>
\n q.<\/span>pop<\/span>(<\/span>)<\/span>;<\/span>
\n if<\/span> (<\/span>dis[<\/span>now]<\/span> <<\/span> d)<\/span> continue<\/span>;<\/span>
\n for<\/span> (<\/span>auto<\/span> p :<\/span> ve[<\/span>now]<\/span>)<\/span> {<\/span>
\n int<\/span> w =<\/span> p.<\/span>first,<\/span> nxt =<\/span> p.<\/span>second;<\/span>
\n if<\/span> (<\/span>dis[<\/span>nxt]<\/span> ><\/span> dis[<\/span>now]<\/span> +<\/span> w)<\/span> {<\/span>
\n dis[<\/span>nxt]<\/span> =<\/span> dis[<\/span>now]<\/span> +<\/span> w;<\/span>
\n q.<\/span>push<\/span>(<\/span>{<\/span>dis[<\/span>nxt]<\/span>,<\/span> nxt}<\/span>)<\/span>;<\/span>
\n }<\/span>
\n }<\/span>
\n }<\/span>
\n}<\/span><\/p>\n

\u4ed6\u4eec\u7684\u65f6\u7a7a\u590d\u6742\u5ea6\u5206\u522b\u662f\u591a\u5c11？<\/p>\n

\u6734\u7d20 \u65f6\u95f4O(n2+m)O(n^2+m)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>m<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5806\u4f18\u5316 \u65f6\u95f4O((n+m)logm)O((n+m)logm)<\/span><\/span>O<\/span>((<\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>m<\/span>)<\/span>l<\/span>o<\/span>g<\/span>m<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5\u4ec0\u4e48\u60c5\u51b5\u4e0b\u7528\u6734\u7d20dij，\u4ec0\u4e48\u60c5\u51b5\u4e0b\u7528\u5806\u4f18\u5316dij？<\/p>\n

\u7a00\u758f\u56fe \u5806\u4f18\u5316dij<\/p>\n

\u7a20\u5bc6\u56fe \u6734 \u7d20 dij<\/p>\n

\u4e00\u822c\u56fe<\/h3>\n

\u5982\u679c\u56fe\u4e0a\u6709\u8d1f\u6743\u8fb9，dij\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u5c31\u4e0d\u80fd\u4fdd\u8bc1\u4e86，\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528bellman-ford\u7b97\u6cd5\u6765\u6c42\u89e3\u5355\u6e90\u6700\u77ed\u8def，\u6211\u4eec\u5e38\u8bf4\u7684SPFA\u7b97\u6cd5\u5c31\u662fbellman\u7684\u961f\u5217\u4f18\u5316\u3002<\/p>\n

\u8fc7\u7a0b<\/h4>\n

bellman\u7b97\u6cd5\u662f\u4e00\u79cd\u8fb9\u6838\u5fc3\u7684\u6700\u77ed\u8def\u7b97\u6cd5\u3002<\/p>\n

\u548c\u70b9\u6838\u5fc3\u7684dij\u4e0d\u540c，bellman\u5728\u677e\u5f1b\u65f6\u4e0d\u662f\u4ee5\u70b9\u4f5c\u4e3a\u5355\u4f4d\u6765\u677e\u5f1b，\u800c\u662f\u5728\u4e00\u8f6e\u677e\u5f1b\u4e2d\u4f9d\u6b21\u677e\u5f1b\u6240\u6709\u8fb9，\u76f4\u5230\u627e\u5230\u6240\u6709\u70b9\u7684\u6700\u77ed\u8def（\u6216\u53d1\u73b0\u6709\u8d1f\u73af）\u4e3a\u6b62\u3002<\/p>\n

