[ZOJ2112][可持久化线段树（主席树）][树状数组]Dynamic Rankings[好题]

题意：

    给出一个序列，求区间第k小。要求支持单点修改。

题解：

不谈修改的时候，地球人都会做。
主要考虑如何维护修改。如果直接在建好的线段树上修改，每次需要新建$O(n)$棵线段树，显然吃不消。
请诸君一定要有一个思想，所谓“函数式线段树”，本质是“化‘树’为‘数’”，让“线段树”这一复杂的结构变得像“数字”一样可以加加减减。为什么区间第k大要用主席树做？因为我们要求区间内小于某一个数的数字个数。这显然是可加的。我们不如就类比做区间求和。带修改的区间求和如何做？树状数组可做。那树状数组每个节点维护的东西变成主席树，就能维护题目中的修改了。

（不要随便看代码。不要随便看代码。不要随便看代码。重要的事情说三遍。）

指针版：

#ifdef _MSC_VER
# include "stdafx.h"
# pragma warning(disable:4996)
#endif

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <iostream>
#include <algorithm>
using namespace std;

//Global Variables & Definitions
#define lowbit(x) ((x) & (-(x)))

int X, M, N;

#define MAXN 50050
#define MAXM 10010
#define MAXT (MAXN + MAXM)

int A[MAXN];
//End Global Variables & Definitions

//Discretization
int w[MAXT];
int wcnt;

void Discre(int cnt) {
    sort(w + 1, w + cnt + 1);
    wcnt = unique(w + 1, w + cnt + 1) - w - 1;
}

int getid(int ww) {
    return lower_bound(w + 1, w + wcnt + 1, ww) - w;
}
//End Discretization

//Chairman Tree
#define DEFINE_MID int mid = (l + r) >> 1
#define lson (u -> lc)
#define rson (u -> rc)

namespace CMT {
    struct node {
        node *lc, *rc;
        int sum;
    } T[1750250];

    int tcnt = 0;
    inline void Init() {
        tcnt = 0;
    }

    inline node* newnode() {
        //return new node();
        return &T[tcnt++];
    }

    inline node* Copy(node *u) {
        node *temp = newnode();
        *temp = *u;
        return temp;
    }

    inline void PushUp(node *u) {
        u->sum = lson->sum + rson->sum;
    }

    void Build(node *u, int l, int r) {
        u->sum = 0; u->lc = u->rc = NULL;

        if (l == r) return;
        DEFINE_MID;

        lson = newnode();
        Build(lson, l, mid);
        rson = newnode();
        Build(rson, mid + 1, r);
    }

    void Insert(node *u, int l, int r, int p, int v = 1) {
        if (l == r) { u->sum += v; return; }

        DEFINE_MID;
        if (p <= mid) {
            lson = Copy(lson);
            Insert(lson, l, mid, p, v);
        }
        else {
            rson = Copy(rson);
            Insert(rson, mid + 1, r, p, v);
        }

        PushUp(u);
    }

    int Query(node *u, int l, int r, int p) {
        if (p >= r) return u->sum;

        DEFINE_MID;

        if (p == mid) return lson->sum;
        else if (p < mid) return Query(lson, l, mid, p);
        else return lson->sum + Query(rson, mid + 1, r, p);
    }
}

using CMT::node;

node* root[MAXN];
//End Chairman Tree

//Fenwick Tree
node* ftr[MAXN];

void Modify(int x, int p, int v) {
    while (x <= N) {
        ftr[x] = CMT::Copy(ftr[x]);
        CMT::Insert(ftr[x], 1, wcnt, p, v);
        x += lowbit(x);
    }
}

int sum(int x, int p) {
    int temp = 0;
    while (x) {
        temp += CMT::Query(ftr[x], 1, wcnt, p);
        x -= lowbit(x);
    }

    return temp;
}
//End Fenwick Tree

//Main Structure
inline int getans(int l, int r, int v) {
    int ansa = CMT::Query(root[r], 1, wcnt, v) - CMT::Query(root[l], 1, wcnt, v);
    ansa += sum(r, v);
    ansa -= sum(l, v);
    return ansa;
}

int Query(int l, int r, int k) {
    int L = 1, R = wcnt;

    while (L < R) {
        int mid = (L + R) >> 1;
        int tans = getans(l - 1, r, mid);

        if (tans >= k) R = mid;
        else L = mid + 1;
    }

    return w[L];
}

char Act[MAXM];
int I[MAXM], J[MAXM], K[MAXM];

void ir() {
    //Read
    scanf("%d%d", &N, &M);

    for (int i = 1; i <= N; ++i) {
        scanf("%d", &A[i]);
        w[i] = A[i];
    }

    wcnt = N;
    char temp[20]; int ii, jj, kk;
    for (int i = 0; i < M; ++i) {
        scanf("%s%d", temp, &ii);

        if ((Act[i] = temp[0]) == 'Q') {
            scanf("%d%d", &jj, &kk);

            I[i] = ii;
            J[i] = jj;
            K[i] = kk;
        }
        else {
            scanf("%d", &jj);

            I[i] = ii;
            w[++wcnt] = J[i] = jj;
        }
    }

    Discre(wcnt);

