【BZOJ1036】树的统计Count（树链剖分，LCT）

题意：一棵树上有n个节点，编号分别为1到n，每个节点都有一个权值w。我们将以下面的形式来要求你对这棵树完成
一些操作：

I. CHANGE u t : 把结点u的权值改为t

II. QMAX u v: 询问从点u到点v的路径上的节点的最大权值

I

II. QSUM u v: 询问从点u到点v的路径上的节点的权值和

注意：从点u到点v的路径上的节点包括u和v本身

1<=n<=30000，0<=q<=200000；中途操作中保证每个节点的权值w在-30000到30000之间。

思路：树链剖分，单点修改，区间查询和与最大值。

 var t1,t2,next,vet,head,dep,flag,tid,fa,top,size,id,son,a
    :array[..]of longint;
     n,m,i,j,k,k1,t,x,y,tot,time,q,len,f:longint;
     ch:string;

 procedure add(a,b:longint);
 begin
  inc(tot);
  next[tot]:=head[a];
  vet[tot]:=b;
  head[a]:=tot;
 end;

 function max(x,y:longint):longint;
 begin
  if x>y then exit(x);
  exit(y);
 end;

 procedure dfs1(u,fath,depth:longint);
 var e,v,maxsize:longint;
 begin
  flag[u]:=; dep[u]:=depth; fa[u]:=fath;
  e:=head[u];
  flag[u]:=; size[u]:=;
  maxsize:=; son[u]:=;
  while e<> do
  begin
   v:=vet[e];
   if flag[v]= then
   begin
   // h[e]:=;
    dfs1(v,u,depth+);
    size[u]:=size[u]+size[v];
    if size[v]>maxsize then
    begin
     maxsize:=size[v];
     son[u]:=v;
    end;
   end;
   e:=next[e];
  end;
 end;

 procedure dfs2(u,ance:longint);
 var e,v:longint;
 begin
  flag[u]:=; inc(time); tid[u]:=time; id[time]:=u; top[u]:=ance;
  if son[u]> then dfs2(son[u],ance);
  e:=head[u];
  while e<> do
  begin
   v:=vet[e];
   if flag[v]= then dfs2(v,v);
   e:=next[e];
  end;
 end;

 procedure build(l,r,p:longint);
 var mid:longint;
 begin
  if l=r then
  begin
   t1[p]:=a[id[l]];
   t2[p]:=a[id[l]];
   exit;
  end;
  mid:=(l+r)>>;
  build(l,mid,p<<);
  build(mid+,r,p<<+);
  t1[p]:=max(t1[p<<],t1[p<<+]);
  t2[p]:=t2[p<<]+t2[p<<+];
 end;

 procedure update(l,r,x,v,p:longint);
 var mid:longint;
 begin
  if l=r then
  begin
   t1[p]:=v; t2[p]:=v;
   exit;
  end;
  mid:=(l+r)>>;
  if x<=mid then update(l,mid,x,v,p<<);
  if x>mid then update(mid+,r,x,v,p<<+);
  t1[p]:=max(t1[p<<],t1[p<<+]);
  t2[p]:=t2[p<<]+t2[p<<+];
 end;

 function querymax(l,r,x,y,p:longint):longint;
 var mid,t:longint;
 begin
  if (l=x)and(r=y) then exit(t1[p]);
  mid:=(l+r)>>;
  t:=-maxlongint;
  if y<=mid then t:=querymax(l,mid,x,y,p<<)
   else if x>mid then t:=querymax(mid+,r,x,y,p<<+)
   else t:=max(querymax(l,mid,x,mid,p<<),querymax(mid+,r,mid+,y,p<<+));
  exit(t);
 end;

 function querysum(l,r,x,y,p:longint):longint;
 var mid,t:longint;
 begin
  if (l=x)and(r=y) then exit(t2[p]);
  mid:=(l+r)>>;
  t:=;
  if y<=mid then t:=querysum(l,mid,x,y,p<<)
   else if x>mid then t:=querysum(mid+,r,x,y,p<<+)
    else t:=querysum(l,mid,x,mid,p<<)+querysum(mid+,r,mid+,y,p<<+);
  exit(t);
 end;

 procedure swap(var x,y:longint);
 var t:longint;
 begin
  t:=x; x:=y; y:=t;
 end;

