【SDOI2017】相关分析（线段树）

Description

你需要维护一个长度为 \(n\) 的实数对的序列，第 \(i\) 个元素为 \((x_i, y_i)\)。现有 \(m\) 次操作：

• \(\texttt{1 L R}\)：设区间 \([L, R]\) 的平均数 \(\bar x = \frac{\sum_{i=L}^R x_i}{R-L+1},\bar y = \frac{\sum_{i=L}^R y_i}{R-L+1}\)，求

\[a = \dfrac{\sum_{i=L}^R (x_i - \bar x)(y_i - \bar y)}{\sum_{i=L}^R (x_i - \bar x)^2}
\]

的值。误差不超过 \(10^{-5}\)。

• \(\texttt{2 L R S T}\)：进行如下修改：

  \(\forall i\in[L, R],\quad x_i \leftarrow x_i + S\)

  \(\forall i\in[L, R],\quad y_i \leftarrow y_i + T\)

• \(\texttt{3 L R S T}\)：进行如下修改：

  \(\forall i \in [L, R],\quad x_i \leftarrow i + S\)

  \(\forall i \in [L, R],\quad y_i \leftarrow i + T\)

Hint

\(1\le n, m\le 10^5, 0\le |S|, |T|\le 10^5, 0\le |x_i|, |y_i|\le 10^5\)

Solution

一道细节很多的线段树练手题。

在考虑怎么支持这些操作时不妨先看看我们需要维护些什么东西。拆开 \(a\) 这个柿子：

（下面用 \(\sum\) 代替 \(\sum_{i=L}^R\)，用 \(n\) 代替 \(R - L + 1\)）

\[\begin{aligned}
a = & \dfrac{\sum (x_i - \bar x)(y_i - \bar y)}{\sum (x_i - \bar x)^2} = \dfrac A B \\ \\
A = & \sum (x_i y_i - x_i \bar y - y_i \bar x_i + \bar x \bar y) \\
= & \sum x_i y_i - \bar y\sum x_i - \bar x\sum y_i + n\bar x \bar y \\
= & \sum x_i y_i - \tfrac{1}{n}\sum x_i \sum y_i \\ \\
B = & \sum (x_i^2 + {\bar x}^2 - 2x_i\bar x) \\
= & \sum x_i^2 +n{\bar x}^2 - 2\bar x\sum x_i \\
= & \sum x_i^2 -\tfrac{1}{n}\left(\sum x_i\right)^2\\
\end{aligned}
\]

将 \(\bar x, \bar y\) 展开的话，可以发现我们需要维护四个东西：

\[\begin{aligned}
A = & \boxed{\sum x_i y_i} - \tfrac{1}{n}\boxed{\sum x_i}\times \boxed{\sum y_i} \\
B = & \boxed{\sum x_i^2} -\tfrac{1}{n}\left(\sum x_i\right)^2\\
\end{aligned}
\]

整坨柿子瞬间简洁可做了。

---

那么如何维护这四个信息呢？为了方便，我们用 \(v_1, v_2, v_3, v_4\) 分别代表 \(\sum x_i, \sum y_i,\sum x_i^2,\sum x_i y_i\)。

考虑修改操作之后这四个信息的变化：

• 操作 \(2\)：

\[\begin{aligned}
&\sum x_i\to\sum(x_i+S)=\sum x_i+nS & v_1\to v_1 + nS \\
&\sum y_i\to\sum(y_i+T)=\sum y_i+nT & v_2\to v_2 + nT \\
&\sum x_i^2\to\sum(x_i+S)^2=\sum x_i^2+nS^2+2S\sum x_i& v_3\to v_3+nS^2+2Sv_1\\
&\sum x_i y_i \to\sum(x_i+S)(y_i+T)=\sum x_i y_i+T\sum x_i+S\sum y_i+nST & v_4 \to v_4+Tv_1+Sv_2+nST\\
\end{aligned}
\]

• 操作 \(3\)：
• 记 \(s_1 = \sum_{i=L}^R i, s_2 = \sum_{i=L}^R i^2\)。

\[\begin{aligned}
&\sum x_i\to\sum(i+S)=s_1+nS & v_1\to s_1 + nS \\
&\sum y_i\to\sum(i+T)=s_1+nT & v_2\to s_1 + nT \\
&\sum x_i^2\to\sum(i+S)^2=s_2+nS^2+2Ss_1& v_3\to s_2+nS^2+2Ss_1\\
&\sum x_i y_i \to\sum(i+S)(i+T)=s_2+(T+S)s_1+nST & v_4 \to s_2+(T+S)s_1+nST\\
\end{aligned}
\]

于是就这样维护就行了，注意操作二的更新顺序。

其中 \(s_1, s_2\) 的计算可以预处理，也可以直接使用公式：

• \(\sum_{j=1}^i j = \frac{i(i+1)}{2}\)
• \(\sum_{j=1}^i j^2 = \frac{i(i+1)(2i+1)}{6}\)

