[转]主席树介绍

　　转自：http://blog.sina.com.cn/s/blog_7812e98601012cim.html

函数式（现在又称主席式。。。）数据结构从来都没写过，感觉这个东西可以挖掘出不少东西出来，于是开一组专题。

先根据 Seter 留下的文本做一些记录。。

主席树大概是一种离线结构，我以前反正没看到过这东西，所以就自己给他起名字了！如果谁知道这东西的真名，请告诉我！

现在我们知道，主席树的全名应该是 函数式版本的线段树。加上附带的一堆 technology。。
。。总之由于原名字太长了，而且 “主席” 两个字念起来冷艳高贵，以后全部称之为主席树好了。。。

主席树的主体是线段树，准确的说，是很多棵线段树，存的是一段数字区间出现次数（所以要先离散化可能出现的数字）。举个例子，假设我每次都要求整个序列内的第 k 小，那么对整个序列构造一个线段树，然后在线段树上不断找第 k 小在当前数字区间的左半部分还是右半部分。这个操作和平衡树的 Rank 操作一样，只是这里将离散的数字搞成了连续的数字。

先假设没有修改操作：

对于每个前缀 S1…i，保存这样一个线段树 Ti，组成主席树。这样不是会 MLE 么？最后再讲。

注意，这个线段树对一条线段，保存的是这个数字区间的出现次数，所以是可以互相加减的！还有，由于每棵线段树都要保存同样的数字，所以它们的大小、形态也都是一样的！这实在是两个非常好的性质，是平衡树所不具备的。

对于询问 (i,j)，我只要拿出 Tj 和 Ti-1，对每个节点相减就可以了。说的通俗一点，询问 i..j 区间中，一个数字区间的出现次数时，就是这些数字在 Tj 中出现的次数减去在 Ti-1 中出现的次数。

那么有修改操作怎么办呢？

如果将询问看成求一段序列的数字和，那么上面那个相当于求出了前缀和。加入修改操作后，就要用树状数组等来维护前缀和了。于是那个 “很好的性质” 又一次发挥了作用，由于主席树可以互相加减，所以可以用树状数组来套上它。做法和维护前缀和长得基本一样，不说了。

这段指出了主席树的主要性质。。

  • 线段树的每个结点，保存的是这个区间含有的数字的个数。
  • 主席树的每个结点，也就是每颗线段树的大小和形态也是一样的，也就是主席树之间可以相互进行加减运算。。

同时我们也枚举一下主席树的一些局限：

  • 主席树是一种离线结构。。（必须预先知道所有数字的范围。。这在一些应用中会成为障碍。。。
  • 存在 MLE 问题。。（如果按照定义里面的方法去写，对每个结点都需要开一整可线段树，至少都是 O(n2) 级别的空间。。

——————————————————
开始填坑。由于每棵线段树的大小形态都是一样的，而且初始值全都是 0，那每个线段树都初始化不是太浪费了？所以一开始只要建一棵空树即可。

然后是在某棵树上修改一个数字，由于和其他树相关联，所以不能在原来的树上改，必须弄个新的出来。难道要弄一棵新树？不是的，由于一个数字的更改只影响了一条从这个叶子节点到根的路径，所以只要只有这条路径是新的，另外都没有改变。比如对于某个节点，要往右边走，那么左边那些就不用新建，只要用个指针链到原树的此节点左边就可以了，这个步骤的前提也是线段树的形态一样。

假设s是数字个数，这个步骤的空间复杂度显然是 O(logs)。用树状数组去套它，共有 2logn 棵树被修改，m 个操作再加上一开始的空树和 n 个数字，总共就是 O((n+m)lognlogs)。Fotile 大神说如果加上垃圾回收的话，可以去掉一个 log…… ym

至此主席树结构已经分析清楚了。。

const int N = 50009, M = 10009, NN = 2500009;  
   
int l[NN], r[NN], c[NN], total;  
   
PII A[N+M]; int B[N+M], Q[M][3];  
int S[N], C[N], Null;  
int n, m, An, Tn;  
.. .

这里仍然用池子法保存线段树的结点，NN 是一个估计值。。（大概是 nlogn 。。。
l[], r[], c[], total 分别是左孩子下标，右孩子下标，该区间的数字个数和当前已分配的结点数。
（线段树的区间可以在递归的时候实时计算，可以不予保存。。。

PII A[]; int B[]; A、B 用于离散化。。。 B 记录相应数字的 Rank 值。
int Q[]; 记录询问。。。

int S[], C[], Null; 是主席树下标。。
分别对应前缀和，树状数组 和 初始时的空树。

An 是 A 数组的长度，由初始 n 个数字和修改后的数字决定，
Tn 是所有出现的不同数字的个数，用于建立线段树，由对 A 数组去重后得到。

#define lx l[x]  
#define rx r[x]  
#define ly l[y]  
#define ry r[y]  
#define cx c[x]  
#define cy c[y]  
   
