POJ 2728 Desert King(最优比率生成树， 01分数规划)

题意：

给定n个村子的坐标（x,y）和高度z， 求出修n-1条路连通所有村子， 并且让 修路花费/修路长度 最少的值

两个村子修一条路， 修路花费 = abs(高度差), 修路长度 = 欧氏距离

分析：

01分数划分的题目， 构造出 d[i] = 修路花费 - L * 修路长度， 这个L值我们可以二分（这道题看数据范围的话二分上限其实挺大的， 但其实上限取到100就可以过）， 也可以用Dinkelbach迭代出来。

  二分（1422ms）

#include <stdio.h>
#include <string.h>
#include <iostream>
#include <iostream>
#include <algorithm>
#include <vector>
#include <queue>
#include <set>
#include <map>
#include <cstring>
#include <cmath>
#define rep(i,a,b) for(int i = a; i < b;i++)
#define _rep(i,a,b) for(int i = a; i <= b;i++)
using namespace std;
const int maxn = 1000 + 7;
const double inf = 1e9 + 7;
const double eps = 1e-5;
int n;

double G[maxn][maxn];
double x[maxn], y[maxn], z[maxn];

inline double p2p_dis(double x1, double y1, double x2, double y2){
    return sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
}
double prim(double L){
    double tree_dis = 0;
    double dis[maxn];
    int vis[maxn];
    fill(dis, dis + maxn, inf);
    memset(vis, 0 ,sizeof(vis));

    dis[0] = 0;
    vis[0] = 1;

    for(int i = 1; i < n; i++) dis[i] =  abs(z[0] - z[i]) - L * G[0][i];


    for(int times = 0; times < n - 1; times++){
        int picked = -1;
        double min_dis = inf;
        for(int i = 0; i < n; i++){
            if(!vis[i] && dis[i] < min_dis){
                min_dis = dis[i];
                picked = i;
            }
        }
        tree_dis += min_dis;
        vis[picked] = 1;
        for(int i = 0; i < n; i++)
            if(!vis[i])
                dis[i] = min(dis[i] , abs(z[picked] - z[i]) - L * G[picked][i]) ;
    }
    return tree_dis;
}

int main(){
//    freopen("1.txt","r", stdin);
    while(cin >> n && n){
        rep(i,0,n) cin >> x[i] >> y[i] >> z[i];
        rep(i,0,n) rep(j,i+1,n) G[i][j] = G[j][i] = p2p_dis(x[i],y[i],x[j],y[j]);
        double l = 0, r = 14143000;
        while(abs(r - l) > eps){
            double mid = (l + r) / 2;
//            printf("%.3f
", mid);
            if(prim(mid) >= 0){
                l = mid;
            }else{
                r = mid;
            }
        }
        printf("%.3f
", l);
    }
    return 0;
}

Dinkelbach（204ms）

#include <stdio.h>
#include <string.h>
#include <iostream>
#include <iostream>
#include <algorithm>
#include <vector>
#include <queue>
#include <set>
#include <map>
#include <cstring>
#include <cmath>
#define rep(i,a,b) for(int i = a; i < b;i++)
#define _rep(i,a,b) for(int i = a; i <= b;i++)
using namespace std;
const int maxn = 1000 + 7;
const double inf = 1e9 + 7;
const double eps = 1e-5;
int n;

double G[maxn][maxn];
double x[maxn], y[maxn], z[maxn];

inline double p2p_dis(double x1, double y1, double x2, double y2){
    return sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
}
double prim(double L){
    double ele = 0, deno = 0; //分子分母
    double dis[maxn];
    int near[maxn]; //开一个数组记录每次选中是哪一条边

    int vis[maxn];
    fill(dis, dis + maxn, inf);
    memset(vis, 0 ,sizeof(vis));
    memset(near, -1, sizeof(near));
    dis[0] = 0;
    vis[0] = 1;

    for(int i = 1; i < n; i++) dis[i] =  abs(z[0] - z[i]) - L * G[0][i], near[i] = 0; //每个点都是由0点更新的， 所以near都是0

    for(int times = 0; times < n - 1; times++){
        int picked = -1;
        double min_dis = inf;
        for(int i = 0; i < n; i++){
            if(!vis[i] && dis[i] < min_dis){
                min_dis = dis[i];
                picked = i;
            }
        }
        ele += abs(z[near[picked]] - z[picked]);//分子是选的路程
        deno += G[near[picked]][picked]; //分母是真实的花费
        vis[picked] = 1;
        for(int i = 0; i < n; i++)
            if(!vis[i] && dis[i] > abs(z[picked] - z[i]) - L * G[picked][i]){
                dis[i] = abs(z[picked] - z[i]) - L * G[picked][i];
                near[i] = picked;//如果通过picked点更新dis，那么该位置near就是picked
            }

    }

    return ele/deno; //注意返回的跟二分不一样， 应该返回当前L下真正题目的所求
}

int main(){
//    freopen("1.txt","r", stdin);
    while(cin >> n && n){
        rep(i,0,n) cin >> x[i] >> y[i] >> z[i];
        rep(i,0,n) rep(j,i+1,n) G[i][j] = G[j][i] = p2p_dis(x[i],y[i],x[j],y[j]);
        double L, ans = 0;
        while(1){
            L = ans;
            ans = prim(L); //这是一个真实值， 不是含L的变式
            if(abs(ans - L)< eps) break;
        }
        printf("%.3f
", ans);
    }
    return 0;
}