Farmer John has a number of pastures on his farm. Cow paths connect some pastures with certain other pastures, forming a field. But, at the present time, you can find at least two pastures that cannot be connected by any sequence of cow paths, thus partitioning Farmer John's farm into multiple fields.
Farmer John would like add a single a cow path between one pair of pastures using the constraints below.
A field's `diameter' is defined to be the largest distance of all the shortest walks between any pair of pastures in the field. Consider the field below with five pastures, located at the points shown, and cow paths marked by lines:
15,15 20,15
D E
*-------*
| _/|
| _/ |
| _/ |
|/ |
*--------*-------*
A B C
10,10 15,10 20,10
The `diameter' of this field is approximately 12.07106, since the longest of the set of shortest paths between pairs of pastures is the path from A to E (which includes the point set {A,B,E}). No other pair of pastures in this field is farther apart when connected by an optimal sequence of cow paths.
Suppose another field on the same plane is connected by cow paths as follows:
*F 30,15
/
_/
_/
/
*------
G H
25,10 30,10
In the scenario of just two fields on his farm, Farmer John would add a cow path between a point in each of these two fields (namely point sets {A,B,C,D,E} and {F,G,H}) so that the joined set of pastures {A,B,C,D,E,F,G,H} has the smallest possible diameter.
Note that cow paths do not connect just because they cross each other; they only connect at listed points.
The input contains the pastures, their locations, and a symmetric "adjacency" matrix that tells whether pastures are connected by cow paths. Pastures are not considered to be connected to themselves. Here's one annotated adjacency list for the pasture {A,B,C,D,E,F,G,H} as shown above:
A B C D E F G H
A 0 1 0 0 0 0 0 0
B 1 0 1 1 1 0 0 0
C 0 1 0 0 1 0 0 0
D 0 1 0 0 1 0 0 0
E 0 1 1 1 0 0 0 0
F 0 0 0 0 0 0 1 0
G 0 0 0 0 0 1 0 1
H 0 0 0 0 0 0 1 0
Other equivalent adjacency lists might permute the rows and columns by using some order other than alphabetical to show the point connections. The input data contains no names for the points.
The input will contain at least two pastures that are not connected by any sequence of cow paths.
Find a way to connect exactly two pastures in the input with a cow path so that the new combined field has the smallest possible diameter of any possible pair of connected pastures. Output that smallest possible diameter.
PROGRAM NAME: cowtour
INPUT FORMAT
| Line 1: | An integer, N (1 <= N <= 150), the number of pastures |
| Line 2-N+1: | Two integers, X and Y (0 <= X ,Y<= 100000), that denote that X,Y grid location of the pastures; all input pastures are unique. |
| Line N+2-2*N+1: | lines, each containing N digits (0 or 1) that represent the adjacency matrix as described above, where the rows' and columns' indices are in order of the points just listed. |
SAMPLE INPUT (file cowtour.in)
8 10 10 15 10 20 10 15 15 20 15 30 15 25 10 30 10 01000000 10111000 01001000 01001000 01110000 00000010 00000101 00000010
OUTPUT FORMAT
The output consists of a single line with the diameter of the newly joined pastures. Print the answer to exactly six decimal places. Do not perform any special rounding on your output.
SAMPLE OUTPUT (file cowtour.out)
22.071068
思路：先把不同的牧场分开（我使用并查集），依次算出所有点所在牧场与当前点的最大距离（最短路），依次连接两个不同的牧场。答案就是MAX（所有点所在牧场的最大距离值，连接后的牧场的最小距离值)；
失误点：在做题之前有考虑过可能单个牧场的最大距离值会大于两个牧场连接后的最小距离值，可是在敲代码的时候给忘了，果断WA了一次，还去调试，真无语，最后实验室关门了，在回去的路上在发现落下了这个条件，今天补上，果断AC了速度还行54MS。
下面看代码吧！
/*
ID:nealgav1
LANG:C++
PROG:cowtour
*/
#include<fstream>
#include<iomanip>
#include<algorithm>
#include<cmath>
using namespace std;
ifstream cin("cowtour.in");
ofstream cout("cowtour.out");
const int mm=210;
const int oo=1e9;
const double klen=0.00001;//大于0的极小值
char vis[mm][mm];
int root[mm],rank[mm];//并查集区分不同牧场，rank[]在这没用
double len[mm][mm];//任意两个联通点的最短距离
double Mlen[mm];//点与所在牧场离当前点最远点的最小距离
int m;
class node//记录点坐标
{public:
double x,y;
}d[mm];
double lenth(int a,int b)//两点距离
{
return sqrt((d[a].x-d[b].x)*(d[a].x-d[b].x)+(d[a].y-d[b].y)*(d[a].y-d[b].y));
}/*并查集*/
void creat(int n)
{
for(int i=0;i<n;i++)root[i]=i,rank[i]=1;
}
int find(int n)
{
if(root[n]!=n)root[n]=find(root[n]);
return root[n];
}
void unio(int x,int y)