#define mid ((ll+rr)>>1)  
#define lc lx, ll, mid  
#define rc rx, mid+1, rr  
   
   
void Build(int &x, int ll, int rr){  
    x = ++total; if (ll < rr) Build(lc), Build(rc);  
}  
   
int Insert(int y, int p, int d){  
   
    int x = ++total, root = x;  
   
    c[x] = c[y] + d; int ll = 0, rr = Tn;  
   
    while (ll < rr){  
        if (p <= mid){  
            lx = ++total, rx = ry;  
            x = lx, y = ly, rr = mid;  
        }  
        else {  
            lx = ly, rx = ++total;  
            x = rx, y = ry, ll = mid + 1;  
        }  
        c[x] = c[y] + d;  
    }  
   
    return root;  
}

这里是主席树主要支持的操作。。前面一堆 Macro 方便敲代码。。。
注意由于 l[], r[], 还有 m 全部有冲突干脆用 ll, rr, mid 代替区间下标。。。

这里 Insert() 过程就是前面所说的插入单链的操作。。。以 y 为基础，形成一棵新的主席树 x。
。。。没有修改的链仍然指向 y 的相对应部分。复杂度为树高。。。。

。。。下面是 int main(); 函数代码了。。。完整程序在最后。。。。

int main(){  
   
#ifndef ONLINE_JUDGE  
    freopen("in.txt", "r", stdin);  
    //freopen("out.txt", "w", stdout);  
#endif  
   
#define key first  
#define id second  
   
    RD(n, m); REP(i, n) A[i] = MP(RD(), i);  
   
    An = n; char cmd; REP(i, m){  
        RC(cmd); if(cmd == 'Q') RD(Q[i][0], Q[i][1], Q[i][2]);  
        else RD(Q[i][0]), Q[i][2] = 0, A[An++] = MP(RD(), An);  
    }  
   
    sort(A, A + An), B[A[0].id] = Tn = 0;  
    FOR(i, 1, An){  
        if(A[i].key != A[i-1].key) A[++Tn].key = A[i].key;  
        B[A[i].id] = Tn;  
    }  
   
    Build(Null, 0, Tn); REP_1(i, n) C[i] = Null;  
   
    S[0] = Null; REP(i, n){  
        S[i+1] = Insert(S[i], B[i], 1);  
    }  
   
    An = n;  
   
    REP(i, m) if (Q[i][2]){  
        OT(A[Query(Q[i][0], Q[i][1], Q[i][2])].key);  
    }else{  
        Modify(Q[i][0], B[Q[i][0]-1], -1);  
        Modify(Q[i][0], B[Q[i][0]-1] = B[An++], 1);  
    }  
}

待讨论问题：

  1. …

    由于每棵线段树的大小形态都是一样的，而且初始值全都是 0，那每个线段树都初始化不是太浪费了？所以一开始只要建一棵空树即可。

    事实上由于当一个结点的 c[] 结果为 0 时。。无论它向左还是向右递归下去结果都会是0 。。。
    所以初始时甚至不是建立空树。。而是只需要一个空结点即可。。。（如何实现这层。。。

  2. 仍然没有将 Lazy Evalutaion 的理念做到完美。。。这题中的 Lazy Evaluation 体现在两个方面。。
    主席树的单链修改和主席树与主席树之间的运算。。

    主席树与主席树之间的运算如果全部做的话一次是 O(n)。。。但是因为询问时只需要计算一条到叶子的路径。。。
    。。。上面的代码中，这种运算是通过 Struct Pack; 结构体现的。。。一种多颗主席树结点的打包。。
    。。全部在同一层的同一个位置。。向同一个方向递归下降。。一个 Pack; 的值为该结构内所有结点值的和。。

    。。现在实现的不足之处：

    • 无法很好的处理减运算。。因此这里才还要保存了 S[] 前缀和，让 C[] 逐个逐个修改。。。
      。。。然后每次回答询问的时候还要一并带上 S[] 。。Orz。（虽然说有助于预处理。。
    • 对计算过后结果无法有效利用。。（换句话说没有体现出主席式 Memorization 的一面。。。
    • 对单个链的插入操作也不是要即时处理的。。（。。如何用数据结构把所有操作先 Hold 下来 。。?。。这样的好处是有一些操作对结果没有影响是可以忽略。。另外把一堆操作放在一起当做一个操作处理的话有时也能得到 Bonus 。。。
  3. …

    Fotile 大神说如果加上垃圾回收的话，可以去掉一个 log…… ym

    （另外正一个误。。。加上回收并不能去掉一个 log 。。。。

完整代码：树状数组套主席树

#include <algorithm>  
#include <iostream>  
#include <iomanip>  
#include <sstream>  
#include <cstring>  
#include <cstdio>  
#include <string>  
#include <vector>  
#include <bitset>  
#include <queue>  
#include <stack>  
#include <cmath>  
#include <ctime>  
#include <list>  
#include <set>  
#include <map>  
   
using namespace std;  
   