\u677e\u5f1b\u64cd\u4f5c\u4e0edij\u4e2d\u76f8\u540c，\u5047\u8bbe\u6709\u4e00\u6761\u4eceuu<\/span><\/span>u<\/span><\/span><\/span><\/span><\/span>\u5230vv<\/span><\/span>v<\/span><\/span><\/span><\/span><\/span>，\u8fb9\u6743\u4e3aww<\/span><\/span>w<\/span><\/span><\/span><\/span><\/span>\u7684\u8fb9，\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u5c1d\u8bd5\u7528dis[u]+wdis[u] + w<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>u<\/span>]<\/span><\/span>+<\/span><\/span><\/span><\/span>w<\/span><\/span><\/span><\/span><\/span>\u6765\u66f4\u65b0dis[v]dis[v]<\/span><\/span>d<\/span>i<\/span>s<\/span>[<\/span>v<\/span>]<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u5728\u6bcf\u4e00\u8f6e\u5faa\u73af\u4e2d，\u6211\u4eec\u5c1d\u8bd5\u7528\u6bcf\u4e00\u6761\u8fb9\u66f4\u65b0\u8fd9\u6761\u8fb9\u7684\u7ec8\u70b9\u7684\u6700\u77ed\u8def，\u590d\u6742\u5ea6O(m)O(m)<\/span><\/span>O<\/span>(<\/span>m<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c\u8d77\u70b9\u4e0d\u80fd\u5230\u8fbe\u8d1f\u73af，\u5b58\u5728\u6700\u77ed\u8def，\u90a3\u4e48\u6700\u77ed\u8def\u6700\u591a\u5305\u62ecn\u22121n-1<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u6761\u8fb9<\/p>\n

\u6bcf\u8f6e\u677e\u5f1b\u90fd\u4f1a\u4f7f\u6700\u77ed\u8def\u81f3\u5c11\u591a\u4e00\u6761\u8fb9\u3002<\/p>\n

\u6240\u4ee5\u6700\u591a\u9700\u8981\u8fdb\u884cn\u22121n-1<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u8f6e\u5faa\u73af，\u6700\u7ec8\u590d\u6742\u5ea6O(nm)O(nm)<\/span><\/span>O<\/span>(<\/span>nm<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5982\u679c\u9700\u8981\u5224\u65ad\u662f\u5426\u5b58\u5728\u8d1f\u73af，\u6bd4\u8f83\u7b80\u5355\u7684\u505a\u6cd5\u662f\u65b0\u52a0\u4e00\u4e2a\u8d85\u7ea7\u6e90\u70b9S，\u4eceS\u51fa\u53d1\u5efann<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u6761\u8fb9\u6743\u4e3a00<\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u7684\u8fb9\u8fde\u5411\u5176\u4ed6nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u4e2a\u8282\u70b9，\u7136\u540e\u4ee5S\u4f5c\u4e3a\u8d77\u70b9\u6267\u884cbellman\u3002<\/p>\n

\u5b9e\u73b0<\/h4>\n

vector <<\/span>array<<\/span>int<\/span>,<\/span> 3<\/span>><\/span> ><\/span> path;<\/span>
\nbool<\/span> bellman<\/span>(<\/span>int<\/span> st)<\/span> {<\/span>
\n dis[<\/span>st]<\/span> =<\/span> 0<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 1<\/span>;<\/span> i <=<\/span> n;<\/span> i++<\/span>)<\/span> {<\/span>
\n bool<\/span> ed =<\/span> 1<\/span>;<\/span>
\n for<\/span> (<\/span>auto<\/span> p :<\/span> path)<\/span> {<\/span>
\n int<\/span> u =<\/span> p[<\/span>0<\/span>]<\/span>,<\/span> v =<\/span> p[<\/span>1<\/span>]<\/span>,<\/span> w =<\/span> p[<\/span>2<\/span>]<\/span>;<\/span>
\n if<\/span> (<\/span>dis[<\/span>u]<\/span> ==<\/span> inf)<\/span> continue<\/span>;<\/span>
\n if<\/span> (<\/span>dis[<\/span>v]<\/span> ><\/span> dis[<\/span>u]<\/span> +<\/span> w)<\/span> {<\/span>
\n dis[<\/span>v]<\/span> =<\/span> dis[<\/span>u]<\/span> +<\/span> w;<\/span>
\n ed =<\/span> 0<\/span>;<\/span>
\n }<\/span>
\n }<\/span>
\n if<\/span> (<\/span>ed)<\/span> return<\/span> 0<\/span>;<\/span>
\n }<\/span>
\n return<\/span> 1<\/span>;<\/span>
\n}<\/span><\/p>\n

