题解 P4546/LOJ 2289【[THUWC2017]在美妙的数学王国中畅游】

简化题面

维护四种操作和一颗树

\(1\)、断开\(u\)和\(v\)之间的边
\(2\)、在\(u\)和\(v\)之间连一条边
\(3\)、讲节点\(c\)更换为\(i\)种函数，且系数和常数项为\(a\)和\(b\)
\(4\)、查询\(u\)到\(v\)路径上所有点上函数带入等于\(x\)时的函数值之和

\(n\le 10^5\)，\(m\le 2\times 10^5\)

解题思路

首先\(1,2\)号操作告诉我们这题肯定要用\(\mathcal{LCT}\)来做，但是考虑怎么维护函数和函数值，根据小绿本高等数学上有个叫泰勒展开的玩意，我们对于每个节点维护其函数的泰勒展开来代表其近似值，因为三个函数\(x\)的幂数都很小因此我们维护大概展开\(10\)次的函数即可，最后查询的时候直接代入即可，具体看代码。

\(\mathcal{Code}\)

// Author: Ame__
#include<bits/stdc++.h>
#define _ 0
//#define AME__
#define AME__DEBUG
#define bomb exit(0)
#define LOG(FMT...) fprintf(stderr , FMT)
#define TOWA(FMT...) fprintf(stdout , FMT)
using namespace std;
/*Grievous Lady*/
    
typedef int32_t i32;
typedef int64_t i64;
typedef double qwq

const int BUF_SIZE = 1 << 12;
char buf[BUF_SIZE] , *buf_s = buf , *buf_t = buf + 1;
    
#define PTR_NEXT() \
{ \
    buf_s ++; \
    if(buf_s == buf_t) \
    { \
        buf_s = buf; \
        buf_t = buf + fread(buf , 1 , BUF_SIZE , stdin); \
    } \
}
    
#define mians(_s_) \
{ \
    while(!isgraph(*buf_s)) PTR_NEXT();\
    char register *_ptr_ = (_s_); \
    while(isgraph(*buf_s) || *buf_s == '-') \
    { \
        *(_ptr_ ++) = *buf_s; \
        PTR_NEXT(); \
    } \
    (*_ptr_) = '\0'; \
}
    
template <typename _n_> void mian(_n_ & _x_){
    char buf_s; while(buf_s != '-' && !isdigit(buf_s)) buf_s = getchar();
    bool register _nega_ = false; if(buf_s == '-'){ _nega_ = true; buf_s = getchar(); }
    _x_ = 0; while(isdigit(buf_s)){  _x_ = _x_ * 10 + buf_s - '0'; buf_s = getchar(); } if(_nega_) _x_ = -_x_;
}
    
const i32 kato = 2e5 + 10;

template <typename _n_> bool cmax(_n_ &a , const _n_ &b){ return a < b ? a = b , 1 : 0; }
template <typename _n_> bool cmin(_n_ &a , const _n_ &b){ return a > b ? a = b , 1 : 0; }
    
i32 n , m , op , x , y;
qwq a , b;
qwq qaq[2] , opt[20];

namespace towa{
    struct Tylar{
        qwq a[12];
        void clear(){ memset(a , 0 , sizeof a); }
        Tylar(){ clear(); }
        Tylar(i32 opt , qwq x , qwq y){
        	y += x * 0.5;
            if(opt == 1){
                qwq res = 1 , f = sin(y) , g = cos(y);
                for(int i = 1;i <= 11;i ++ , res = res * x , swap(f = -f , g)) this -> a[i] = res * f;
            }else if(opt == 2){
                qwq res = exp(y);
                for(int i = 1;i <= 11;i ++ , res *= x) this -> a[i] = res;
            }else *this = Tylar() , this -> a[1] = y , this -> a[2] = x;
        }
        friend Tylar operator + (const Tylar &a , const Tylar &b){
            Tylar c;
            for(i32 i = 1;i <= 11;i ++) c.a[i] = a.a[i] + b.a[i];
            return c;
        }
        inline qwq get_ans(qwq x){
            qwq res = 0; x -= 0.5;
            for(i32 i = 11 ; i ; i --) res = this -> a[i] + res * x / i;
            return res;
        }
    };