#define REP(i, n) for (int i=0;i<int(n);++i)  
#define FOR(i, a, b) for (int i=int(a);i<int(b);++i)  
#define DWN(i, b, a) for (int i=int(b-1);i>=int(a);--i)  
#define REP_1(i, n) for (int i=1;i<=int(n);++i)  
#define FOR_1(i, a, b) for (int i=int(a);i<=int(b);++i)  
#define DWN_1(i, b, a) for (int i=int(b);i>=int(a);--i)  
#define REP_C(i, n) for (int n____=int(n),i=0;i<n____;++i)  
#define FOR_C(i, a, b) for (int b____=int(b),i=a;i<b____;++i)  
#define DWN_C(i, b, a) for (int a____=int(a),i=b-1;i>=a____;--i)  
#define REP_N(i, n) for (i=0;i<int(n);++i)  
#define FOR_N(i, a, b) for (i=int(a);i<int(b);++i)  
#define DWN_N(i, b, a) for (i=int(b-1);i>=int(a);--i)  
#define REP_1_C(i, n) for (int n____=int(n),i=1;i<=n____;++i)  
#define FOR_1_C(i, a, b) for (int b____=int(b),i=a;i<=b____;++i)  
#define DWN_1_C(i, b, a) for (int a____=int(a),i=b;i>=a____;--i)  
#define REP_1_N(i, n) for (i=1;i<=int(n);++i)  
#define FOR_1_N(i, a, b) for (i=int(a);i<=int(b);++i)  
#define DWN_1_N(i, b, a) for (i=int(b);i>=int(a);--i)  
#define REP_C_N(i, n) for (n____=int(n),i=0;i<n____;++i)  
#define FOR_C_N(i, a, b) for (b____=int(b),i=a;i<b____;++i)  
#define DWN_C_N(i, b, a) for (a____=int(a),i=b-1;i>=a____;--i)  
#define REP_1_C_N(i, n) for (n____=int(n),i=1;i<=n____;++i)  
#define FOR_1_C_N(i, a, b) for (b____=int(b),i=a;i<=b____;++i)  
#define DWN_1_C_N(i, b, a) for (a____=int(a),i=b;i>=a____;--i)  
   
#define ECH(it, A) for (typeof(A.begin()) it=A.begin(); it != A.end(); ++it)  
#define DO(n) while(n--)  
#define DO_C(n) int n____ = n; while(n____--)  
#define TO(i, a, b) int s_=a<b?1:-1,b_=b+s_;for(int i=a;i!=b_;i+=s_)  
#define TO_1(i, a, b) int s_=a<b?1:-1,b_=b;for(int i=a;i!=b_;i+=s_)  
#define SQZ(i, j, a, b) for (int i=int(a),j=int(b)-1;i<j;++i,--j)  
#define SQZ_1(i, j, a, b) for (int i=int(a),j=int(b);i<=j;++i,--j)  
#define REP_2(i, j, n, m) REP(i, n) REP(j, m)  
#define REP_2_1(i, j, n, m) REP_1(i, n) REP_1(j, m)  
   
#define ALL(A) A.begin(), A.end()  
#define LLA(A) A.rbegin(), A.rend()  
#define CPY(A, B) memcpy(A, B, sizeof(A))  
#define INS(A, P, B) A.insert(A.begin() + P, B)  
#define ERS(A, P) A.erase(A.begin() + P)  
#define BSC(A, X) find(ALL(A), X) // != A.end()  
#define CTN(T, x) (T.find(x) != T.end())  
#define SZ(A) int(A.size())  
#define PB push_back  
#define MP(A, B) make_pair(A, B)  
   
#define Rush int T____; RD(T____); DO(T____)  
#pragma comment(linker, "/STACK:36777216")  
//#pragma GCC optimize ("O2")  
#define Ruby system("ruby main.rb")  
#define Haskell system("runghc main.hs")  
#define Pascal system("fpc main.pas")  
   
typedef long long LL;  
typedef double DB;  
typedef unsigned UINT;  
typedef unsigned long long ULL;  
   
typedef vector<int> VI;  
typedef vector<char> VC;  
typedef vector<string> VS;  
typedef vector<LL> VL;  
typedef vector<DB> VD;  
typedef set<int> SI;  
typedef set<string> SS;  
typedef set<LL> SL;  
typedef set<DB> SD;  
typedef map<int, int> MII;  
typedef map<string, int> MSI;  
typedef map<LL, int> MLI;  
typedef map<DB, int> MDI;  
typedef map<int, bool> MIB;  
typedef map<string, bool> MSB;  
typedef map<LL, bool> MLB;  
typedef map<DB, bool> MDB;  
typedef pair<int, int> PII;  
typedef pair<int, bool> PIB;  
typedef vector<PII> VII;  
typedef vector<VI> VVI;  
typedef vector<VII> VVII;  
typedef set<PII> SII;  
typedef map<PII, int> MPIII;  
typedef map<PII, bool> MPIIB;  
   
