Qin Shi Huang's National Road System

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 6137    Accepted Submission(s): 2143

Problem Description
During the Warring States Period of ancient China(476 BC to 221 BC), there were seven kingdoms in China ---- they were Qi, Chu, Yan, Han, Zhao, Wei and Qin. Ying Zheng was the king of the kingdom Qin. Through 9 years of wars, he finally conquered all six other kingdoms and became the first emperor of a unified China in 221 BC. That was Qin dynasty ---- the first imperial dynasty of China(not to be confused with the Qing Dynasty, the last dynasty of China). So Ying Zheng named himself "Qin Shi Huang" because "Shi Huang" means "the first emperor" in Chinese.

Qin Shi Huang undertook gigantic projects, including the first version of the Great Wall of China, the now famous city-sized mausoleum guarded by a life-sized Terracotta Army, and a massive national road system. There is a story about the road system:
There were n cities in China and Qin Shi Huang wanted them all be connected by n-1 roads, in order that he could go to every city from the capital city Xianyang.
Although Qin Shi Huang was a tyrant, he wanted the total length of all roads to be minimum,so that the road system may not cost too many people's life. A daoshi (some kind of monk) named Xu Fu told Qin Shi Huang that he could build a road by magic and that magic road would cost no money and no labor. But Xu Fu could only build ONE magic road for Qin Shi Huang. So Qin Shi Huang had to decide where to build the magic road. Qin Shi Huang wanted the total length of all none magic roads to be as small as possible, but Xu Fu wanted the magic road to benefit as many people as possible ---- So Qin Shi Huang decided that the value of A/B (the ratio of A to B) must be the maximum, which A is the total population of the two cites connected by the magic road, and B is the total length of none magic roads.
Would you help Qin Shi Huang?
A city can be considered as a point, and a road can be considered as a line segment connecting two points.
 
Input
The first line contains an integer t meaning that there are t test cases(t <= 10).
For each test case:
The first line is an integer n meaning that there are n cities(2 < n <= 1000).
Then n lines follow. Each line contains three integers X, Y and P ( 0 <= X, Y <= 1000, 0 < P < 100000). (X, Y) is the coordinate of a city and P is the population of that city.
It is guaranteed that each city has a distinct location.
 
Output
For each test case, print a line indicating the above mentioned maximum ratio A/B. The result should be rounded to 2 digits after decimal point.
 
Sample Input
2
4
1 1 20
1 2 30
200 2 80
200 1 100
3
1 1 20
1 2 30
2 2 40
 
Sample Output
65.00
70.00
 
Source

题意:给你n个点的坐标和每个城市的人数p,现在要修路,两点之间距离就是欧拉距离,要使得所有的n个城市都联通,且可以免费修一条路(u,v),求使得A =pu + pv,  B = 要修的其他路的总长度,  求A/B的最大值

因为要使得n个城市联通,那我们只需要修n-1条路,可以枚举免费修的路(u,v),只需要得出:包含边(u,v)的最小生成树mst(u,v)

如何得出包含边(u,v)的最小生成树呢?

我们先求一次原图的最小生成树MST,在该生成树上添加边(u,v),就会形成一个环,我们要做的就是在树中去掉u到v唯一路径上的最大边权

如何求出u到v唯一路径上的最大边呢?

mc[u][v]表示节点u到v路径上的最大边,用数组记录访问过的节点x, 有mc[v][x] = max(mc[v][u], mc[u][x])  其中u是v的父亲

在CF上有一题就是求包含每一条边的最小生成树,我们也可以在原图的最小生成树上用LCA预处理出fa[u][i]表示节点u到其第2^i个祖先的路径上的最大边权,每次可以log得出u到v的最大边

#include <bits/stdc++.h>
using namespace std;
const int N = ;
struct Edge {
int u, v;
double w;
Edge() {}
Edge(int u, int v, double w):u(u),v(v),w(w) {}
friend bool operator < (Edge a, Edge b) {
return a.w < b.w;
}
};
Edge e[N * N * ];
int x[N], y[N], fa[N];
int p[N];
vector<Edge> g[N];
double dist(int i, int j) {
return sqrt((x[i] - x[j]) * (x[i] - x[j]) + (y[i] - y[j]) * (y[i] - y[j]) * 1.0);
}
int findx(int x) {
return fa[x] == x ? x:fa[x] = findx(fa[x]);
}
double MST(int m, int n) {
for(int i = ; i < n; ++i) fa[i] = i;
sort(e, e + m);
double ans = ;
int num = ;
for(int i = ; i < m; ++i)
{
int fu = findx(e[i].u);
int fv = findx(e[i].v);
if(fu == fv) continue;
num++;
fa[fu] = fv;
ans += e[i].w;
g[e[i].u].push_back(Edge(-,e[i].v, e[i].w));
g[e[i].v].push_back(Edge(-,e[i].u, e[i].w));
if(num == n - ) break;
}
return ans;
}
double mc[N][N];
int vis[N], cur[N], c;
void DFS(int u) {
vis[u] = ; cur[c++] = u;
int sx = g[u].size();
for(int i = ; i < sx; ++i) {
int v = g[u][i].v;
double w = g[u][i].w;
if(vis[v]) continue;
for(int j = ; j < c; ++j) {
int x = cur[j];
mc[v][x] = mc[x][v] = max(w, mc[x][u]);
}
DFS(v);
}
}
double solve(int n, double B) {
double ans = ;
for(int i = ; i < n; ++i) {
for(int j = i + ; j < n; ++j)
ans = max(ans, (p[i] + p[j]) * 1.0 / (B - mc[i][j]));
}
return ans;
}
int main()
{
int _; scanf("%d", &_);
while(_ --) {
int n; scanf("%d", &n);
for(int i = ; i < n; ++i) g[i].clear();
for(int i = ; i < n; ++i) scanf("%d%d%d", &x[i], &y[i], &p[i]);
int m = ;
for(int i = ; i < n; ++i)
for(int j = i + ; j < n; ++j) {
e[m++] = Edge(i, j, dist(i, j));
}
// for(int i = 0; i < m; ++i) printf("%d %d %f\n", e[i].u, e[i].v, e[i].w);
c = ;
memset(mc, , sizeof mc);
memset(vis, , sizeof vis);
double res = MST(m, n);
DFS();
printf("%.2f\n", solve(n, res));
}
return ;
}