SPFA<\/h3>\n

SPFA\u662f\u57fa\u4e8ebellman\u7b97\u6cd5\u7684\u961f\u5217\u4f18\u5316，\u5728\u968f\u673a\u56fe\u548c\u4e00\u4e9b\u6709\u7279\u6b8a\u6027\u8d28\u7684\u56fe\u4e0a\u8868\u73b0\u4f18\u79c0，\u4f46\u590d\u6742\u5ea6\u4e0a\u754c\u548cbellman\u76f8\u540c，\u4e5f\u662fO(nm)O(nm)<\/span><\/span>O<\/span>(<\/span>nm<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

SPFA\u662f\u57fa\u4e8e\u8fd9\u6837\u4e00\u79cd\u601d\u60f3：\u53ea\u6709\u4e0a\u4e00\u6b21\u88ab\u677e\u5f1b\u8fc7\u7684\u7ec8\u70b9，\u624d\u6709\u4f5c\u4e3a\u8d77\u70b9\u677e\u5f1b\u7684\u4ef7\u503c\u3002\u6240\u4ee5\u53ef\u4ee5\u7528\u4e00\u4e2a\u961f\u5217\u5b58\u88ab\u677e\u5f1b\u8fc7\u7684\u7ec8\u70b9\u7684\u96c6\u5408，\u6bcf\u6b21\u53d6\u51fa\u961f\u5934\u677e\u5f1b\u8be5\u70b9\u7684\u51fa\u8fb9\u3002<\/p>\n

bool<\/span> spfa<\/span>(<\/span>int<\/span> st)<\/span> {<\/span>
\n queue<<\/span>int<\/span>><\/span> q;<\/span>
\nq.<\/span>push<\/span>(<\/span>st)<\/span>;<\/span>
\nfor<\/span> (<\/span>int<\/span> i =<\/span> 1<\/span>;<\/span> i <=<\/span> n;<\/span> i++<\/span>)<\/span> dis[<\/span>i]<\/span> =<\/span> inf,<\/span> vis[<\/span>i]<\/span> =<\/span> 0<\/span>;<\/span>
\ndis[<\/span>st]<\/span> =<\/span> 0<\/span>;<\/span>
\ncnt[<\/span>st]<\/span> =<\/span> 0<\/span>;<\/span>
\nvis[<\/span>st]<\/span> =<\/span> 1<\/span>;<\/span>
\nwhile<\/span>(<\/span>q.<\/span>size<\/span>(<\/span>)<\/span>)<\/span> {<\/span>
\nint<\/span> now =<\/span> q.<\/span>front<\/span>(<\/span>)<\/span>;<\/span> q.<\/span>pop<\/span>(<\/span>)<\/span>;<\/span>
\nvis[<\/span>now]<\/span> =<\/span> 0<\/span>;<\/span>
\nfor<\/span> (<\/span>auto<\/span> p :<\/span> ve[<\/span>now]<\/span>)<\/span> {<\/span>
\nint<\/span> nxt =<\/span> p.<\/span>first,<\/span> w =<\/span> p.<\/span>second;<\/span>
\nif<\/span> (<\/span>dis[<\/span>nxt]<\/span> ><\/span> dis[<\/span>now]<\/span> +<\/span> w)<\/span> {<\/span>
\ndis[<\/span>nxt]<\/span> =<\/span> dis[<\/span>now]<\/span> +<\/span> w;<\/span>
\ncnt[<\/span>nxt]<\/span> =<\/span> cnt[<\/span>now]<\/span> +<\/span> 1<\/span>;<\/span>
\nif<\/span> (<\/span>cnt[<\/span>nxt]<\/span> >=<\/span> n)<\/span> {<\/span>
\nreturn<\/span> 1<\/span>;<\/span>
\n}<\/span>
\nif<\/span> (<\/span>!<\/span>vis[<\/span>nxt]<\/span>)<\/span> {<\/span>
\nq.<\/span>push<\/span>(<\/span>nxt)<\/span>;<\/span>
\nvis[<\/span>nxt]<\/span> =<\/span> 1<\/span>;<\/span>
\n}<\/span>
\n}<\/span>
\n}<\/span>
\n}<\/span>
\nreturn<\/span> 0<\/span>;<\/span>
\n}<\/span><\/p>\n