/** I/O Accelerator **/  
   
/* ... :" We are I/O Accelerator ... Use us at your own risk ;) ... " .. */  
   
template<class T> inline void RD(T &);  
template<class T> inline void OT(const T &);  
   
inline int RD(){ int x; RD(x); return x;}  
template<class T> inline T& _RD(T &x){ RD(x); return x;}  
inline void RC(char &c){scanf(" %c", &c);}  
inline void RS(char *s){scanf("%s", s);}  
   
template<class T0, class T1> inline void RD(T0 &x0, T1 &x1){RD(x0), RD(x1);}  
template<class T0, class T1, class T2> inline void RD(T0 &x0, T1 &x1, T2 &x2){RD(x0), RD(x1), RD(x2);}  
template<class T0, class T1, class T2, class T3> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3){RD(x0), RD(x1), RD(x2), RD(x3);}  
template<class T0, class T1, class T2, class T3, class T4> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4);}  
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5);}  
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RD(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){RD(x0), RD(x1), RD(x2), RD(x3), RD(x4), RD(x5), RD(x6);}  
template<class T0, class T1> inline void OT(T0 &x0, T1 &x1){OT(x0), OT(x1);}  
template<class T0, class T1, class T2> inline void OT(T0 &x0, T1 &x1, T2 &x2){OT(x0), OT(x1), OT(x2);}  
template<class T0, class T1, class T2, class T3> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3){OT(x0), OT(x1), OT(x2), OT(x3);}  
template<class T0, class T1, class T2, class T3, class T4> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4);}  
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5);}  
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void OT(T0 &x0, T1 &x1, T2 &x2, T3 &x3, T4 &x4, T5 &x5, T6 &x6){OT(x0), OT(x1), OT(x2), OT(x3), OT(x4), OT(x5), OT(x6);}  
   
template<class T> inline void RST(T &A){memset(A, 0, sizeof(A));}  
template<class T0, class T1> inline void RST(T0 &A0, T1 &A1){RST(A0), RST(A1);}  
template<class T0, class T1, class T2> inline void RST(T0 &A0, T1 &A1, T2 &A2){RST(A0), RST(A1), RST(A2);}  
template<class T0, class T1, class T2, class T3> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3){RST(A0), RST(A1), RST(A2), RST(A3);}  
template<class T0, class T1, class T2, class T3, class T4> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4);}  
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5);}  
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void RST(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){RST(A0), RST(A1), RST(A2), RST(A3), RST(A4), RST(A5), RST(A6);}  
   
   
template<class T> inline void CLR(priority_queue<T, vector<T>, less<T> > &Q){  
    while (!Q.empty()) Q.pop();  
}  
   
template<class T> inline void CLR(priority_queue<T, vector<T>, greater<T> > &Q){  
    while (!Q.empty()) Q.pop();  
}  
   
template<class T> inline void CLR(T &A){A.clear();}  
template<class T0, class T1> inline void CLR(T0 &A0, T1 &A1){CLR(A0), CLR(A1);}  
template<class T0, class T1, class T2> inline void CLR(T0 &A0, T1 &A1, T2 &A2){CLR(A0), CLR(A1), CLR(A2);}  
template<class T0, class T1, class T2, class T3> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3){CLR(A0), CLR(A1), CLR(A2), CLR(A3);}  
template<class T0, class T1, class T2, class T3, class T4> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4);}  
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5);}  
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void CLR(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){CLR(A0), CLR(A1), CLR(A2), CLR(A3), CLR(A4), CLR(A5), CLR(A6);}  
template<class T> inline void CLR(T &A, int n){REP(i, n) CLR(A[i]);}  
template<class T> inline void FLC(T &A, int x){memset(A, x, sizeof(A));}  
template<class T0, class T1> inline void FLC(T0 &A0, T1 &A1, int x){FLC(A0, x), FLC(A1, x);}  
template<class T0, class T1, class T2> inline void FLC(T0 &A0, T1 &A1, T2 &A2){FLC(A0), FLC(A1), FLC(A2);}  
template<class T0, class T1, class T2, class T3> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3){FLC(A0), FLC(A1), FLC(A2), FLC(A3);}  
template<class T0, class T1, class T2, class T3, class T4> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4);}  
template<class T0, class T1, class T2, class T3, class T4, class T5> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5);}  
template<class T0, class T1, class T2, class T3, class T4, class T5, class T6> inline void FLC(T0 &A0, T1 &A1, T2 &A2, T3 &A3, T4 &A4, T5 &A5, T6 &A6){FLC(A0), FLC(A1), FLC(A2), FLC(A3), FLC(A4), FLC(A5), FLC(A6);}  
   
template<class T> inline void SRT(T &A){sort(ALL(A));}  
template<class T, class C> inline void SRT(T &A, C B){sort(ALL(A), B);}  
   
/** Add - On **/  
   
const int MOD = 1000000007;  
const int INF = 1000000000;  
const DB EPS = 1e-2;  
const DB OO = 1e15;  
const DB PI = 3.14159265358979323846264; //M_PI;  
   
