【BZOJ1901】Dynamic Rankings（树套树，树状数组，主席树）

题意：给定一个N个数的序列，要求维护一个数据结构支持以下两种操作：

1：将第X个数改成Y

2：查询第X到第Y个数里第K小的数是多少

n,m<=10000,a[i]<=10^9

思路：单点修改版本的主席树

对于没有修改的主席树，我们直接在对应的节点上每个点继承上个点的状态，再用链表创建logn个点来表示这个数对t数组的修改

而单点修改不可能把后面的前缀和都加上logn个点

所以就利用树状数组的思想

t是一个树状数组，里面的每个点都是一棵主席树

更改时没有区别

询问时最多可能有logn段区间相减

时间复杂度(Nlogn^2)

 var t:array[..]of record
 l,r,s:longint;
 end;
 a,b,root,hash,left,right,flag:array[..]of longint;
 d:array[..,..]of longint;
 n,m,i,j,q,u,cnt,la,lb,s,tmp:longint;
 ch:string;

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

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

 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;

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

 begin
 assign(input,'data.in'); reset(input);
 assign(output,'bzoj1901.out'); rewrite(output);
 readln(n,m);
 q:=;
 for i:= to n do
 begin
 read(a[i]);
 inc(q); b[q]:=a[i];
 end;
 readln;
 for i:= to m do
 begin
 readln(ch); s:=;
 if ch[]='C' then
 begin
 inc(q);
 for j:= to length(ch) do
 begin
 if ch[j]=' ' then begin inc(s); continue; end;
 d[i,s+]:=d[i,s+]*+ord(ch[j])-ord('');
 end;
 b[q]:=d[i,];
 end;
 if ch[]='Q' then
 begin
 flag[i]:=;
 for j:= to length(ch) do
 begin
 if ch[j]=' ' then begin inc(s); continue; end;
 d[i,s+]:=d[i,s+]*+ord(ch[j])-ord('');
 end;
 end;
 end;
 qsort(,q);
 hash[]:=b[]; u:=;
 for i:= to q do
 if b[i]<>b[i-] then begin inc(u); hash[u]:=b[i]; end;

 for i:= to n do
 begin
 tmp:=find(a[i]);
 j:=i;
 while j<=n do
 begin
 update(,u,root[j],tmp,);
 j:=j+lowbit(j);
 end;
 end;

 for i:= to m do
 if flag[i]= then
 begin
 la:=; lb:=; j:=d[i,]-;
 while j> do
 begin
 inc(la); left[la]:=root[j];
 j:=j-lowbit(j);
 end;
 j:=d[i,];
 while j> do
 begin
 inc(lb); right[lb]:=root[j];
 j:=j-lowbit(j);
 end;
 writeln(hash[query(,u,d[i,])]);
 end
 else
 begin
 tmp:=find(a[d[i,]]);
 j:=d[i,];
 while j<=n do
 begin
 update(,u,root[j],tmp,-);
 j:=j+lowbit(j);
 end;
 tmp:=find(d[i,]);
 j:=d[i,];
 while j<=n do
 begin
 update(,u,root[j],tmp,);
 j:=j+lowbit(j);
 end;
 a[d[i,]]:=d[i,];
 end;
 close(input);
 close(output);
 end.

UPD（2018.9.21）：C++

 #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   3100000
 #define M   30000
 #define MOD 1000000007
 #define eps 1e-8
 #define pi acos(-1)
 #define oo 1e9

 char ch[];
 struct arr
 {
     int l,r,s;
 }t[N];
 int a[M],b[M],root[M],
     L[],R[],A[M],B[M],K[M],flag[M],
     n,m,cnt,l1,l2,mx; 

 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;
 }

 int lowbit(int x)
 {
     return x&(-x);
 }

 int lsh(int x)
 {
     int l=;
     int r=mx;
     while(l<=r)
     {
         int mid=(l+r)>>;
         if(b[mid]==x) return mid;
         if(b[mid]<x) l=mid+;
          else r=mid-;
     }
 }

 void update(int l,int r,int x,int v,int &p)
 {
     t[++cnt].l=t[p].l;
     t[cnt].r=t[p].r;
     t[cnt].s=t[p].s;
     p=cnt;
     t[p].s+=v;
     if(l==r) return;
     int mid=(l+r)>>;
     if(x<=mid) update(l,mid,x,v,t[p].l);
      else update(mid+,r,x,v,t[p].r);
 }

 int query(int l,int r,int k)
 {
     //printf("%d %d %d ",l,r,k);
     if(l==r) return l;
     int s1=;
     int s2=;
     for(int i=;i<=l1;i++) s1+=t[t[L[i]].l].s;
     for(int i=;i<=l2;i++) s2+=t[t[R[i]].l].s;
     int tmp=s2-s1;
     //printf("%d\n",tmp);
     int mid=(l+r)>>;
     if(tmp>=k)
     {
          for(int i=;i<=l1;i++) L[i]=t[L[i]].l;
          for(int i=;i<=l2;i++) R[i]=t[R[i]].l;
          return query(l,mid,k);
     }
      else
      {
          for(int i=;i<=l1;i++) L[i]=t[L[i]].r;
          for(int i=;i<=l2;i++) R[i]=t[R[i]].r;
          return query(mid+,r,k-tmp);
      }
 }

 int main()
 {
     freopen("data.in","r",stdin);
     freopen("bzoj1901.out","w",stdout);
     int q;
     scanf("%d%d",&n,&q);
     m=;
     for(int i=;i<=n;i++)
     {
         scanf("%d",&a[i]);
         b[++m]=a[i];
     }
     for(int i=;i<=q;i++)
     {
         scanf("%s",ch);
         scanf("%d%d",&A[i],&B[i]);
         if(ch[]=='Q'){scanf("%d",&K[i]); flag[i]=;}
          else b[++m]=B[i];
     }
     sort(b+,b+m+);
     cnt=;
     mx=;
     for(int i=;i<=m;i++)
      if(b[i]!=b[mx]) b[++mx]=b[i];
     for(int i=;i<=n;i++)
     {
         int tmp=lsh(a[i]);
         int j=i;
         while(j<=n)
         {
              update(,mx,tmp,,root[j]);
              j+=lowbit(j);
         }
     }

     for(int i=;i<=q;i++)
      if(flag[i])
      {
          l1=; l2=; A[i]--;
          int j=A[i];
          while(j)
          {
              L[++l1]=root[j];
              j-=lowbit(j);
          }
          j=B[i];
          while(j)
          {
              R[++l2]=root[j];
              j-=lowbit(j);
          }
          printf("%d\n",b[query(,mx,K[i])]);
      }
      else
      {
          int x=lsh(a[A[i]]);
          int j=A[i];
          while(j<=n)
          {
              update(,mx,x,-,root[j]);
              j+=lowbit(j);
          }
          a[A[i]]=B[i];
          x=lsh(B[i]);
          j=A[i];
          while(j<=n)
          {
              update(,mx,x,,root[j]);
             j+=lowbit(j);
          }
      }

 }