HDU4418 Time travel(期望dp 高斯消元)
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2022-12-22 21:11:39
题意 "题目链接" Sol mdzz这题真的太恶心了。。 首先不难看出这就是个高斯消元解方程的板子题 $f[x] = \sum_{i = 1}^n f[to(x + i)] p[i] + ave$ $ave$表示每次走的期望路程 然后一件很恶心的事情是可以来回走,而且会出现$M N$的情况(因为这个 ......
题意
sol
mdzz这题真的太恶心了。。
首先不难看出这就是个高斯消元解方程的板子题
\(f[x] = \sum_{i = 1}^n f[to(x + i)] * p[i] + ave\)
\(ave\)表示每次走的期望路程
然后一件很恶心的事情是可以来回走,而且会出现\(m > n\)的情况(因为这个调了两个小时。。)
一种简单的解决方法是在原序列的后面接一段翻转后的序列
比如\(1 \ 2 \ 3 \ 4\)可以写成\(1 2 3 4 3 2\)
然后列式子解方程就行了
附送一个数据生成器
#include<bits/stdc++.h> using namespace std; int main() { freopen("a.in", "w", stdout); srand((unsigned)time(null)); int t = 30; printf("%d\n", t); while(t--) { int n = rand() % 100 + 1, m = rand() % 20 + 1, y = rand() % n, x = rand() % n, d = rand() % 2; if(x == 0 || x == n - 1) d = -1; printf("%d %d %d %d %d\n", n, m, y, x, d); int res = 100; for(int i = 1; i <= m - 1; i++) { int rd; if(res == 0) rd = 0; else rd = rand() % res + 1; printf("%d ", rd); res -= rd; } printf("%d\n", res); } return 0; }
#include<bits/stdc++.h> #define ll long long using namespace std; const int maxn = 1001, mod = 998244353; const double eps = 1e-9; inline int read() { char c = getchar(); int x = 0, f = 1; while(c < '0' || c > '9') {if(c == '-') f = -1; c = getchar();} while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar(); return x * f; } int n, m, x, y, d, lim, vis[maxn]; double g[maxn][maxn], p[maxn], ave; int gauss() { for(int i = 1; i < lim; i++) { int mx = i; for(int j = i + 1; j < lim; j++) if(fabs(g[j][i]) > fabs(g[mx][i])) mx = j; swap(g[i], g[mx]); //assert(g[i][i] > eps); if(fabs(g[i][i]) < eps) return -1; for(int j = i + 1; j < lim; j++) { double p = g[j][i] / g[i][i]; for(int k = i + 1; k <= lim; k++) g[j][k] -= g[i][k] * p; } } for(int i = 1; i < lim; i++) if(fabs(g[i][i]) < eps) return -1; for(int i = lim - 1; i >= 1; i--) { g[i][i] = g[i][lim] / g[i][i]; for(int j = i - 1; j >= 1; j--) g[j][lim] -= g[j][i] * g[i][i], g[j][i] = 0; } } int walk(int a, int b) { b %= (lim - 1); int x = a + b; if(x <= lim - 1) return x; return x % (lim - 1); } void init() { memset(g, 0, sizeof(g)); memset(vis, 0, sizeof(vis)); ave = 0; } void bfs() { queue<int> q; q.push(x); vis[x] = 1; while(!q.empty()) { int x = q.front(); q.pop(); for(int i = 1; i <= m; i++) { if(p[i] > eps) { int t = walk(x, i); if(!vis[t]) q.push(t), vis[t] = 1; } } } } void solve() { init(); n = read(); m = read(); y = read() + 1; x = read() + 1; d = read(); lim = (n << 1) - 1; for(int i = 1; i <= m; i++) p[i] = (double) read() / 100, ave += (double) i * p[i]; if(x == y) {puts("0.00"); return;} if(d > 0 || (d == -1 && x > y)) x = n - x + 1, y = n - y + 1; bfs(); for(int i = 1; i <= 2 * n - 2; i++) { g[i][i] = 1; if(!vis[i]) {g[i][lim] = 3e18; continue;} if(i == y || (lim - i + 1 == y)) continue; g[i][lim] = ave; for(int j = 1, t; j <= m; j++) { t = walk(i, j); g[i][t] -= p[j]; } } if((!vis[y] && !vis[lim - y + 1]) || (gauss() == -1)) puts("impossible !"); else printf("%.2lf\n", g[x][x]); } int main() { //freopen("a.in", "r", stdin); //freopen("b.out", "w", stdout); for(int t = read(); t; t--, solve()); return 0; }
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