Code

/*
 * Author : _Wallace_
 * Source : https://www.cnblogs.com/-Wallace-/
 * Problem : SDOI2017 相关分析
 */
#include <algorithm>
#include <cmath>
#include <cstdio>
using namespace std;
const int N = 1e5 + 5;
const int S = N << 2;
namespace calculator {
    inline double sum(const double& l, const double& r) {
        return (l + r) * (r - l + 1) / 2;
    }
    inline double sqrs(const double& p) {
        return p * (p + 1) * (2 * p + 1) / 6;
    }
}
struct dataType {
    double sumx, sumy, sqrs, muls;
    inline dataType()
    : sumx(0.0), sumy(0.0), sqrs(0.0), muls(0.0) { }
    inline dataType(const double& a, const double& b,
                    const double& c, const double& d)
    : sumx(a), sumy(b), sqrs(c), muls(d) { }
    inline dataType(const double& x, const double& y)
    : sumx(x), sumy(y), sqrs(x * x), muls(x * y) { }
    inline dataType operator + (const dataType& rhs) const {
        return dataType(this->sumx + rhs.sumx, this->sumy + rhs.sumy,
                        this->sqrs + rhs.sqrs, this->muls + rhs.muls);
    }
    inline void update(const dataType& rhs) {
        this->sumx += rhs.sumx, this->sumy += rhs.sumy;
        this->sqrs += rhs.sqrs, this->muls += rhs.muls;
    }
};
struct tagType {
    double S, T;
    inline tagType()
    : S(0.0), T(0.0) { }
    inline tagType(const double& s, const double& t)
    : S(s), T(t) { }
    inline bool operator == (const tagType& rhs) const {
        return (fabs(this->S - rhs.S) <= 1e-8) && (fabs(this->T - rhs.T) <= 1e-8);
    }
    inline bool operator != (const tagType& rhs) const {
        return !(*this == rhs);
    }
};
const tagType std_cov(-1e18, -1e18);
const tagType std_add(0, 0);
int L[S], R[S];
dataType tr[S];
tagType cov[S];
tagType add[S];
#define mid ((L[x] + R[x]) / 2)
#define len double(R[x] - L[x] + 1)
inline void pushup(int x) {
    tr[x] = tr[x << 1] + tr[x << 1 | 1];
}
inline void setCov(int x, const tagType& v) {
    double s1 = calculator::sum(L[x], R[x]);
    double s2 = calculator::sqrs(R[x]) - calculator::sqrs(L[x] - 1);
    tr[x].sqrs = s2 + len * v.S * v.S + 2 * v.S * s1;
    tr[x].muls = s2 + (v.S + v.T) * s1 + len * v.S * v.T;
    tr[x].sumx = s1 + len * v.S;
    tr[x].sumy = s1 + len * v.T;
    cov[x] = v, add[x] = std_add;
}
inline void setAdd(int x, const tagType& v) {
    tr[x].sqrs += v.S * v.S * len + 2 * v.S * tr[x].sumx;
    tr[x].muls += v.S * tr[x].sumy + v.T * tr[x].sumx + len * v.S * v.T;
    tr[x].sumx += v.S * len, tr[x].sumy += v.T * len;
    add[x].S += v.S, add[x].T += v.T;
}
inline void pushdown(int x) {
    if (cov[x] != std_cov) {
        setCov(x << 1, cov[x]);
        setCov(x << 1 | 1, cov[x]);
        cov[x] = std_cov;
    }
    if (add[x] != std_add) {
        setAdd(x << 1, add[x]);
        setAdd(x << 1 | 1, add[x]);
        add[x] = std_add;
    }
}
void build(int x, int l, int r, double* datx, double* daty) {
    L[x] = l, R[x] = r;
    cov[x] = std_cov, add[x] = std_add;
    if (l == r) {
        tr[x] = dataType(datx[l], daty[l]);
        return;
    }
    build(x << 1, l, mid, datx, daty);
    build(x << 1 | 1, mid + 1, r, datx, daty);
    pushup(x);
}
void modify(int x, int l, int r, void(*update)(int, const tagType&), const tagType& v) {
    if (l <= L[x] && R[x] <= r) return update(x, v);
    pushdown(x);
    if (l <= mid) modify(x << 1, l, r, update, v);
    if (r > mid) modify(x << 1 | 1, l, r, update, v);
    pushup(x);
}
void query(int x, int l, int r, dataType& ret) {
    if (l <= L[x] && R[x] <= r) return ret.update(tr[x]);
    if (l > R[x] || L[x] > r) return;
    pushdown(x), query(x << 1, l, r, ret), query(x << 1 | 1, l, r, ret);
}
#undef mid
#undef len
int n, Q;
double x[N], y[N];
signed main() {
    scanf("%d%d", &n, &Q);
    for (int i = 1; i <= n; i++) scanf("%lf", x + i);
    for (int i = 1; i <= n; i++) scanf("%lf", y + i);
    build(1, 1, n, x, y);
    while (Q--) {
        int opt, l, r;
        scanf("%d%d%d", &opt, &l, &r);
        if (opt == 1) {
            dataType res; query(1, l, r, res);
            double len = r - l + 1;
            double avex = res.sumx / len;
            double avey = res.sumy / len;
            double A = res.muls - res.sumx * res.sumy / len;
            double B = res.sqrs - res.sumx * res.sumx / len;
            printf("%lf\n", A / B);
        } else {
            tagType val; scanf("%lf%lf", &val.S, &val.T);
            modify(1, l, r, (opt == 2 ? setAdd : setCov), val);
        }
    }
}