 function askmax(x,y:longint):longint;
 var f1,f2,t:longint;
 begin
  t:=-maxlongint;
  f1:=top[x]; f2:=top[y];
  while f1<>f2 do
  begin
   if dep[f1]<dep[f2] then
   begin
    swap(f1,f2); swap(x,y);
   end;
   t:=max(t,querymax(,n,tid[f1],tid[x],));
   x:=fa[f1]; f1:=top[x];
  end;
  if dep[x]>dep[y] then swap(x,y);
  t:=max(t,querymax(,n,tid[x],tid[y],));
  exit(t);
 end;

 function asksum(x,y:longint):longint;
 var f1,f2,t:longint;
 begin
  t:=;
  f1:=top[x]; f2:=top[y];
  while f1<>f2 do
  begin
   if dep[f1]<dep[f2] then
   begin
    swap(f1,f2); swap(x,y);
   end;
   t:=t+querysum(,n,tid[f1],tid[x],);
   x:=fa[f1]; f1:=top[x];
  end;
  if dep[x]>dep[y] then swap(x,y);
  t:=t+querysum(,n,tid[x],tid[y],);
  exit(t);
 end;

 begin
  //assign(input,'bzoj1036.in'); reset(input);
  //assign(output,'bzoj1036.out'); rewrite(output);
  readln(n);
  for i:= to n- do
  begin
   readln(x,y);
   add(x,y);
   add(y,x);
  end;
  for i:= to n do read(a[i]);
  dfs1(,-,);
  fillchar(flag,sizeof(flag),);
  dfs2(,);
  fillchar(t1,sizeof(t1),$8f);
  build(,n,);
  readln(q);
  for i:= to q do
  begin
   readln(ch);
   x:=; y:=;
   for j:= to length(ch) do
    if ch[j]=' ' then break;
   while ch[j]=' ' do inc(j);
   f:=;
   while ch[j]<>' ' do
   begin
   // if ch[j]='-' then f:=-;
    if (ch[j]>='')and(ch[j]<='') then x:=x*+ord(ch[j])-ord('');
    inc(j);
   end;

   while ch[j]=' ' do inc(j);
   while (j<=length(ch))and(ch[j]<>' ') do
   begin
    if ch[j]='-' then f:=-;
    if (ch[j]>='')and(ch[j]<='') then y:=y*+ord(ch[j])-ord('');
    inc(j);
   end;
   if f=- then y:=-y;

   if ch[]='C' then update(,n,tid[x],y,);
   if (ch[]='Q')and(ch[]='M') then writeln(askmax(x,y));
   if (ch[]='Q')and(ch[]='S') then writeln(asksum(x,y));
  {  if ch[1]='C' then
   begin
    delete(ch,1,7);
    val(copy(ch,1,pos(' ',ch)-1),x);
    val(copy(ch,pos(' ',ch)+1,length(ch)-pos(' ',ch)),y);
    update(1,n,tid[x],y,1);
   end
    else if ch[2]='M' then
    begin
     delete(ch,1,5);
     val(copy(ch,1,pos(' ',ch)-1),x);
     val(copy(ch,pos(' ',ch)+1,length(ch)-pos(' ',ch)),y);
     writeln(askmax(x,y));
    end
     else
     begin
      delete(ch,1,5);
      val(copy(ch,1,pos(' ',ch)-1),x);
      val(copy(ch,pos(' ',ch)+1,length(ch)-pos(' ',ch)),y);
      writeln(asksum(x,y));

     end;    }

  end;
  //close(input);
  //close(output);
 end.

这是LCT写法，这种没有结构变化的树上操作还是写树剖吧，BZOJ上LCT只能卡着时限过

 var c:array[..,..]of longint;
     mx,sum,w:array[..]of int64;
     fa,rev,q,a,b:array[..]of longint;
     n,m,i,x,y,k,que,j,f,top,t:longint;
     ch:string;

 procedure update(x:longint);
 var l,r:longint;
 begin
  l:=c[x,]; r:=c[x,];
  sum[x]:=sum[l]+sum[r]+w[x];
  mx[x]:=w[x];
  if mx[l]>mx[x] then mx[x]:=mx[l];
  if mx[r]>mx[x] then mx[x]:=mx[r];
 //mx[x]:=max(w[x],max(mx[l],mx[r]));
 end;

 function isroot(x:longint):boolean;
 begin
  if (c[fa[x],]<>x)and(c[fa[x],]<>x) then exit(true);
  exit(false);
 end;

