Zoj 1718 poj 2031 building a space station building space station least Spanning Tree Kruskal Algorithm

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Building a space station Time Limit: 2 seconds memory limit: 65536 KB 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 shocould 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) shocould 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 shoshould 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 are 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 shocould be printed, each in a separate line. the printed values shoshould 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

Question:

If you are a member of the space station engineering team, you are assigned to the space station construction task and want to write a program to complete this task.

The Space Station consists of many units called single rooms. All single rooms are spherical and the sizes are not the same. After the space station successfully enters the orbit, each single room is fixed at a predetermined position. It is strange that two single rooms can be connected or even overlapped. In extreme cases, one single room can even completely include another single room.

All single rooms must be connected because astronauts can move from one single room to another. In the following circumstances, astronauts can go from single room A to single room B.

(1) contact a and B, or overlap.

(2) A and B are connected using a corridor.

(3) There is a single room C. Yes, from A to C, or from B to C.

If you want to design the structure of the space station, that is, to arrange which single rooms need to be connected by corridor. There is a certain degree of freedom in designing the corridor structure. The cost of building a corridor is proportional to its length. Therefore, you must select a solution to minimize the total corridor length. The corridor is built on the surface of two single rooms, and it is assumed that any two corridors do not overlap.

Analysis:

First, determine whether all the single rooms are in contact with each other. If they are in contact with each other, the edge length between them is 0. If they are not in contact with each other, the distance from the center minus the radius of the two single rooms. Then, the Minimum Spanning Tree is obtained by using the Kruskal algorithm.

Code:

#include <iostream>#include <cstdio>#include <cstring>#include <cmath>#include <algorithm>using namespace std;#define maxm 5050#define maxn 110int m, parent[maxn];double ans;struct ball{    double x, y, z, r;}BL[maxn];double dis(ball a, ball b){    return sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y)+(a.z-b.z)*(a.z-b.z))-a.r-b.r;}struct edge{    int u, v;    double w;}EG[maxm];bool cmp(edge a, edge b){    return a.w < b.w;}int Find(int x){    if(parent[x] == -1) return x;    return Find(parent[x]);}void Kruskal(){    memset(parent, -1, sizeof(parent));    ans = 0;    sort(EG, EG+m, cmp);    for(int i = 0; i < m; i++)    {        int t1 = Find(EG[i].u), t2 = Find(EG[i].v);        if(t1 != t2)        {            ans += EG[i].w;            parent[t1] = t2;        }    }}int main(){    int n;    while(scanf("%d", &n), n)    {        for(int i = 0; i < n; ++i)            scanf("%lf%lf%lf%lf", &BL[i].x, &BL[i].y, &BL[i].z, &BL[i].r);        m = 0;        for(int i = 0; i < n; ++i)            for(int j = i+1; j < n; ++j)            {                double d = dis(BL[i], BL[j]);                EG[m].u = i;                EG[m].v = j;                if(d > 0)                    EG[m].w = d;                else                    EG[m].w = 0;                ++m;            }        Kruskal();        printf("%.3lf\n", ans);    }    return 0;}

Zoj 1718 poj 2031 building a space station building space station least Spanning Tree Kruskal Algorithm