\u6bcf\u4e2a\u70b9\u7684\u5165\u961f\u6b21\u6570\u662fO(\u5ea6) \u7684，\u590d\u6742\u5ea6\u4e0a\u754c\u4ecd\u7136\u662fO(nm)O(nm)<\/span><\/span>O<\/span>(<\/span>nm<\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\u6587\u7ae0\u6d4f\u89c8\u9605\u8bfb1.4k\u6b21\uff0c\u70b9\u8d5e40\u6b21\uff0c\u6536\u85cf19\u6b21\u3002\u548c\u70b9\u6838\u5fc3\u7684dij\u4e0d\u540c\uff0cbellman\u5728\u677e\u5f1b\u65f6\u4e0d\u662f\u4ee5\u70b9\u4f5c\u4e3a\u5355\u4f4d\u6765\u677e\u5f1b\uff0c\u800c\u662f\u5728\u4e00\u8f6e\u677e\u5f1b\u4e2d\u4f9d\u6b21\u677e\u5f1b\u6240\u6709\u8fb9\uff0c\u76f4\u5230\u627e\u5230\u6240\u6709\u70b9\u7684\u6700\u77ed\u8def\uff08\u6216\u53d1\u73b0\u6709\u8d1f\u73af\uff09\u4e3a\u6b62\u3002SPFA\u662f\u57fa\u4e8ebellman\u7b97\u6cd5\u7684\u961f\u5217\u4f18\u5316\uff0c\u5728\u968f\u673a\u56fe\u548c\u4e00\u4e9b\u6709\u7279\u6b8a\u6027\u8d28\u7684\u56fe\u4e0a\u8868\u73b0\u4f18\u79c0\uff0c\u4f46\u590d\u6742\u5ea6\u4e0a\u754c\u548cbellman\u76f8\u540c\uff0c\u4e5f\u662f\u3002\u6700\u77ed\u8def\u95ee\u9898\u662f\u4e00\u7c7b\u7ecf\u5178\u7684\u56fe\u8bba\u95ee\u9898\uff0c\u4e3b\u8981\u662f\u6c42\u89e3\u6709\u6743\u56fe\u4e0a\u4e24\u70b9\u95f4\u7684\u6700\u77ed\u8ddd\u79bb\u30022\u3001\u7528\u8d77\u70b9\u5230\u70b9u\u7684\u8ddd\u79bbdis[u]\uff0c\u5c1d\u8bd5\u66f4\u65b0u\u7684\u90bb\u5c45\u4eec\u7684\u6700\u77ed\u8ddd\u79bb\u3002\u5728\u6bcf\u4e00\u8f6e\u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u5c1d\u8bd5\u7528\u6bcf\u4e00\u6761\u8fb9\u66f4\u65b0\u8fd9\u6761\u8fb9\u7684\u7ec8\u70b9\u7684\u6700\u77ed\u8def\uff0c\u590d\u6742\u5ea6\u3002<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[55,427],"topic":[],"class_list":["post-58575","post","type-post","status-publish","format-standard","hentry","category-server","tag-c","tag-427"],"yoast_head":"\n20250812 \u5355\u6e90\u6700\u77ed\u8def\u603b\u7ed3 - 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