// <<= ` 0. Daily Use .,  
   
template<class T> inline void checkMin(T &a,const T b){if (b<a) a=b;}  
template<class T> inline void checkMax(T &a,const T b){if (b>a) a=b;}  
template <class T, class C> inline void checkMin(T& a, const T b, C c){if (c(b,a)) a = b;}  
template <class T, class C> inline void checkMax(T& a, const T b, C c){if (c(a,b)) a = b;}  
template<class T> inline T min(T a, T b, T c){return min(min(a, b), c);}  
template<class T> inline T max(T a, T b, T c){return max(max(a, b), c);}  
template<class T> inline T min(T a, T b, T c, T d){return min(min(a, b), min(c, d));}  
template<class T> inline T max(T a, T b, T c, T d){return max(min(a, b), max(c, d));}  
template<class T> inline T sqr(T a){return a*a;}  
template<class T> inline T cub(T a){return a*a*a;}  
int Ceil(int x, int y){return (x - 1) / y + 1;}  
   
// <<= ` 1. Bitwise Operation .,  
inline bool _1(int x, int i){return x & 1<<i;}  
inline bool _1(LL x, int i){return x & 1LL<<i;}  
inline LL _1(int i){return 1LL<<i;}  
//inline int _1(int i){return 1<<i;}  
inline LL _U(int i){return _1(i) - 1;};  
//inline int _U(int i){return _1(i) - 1;};  
   
   
template<class T> inline T low_bit(T x) {  
    return x & -x;  
}  
   
template<class T> inline T high_bit(T x) {  
    T p = low_bit(x);  
    while (p != x) x -= p, p = low_bit(x);  
    return p;  
}  
   
inline int count_bits(int x){  
    x = (x & 0x55555555) + ((x & 0xaaaaaaaa) >> 1);  
    x = (x & 0x33333333) + ((x & 0xcccccccc) >> 2);  
    x = (x & 0x0f0f0f0f) + ((x & 0xf0f0f0f0) >> 4);  
    x = (x & 0x00ff00ff) + ((x & 0xff00ff00) >> 8);  
    x = (x & 0x0000ffff) + ((x & 0xffff0000) >> 16);  
    return x;  
}  
   
inline int count_bits(LL x){  
    x = (x & 0x5555555555555555LL) + ((x & 0xaaaaaaaaaaaaaaaaLL) >> 1);  
    x = (x & 0x3333333333333333LL) + ((x & 0xccccccccccccccccLL) >> 2);  
    x = (x & 0x0f0f0f0f0f0f0f0fLL) + ((x & 0xf0f0f0f0f0f0f0f0LL) >> 4);  
    x = (x & 0x00ff00ff00ff00ffLL) + ((x & 0xff00ff00ff00ff00LL) >> 8);  
    x = (x & 0x0000ffff0000ffffLL) + ((x & 0xffff0000ffff0000LL) >> 16);  
    x = (x & 0x00000000ffffffffLL) + ((x & 0xffffffff00000000LL) >> 32);  
    return x;  
}  
   
int reverse_bits(int x){  
    x = ((x >> 1) & 0x55555555) | ((x << 1) & 0xaaaaaaaa);  
    x = ((x >> 2) & 0x33333333) | ((x << 2) & 0xcccccccc);  
    x = ((x >> 4) & 0x0f0f0f0f) | ((x << 4) & 0xf0f0f0f0);  
    x = ((x >> 8) & 0x00ff00ff) | ((x << 8) & 0xff00ff00);  
    x = ((x >>16) & 0x0000ffff) | ((x <<16) & 0xffff0000);  
    return x;  
}  
   
LL reverse_bits(LL x){  
    x = ((x >> 1) & 0x5555555555555555LL) | ((x << 1) & 0xaaaaaaaaaaaaaaaaLL);  
    x = ((x >> 2) & 0x3333333333333333LL) | ((x << 2) & 0xccccccccccccccccLL);  
    x = ((x >> 4) & 0x0f0f0f0f0f0f0f0fLL) | ((x << 4) & 0xf0f0f0f0f0f0f0f0LL);  
    x = ((x >> 8) & 0x00ff00ff00ff00ffLL) | ((x << 8) & 0xff00ff00ff00ff00LL);  
    x = ((x >>16) & 0x0000ffff0000ffffLL) | ((x <<16) & 0xffff0000ffff0000LL);  
    x = ((x >>32) & 0x00000000ffffffffLL) | ((x <<32) & 0xffffffff00000000LL);  
    return x;  
}  
   