 procedure pushdown(x:longint);
 var l,r:longint;
 begin
  l:=c[x,]; r:=c[x,];
  if rev[x]= then
  begin
   rev[x]:=rev[x] xor ; rev[l]:=rev[l] xor ; rev[r]:=rev[r] xor ;
   //swap(c[x,],c[x,]);
   t:=c[x,]; c[x,]:=c[x,]; c[x,]:=t;
  end;
 end;

 procedure rotate(x:longint);
 var y,z,l,r:longint;
 begin
  y:=fa[x]; z:=fa[y];
  if c[y,]=x then l:=
   else l:=; r:=-l;
  if not((c[fa[y],]<>y)and(c[fa[y],]<>y)) then
   if c[z,]=y then c[z,]:=x
    else c[z,]:=x;
  fa[c[x,r]]:=y; fa[y]:=x; fa[x]:=z;
  c[y,l]:=c[x,r]; c[x,r]:=y;
  update(y);
  update(x);
 end;

 procedure splay(x:longint);
 var k,y,z,i:longint;
 begin
  inc(top); q[top]:=x;
  k:=x;
  while not((c[fa[k],]<>k)and(c[fa[k],]<>k)) do
  begin
   inc(top); q[top]:=fa[k];
   k:=fa[k];
  end;
  while top> do
  begin
   pushdown(q[top]);
   dec(top);
  end;

  while not((c[fa[x],]<>x)and(c[fa[x],]<>x)) do
  begin
   y:=fa[x]; z:=fa[y];
   if not((c[fa[y],]<>y)and(c[fa[y],]<>y)) then
   begin
    if (c[y,]=x)xor(c[z,]=y) then rotate(x)
     else rotate(y);
   end;
   rotate(x);
  end;
 end;

 procedure access(x:longint);
 var t:longint;
 begin
  t:=;
  while x> do
  begin
   splay(x); c[x,]:=t; update(x);
   t:=x; x:=fa[x];
  end;
 end;

 procedure makeroot(x:longint);
 begin
  access(x); splay(x); rev[x]:=rev[x] xor ;
 end;

 procedure link(x,y:longint);
 begin
  makeroot(x); fa[x]:=y;
 end;

 procedure split(x,y:longint);
 begin
  makeroot(x); access(y); splay(y);
 end;

 begin
  assign(input,'bzoj1036.in'); reset(input);
  assign(output,'bzoj1036.out'); rewrite(output);
  readln(n);
  mx[]:=-maxlongint div ;

  for i:= to n- do readln(a[i],b[i]);
  for i:= to n do
  begin
   read(w[i]); sum[i]:=w[i]; mx[i]:=w[i];
  end;
  for i:= to n- do link(a[i],b[i]);
  readln(que);
  for i:= to que do
  begin
   readln(ch);
   x:=; y:=;
   for j:= to length(ch) do
    if ch[j]=' ' then break;
   while ch[j]=' ' do inc(j);
   f:=;
   while ch[j]<>' ' do
   begin
    if (ch[j]>='')and(ch[j]<='') then x:=x*+ord(ch[j])-ord('');
    inc(j);
   end;

   while ch[j]=' ' do inc(j);
   while (j<=length(ch))and(ch[j]<>' ') do
   begin
    if ch[j]='-' then f:=-;
    if (ch[j]>='')and(ch[j]<='') then y:=y*+ord(ch[j])-ord('');
    inc(j);
   end;
   if f=- then y:=-y;

   if ch[]='C' then
   begin
    splay(x);
    w[x]:=y;
    update(x);
   end;
   if (ch[]='Q')and(ch[]='M') then
   begin
    split(x,y);
    writeln(mx[y]);
   end;

   if (ch[]='Q')and(ch[]='S') then
   begin
    split(x,y);
    writeln(sum[y]);
   end;

  end;
  close(input);
  close(output);
 end.

UPD（2018.9.19）：C++ 树链剖分写法

 #include<cstdio>
 #include<cstring>
 #include<string>
 #include<cmath>
 #include<iostream>
 #include<algorithm>
 #include<map>
 #include<set>
 #include<queue>
 #include<vector>
 using namespace std;
 typedef long long ll;
 typedef unsigned int uint;
 typedef unsigned long long ull;
 typedef pair<int,int> PII;
 typedef vector<int> VI;
 #define fi first
 #define se second
 #define MP make_pair
 #define N   210000
 #define MOD 1000000007
 #define eps 1e-8
 #define pi acos(-1)
 #define oo 1e9