    struct node{
        node *ch[2] , *fa;
        static queue<node*> q;
        i32 rev;
        Tylar val , tot;
        node(node *fa = 0x0 , i32 rev = 0): fa(fa) , rev(rev){
            ch[0] = ch[1] = 0x0;
        }
        inline bool ntr(){
            return fa && (fa -> ch[0] == this || fa -> ch[1] == this);
        }
        inline bool isr(){
            return this == fa -> ch[1];
        }
        inline void Modify_rev(){
            rev ^= 1; swap(ch[0] , ch[1]);
        }
        inline void up(){
            tot = this -> val;
            if(ch[0]) this -> tot = this -> tot + ch[0] -> tot;
            if(ch[1]) this -> tot = this -> tot + ch[1] -> tot;
        }
        inline void down(){
            if(rev){
                if(ch[0]) ch[0] -> Modify_rev(); if(ch[1]) ch[1] -> Modify_rev();
                rev = 0;
            }
        }
        void *operator new(size_t){
            static node *S = 0x0 , *T = 0x0; node *tmp;
            return q.empty() ? (S == T && (T = (S = new node[1024]) + 1024 , S == T) ? 0x0 : S ++) : (tmp = q.front() , q.pop() , tmp);
        }
        void operator delete(void *qaq){ q.push(static_cast<node*>(qaq)); }
    }pool[kato];

    queue<node*> node::q;

    inline void split(node *x){
        node *y = x -> fa , *z = y -> fa;
        y -> down() , x -> down();
        int k = x -> isr(); node *w = x -> ch[!k];
        if(y -> ntr()) z -> ch[y -> isr()] = x;
        x -> ch[!k] = y , y -> ch[k] = w;
        y -> fa = x , x -> fa = z;
        if(w) w -> fa = y;
        y -> up() , x -> up();
    }

    inline void splay(node *o){
        static node *tree[kato]; int top = 0;
        tree[top = 1] = o;
        while(tree[top] -> ntr()) tree[top + 1] = tree[top] -> fa , top ++;
        while(top) tree[top --] -> down();
        while(o -> ntr()){
            if(o -> fa -> ntr()) split(o -> isr() ^ o -> fa -> isr() ? o : o -> fa);
            split(o);
        }
    }

    inline void access(node *x){
        for(node *y = 0x0 ; x ; x = (y = x) -> fa){
            splay(x) , x -> ch[1] = y , x -> up();
        }
    }

    inline void Move_to_root(node *x){
        access(x) , splay(x) , x -> Modify_rev();
    }

    inline node *find_root(node *x){
        access(x) , splay(x);
        while(x -> down() , x -> ch[0]) x = x -> ch[0];
        return splay(x) , x;
    }

    inline void link(node *x , node *y){
        if(find_root(x) == find_root(y)) return;
        Move_to_root(x); x -> fa = y;
    }

    inline void cut(node *x , node *y){
        Move_to_root(x); access(y); splay(y);
        y -> ch[0] = x -> fa = 0x0 , y -> up();
    }

    inline void change(int id , int tp , qwq a , qwq b){
        node *o = pool + id;
        splay(o);
        o -> val = Tylar(tp , a , b);
        o -> up();
    }

    inline Tylar query(node *x , node *y){ 
        Move_to_root(x) , access(y) , splay(y);
        return y -> tot;
    }

    inline void link(int x , int y){
        link(pool + x , pool + y);
    }

    inline void cut(int x , int y){
        cut(pool + x , pool + y);
    }

    inline void get_ans(int x , int y , qwq z){
        if(find_root(pool + x) == find_root(pool + y)) TOWA("%.10f\n" , query(pool + x , pool + y).get_ans(z));
        else TOWA("unreachable\n");
    }
}

inline int Ame_(){
#ifdef AME__
    freopen("b1.in" , "r" , stdin); freopen("1.out" , "w" , stdout); int nol_cl = clock();
#endif
    mian(n) , mian(m); scanf("%s" , qaq);
    for(int i = 0;i < n;i ++) mian(op) , scanf("%lf%lf" , &a , &b) , towa::change(i , op , a , b);
    for(; m --> 0 ;){
        scanf("%s" , opt);
        if(opt[0] == 'a') mian(x) , mian(y) , towa::link(x , y);
        if(opt[0] == 'm') mian(x) , mian(y) , scanf("%lf%lf" , &a , &b) , towa::change(x , y , a , b);
        if(opt[0] == 'd') mian(x) , mian(y) , towa::cut(x , y);
        if(opt[0] == 't') mian(x) , mian(y) , scanf("%lf" , &a) , towa::get_ans(x , y , a);
    }
#ifdef AME__TIME
    LOG("Time: %dms\n", int((clock() - nol_cl) / (qwq)CLOCKS_PER_SEC * 1000));
#endif
    return ~~(0^_^0); /*さようならプログラム*/
}
    
int Ame__ = Ame_();
    
int main(){;}