// <<= ` 2. Modular Arithmetic Basic .,  
   
inline void INC(int &a, int b){a += b; if (a >= MOD) a -= MOD;}  
inline int sum(int a, int b){a += b; if (a >= MOD) a -= MOD; return a;}  
inline void DEC(int &a, int b){a -= b; if (a < 0) a += MOD;}  
inline int dff(int a, int b){a -= b; if (a < 0) a  += MOD; return a;}  
inline void MUL(int &a, int b){a = (LL)a * b % MOD;}  
inline int pdt(int a, int b){return (LL)a * b % MOD;}  
   
inline int pow(int a, int b){  
    int c = 1;  
    while (b) {  
        if (b&1) MUL(c, a);  
        MUL(a, a), b >>= 1;  
    }  
    return c;  
}  
   
template<class T>  
inline int pow(T a, int b){  
    T c(1);  
    while (b) {  
        if (b&1) MUL(c, a);  
        MUL(a, a), b >>= 1;  
    }  
    return c;  
}  
   
inline int _I(int b){  
    int a = MOD, x1 = 0, x2 = 1, q;  
    while (true){  
        q = a / b, a %= b;  
        if (!a) return (x2 + MOD) % MOD;  
        DEC(x1, pdt(q, x2));  
   
        q = b / a, b %= a;  
        if (!b) return (x1 + MOD) % MOD;  
        DEC(x2, pdt(q, x1));  
    }  
}  
   
inline void DIV(int &a, int b){MUL(a, _I(b));}  
inline int qtt(int a, int b){return pdt(a, _I(b));}  
   
inline int sum(int a, int b, int MOD){  
    a += b; if (a >= MOD) a -= MOD;  
    return a;  
}  
   
inline int phi(int n){  
    int res = n;  
    for (int i=2;sqr(i)<=n;++i) if (!(n%i)){  
        DEC(res, qtt(res, i));  
        do{n /= i;} while(!(n%i));  
    }  
    if (n != 1)  
        DEC(res, qtt(res, n));  
    return res;  
}  
   
// <<= '9. Comutational Geometry .,  
   
struct Po; struct Line; struct Seg;  
   
inline int sgn(DB x){return x < -EPS ? -1 : x > EPS;}  
inline int sgn(DB x, DB y){return sgn(x - y);}  
   
struct Po{  
    DB x, y;  
    Po(DB _x = 0, DB _y = 0):x(_x), y(_y){}  
   
    friend istream& operator >>(istream& in, Po &p){return in >> p.x >> p.y;}  
    friend ostream& operator <<(ostream& out, Po p){return out << "(" << p.x << ", " << p.y << ")";}  
   
    friend bool operator ==(Po, Po);  
    friend bool operator !=(Po, Po);  
    friend Po operator +(Po, Po);  
    friend Po operator -(Po, Po);  
    friend Po operator *(Po, DB);  
    friend Po operator /(Po, DB);  
   
    bool operator < (const Po &rhs) const{return sgn(x, rhs.x) < 0 || sgn(x, rhs.x) == 0 && sgn(y, rhs.y) < 0;}  
    Po operator-() const{return Po(-x, -y);}  
    Po& operator +=(Po rhs){x += rhs.x, y += rhs.y; return *this;}  
    Po& operator -=(Po rhs){x -= rhs.x, y -= rhs.y; return *this;}  
    Po& operator *=(DB k){x *= k, y *= k; return *this;}  
    Po& operator /=(DB k){x /= k, y /= k; return *this;}  
   
    DB length_sqr(){return sqr(x) + sqr(y);}  
    DB length(){return sqrt(length_sqr());}  
   
    DB atan(){  
        return atan2(y, x);  
    }  
   
    void input(){  
        scanf("%lf %lf", &x, &y);  
    }  
};  
   
bool operator ==(Po a, Po b){return sgn(a.x - b.x) == 0 && sgn(a.y - b.y) == 0;}  
bool operator !=(Po a, Po b){return sgn(a.x - b.x) != 0 || sgn(a.y - b.y) != 0;}  
Po operator +(Po a, Po b){return Po(a.x + b.x, a.y + b.y);}  
Po operator -(Po a, Po b){return Po(a.x - b.x, a.y - b.y);}  
Po operator *(Po a, DB k){return Po(a.x * k, a.y * k);}  
Po operator *(DB k, Po a){return a * k;}  
Po operator /(Po a, DB k){return Po(a.x / k, a.y / k);}  
   
struct Line{  
    Po a, b;  
    Line(Po _a = Po(), Po _b = Po()):a(_a), b(_b){}  
    Line(DB x0, DB y0, DB x1, DB y1):a(Po(x0, y0)), b(Po(x1, y1)){}  
    Line(Seg);  
   
    friend ostream& operator <<(ostream& out, Line p){return out << p.a << "-" << p.b;}  
};  
   
struct Seg{  
    Po a, b;  
    Seg(Po _a = Po(), Po _b = Po()):a(_a), b(_b){}  
    Seg(DB x0, DB y0, DB x1, DB y1):a(Po(x0, y0)), b(Po(x1, y1)){}  
    Seg(Line l);  
   
    friend ostream& operator <<(ostream& out, Seg p){return out << p.a << "-" << p.b;}  
    DB length(){return (b - a).length();}  
};  
   
