[CF895E] Eyes Closed(线段树,期望)

Posted 2022-10-18 qrsikno

Desctiption

传送门:Portal

大致题意: 给你一个序列, 支持两种操作:

1. 1 l1 r1 l2 y2 在\([l1, r1]\)随机选择一个数a, \([l2, r2]\) 内随机选择一个数b, 交换a, b.
2. 2 l r 询问一个区间的期望.

\[ n \leq 200000; a_i \leq 1e9 \]

Solution

根据==期望线性性==,只需要维护出==每个位置元素值的期望==就可以.

==区间操作期望==问题的经典套路是==维护出每个位置的期望==.

考虑一个数字\(a_i\)

他有\(\fraclen - 1len\) 的概率保持原数, 否则,根据全期望公式, 它会变成\(E(b_i) = \sum\fracb_ilenB\)

那么有:
\[ A_i = \frac(lenA - 1)A_i + \sum \fracb_ilenBlenA \]
\(b_i\)也同理.

所以我们只需要维护一个区间加/乘, 区间求和的线段树就可以了.

Code

#include<bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for(int i = (a), i##_end_ = (b); i <= i##_end_; ++i)
#define drep(i, a, b) for(int i = (a), i##_end_ = (b); i >= i##_end_; --i)
#define clar(a, b) memset((a), (b), sizeof(a))
#define debug(...) fprintf(stderr, __VA_ARGS__)
typedef long long LL;
typedef long double LD;
int read() 
    char ch = getchar();
    int x = 0, flag = 1;
    for (;!isdigit(ch); ch = getchar()) if (ch == '-') flag *= -1;
    for (;isdigit(ch); ch = getchar()) x = x * 10 + ch - 48;
    return x * flag;

void write(int x) 
    if (x < 0) putchar('-'), x = -x;
    if (x >= 10) write(x / 10);
    putchar(x % 10 + 48);


const int Maxn = 200009;
int a[Maxn];

namespace SGMTtree 
    double tree[Maxn << 3], addTag[Maxn << 3], mulTag[Maxn << 3];
#define lc(x) ((x) << 1)
#define rc(x) (((x) << 1) | 1)
#define ls rt << 1, l, mid
#define rs (rt << 1) | 1, mid + 1, r
    void setAdd(int root, int l, int r, double v) 
        tree[root] += v * (r - l + 1.0); addTag[root] += v;
    
    void setMul(int root, int l, int r, double v) 
        tree[root] *= v;
        mulTag[root] *= v; addTag[root] *= v;
    
    void (*setTag)(int, int, int, double);
    void set(int p) 
        if(p == 1) setTag = setAdd;
        if(p == 2) setTag = setMul;
    

    void pushdown(int rt, int l, int r) 
        int mid = (l + r) >> 1;
        if (mulTag[rt] != 1.0) 
            setMul(ls, mulTag[rt]), setMul(rs, mulTag[rt]);
            mulTag[rt] = 1;
        
        if (addTag[rt] != 0.0) 
            setAdd(ls, addTag[rt]); setAdd(rs, addTag[rt]); 
            addTag[rt] = 0;
        
    
    void pushup(int rt)  tree[rt] = tree[lc(rt)] + tree[rc(rt)]; 

    void build(int rt, int l, int r) 
        addTag[rt] = 0, mulTag[rt] = 1;
        if (l == r) 
            tree[rt] = a[l];
            return ;
        
        int mid = (l + r) >> 1;
        build(ls), build(rs);
        pushup(rt);
    
    void modify(int rt, int l, int r, int p, int q, double v) 
        if (p > q) return ;
        if (p <= l && r <= q) 
            setTag(rt, l, r, v);
            return ;
        
        int mid = (l + r) >> 1; pushdown(rt, l, r);
        if (q <= mid) modify(ls, p, q, v);
        else if (p > mid) modify(rs, p, q, v);
        else modify(ls, p, q, v), modify(rs, p, q, v);
        pushup(rt);
    
    double query(int rt, int l, int r, int p, int q) 
        if (p <= l && r <= q) return tree[rt];
        int mid = (l + r) >> 1; pushdown(rt, l, r);
        if (q <= mid) return query(ls, p, q);
        else if (p > mid) return query(rs, p, q);
        else return query(ls, p, q) + query(rs, p, q);
    
#undef lc
#undef rc
#undef ls
#undef rs
;

int n, q;

void init() 
    n = read(); q = read();
    rep (i, 1, n) a[i] = read();
    SGMTtree :: build(1, 1, n);


void solve() 
    rep (i, 1, q) 
        int opt = read();
        if (opt == 1) 
            int l1 = read(), r1 = read(), l2 = read(), r2 = read();
            double c = r1 - l1 + 1.0, d = r2 - l2 + 1.0, tmp1 = SGMTtree :: query(1, 1, n, l1, r1), tmp2 = SGMTtree :: query(1, 1, n, l2, r2);
            SGMTtree :: set(2), SGMTtree :: modify(1, 1, n, l1, r1, c - 1.0);
            SGMTtree :: set(2), SGMTtree :: modify(1, 1, n, l2, r2, d - 1.0);
            SGMTtree :: set(1), SGMTtree :: modify(1, 1, n, l1, r1, tmp2 / d);
            SGMTtree :: set(1), SGMTtree :: modify(1, 1, n, l2, r2, tmp1 / c);
            SGMTtree :: set(2), SGMTtree :: modify(1, 1, n, l1, r1, 1 / c);
            SGMTtree :: set(2), SGMTtree :: modify(1, 1, n, l2, r2, 1 / d);
        
        if (opt == 2) 
            int l = read(), r = read();
            printf("%.7lf\n", SGMTtree :: query(1, 1, n, l, r));
        
    


int main() 
//  freopen("CF895E.in", "r", stdin);
//  freopen("CF895E.out", "w", stdout);

    init();
    solve();

#ifdef Qrsikno
    debug("\nRunning time: %.3lf(s)\n", clock() * 1.0 / CLOCKS_PER_SEC);
#endif
    return 0;

调试:

线段树函数指针必须要用namespace.