「題解」靈大會議
阿新 • • 發佈：2021-06-28
本文將同步釋出於：
題目
題意簡述
給定一棵樹，兩種操作：
- 詢問一條鏈的帶權重心；
- 修改一個點的權值。
\(1\leq n,q\leq 152501\)。
題解
重心
顯然可以通過在路徑上二分求得。
求答案
用樹鏈剖分加線段樹即可。
參考程式
#pragma GCC optimize("Ofast") #include<bits/stdc++.h> using namespace std; #define reg register typedef long long ll; bool st; #define getchar() (p1==p2&&(p2=(p1=buf)+fread(buf,1,1<<21,stdin),p1==p2)?EOF:*p1++) static char buf[1<<21],*p1=buf,*p2=buf; #define flush() (fwrite(wbuf,1,wp1,stdout),wp1=0) #define putchar(c) (wp1==wp2&&(flush(),0),wbuf[wp1++]=c) static char wbuf[1<<21];int wp1;const int wp2=1<<21; inline int read(void){ reg char ch=getchar(); reg int res=0; while(!isdigit(ch)) ch=getchar(); while(isdigit(ch)) res=10*res+(ch^'0'),ch=getchar(); return res; } inline void writeln(reg ll x){ static char buf[32]; reg int p=-1; if(!x) putchar('0'); else while(x) buf[++p]=(x%10)^'0',x/=10; while(~p) putchar(buf[p--]); putchar('\n'); return; } const int MAXN=152501+5; const int MAXLOG2N=18+1; const int MAXQ=152501+5; int n; int a[MAXN]; struct Edge{ int v,w; inline Edge(reg int v=0,reg int w=0):v(v),w(w){ return; } }; vector<Edge> G[MAXN]; inline void Add_Edge(reg int u,reg int v,reg int w){ G[u].push_back(Edge(v,w)); return; } int fa[MAXN],dep[MAXN]; int siz[MAXN],wson[MAXN]; int anc[MAXN][MAXLOG2N]; int Maxdep[MAXN],lson[MAXN]; ll dis[MAXN]; inline void dfs1(reg int u,reg int father){ siz[u]=1; fa[u]=father; anc[u][0]=father; dep[u]=dep[father]+1; Maxdep[u]=dep[u]; for(reg int i=1;anc[u][i-1];++i) anc[u][i]=anc[anc[u][i-1]][i-1]; for(Edge e:G[u]) if(e.v!=father){ dis[e.v]=dis[u]+e.w; dfs1(e.v,u); siz[u]+=siz[e.v]; if(siz[wson[u]]<siz[e.v]) wson[u]=e.v; if(Maxdep[e.v]>Maxdep[u]){ Maxdep[u]=Maxdep[e.v]; lson[u]=e.v; } } return; } int tim,dfn[MAXN],rnk[MAXN]; int wtop[MAXN]; inline void dfs2(reg int u,reg int father,reg int topf){ wtop[u]=topf; rnk[dfn[u]=++tim]=u; if(!wson[u]) return; dfs2(wson[u],u,topf); for(Edge e:G[u]) if(e.v!=father&&e.v!=wson[u]) dfs2(e.v,u,e.v); return; } int ltop[MAXN]; vector<int> up[MAXN],dn[MAXN]; inline void dfs3(reg int u,reg int father,reg int topf){ ltop[u]=topf; if(u==topf){ up[u].resize(Maxdep[u]-dep[u]+1),dn[u].resize(Maxdep[u]-dep[u]+1); for(reg int i=0,v=u;i<=Maxdep[u]-dep[u];++i) up[u][i]=v,v=anc[v][0]; for(reg int i=0,v=u;i<=Maxdep[u]-dep[u];++i) dn[u][i]=v,v=wson[v]; } if(!wson[u]) return; dfs3(wson[u],u,topf); for(Edge e:G[u]) if(e.v!=father&&e.v!=wson[u]) dfs3(e.v,u,e.v); return; } namespace SegmentTree{ #define lson ( (k) << 1 ) #define rson ( (k) << 1 | 1 ) #define mid ( ( (l) + (r) ) >> 1 ) struct Node{ ll cnt,sum; #define cnt(x) unit[(x)].cnt #define sum(x) unit[(x)].sum }; Node unit[MAXN<<2]; inline void pushup(reg int k){ cnt(k)=cnt(lson)+cnt(rson); sum(k)=sum(lson)+sum(rson); return; } inline void build(reg int k,reg int l,reg int r,reg int a[],reg ll b[]){ if(l==r){ cnt(k)=a[l],sum(k)=b[l]; return; } build(lson,l,mid,a,b),build(rson,mid+1,r,a,b); pushup(k); return; } inline void update(reg int k,reg int l,reg int r,reg int pos,reg int vala,reg ll valb){ if(l==r){ cnt(k)=vala,sum(k)=valb; return; } if(pos<=mid) update(lson,l,mid,pos,vala,valb); else update(rson,mid+1,r,pos,vala,valb); pushup(k); return; } inline ll queryCnt(reg int k,reg int l,reg int r,reg int L,reg int R){ if(L<=l&&r<=R) return cnt(k); if(L<=mid&&mid<R) return queryCnt(lson,l,mid,L,R)+queryCnt(rson,mid+1,r,L,R); else