 int mx[N],sum[N],a[N],head[N],vet[N],nxt[N],top[N],tid[N],id[N],
     fa[N],size[N],son[N],dep[N],flag[N],n,cnt,tot;
 char ch[];

 int read()
 {
    int v=,f=;
    char c=getchar();
    while(c<||<c) {if(c=='-') f=-; c=getchar();}
    while(<=c&&c<=) v=(v<<)+v+v+c-,c=getchar();
    return v*f;
 }

 void add(int a,int b)
 {
     nxt[++tot]=head[a];
     vet[tot]=b;
     head[a]=tot;
 }

 void dfs1(int u)
 {
     flag[u]=; size[u]=;
     int maxsize=; son[u]=;
     int e=head[u];
     while(e)
     {
         int v=vet[e];
         if(!flag[v])
         {
             fa[v]=u;
             dep[v]=dep[u]+;
             dfs1(v);
             size[u]+=size[v];
             if(size[v]>maxsize)
             {
                 maxsize=size[v];
                 son[u]=v;
             }
         }
         e=nxt[e];
     }
 }

 void dfs2(int u,int ance)
 {
      flag[u]=;
      tid[u]=++cnt; id[cnt]=u; top[u]=ance;
      if(son[u]) dfs2(son[u],ance);
      int e=head[u];
      while(e)
      {
          int v=vet[e];
          if(!flag[v]) dfs2(v,v);
          e=nxt[e];
      }
 }

 void pushup(int p)
 {
     mx[p]=max(mx[p<<],mx[p<<|]);
     sum[p]=sum[p<<]+sum[p<<|];
 }

 void build(int l,int r,int p)
 {
     if(l==r)
     {
         mx[p]=sum[p]=a[id[l]];
         return;
     }
     int mid=(l+r)>>;
     build(l,mid,p<<);
     build(mid+,r,p<<|);
     pushup(p);
 }

 void update(int l,int r,int x,int v,int p)
 {
     if(l==r)
     {
         mx[p]=sum[p]=v;
         return;
     }
     int mid=(l+r)>>;
     if(x<=mid) update(l,mid,x,v,p<<);
      else update(mid+,r,x,v,p<<|);
     pushup(p);
 }

 int querymax(int l,int r,int x,int y,int p)
 {
     if(x<=l&&r<=y) return mx[p];
     int mid=(l+r)>>;
     int ans=-oo;
     if(x<=mid) ans=max(ans,querymax(l,mid,x,y,p<<));
     if(y>mid) ans=max(ans,querymax(mid+,r,x,y,p<<|));
     return ans;
 }

 int querysum(int l,int r,int x,int y,int p)
 {
     if(x<=l&&r<=y) return sum[p];
     int mid=(l+r)>>;
     int ans=;
     if(x<=mid) ans+=querysum(l,mid,x,y,p<<);
     if(y>mid) ans+=querysum(mid+,r,x,y,p<<|);
     return ans;
 }

 int qmax(int x,int y)
 {
     int ans=-oo;
     int f1=top[x];
     int f2=top[y];
     while(f1!=f2)
     {
         if(dep[f1]<dep[f2])
         {
             swap(f1,f2);
             swap(x,y);
         }
         ans=max(ans,querymax(,n,tid[f1],tid[x],));
         x=fa[f1];
         f1=top[x];
     }
     if(dep[x]>dep[y]) swap(x,y);
     ans=max(ans,querymax(,n,tid[x],tid[y],));
     return ans;
 }

 int qsum(int x,int y)
 {
     int ans=;
     int f1=top[x];
     int f2=top[y];
     while(f1!=f2)
     {
         if(dep[f1]<dep[f2])
         {
             swap(f1,f2);
             swap(x,y);
         }
         ans+=querysum(,n,tid[f1],tid[x],);
         x=fa[f1];
         f1=top[x];
     }
     if(dep[x]>dep[y]) swap(x,y);
     ans+=querysum(,n,tid[x],tid[y],);
     return ans;
 }

 int main()
 {
     freopen("bzoj1036.in","r",stdin);
     freopen("bzoj1036.out","w",stdout);
     scanf("%d",&n);
     for(int i=;i<=n-;i++)
     {
         int x,y;
         scanf("%d%d",&x,&y);
         add(x,y);
         add(y,x);
     }
     for(int i=;i<=n;i++) scanf("%d",&a[i]);

     cnt=;
     dfs1();
     memset(flag,,sizeof(flag));
     dfs2(,);
     for(int i=;i<=N-;i++) mx[i]=-oo;
     build(,n,);
     int q;
     scanf("%d",&q);
     //q=0;
     for(int i=;i<=q;i++)
     {
         int x,y;
         scanf("%s%d%d",ch,&x,&y);
         if(ch[]=='C') update(,n,tid[x],y,);
         if(ch[]=='M')
         {
             int ans=qmax(x,y);
             printf("%d\n",ans);
         }
         if(ch[]=='S')
         {
             int ans=qsum(x,y);
             printf("%d\n",ans);
         }
     }
 }

