【Tyvj2133&BZOJ1146】网络管理Network（树套树，DFS序，树状数组，主席树，树上差分）

题意：有一棵N个点的树，每个点有一个点权a[i]，要求在线实现以下操作：

1：将X号点的点权修改为Y

2：查询X到Y的路径上第K大的点权

n,q<=80000 a[i]<=10^8

思路：此题明显地体现了我对主席树理解不深

树上路径K大可以直接用树剖+二分答案+树做

但DFS序+主席树也可以

对于点U，它能影响DFS序上的区间(st[u],ed[u])

所以维护方法就是类似序列K大一样 s[st[u]]++ s[ed[u]+1]--

对于路径(x,y)，信息为s[x]+s[y]-s[lca(x,y)]-s[fa[lca(x,y)]]

但因为是在树上的操作，所以每个点是从它在树上的父亲继承下来的，并不是DFS序上的前一个继承的

所以在DFS时就可以建立一棵没有修改过的原树

对于询问，再建立一棵树只做更改，询问时加上原树的信息即可

不开两棵树可以，不过需要主席树合并

 var t:array[..]of record
                            l,r,s:longint;
                           end;
     f:array[..,..]of longint;
     head,vet,next,st,ed,dep,flag,hash,root1,root2,b:array[..]of longint;
     d:array[..,..]of longint;
     a:array[..]of longint;
     q1,q2:array[..]of longint;
     n,m,i,j,tot,time,up,la,lb,cnt,q,x,y,tmp,p,sum,z,f1,f2,f3,f4:longint;

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

 function find(x:longint):longint;
 var l,r,mid:longint;
 begin
  l:=; r:=up;
  while l<=r do
  begin
   mid:=(l+r)>>;
   if hash[mid]=x then exit(mid);
   if hash[mid]<x then l:=mid+
    else r:=mid-;
  end;
 end;

 procedure update(l,r:longint;var p:longint;v,x:longint);
 var mid:longint;
 begin
  inc(cnt); t[cnt]:=t[p];
  p:=cnt; t[p].s:=t[p].s+x;
  if l=r then exit;
  mid:=(l+r)>>;
  if v<=mid then update(l,mid,t[p].l,v,x)
   else update(mid+,r,t[p].r,v,x);
 end;

 procedure dfs(u:longint);
 var e,v,i,tmp:longint;
 begin
  for i:= to  do
  begin
   if dep[u]<(<<i) then break;
   f[u,i]:=f[f[u,i-],i-];
  end;
  tmp:=find(a[u]);
  root1[u]:=root1[f[u,]];
  update(,up,root1[u],tmp,);
  inc(time); st[u]:=time; ed[u]:=time;
  flag[u]:=;
  e:=head[u];
  while e<> do
  begin
   v:=vet[e];
   if flag[v]= then
   begin
    f[v,]:=u;
    dep[v]:=dep[u]+;
    dfs(v);
    ed[u]:=max(ed[u],ed[v]);
   end;
   e:=next[e];
  end;
 end;

 function lca(x,y:longint):longint;
 var d,i:longint;
 begin
  if dep[x]<dep[y] then swap(x,y);
  d:=dep[x]-dep[y];
  for i:= to  do
   if d and (<<i)> then x:=f[x,i];
  for i:= downto  do
   if f[x,i]<>f[y,i] then
   begin
    x:=f[x,i]; y:=f[y,i];
   end;
  if x=y then exit(x);
  exit(f[x,]);
 end;

 procedure qsort(l,r:longint);
 var i,j,mid:longint;
 begin
  i:=l; j:=r; mid:=b[(l+r)>>];
  repeat
   while mid>b[i] do inc(i);
   while mid<b[j] do dec(j);
   if i<=j then
   begin
    swap(b[i],b[j]);
    inc(i); dec(j);
   end;
  until i>j;
  if l<j then qsort(l,j);
  if i<r then qsort(i,r);
 end;