    //Build
    CMT::Init();
    root[0] = CMT::newnode();
    CMT::Build(root[0], 1, wcnt);
    for (int i = 1; i <= N; ++i) {
        root[i] = CMT::Copy(root[i - 1]);
        CMT::Insert(root[i], 1, wcnt, getid(A[i]));
    }

    for (int i = 1; i <= N; ++i) ftr[i] = root[0];
}

void solve() {
    ir();

    for (int i = 0; i < M; ++i) {
        if (Act[i] == 'Q') {
            printf("%d\n", Query(I[i], J[i], K[i]));
        }
        else {
            Modify(I[i], getid(A[I[i]]), -1);
            Modify(I[i], getid(A[I[i]] = J[i]), 1);
        }
    }
}

inline void g_ir() {
    scanf("%d", &X);
}

int main() {
    g_ir();

    while (X--) solve();
    return 0;
}

数组版：

#ifdef _MSC_VER
# include "stdafx.h"
# pragma warning(disable:4996)
#endif

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <iostream>
#include <algorithm>
using namespace std;

//Global Variables & Definitions
#define lowbit(x) ((x) & (-(x)))

int X, M, N;

#define MAXN 50050
#define MAXM 10010
#define MAXT (MAXN + MAXM)

int A[MAXN];
//End Global Variables & Definitions

//Discretization
int w[MAXT];
int wcnt;

void Discre(int cnt) {
    sort(w + 1, w + cnt + 1);
    wcnt = unique(w + 1, w + cnt + 1) - w - 1;
}

int getid(int ww) {
    return lower_bound(w + 1, w + wcnt + 1, ww) - w;
}
//End Discretization

//Chairman Tree
#define DEFINE_MID int mid = (l + r) >> 1
#define lson T[u].lc
#define rson T[u].rc

namespace CMT {
    struct node {
        int lc, rc;
        int sum;
    } T[2500250];

    int tcnt = 0;
    inline void Init() {
        tcnt = 0;
    }

    inline int newnode() {
        return tcnt++;
    }

    inline int Copy(int u) {
        int temp = newnode();
        T[temp] = T[u];
        return temp;
    }

    inline void PushUp(int u) {
        T[u].sum = T[lson].sum + T[rson].sum;
    }

    void Build(int u, int l, int r) {
        T[u].sum = 0; T[u].lc = T[u].rc = 0;

        if (l == r) return;
        DEFINE_MID;

        lson = newnode();
        Build(lson, l, mid);
        rson = newnode();
        Build(rson, mid + 1, r);
    }

    void Insert(int u, int l, int r, int p, int v = 1) {
        if (l == r) { T[u].sum += v; return; }

        DEFINE_MID;
        if (p <= mid) {
            lson = Copy(lson);
            Insert(lson, l, mid, p, v);
        }
        else {
            rson = Copy(rson);
            Insert(rson, mid + 1, r, p, v);
        }

        PushUp(u);
    }

    int Query(int u, int l, int r, int p) {
        if (p >= r) return T[u].sum;

        DEFINE_MID;

        if (p == mid) return T[lson].sum;
        else if (p < mid) return Query(lson, l, mid, p);
        else return T[lson].sum + Query(rson, mid + 1, r, p);
    }
}

using CMT::node;

int root[MAXN];
//End Chairman Tree

//Fenwick Tree
int ftr[MAXN];

void Modify(int x, int p, int v) {
    while (x <= N) {
        ftr[x] = CMT::Copy(ftr[x]);
        CMT::Insert(ftr[x], 1, wcnt, p, v);
        x += lowbit(x);
    }
}

int sum(int x, int p) {
    int temp = 0;
    while (x) {
        temp += CMT::Query(ftr[x], 1, wcnt, p);
        x -= lowbit(x);
    }

    return temp;
}
//End Fenwick Tree

//Main Structure
inline int getans(int l, int r, int v) {
    int ansa = CMT::Query(root[r], 1, wcnt, v) - CMT::Query(root[l], 1, wcnt, v);
    ansa += sum(r, v);
    ansa -= sum(l, v);
    return ansa;
}

int Query(int l, int r, int k) {
    int L = 1, R = wcnt;

    while (L < R) {
        int mid = (L + R) >> 1;
        int tans = getans(l - 1, r, mid);

        if (tans >= k) R = mid;
        else L = mid + 1;
    }

    return w[L];
}

char Act[MAXM];
int I[MAXM], J[MAXM], K[MAXM];

void ir() {
    //Read
    scanf("%d%d", &N, &M);

    for (int i = 1; i <= N; ++i) {
        scanf("%d", &A[i]);
        w[i] = A[i];
    }

    wcnt = N;
    char temp[20]; int ii, jj, kk;
    for (int i = 0; i < M; ++i) {
        scanf("%s%d", temp, &ii);

        if ((Act[i] = temp[0]) == 'Q') {
            scanf("%d%d", &jj, &kk);

            I[i] = ii;
            J[i] = jj;
            K[i] = kk;
        }
        else {
            scanf("%d", &jj);

            I[i] = ii;
            w[++wcnt] = J[i] = jj;
        }
    }

    Discre(wcnt);

    //Build
    CMT::Init();
    root[0] = CMT::newnode();
    CMT::Build(root[0], 1, wcnt);
    for (int i = 1; i <= N; ++i) {
        root[i] = CMT::Copy(root[i - 1]);
        CMT::Insert(root[i], 1, wcnt, getid(A[i]));
    }

    for (int i = 1; i <= N; ++i) ftr[i] = root[0];
}

void solve() {
    ir();

    for (int i = 0; i < M; ++i) {
        if (Act[i] == 'Q') {
            printf("%d\n", Query(I[i], J[i], K[i]));
        }
        else {
            Modify(I[i], getid(A[I[i]]), -1);
            Modify(I[i], getid(A[I[i]] = J[i]), 1);
        }
    }
}

inline void g_ir() {
    scanf("%d", &X);
}

int main() {
    g_ir();

    while (X--) solve();
    return 0;
}

（当你看到这里我才会告诉你ZOJ上指针版没有WA但是会MLE）