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[SCOI2010]传送带

news 来源：原创 2024/5/4 22:11:00

嘟嘟嘟

首先令E点表示在E点离开传送带AB，F点表示在F点上传送带CD，则总用时为：dis(A, E) / p + dis(E, F) / r + dis(F, D) / q。然而这是一个有两个变量的函数。于是有一个不错的方法：把E成参数，然后就变成了一个形如f(x) = √(t² + x²) - x的函数，虽然我证不出来是一个单峰函数，但是几何画板告诉我是。于是我们可以三分求F。至于E，再三分求。

所以说，这是一道二维的三分套三分的题，三分求E，假设E已知的情况下三分求F。

代码很好懂。

 1 #include<cstdio>
 2 #include<iostream>
 3 #include<cmath>
 4 #include<algorithm>
 5 #include<cstring>
 6 #include<cstdlib>
 7 #include<cctype>
 8 #include<vector>
 9 #include<stack>
10 #include<queue>
11 using namespace std;
12 #define enter puts("") 
13 #define space putchar(' ')
14 #define Mem(a, x) memset(a, x, sizeof(a))
15 #define rg register
16 typedef long long ll;
17 typedef double db;
18 const int INF = 0x3f3f3f3f;
19 const db eps = 1e-8;
20 //const int maxn = ;
21 inline ll read()
22 {
23   ll ans = 0;
24   char ch = getchar(), last = ' ';
25   while(!isdigit(ch)) {last = ch; ch = getchar();}
26   while(isdigit(ch)) {ans = ans * 10 + ch - '0'; ch = getchar();}
27   if(last == '-') ans = -ans;
28   return ans;
29 }
30 inline void write(ll x)
31 {
32   if(x < 0) x = -x, putchar('-');
33   if(x >= 10) write(x / 10);
34   putchar(x % 10 + '0');
35 }
36 
37 int ax, ay, bx, by, cx, cy, dx, dy;
38 int p, q, r;
39 
40 db dis(db lx, db ly, db rx, db ry)
41 {
42   return sqrt((rx - lx) * (rx - lx) + (ry - ly) * (ry - ly));
43 }
44 
45 db calcF(db Ex, db Ey)
46 {
47   db Lx = cx, Ly = cy, Rx = dx, Ry = dy;
48   while(dis(Lx, Ly, Rx, Ry) > eps)
49     {    
50       db Dx = (Rx - Lx) / 3.00, Dy = (Ry - Ly) / 3.00;      
51       db m1x = Lx + Dx, m2x = Rx - Dx;
52       db m1y = Ly + Dy, m2y = Ry - Dy;
53       db ans1 = dis(Ex, Ey, m1x, m1y) / r + dis(m1x, m1y, dx, dy) / q;
54       db ans2 = dis(Ex, Ey, m2x, m2y) / r + dis(m2x, m2y, dx, dy) / q;
55       if(ans1 < ans2 + eps) Rx = m2x, Ry = m2y;
56       else Lx = m1x, Ly = m1y;   
57     }
58   return dis(Ex, Ey, Lx, Ly) / r + dis(Lx, Ly, dx, dy) / q;
59 }
60 db calcE()
61 {
62   db Lx = ax, Ly = ay, Rx = bx, Ry = by;
63   while(dis(Lx, Ly, Rx, Ry) > eps)
64     {
65       db Dx = (Rx - Lx) / 3.00, Dy = (Ry - Ly) / 3.00;
66       db m1x = Lx + Dx, m2x = Rx - Dx;
67       db m1y = Ly + Dy, m2y = Ry - Dy;
68       db ans1 = calcF(m1x, m1y) + dis(ax, ay, m1x, m1y) / p;
69       db ans2 = calcF(m2x, m2y) + dis(ax, ay, m2x, m2y) / p;
70       if(ans1 < ans2 + eps) Rx = m2x, Ry = m2y;
71       else Lx = m1x, Ly = m1y;
72     }
73   return calcF(Lx, Ly) + dis(ax, ay, Lx, Ly) / p;
74 }
75 
76 int main()
77 {
78   ax = read(), ay = read(); bx = read(), by = read();
79   cx = read(), cy = read(), dx = read(), dy = read();
80   p = read(), q = read(), r = read();
81   printf("%.2lf\n", calcE());
82   return 0;
83 }