 function lowbit(x:longint):longint;
 begin
  exit(x and (-x));
 end;

 function query(l,r,k:longint):longint;
 var s1,s2,i,mid:longint;
 begin
  if l=r then exit(l);
  mid:=(l+r)>>;
  s1:=; s2:=;
  for i:= to la do s1:=s1+t[t[q1[i]].l].s;
  for i:= to lb do s2:=s2+t[t[q2[i]].l].s;
  s2:=s2+t[t[f1].l].s+t[t[f2].l].s;
  s1:=s1+t[t[f3].l].s+t[t[f4].l].s;
  if s2-s1>=k then
  begin
   f1:=t[f1].l; f2:=t[f2].l;
   f3:=t[f3].l; f4:=t[f4].l;
   for i:= to la do q1[i]:=t[q1[i]].l;
   for i:= to lb do q2[i]:=t[q2[i]].l;
   exit(query(l,mid,k));
  end
   else
   begin
    f1:=t[f1].r; f2:=t[f2].r;
    f3:=t[f3].r; f4:=t[f4].r;
    for i:= to la do q1[i]:=t[q1[i]].r;
    for i:= to lb do q2[i]:=t[q2[i]].r;
    exit(query(mid+,r,k-(s2-s1)));
   end;
 end;

 begin
  assign(input,'data.in'); reset(input);
  assign(output,'tyvj2133.out'); rewrite(output);
  read(n,m);
  for i:= to n do
  begin
   read(a[i]); b[i]:=a[i];
  end;
  q:=n;
  for i:= to n- do
  begin
   read(x,y);
   add(x,y);
   add(y,x);
  end;

  for i:= to m do
   for j:= to  do read(d[i,j]);
  for i:= to m do
   if d[i,]= then
   begin
    inc(q); b[q]:=d[i,];
   end;
  qsort(,q);
  hash[]:=b[]; up:=;
  for i:= to q do
   if b[i]<>b[i-] then begin inc(up); hash[up]:=b[i]; end;
  dfs();

  for i:= to m do
   if d[i,]= then
   begin
    tmp:=find(a[d[i,]]);
    j:=st[d[i,]];
    while j<=n do
    begin
     update(,up,root2[j],tmp,-);
     j:=j+lowbit(j);
    end;
    j:=ed[d[i,]]+;
    while j<=n do
    begin
     update(,up,root2[j],tmp,);
     j:=j+lowbit(j);
    end;
    tmp:=find(d[i,]);
    j:=st[d[i,]];
    while j<=n do
    begin
     update(,up,root2[j],tmp,);
     j:=j+lowbit(j);
    end;
    j:=ed[d[i,]]+;
    while j<=n do
    begin
     update(,up,root2[j],tmp,-);
     j:=j+lowbit(j);
    end;

    a[d[i,]]:=d[i,];
   end
    else
    begin
     la:=; lb:=; p:=lca(d[i,],d[i,]);
     j:=st[d[i,]];
     while j> do
     begin
      inc(lb); q2[lb]:=root2[j];
      j:=j-lowbit(j);
     end;

     j:=st[d[i,]];
     while j> do
     begin
      inc(lb); q2[lb]:=root2[j];
      j:=j-lowbit(j);
     end;

     j:=st[p];
     while j> do
     begin
      inc(la); q1[la]:=root2[j];
      j:=j-lowbit(j);
     end;

     j:=st[f[p,]];
     while j> do
     begin
      inc(la); q1[la]:=root2[j];
      j:=j-lowbit(j);
     end;

     f1:=root1[d[i,]]; f2:=root1[d[i,]];
     f3:=root1[p]; f4:=root1[f[p,]];
     sum:=dep[d[i,]]+dep[d[i,]]-dep[p]*+;
     z:=sum-d[i,]+;
     if sum<d[i,] then writeln('invalid request!')
      else writeln(hash[query(,up,z)]);
    end;

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