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。。
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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：
- …
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。。。 - 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 。。。
- 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。。。
- …
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\n", 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); } }