Hdu 4081 最小生成树的更多相关文章

  1. hdu 4081 最小生成树+树形dp

    思路:直接先求一下最小生成树,然后用树形dp来求最优值.也就是两遍dfs. #include<iostream> #include<algorithm> #include< ...

  2. hdu 4081 最小生成树变形

    /*关于最小生成树的等效边,就是讲两个相同的集合连接在一起 先建立一个任意最小生成树,这条边分开的两个子树的节点最大的一个和为A,sum为最小生成树的权值和,B为sum-当前边的权值 不断枚举最小生成 ...

  3. Qin Shi Huang's National Road System HDU - 4081(树形dp+最小生成树)

    Qin Shi Huang's National Road System HDU - 4081 感觉这道题和hdu4756很像... 求最小生成树里面删去一边E1 再加一边E2 求该边两顶点权值和除以 ...

  4. HDU 1233(最小生成树)

    HDU 1233(最小生成树 模板) #include <iostream> #include <algorithm> #include <cstdio> usin ...

  5. HDU 4081 Qin Shi Huang's National Road System 最小生成树+倍增求LCA

    原题链接:http://acm.hdu.edu.cn/showproblem.php?pid=4081 Qin Shi Huang's National Road System Time Limit: ...

  6. hdu 4081 Qin Shi Huang's National Road System(最小生成树+dp)2011 Asia Beijing Regional Contest

    同样是看别人题解才明白的 题目大意—— 话说秦始皇统一六国之后,打算修路.他要用n-1条路,将n个城市连接起来,并且使这n-1条路的距离之和最短.最小生成树是不是?不对,还有呢.接着,一个自称徐福的游 ...

  7. HDU 4081 Qin Shi Huang's National Road System 最小生成树

    分析:http://www.cnblogs.com/wally/archive/2013/02/04/2892194.html 这个题就是多一个限制,就是求包含每条边的最小生成树,这个求出原始最小生成 ...

  8. HDU 4081 Qin Shi Huang&#39;s National Road System(最小生成树/次小生成树)

    题目链接:传送门 题意: 有n坐城市,知道每坐城市的坐标和人口.如今要在全部城市之间修路,保证每一个城市都能相连,而且保证A/B 最大.全部路径的花费和最小,A是某条路i两端城市人口的和,B表示除路i ...

  9. HDU 4081 Peach Blossom Spring (最小生成树+dfs)

    题意:给定一个 n 个点和相应的权值,要求你用 n-1 条边连接起来,其中一条边是魔法边,不用任何费用,其他的边是长度,求该魔法边的两端的权值与其他边费用的尽量大. 析:先求出最小生成树,然后再枚举每 ...

随机推荐

  1. 【原创】自己动手写工具----签到器[Beta 1.0]

    一.写在前面 最近公司没有什么项目,想通过项目练练手的机会也没有,只能自己学习了,因此空下来的时间也挺多的,就打开网页看看吧,哎,一打开就让签到(像什么百度知道啊.百度云盘啊之类的),我签到的目的是获 ...

  2. Java构造方法

  3. Python学习笔记——列表

    1.创建列表类型数据并给其赋值 >>> aList = [123,'abc',4.56,['inner','list'],7-9j] >>> aList [123, ...

  4. tomcat共享lib里面的jar包

    参考:http://shitouququ.blog.51cto.com/24569/1255094 1.在tomcat根目录下新建shared/lib目录结构,将项目的jar包放在此目录下,记得将项目 ...

  5. entiryFramework 事务控制

    1.在项目中添加System.Transactions命名空间 2.在代码中编写如下代码段: using (var trans = new TransactionScope()) { EF的代码 tr ...

  6. [Head First设计模式]面向对象的3特征5原则

    系列文章 [Head First设计模式]山西面馆中的设计模式——装饰者模式 [Head First设计模式]山西面馆中的设计模式——观察者模式 [Head First设计模式]山西面馆中的设计模式— ...

  7. [NHibernate]基本配置与测试

    目录 写在前面 nhibernate文档 搭建项目 映射文件 持久化类 辅助类 数据库设计与连接配置 测试 总结 写在前面 一年前刚来这家公司,发现项目中使用的ORM是Nhibernate,这个之前确 ...

  8. js自适应屏幕高度

    //自适应屏幕高度 $(window).resize(function() { hightChange(); }); function hightChange(){ ; $();// iframe i ...

  9. 《征服 C 指针》摘录6:解读 C 的声明

    一.混乱的声明——如何自然地理解 C 的声明? 通常,C 的声明 int hoge; 这样,使用“类型 变量名;”的形式进行书写. 可是,像“指向 int 的指针”类型的变量,却要像下面这样进行声明: ...

  10. 给CentOS6.3 + PHP5.3 安装PHP性能测试工具 XHProf-0.9.2

    一.什么是XHProf XHProf官网:http://pecl.php.net/package/xhprof XHProf是一个分层PHP性能分析工具.它报告函数级别的请求次数和各种指标,包括 阻塞 ...