UPD（2018.9.20）：C++ LCT写法

 #include<cstdio>
 #include<cstring>
 #include<string>
 #include<cmath>
 #include<iostream>
 #include<algorithm>
 #include<map>
 #include<set>
 #include<queue>
 #include<vector>
 using namespace std;
 typedef long long ll;
 typedef unsigned int uint;
 typedef unsigned long long ull;
 typedef pair<int,int> PII;
 typedef vector<int> VI;
 #define fi first
 #define se second
 #define MP make_pair
 #define N   210000
 #define MOD 1000000007
 #define eps 1e-8
 #define pi acos(-1)
 #define oo 1e9

 char ch[];
 int t[N][],fa[N],a[N],b[N],q[N],rev[N],n,top;
 ll w[N],sum[N],mx[N];

 int read()
 {
    int v=,f=;
    char c=getchar();
    while(c<||<c) {if(c=='-') f=-; c=getchar();}
    while(<=c&&c<=) v=(v<<)+v+v+c-,c=getchar();
    return v*f;
 }

 bool isroot(int x)
 {
     return t[fa[x]][]!=x&&t[fa[x]][]!=x;
 }

 void pushup(int x)
 {
     int l=t[x][];
     int r=t[x][];
     sum[x]=sum[l]+sum[r]+w[x];
     mx[x]=max(w[x],max(mx[l],mx[r]));
 }

 void pushdown(int x)
 {
     int l=t[x][];
     int r=t[x][];
     if(rev[x])
     {
         rev[x]^=;
         rev[l]^=;
         rev[r]^=;
         swap(t[x][],t[x][]);
     }
 }

 void rotate(int x)
 {
      int y=fa[x];
      int z=fa[y];
      int l=(t[y][]==x);
      int r=l^;
      if(!isroot(y)) t[z][t[z][]==y]=x;
      fa[t[x][r]]=y; fa[y]=x; fa[x]=z;
      t[y][l]=t[x][r]; t[x][r]=y;
      pushup(y);
      pushup(x);
 }

 void splay(int x)
 {
     q[++top]=x;
     for(int i=x;!isroot(i);i=fa[i]) q[++top]=fa[i];
     while(top) pushdown(q[top--]);
     while(!isroot(x))
     {
         int y=fa[x];
         int z=fa[y];
         if(!isroot(y))
         {
             if(t[y][]==x^t[z][]==y) rotate(x);
              else rotate(y);
         }
         rotate(x);
     }
 }

 void access(int x)
 {
     for(int k=;x;k=x,x=fa[x])
     {
         splay(x);
         t[x][]=k;
         pushup(x);
     }
 }

 void makeroot(int x)
 {
     access(x);
     splay(x);
     rev[x]^=;
 }

 void link(int x,int y)
 {
     makeroot(x);
     fa[x]=y;
 }

 void split(int x,int y)
 {
     makeroot(x);
     access(y);
     splay(y);
 }

 int main()
 {
     //freopen("bzoj1036.in","r",stdin);
     //freopen("bzoj1036.out","w",stdout);
     int n;
     scanf("%d",&n);
     mx[]=-oo;
     for(int i=;i<=n-;i++) scanf("%d%d",&a[i],&b[i]);
     for(int i=;i<=n;i++)
     {
         scanf("%lld",&w[i]);
         mx[i]=sum[i]=w[i];
     }
     for(int i=;i<=n-;i++) link(a[i],b[i]);
     int q;
     scanf("%d",&q);
     for(int i=;i<=q;i++)
     {
         int x,y;
         scanf("%s%d%d",ch,&x,&y);
         if(ch[]=='C')
         {
             splay(x);
             w[x]=y;
             pushup(x);
         }
         if(ch[]=='M')
         {
             split(x,y);
             ll ans=mx[y];
             printf("%lld\n",ans);
         }
         if(ch[]=='S')
         {
             split(x,y);
             ll ans=sum[y];
             printf("%lld\n",ans);
         }
     }
 }