POJ 2031 Building a Space Station【最小生成树+简单计算几何】

You are a member of the space station engineering team, and are assigned a task in the construction process of the station. You are expected to write a computer program to complete the task. 
The space station is made up with a number of units, called cells. All cells are sphere-shaped, but their sizes are not necessarily uniform. Each cell is fixed at its predetermined position shortly after the station is successfully put into its orbit. It is quite strange that two cells may be touching each other, or even may be overlapping. In an extreme case, a cell may be totally enclosing another one. I do not know how such arrangements are possible.

All the cells must be connected, since crew members should be able to walk from any cell to any other cell. They can walk from a cell A to another cell B, if, (1) A and B are touching each other or overlapping, (2) A and B are connected by a `corridor', or (3) there is a cell C such that walking from A to C, and also from B to C are both possible. Note that the condition (3) should be interpreted transitively.

You are expected to design a configuration, namely, which pairs of cells are to be connected with corridors. There is some freedom in the corridor configuration. For example, if there are three cells A, B and C, not touching nor overlapping each other, at least three plans are possible in order to connect all three cells. The first is to build corridors A-B and A-C, the second B-C and B-A, the third C-A and C-B. The cost of building a corridor is proportional to its length. Therefore, you should choose a plan with the shortest total length of the corridors.

You can ignore the width of a corridor. A corridor is built between points on two cells' surfaces. It can be made arbitrarily long, but of course the shortest one is chosen. Even if two corridors A-B and C-D intersect in space, they are not considered to form a connection path between (for example) A and C. In other words, you may consider that two corridors never intersect.

Input

The input consists of multiple data sets. Each data set is given in the following format.

n 
x1 y1 z1 r1 
x2 y2 z2 r2 
... 
xn yn zn rn

The first line of a data set contains an integer n, which is the number of cells. n is positive, and does not exceed 100.

The following n lines are descriptions of cells. Four values in a line are x-, y- and z-coordinates of the center, and radius (called r in the rest of the problem) of the sphere, in this order. Each value is given by a decimal fraction, with 3 digits after the decimal point. Values are separated by a space character.

Each of x, y, z and r is positive and is less than 100.0.

The end of the input is indicated by a line containing a zero.

Output

For each data set, the shortest total length of the corridors should be printed, each in a separate line. The printed values should have 3 digits after the decimal point. They may not have an error greater than 0.001.

Note that if no corridors are necessary, that is, if all the cells are connected without corridors, the shortest total length of the corridors is 0.000.

Sample Input

3
10.000 10.000 50.000 10.000
40.000 10.000 50.000 10.000
40.000 40.000 50.000 10.000
2
30.000 30.000 30.000 20.000
40.000 40.000 40.000 20.000
5
5.729 15.143 3.996 25.837
6.013 14.372 4.818 10.671
80.115 63.292 84.477 15.120
64.095 80.924 70.029 14.881
39.472 85.116 71.369 5.553
0

Sample Output

20.000
0.000
73.834

#include<cstdio>
#include<string>
#include<cstdlib>
#include<cmath>
#include<iostream>
#include<cstring>
#include<set>
#include<queue>
#include<algorithm>
#include<vector>
#include<map>
#include<cctype>
#include<stack>
#include<sstream>
#include<list>
#include<assert.h>
#include<bitset>
#include<numeric>
#define debug() puts("++++")
#define gcd(a,b) __gcd(a,b)
#define lson l,m,rt<<1
#define rson m+1,r,rt<<1|1
#define fi first
#define se second
#define pb push_back
#define sqr(x) ((x)*(x))
#define ms(a,b) memset(a,b,sizeof(a))
#define sz size()
#define be begin()
#define pu push_up
#define pd push_down
#define cl clear()
#define lowbit(x) -x&x
#define all 1,n,1
#define rep(i,x,n) for(int i=(x); i<=(n); i++)
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int,int> P;
const int INF = 0x3f3f3f3f;
const LL LNF = 1e18;
const int maxm = 1e6 + ;
const double PI = acos(-1.0);
const double eps = 1e-;
const int dx[] = {-,,,,,,-,-};
const int dy[] = {,,,-,,-,,-};
int dir[][] = {{,},{,-},{-,},{,}};
const int mon[] = {, , , , , , , , , , , , };
const int monn[] = {, , , , , , , , , , , , };
const int mod = ;
#define inf 0x3f3f3f3f
#define ll long long
const int maxn = ;

int u,v,w;
int n,m,ans,k,cnt;
double sum=0.0;
struct point
{
    double x,y,z,r;
}a[maxn];
struct node
{
    int u,v;
    double w;
}e[maxn*maxn];
double dis(point a,point b)
{
    return sqrt((b.x-a.x)*(b.x-a.x)+(b.y-a.y)*(b.y-a.y)+(b.z-a.z)*(b.z-a.z)) -a.r - b.r;
}
int fa[maxn];
int Find(int x)
{
    if(fa[x]!=x)
        fa[x]=Find(fa[x]);
    return fa[x];
}
void join(int x,int y)
{
    int xx = Find(x);
    int yy = Find(y);
    fa[xx]=yy;
}
bool cmp(node a,node b)
{
    return a.w < b.w;
}
void kruskal()
{
    cnt=,sum=;
    for(int i=;i<m;i++)
    {
        int xx=Find(e[i].u);
        int yy=Find(e[i].v);
        if(xx != yy)
        {
            join(xx,yy);//
            cnt++;
            sum += e[i].w;
        }
        if(cnt == n-) break;
    }
    printf("%.3f\n",sum);
}
int main()
{
    while(~scanf("%d",&n) && n)
    {
        sum=0.0, cnt=,m=;
        rep(i,,n-)
          fa[i]=i;
        for(int i=;i<n;i++)
            scanf("%lf%lf%lf%lf",&a[i].x, &a[i].y, &a[i].z, &a[i].r);

        for(int i=;i<n;i++) //n-1!
        {
            for(int j=i+; j<n; j++)
            {
                double d = dis(a[i],a[j]);
                e[m].u = i;
                e[m].v = j;
                e[m].w = d<?:d;
                m++;
            }
        }

        sort(e, e+m, cmp);

        kruskal();
    }
}
/*
【题意】
给出三维坐标系上的一些球的球心坐标和其半径，搭建通路，使得他们能够相互连通。
如果两个球有重叠的部分则算为已连通，无需再搭桥。求搭建通路的最小费用（费用就是边权，就是两个球面之间的距离）。

【类型】
最小生成树

【分析】
建图是关键

【时间复杂度&&优化】

【trick】
*/