HDU - 6393 Traffic Network in Numazu (基环树+树链剖分/LCA)

题意:给定一个带权边无向基环树，有两种操作，一种是改变某个边的权值，另一种是询问两点间的最短路径。

可以对环进行缩点，以环为根建立一棵新树，并记录与环相连的所有点和环上的哪个点相连，将路径分为环外和环内的两类进行处理。环外的路径可以用树剖+树状数组维护，环内的路径复制一倍，用另一个树状数组维护。

 #include<bits/stdc++.h>
 using namespace std;
 typedef long long ll;
 const int N=1e5+;
 int hd[N],n,m,ne,sta[N],tp,vis[N],id[N],anc[N],lid[N],nl;
 int fa[N],son[N],siz[N],dep[N],top[N],bg[N],ed[N],rnk[N],tot;
 struct E2 {int u,v,c;} e2[N];
 struct E {int v,c,nxt;} e[N<<];
 struct BIT {
     ll c[N<<];
     int n;
     int lb(int x) {return x&-x;}
     void add(int u,int x) {for(; u<=n; u+=lb(u))c[u]+=x;}
     ll get(int u) {ll ret=; for(; u; u-=lb(u))ret+=c[u]; return ret;}
     ll sum(int l,int r) {return get(r)-get(l-);}
     void init(int _n) {memset(c,,sizeof c),n=_n;}
 } tr1,tr2;
 void addedge(int u,int v,int c) {e[ne]= {v,c,hd[u]},hd[u]=ne++;}
 bool isloop(int u) {return id[u]==n+;}
 bool dfs_loop(int u,int f) {
     if(vis[u]) {
         while(!lid[u]) {
             int v=sta[tp--];
             id[v]=n+,lid[v]=++nl;
         }
         return ;
     }
     vis[u]=,sta[++tp]=u;
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==f)continue;
         if(dfs_loop(v,u))return ;
     }
     tp--;
     return ;
 }
 void dfs_anc(int u,int f,int a) {
     anc[u]=a;
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==f||isloop(v))continue;
         dfs_anc(v,u,a);
     }
 }
 void dfs_chain_1(int u,int f,int d) {
     fa[u]=f,son[u]=,siz[u]=,dep[u]=d;
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==fa[u]||isloop(v))continue;
         dfs_chain_1(v,u,d+),siz[u]+=siz[v];
         if(siz[v]>siz[son[u]])son[u]=v;
     }
 }
 void dfs_chain_2(int u,int tp) {
     top[u]=tp,bg[u]=++tot,rnk[bg[u]]=u;
     if(son[u])dfs_chain_2(son[u],tp);
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==fa[u]||v==son[u]||isloop(v))continue;
         dfs_chain_2(v,v);
     }
     ed[u]=tot;
 }
 void upd(int u,int v,int x) {
     if(isloop(u)&&isloop(v)) {
         int l=lid[u],r=lid[v];
         if(l>r)swap(l,r);
         if(r-l!=)swap(l,r);
         tr2.add(r,-tr2.sum(r,r)),tr2.add(r+nl,-tr2.sum(r+nl,r+nl));
         tr2.add(r,x),tr2.add(r+nl,x);
     } else {
         int l=bg[id[u]],r=bg[id[v]];
         if(l>r)swap(l,r);
         tr1.add(r,-tr1.sum(r,r)),tr1.add(r,x);
     }
 }
 ll qry_loop(int u,int v) {
     int l=lid[u],r=lid[v];
     if(l>r)swap(l,r);
     return min(tr2.sum(l+,r),tr2.sum(r+,l+nl));
 }
 ll qry_tree(int u,int v) {
     int ancu=anc[u],ancv=anc[v];
     u=id[u],v=id[v];
     ll ret=;
     for(; top[u]!=top[v]; u=fa[top[u]]) {
         if(dep[top[u]]<dep[top[v]])swap(u,v);
         ret+=tr1.sum(bg[top[u]],bg[u]);
     }
     if(dep[u]<dep[v])swap(u,v);
     ret+=tr1.sum(bg[v]+,bg[u]);
     ret+=qry_loop(ancu,ancv);
     return ret;
 }
 int main() {
     int T;
     for(scanf("%d",&T); T--;) {
         memset(hd,-,sizeof hd),ne=tp=nl=tot=;
         memset(vis,,sizeof vis);
         scanf("%d%d",&n,&m);
         for(int i=; i<=n; ++i) {
             int u,v,c;
             scanf("%d%d%d",&u,&v,&c);
             addedge(u,v,c),addedge(v,u,c);
             e2[i]= {u,v,c};
         }
         for(int i=; i<=n; ++i)id[i]=i;
         dfs_loop(,),id[n+]=n+;
         for(int u=; u<=n; ++u)if(isloop(u))dfs_anc(u,,u);
         for(int u=; u<=n; ++u)if(isloop(u)) {
                 for(int i=hd[u]; ~i; i=e[i].nxt) {
                     int v=e[i].v;
                     if(!isloop(v))addedge(n+,v,e[i].c);
                 }
             }
         dfs_chain_1(n+,,),dfs_chain_2(n+,n+);
         tr1.init(tot),tr2.init(nl*);
         for(int i=; i<=n; ++i)upd(e2[i].u,e2[i].v,e2[i].c);
         while(m--) {
             int f,a,b;
             scanf("%d%d%d",&f,&a,&b);
             if(f==)upd(e2[a].u,e2[a].v,b);
             else printf("%lld\n",qry_tree(a,b));
         }
     }
     return ;
 }