Line::Line(Seg l):a(l.a), b(l.b){}  
Seg::Seg(Line l):a(l.a), b(l.b){}  
   
#define innerProduct dot  
#define scalarProduct dot  
#define dotProduct dot  
#define outerProduct det  
#define crossProduct det  
   
inline DB dot(DB x1, DB y1, DB x2, DB y2){return x1 * x2 + y1 * y2;}  
inline DB dot(Po a, Po b){return dot(a.x, a.y, b.x, b.y);}  
inline DB dot(Po p0, Po p1, Po p2){return dot(p1 - p0, p2 - p0);}  
inline DB dot(Line l1, Line l2){return dot(l1.b - l1.a, l2.b - l2.a);}  
inline DB det(DB x1, DB y1, DB x2, DB y2){return x1 * y2 - x2 * y1;}  
inline DB det(Po a, Po b){return det(a.x, a.y, b.x, b.y);}  
inline DB det(Po p0, Po p1, Po p2){return det(p1 - p0, p2 - p0);}  
inline DB det(Line l1, Line l2){return det(l1.b - l1.a, l2.b - l2.a);}  
   
template<class T1, class T2> inline DB dist(T1 x, T2 y){return sqrt(dist_sqr(x, y));}  
   
inline DB dist_sqr(Po a, Po b){return sqr(a.x - b.x) + sqr(a.y - b.y);}  
inline DB dist_sqr(Po p, Line l){Po v0 = l.b - l.a, v1 = p - l.a; return sqr(fabs(det(v0, v1))) / v0.length_sqr();}  
inline DB dist_sqr(Po p, Seg l){  
    Po v0 = l.b - l.a, v1 = p - l.a, v2 = p - l.b;  
    if (sgn(dot(v0, v1)) * sgn(dot(v0, v2)) <= 0) return dist_sqr(p, Line(l));  
    else return min(v1.length_sqr(), v2.length_sqr());  
}  
   
inline DB dist_sqr(Line l, Po p){return dist_sqr(p, l);}  
inline DB dist_sqr(Seg l, Po p){return dist_sqr(p, l);}  
   
inline DB dist_sqr(Line l1, Line l2){  
    if (sgn(det(l1, l2)) != 0) return 0;  
    return dist_sqr(l1.a, l2);  
}  
inline DB dist_sqr(Line l1, Seg l2){  
    Po v0 = l1.b - l1.a, v1 = l2.a - l1.a, v2 = l2.b - l1.a; DB c1 = det(v0, v1), c2 = det(v0, v2);  
    return sgn(c1) != sgn(c2) ? 0 : sqr(min(fabs(c1), fabs(c2))) / v0.length_sqr();  
}  
   
bool isIntersect(Seg l1, Seg l2){  
   
    //if (l1.a == l2.a || l1.a == l2.b || l1.b == l2.a || l1.b == l2.b) return true;  
   
    return  
        min(l1.a.x, l1.b.x) <= max(l2.a.x, l2.b.x) &&  
        min(l2.a.x, l2.b.x) <= max(l1.a.x, l1.b.x) &&  
        min(l1.a.y, l1.b.y) <= max(l2.a.y, l2.b.y) &&  
        min(l2.a.y, l2.b.y) <= max(l1.a.y, l1.b.y) &&  
    sgn( det(l1.a, l2.a, l2.b) ) * sgn( det(l1.b, l2.a, l2.b) ) <= 0 &&  
    sgn( det(l2.a, l1.a, l1.b) ) * sgn( det(l2.b, l1.a, l1.b) ) <= 0;  
   
}  
   
inline DB dist_sqr(Seg l1, Seg l2){  
    if (isIntersect(l1, l2)) return 0;  
    else return min(dist_sqr(l1.a, l2), dist_sqr(l1.b, l2), dist_sqr(l2.a, l1), dist_sqr(l2.b, l1));  
}  
   
inline bool isOnExtremePoint(const Po &p, const Seg &l){  
    return p == l.a || p == l.b;  
}  
   
inline bool isOnseg(const Po &p, const Seg &l){  
   
    //if (p == l.a || p == l.b) return false;  
   
    return sgn(det(p, l.a, l.b)) == 0 &&  
        sgn(l.a.x, p.x) * sgn(l.b.x, p.x) <= 0 && sgn(l.a.y, p.y) * sgn(l.b.y, p.y) <= 0;  
}  
   
inline Po intersect(const Line &l1, const Line &l2){  
    return l1.a + (l1.b - l1.a) * (det(l2.a, l1.a, l2.b) / det(l2, l1));  
}  
   
// perpendicular foot  
inline Po intersect(const Po & p, const Line &l){  
    return intersect(Line(p, p + Po(l.a.y - l.b.y, l.b.x - l.a.x)), l);  
}  
   
inline Po rotate(Po p, DB alpha, Po o = Po()){  
    p.x -= o.x, p.y -= o .y;  
    return Po(p.x * cos(alpha) - p.y * sin(alpha), p.y * cos(alpha) + p.x * sin(alpha)) + o;  
}  
   