if(L<=mid) return queryCnt(lson,l,mid,L,R); else return queryCnt(rson,mid+1,r,L,R); } inline ll querySum(reg int k,reg int l,reg int r,reg int L,reg int R){ if(L<=l&&r<=R) return sum(k); if(L<=mid&&mid<R) return querySum(lson,l,mid,L,R)+querySum(rson,mid+1,r,L,R); else if(L<=mid) return querySum(lson,l,mid,L,R); else return querySum(rson,mid+1,r,L,R); } #undef mid #undef lson #undef rson #undef cnt } int lastx,lasty,lastlca; inline int LCA(reg int x,reg int y){ if(x==lastx&&y==lasty) return lastlca; lastx=x,lasty=y; while(wtop[x]!=wtop[y]) if(dep[wtop[x]]>dep[wtop[y]]) x=fa[wtop[x]]; else y=fa[wtop[y]]; lastlca=dep[x]<dep[y]?x:y; return lastlca; } inline int getPathNode(reg int x,reg int y){ reg int lca=LCA(x,y); return dep[x]+dep[y]-dep[lca]-dep[fa[lca]]; } inline ll queryCnt(reg int x,reg int y){ reg ll res=0; while(wtop[x]!=wtop[y]) if(dep[wtop[x]]>dep[wtop[y]]) res+=SegmentTree::queryCnt(1,1,n,dfn[wtop[x]],dfn[x]),x=fa[wtop[x]]; else res+=SegmentTree::queryCnt(1,1,n,dfn[wtop[y]],dfn[y]),y=fa[wtop[y]]; res+=((dep[x]<dep[y])?SegmentTree::queryCnt(1,1,n,dfn[x],dfn[y]):SegmentTree::queryCnt(1,1,n,dfn[y],dfn[x])); return res; } inline ll querySum(reg int x,reg int y){ reg ll res=0; while(wtop[x]!=wtop[y]) if(dep[wtop[x]]>dep[wtop[y]]) res+=SegmentTree::querySum(1,1,n,dfn[wtop[x]],dfn[x]),x=fa[wtop[x]]; else res+=SegmentTree::querySum(1,1,n,dfn[wtop[y]],dfn[y]),y=fa[wtop[y]]; res+=((dep[x]<dep[y])?SegmentTree::querySum(1,1,n,dfn[x],dfn[y]):SegmentTree::querySum(1,1,n,dfn[y],dfn[x])); return res; } int lg[MAXN]; inline int getAnc(reg int u,reg int k){ if(!k) return u; else{ u=anc[u][lg[k]]; k-=(1<<lg[k])+(dep[u]-dep[wtop[u]]); u=wtop[u]; return k>=0?up[u][k]:dn[u][-k]; } } inline int getPoint(reg int x,reg int y,reg int k){ reg int lca=LCA(x,y); if(k<=getPathNode(x,lca)) return getAnc(x,k-1); else return getAnc(y,getPathNode(x,y)-k); } bool ed; int main(void){ n=read(); lg[0]=-1; for(reg int i=1;i<=n;++i) lg[i]=lg[i>>1]+1; for(reg int i=1;i<=n;++i) a[i]=read(); for(reg int i=1;i<n;++i){ static int u,v,w; u=read(),v=read(),w=read(); Add_Edge(u,v,w),Add_Edge(v,u,w); } dfs1(1,0),dfs2(1,0,1),dfs3(1,0,1); static int tmpa[MAXN]; static ll tmpb[MAXN]; for(reg int i=1;i<=n;++i) tmpa[dfn[i]]=a[i],tmpb[dfn[i]]=a[i]*dis[i]; SegmentTree::build(1,1,n,tmpa,tmpb); reg int q=read(); while(q--){ static int opt,x,y; opt=read(),x=read(),y=read(); switch(opt){ case 1:{ reg int lim=(queryCnt(x,y)+1)>>1,len=getPathNode(x,y); reg int __l=1,__r=len,__mid; while(__l<__r){ __mid=(__l+__r)>>1; if(queryCnt(x,getPoint(x,y,__mid))>=lim) __r=__mid; else __l=__mid+1; } reg int pos=getPoint(x,y,__l); reg int lca=LCA(x,y); reg ll ans; if(lca==x){ //puts("S1"); ans=dis[pos]*queryCnt(pos,x)-querySum(pos,x)+querySum(pos,y)-dis[pos]*queryCnt(pos,y); } else if(lca==y){ //puts("S2"); swap(x,y); ans=dis[pos]*queryCnt(pos,x)-querySum(pos,x)+querySum(pos,y)-dis[pos]*queryCnt(pos,y); } else{ //puts("S3"); if(LCA(y,pos)!=lca) swap(x,y); ans=querySum(pos,x)-dis[pos]*queryCnt(pos,x)+dis[pos]*queryCnt(pos,lca)-querySum(pos,lca)+querySum(lca,y)+(dis[pos]-(dis[lca]<<1))*queryCnt(lca,y)-(dis[pos]-dis[lca])*a[lca]; } writeln(ans); break; } case 2:{ a[x]=y; SegmentTree::update(1,1,n,dfn[x],a[x],a[x]*dis[x]); break; } } } flush(); fprintf(stderr,"%.3lf s\n",1.0*clock()/CLOCKS_PER_SEC); fprintf(stderr,"%.3lf MiB\n",(&ed-&st)/1048576.0); return 0; }