也可以LCA+差分维护每个点到根节点的距离，复杂度少了个log(但实际速度差不了多少)

 #include<bits/stdc++.h>
 using namespace std;
 typedef long long ll;
 const int N=1e5+;
 int hd[N],n,m,ne,sta[N],tp,vis[N],id[N],anc[N],lid[N],nl;
 int fa[N],son[N],siz[N],dep[N],top[N],bg[N],ed[N],rnk[N],tot;
 struct E2 {int u,v,c;} e2[N];
 struct E {int v,c,nxt;} e[N<<];
 struct BIT {
     ll c[N<<];
     int n;
     int lb(int x) {return x&-x;}
     void add(int u,int x) {for(; u<=n; u+=lb(u))c[u]+=x;}
     ll get(int u) {ll ret=; for(; u; u-=lb(u))ret+=c[u]; return ret;}
     ll sum(int l,int r) {return get(r)-get(l-);}
     void init(int _n) {memset(c,,sizeof c),n=_n;}
 } tr1,tr2;
 void addedge(int u,int v,int c) {e[ne]= {v,c,hd[u]},hd[u]=ne++;}
 bool isloop(int u) {return id[u]==n+;}
 bool dfs_loop(int u,int f) {
     if(vis[u]) {
         while(!lid[u]) {
             int v=sta[tp--];
             id[v]=n+,lid[v]=++nl;
         }
         return ;
     }
     vis[u]=,sta[++tp]=u;
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==f)continue;
         if(dfs_loop(v,u))return ;
     }
     tp--;
     return ;
 }
 void dfs_anc(int u,int f,int a) {
     anc[u]=a;
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==f||isloop(v))continue;
         dfs_anc(v,u,a);
     }
 }
 void dfs_chain_1(int u,int f,int d) {
     fa[u]=f,son[u]=,siz[u]=,dep[u]=d;
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==fa[u]||isloop(v))continue;
         dfs_chain_1(v,u,d+),siz[u]+=siz[v];
         if(siz[v]>siz[son[u]])son[u]=v;
     }
 }
 void dfs_chain_2(int u,int tp) {
     top[u]=tp,bg[u]=++tot,rnk[bg[u]]=u;
     if(son[u])dfs_chain_2(son[u],tp);
     for(int i=hd[u]; ~i; i=e[i].nxt) {
         int v=e[i].v;
         if(v==fa[u]||v==son[u]||isloop(v))continue;
         dfs_chain_2(v,v);
     }
     ed[u]=tot;
 }
 void upd(int u,int v,int x) {
     if(isloop(u)&&isloop(v)) {
         int l=lid[u],r=lid[v];
         if(l>r)swap(l,r);
         if(r-l!=)swap(l,r);
         tr2.add(r,x),tr2.add(r+nl,x);
     } else {
         u=id[u],v=id[v];
         if(fa[u]==v)swap(u,v);
         tr1.add(bg[v],x),tr1.add(ed[v]+,-x);
     }
 }
 ll qry_loop(int u,int v) {
     int l=lid[u],r=lid[v];
     if(l>r)swap(l,r);
     return min(tr2.sum(l+,r),tr2.sum(r+,l+nl));
 }
 ll qry_tree(int u,int v) {
     int ancu=anc[u],ancv=anc[v];
     u=id[u],v=id[v];
     ll ret=tr1.get(bg[u])+tr1.get(bg[v]);
     for(; top[u]!=top[v]; u=fa[top[u]])
         if(dep[top[u]]<dep[top[v]])swap(u,v);
     if(dep[u]<dep[v])swap(u,v);
     ret-=*tr1.get(bg[v]);
     ret+=qry_loop(ancu,ancv);
     return ret;
 }
 int main() {
     int T;
     for(scanf("%d",&T); T--;) {
         memset(hd,-,sizeof hd),ne=tp=nl=tot=;
         memset(vis,,sizeof vis);
         scanf("%d%d",&n,&m);
         for(int i=; i<=n; ++i) {
             int u,v,c;
             scanf("%d%d%d",&u,&v,&c);
             addedge(u,v,c),addedge(v,u,c);
             e2[i]= {u,v,c};
         }
         for(int i=; i<=n; ++i)id[i]=i;
         dfs_loop(,),id[n+]=n+;
         for(int u=; u<=n; ++u)if(isloop(u))dfs_anc(u,,u);
         for(int u=; u<=n; ++u)if(isloop(u)) {
                 for(int i=hd[u]; ~i; i=e[i].nxt) {
                     int v=e[i].v;
                     if(!isloop(v))addedge(n+,v,e[i].c);
                 }
             }
         dfs_chain_1(n+,,),dfs_chain_2(n+,n+);
         tr1.init(tot),tr2.init(nl*);
         for(int i=; i<=n; ++i)upd(e2[i].u,e2[i].v,e2[i].c);
         while(m--) {
             int f,a,b;
             scanf("%d%d%d",&f,&a,&b);
             if(f==)upd(e2[a].u,e2[a].v,b-e2[a].c),e2[a].c=b;
             else printf("%lld\n",qry_tree(a,b));
         }
     }
     return ;
 }