[THUWC 2017]在美妙的数学王国中畅游

bzoj5020

$$答案误差只要小于 10^{-7}$$

题解

Taylor展开式:

\[若f(x)的n阶导数在[a, b]内连续，则f(x)在x_{0}\in[a, b]可表示为
\]

\[f(x)=\sum_{i=0}^{n} \frac{ f^{(n)}(x_{0})(x-x_{0})^{i} }{i!} + \Theta((x-x_{0})^{n})
\]

\[其中f^{(n)}表示函数f的n阶导数，\Theta((x-x_{0})^{n})为误差
\]

\[对于这道题，令x_{0}=0，求大约12阶导数即可保证误差小于10^{-7}
\]

用Taylor展开式可以直接把路径上的函数合并

直接开12变量个记录每一阶的导数，用LCT维护，统计答案用Taylor展开式计算

# include <bits/stdc++.h>
# define RG register
# define IL inline
# define Fill(a, b) memset(a, b, sizeof(a))
using namespace std;
typedef long long ll;
const int _(400010);
const double E = pow(2, 1.0 / log(2));

IL ll Read(){
	RG char c = getchar(); RG ll x = 0, z = 1;
	for(; c < '0' || c > '9'; c = getchar()) z = c == '-' ? -1 : 1;
	for(; c >= '0' && c <= '9'; c = getchar()) x = (x << 1) + (x << 3) + (c ^ 48);
	return x * z;
}

int n, m;

namespace LCT{
	int ch[2][_], fa[_], rev[_], S[_];
	double w[17][_], sum[17][_];

	IL bool Son(RG int x){  return ch[1][fa[x]] == x;  }

	IL bool Isroot(RG int x){  return ch[0][fa[x]] != x && ch[1][fa[x]] != x;  }

	IL void Update(RG int x){  for(RG int i = 0; i < 16; ++i) sum[i][x] = sum[i][ch[0][x]] + sum[i][ch[1][x]] + w[i][x];  }

	IL void Pushdown(RG int x){  if(!rev[x]) return; rev[x] ^= 1; rev[ch[0][x]] ^= 1; rev[ch[1][x]] ^= 1; swap(ch[0][x], ch[1][x]);  }

	IL void Rot(RG int x){
		RG int y = fa[x], z = fa[y], c = Son(x);
		if(!Isroot(y)) ch[Son(y)][z] = x; fa[x] = z;
		ch[c][y] = ch[!c][x]; fa[ch[c][y]] = y;
		ch[!c][x] = y; fa[y] = x; Update(y);
	}

	IL void Splay(RG int x){
		RG int top = 0; S[++top] = x;
		for(RG int i = x; !Isroot(i); i = fa[i]) S[++top] = fa[i];
		while(top) Pushdown(S[top--]);
		for(RG int y = fa[x]; !Isroot(x); Rot(x), y = fa[x])
			if(!Isroot(y)) Son(x) ^ Son(y) ? Rot(x) : Rot(y);
		Update(x);
	}

	IL void Access(RG int x){  for(RG int y = 0; x; y = x, x = fa[x]) Splay(x), ch[1][x] = y, Update(x);  }

	IL int Findroot(RG int x){  Access(x); Splay(x); while(ch[0][x]) x = ch[0][x]; return x;  }

	IL void Makeroot(RG int x){  Access(x); Splay(x); rev[x] ^= 1;  }

	IL void Split(RG int x, RG int y){  Makeroot(x); Access(y); Splay(y);  }

	IL void Link(RG int x, RG int y){  Makeroot(x); fa[x] = y;  }

	IL void Cut(RG int x, RG int y){  Split(x, y); ch[0][y] = fa[x] = 0;  }

	IL double Query(RG int x, RG int y, RG double xx){
		Split(x, y);
		RG double ans = sum[0][y], fac = 1, xxx = 1;
		for(RG int i = 1; i < 16; i++) fac *= i, xxx *= xx, ans += sum[i][y] / fac * xxx;
		return ans;
	}

	IL void Calc(RG int x, RG int f, RG double a, RG double b){
		for(RG int i = 0; i < 16; ++i) w[i][x] = 0;
		if(f == 3) w[0][x] = b, w[1][x] = a;
		else if(f == 1){
			RG double aa = 1; w[0][x] = sin(b);
			for(RG int i = 1; i < 16; ++i){
				aa *= a;
				if(i % 4 == 1) w[i][x] = aa * cos(b);
				else if(i % 4 == 2) w[i][x] = -aa * sin(b);
				else if(i % 4 == 3) w[i][x] = -aa * cos(b);
				else w[i][x] = aa * sin(b);
			}
		}
		else{
		    w[0][x] = pow(E, b);
			for(RG int i = 1; i < 16; ++i) w[i][x] = w[i - 1][x] * a;
		}
	}

	IL void Work(){
		RG char c; RG int u, v, p, f; RG double x, a, b;
		for(RG int i = 1; i <= n; ++i) f = Read(), scanf("%lf%lf", &a, &b), Calc(i, f, a, b);
		while(m--){
			scanf(" %c", &c);
			if(c == 'a') u = Read() + 1, v = Read() + 1, Link(u, v);
			else if(c == 'd') u = Read() + 1, v = Read() + 1, Cut(u, v);
			else if(c == 'm') p = Read() + 1, f = Read(), scanf("%lf%lf", &a, &b), Calc(p, f, a, b), Splay(p);
			else{
				u = Read() + 1; v = Read() + 1; scanf("%lf", &x);
				if(Findroot(u) != Findroot(v)) puts("unreachable");
				else printf("%.8e\n", Query(u, v, x));
			}
		}
	}
}

int main(RG int argc, RG char *argv[]){
	n = Read(); m = Read(); Read();
	LCT::Work();
	return 0;
}