// <<= ' A. Random Event ..  
   
inline int rand32(){return (bool(rand() & 1) << 30) | (rand() << 15) + rand();}  
inline int random32(int l, int r){return rand32() % (r - l + 1) + l;}  
inline int random(int l, int r){return rand() % (r - l + 1) + l;}  
int dice(){return rand() % 6;}  
bool coin(){return rand() % 2;}  
   
// <<= ' 0. I/O Accelerator interface .,  
   
template<class T> inline void RD(T &x){  
    //cin >> x;  
    scanf("%d", &x);  
    //char c; for (c = getchar(); c < '0'; c = getchar()); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0';  
    //char c; c = getchar(); x = c - '0'; for (c = getchar(); c >= '0'; c = getchar()) x = x * 10 + c - '0';  
}  
   
template<class T> inline void OT(const T &x){  
    printf("%d
", x);  
}  
   
/* .................................................................................................................................. */  
   
const int N = 50009, M = 10009;  
   
const int NN = 2500009;  
   
int l[NN], r[NN], c[NN], total;  
   
PII A[N+M]; int B[N+M], Q[M][3];  
int S[N], C[N], Null;  
int n, m, An, Tn;  
   
#define lx l[x]  
#define rx r[x]  
#define ly l[y]  
#define ry r[y]  
#define cx c[x]  
#define cy c[y]  
   
#define mid ((ll+rr)>>1)  
#define lc lx, ll, mid  
#define rc rx, mid+1, rr  
   
   
void Build(int &x, int ll, int rr){  
    x = ++total; if (ll < rr) Build(lc), Build(rc);  
}  
   
int Insert(int y, int p, int d){  
   
    int x = ++total, root = x;  
   
    c[x] = c[y] + d; int ll = 0, rr = Tn;  
   
    while (ll < rr){  
        if (p <= mid){  
            lx = ++total, rx = ry;  
            x = lx, y = ly, rr = mid;  
        }  
        else {  
            lx = ly, rx = ++total;  
            x = rx, y = ry, ll = mid + 1;  
        }  
        c[x] = c[y] + d;  
    }  
   
    return root;  
}  
   
   
struct Pack{  
    VI L;  
   
    inline Pack(){}  
    inline Pack(int x){L.PB(x);}  
   
    inline void operator += (int x){  
        L.PB(x);  
    }  
   
    inline operator int() const{  
        int res = 0; REP(i, SZ(L)) res += c[l[L[i]]];  
        return res;  
    }  
   
    void lt(){  
        REP(i, SZ(L)) L[i] = l[L[i]];  
    }  
    void rt(){  
        REP(i, SZ(L)) L[i] = r[L[i]];  
    }  
   
};  
   
   
   
void Modify(int x, int p, int d){  
    while (x <= n) C[x] = Insert(C[x], p, d), x += low_bit(x);  
}  
   
Pack Query(int x){  
    Pack res; while (x) res += C[x], x ^= low_bit(x);  
    return res;  
}  
   
int Query(int ll, int rr, int k){  
   
    --ll; Pack a = Query(rr), b = Query(ll), c = S[rr], d = S[ll];  
   
    int t; ll = 0, rr = Tn;  
   
    while (ll < rr){  
        if ((t = a - b + c - d) >= k){  
            a.lt(), b.lt(), c.lt(), d.lt();  
            rr = mid;  
        }  
        else {  
            a.rt(), b.rt(), c.rt(), d.rt();  
            k -= t, ll = mid + 1;  
        }  
    }  
      
    return ll;  
}  
   
   
int main(){  
   
#ifndef ONLINE_JUDGE  
    freopen("in.txt", "r", stdin);  
    //freopen("out.txt", "w", stdout);  
#endif  
   
#define key first  
#define id second  
   
    RD(n, m); REP(i, n) A[i] = MP(RD(), i);  
   
    An = n; char cmd; REP(i, m){  
        RC(cmd); if(cmd == 'Q') RD(Q[i][0], Q[i][1], Q[i][2]);  
        else RD(Q[i][0]), Q[i][2] = 0, A[An++] = MP(RD(), An);  
    }  
   
    sort(A, A + An), B[A[0].id] = Tn = 0;  
    FOR(i, 1, An){  
        if(A[i].key != A[i-1].key) A[++Tn].key = A[i].key;  
        B[A[i].id] = Tn;  
    }  
   
    Build(Null, 0, Tn); REP_1(i, n) C[i] = Null;  
   
    S[0] = Null; REP(i, n){  
        S[i+1] = Insert(S[i], B[i], 1);  
    }  
      
    An = n;  
   
    REP(i, m) if (Q[i][2]){  
        OT(A[Query(Q[i][0], Q[i][1], Q[i][2])].key);  
    }else{  
        Modify(Q[i][0], B[Q[i][0]-1], -1);  
        Modify(Q[i][0], B[Q[i][0]-1] = B[An++], 1);  
    }  
}
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原文地址：https://www.cnblogs.com/zhsl/p/3267090.html