{
if(find(x)!=find(y))
{
if(rank[x]>rank[y])rank[x]+=rank[y],root[y]=x;
else rank[y]+=rank[x],root[x]=y;
}
}
void get_len(int a,int b)//记录两点距离
{ if(len[a][b]<klen)
len[a][b]=len[b][a]=lenth(a,b);
}
void get_lenmax()
{ ///计算任意两个联通点的最小距离
for(int i=0;i<m;i++)
for(int j=0;j<m;j++)
{ if(i==j)continue;
for(int k=0;k<m;k++)
if(len[i][k]>klen&&len[k][j]>klen)///判断len[i][k]，len[k][j]联通
{ if(len[i][j]>klen)
len[i][j]=len[j][i]=min(len[i][j],len[i][k]+len[k][j]);
else len[i][j]=len[j][i]=len[i][k]+len[k][j];
}
}
///计算当前点所在牧场的最大距离
for(int i=0;i<m;i++)
for(int j=0;j<m;j++)
if(i!=j)
Mlen[i]=max(Mlen[i],len[i][j]);
}
double link_min()
{ double m_len=oo;
double _len;
///合并两个牧场，并求最小距离
for(int i=0;i<m;i++)
for(int j=0;j<m;j++)
{ if(find(i)!=find(j))
{
_len=Mlen[i]+Mlen[j]+lenth(i,j);
m_len=min(m_len,_len);
}
}
///原先就是这个没加上，WA了一次
///计算合并后的最小距离与单点所在牧场的最大值取最大值即是答案
for(int i=0;i<m;i++)
m_len=max(m_len,Mlen[i]);
cout.setf(ios::fixed);///C++操作小数点后六位
cout<<setprecision(6)<<m_len<<'\n';
return m_len;
}
void fen()
{
for(int i=0;i<m;i++)
for(int j=0;j<m;j++)
if(vis[i][j]=='1')unio(find(i),find(j)),get_len(i,j);
get_lenmax();
}
int main()
{
cin>>m;
creat(m);
for(int i=0;i<m;i++)
cin>>d[i].x>>d[i].y;
for(int i=0;i<m;i++)
cin>>vis[i];
fen();
link_min();
}
Russ Cox
We do a fair bit of precomputation before choosing the path to add.
First, we calculate the minimum distance between all connected points. Then, we use a recursive depth first search to identify the different fields. Then we fill in diam[i], which is defined to be the distance to the farthest pasture that pasture i is connected to. Fielddiam[j] is the diameter of field j, which is the maximum of diam[i] for all pastures i in the field j.
Once we have all this, selecting a path is simple. If we add a path joining pastures i and j which are in different fields, the diameter of the new field is the maximum of:
- dist to point farthest from i + dist from i to j + dist to point farthest from j.
- old diameter of the field containing pasture i.
- old diameter of the field containing pasture j.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <assert.h>
#include <math.h>
#define INF (1e40)
typedef struct Point Point;
struct Point {
int x, y;
};
#define MAXPASTURE 150
int n;
double dist[MAXPASTURE][MAXPASTURE];
double diam[MAXPASTURE];
double fielddiam[MAXPASTURE];
Point pt[MAXPASTURE];
int field[MAXPASTURE];
int nfield;
double
ptdist(Point a, Point b)
{
return sqrt((double)(b.x-a.x)*(b.x-a.x)+(double)(b.y-a.y)*(b.y-a.y));
}
/* mark the field containing pasture i with number m */
void
mark(int i, int m)
{
int j;
if(field[i] != -1) {
assert(field[i] == m);
return;
}
field[i] = m;
for(j=0; j<n; j++)
if(dist[i][j] < INF/2)
mark(j, m);
}
void
main(void)
{
FILE *fin, *fout;
int i, j, k, c;
double newdiam, d;
fin = fopen("cowtour.in", "r");
fout = fopen("cowtour.out", "w");
assert(fin != NULL && fout != NULL);
fscanf(fin, "%d\n", &n);
for(i=0; i<n; i++)
fscanf(fin, "%d %d\n", &pt[i].x, &pt[i].y);
for(i=0; i<n; i++) {
for(j=0; j<n; j++) {
c = getc(fin);
if(i == j)
dist[i][j] = 0;
else if(c == '0')
dist[i][j] = INF; /* a lot */
else
dist[i][j] = ptdist(pt[i], pt[j]);
}
assert(getc(fin) == '\n');
}
/* Floyd-Warshall all pairs shortest paths */
for(k=0; k<n; k++)
for(i=0; i<n; i++)
for(j=0; j<n; j++)
if(dist[i][k]+dist[k][j] < dist[i][j])
dist[i][j] = dist[i][k]+dist[k][j];
/* mark fields */
for(i=0; i<n; i++)
field[i] = -1;
for(i=0; i<n; i++)
if(field[i] == -1)
mark(i, nfield++);
/* find worst diameters involving pasture i, and for whole field */
for(i=0; i<n; i++) {
for(j=0; j<n; j++)
if(diam[i] < dist[i][j] && dist[i][j] < INF/2)
diam[i] = dist[i][j];
if(diam[i] > fielddiam[field[i]])
fielddiam[field[i]] = diam[i];
}
/* consider a new path between i and j */
newdiam = INF;
for(i=0; i<n; i++)
for(j=0; j<n; j++) {
if(field[i] == field[j])
continue;
d = diam[i]+diam[j]+ptdist(pt[i], pt[j]);
if(d < fielddiam[field[i]])
d = fielddiam[field[i]];
if(d < fielddiam[field[j]])
d = fielddiam[field[j]];
if(d < newdiam)
newdiam = d;
}
fprintf(fout, "%.6lf\n", newdiam);
exit(0);
}
TASK: cowtourLANG: C++Compiling...Compile: OKExecuting... Test 1: TEST OK [0.000 secs, 3752 KB] Test 2: TEST OK [0.000 secs, 3752 KB] Test 3: TEST OK [0.000 secs, 3752 KB] Test 4: TEST OK [0.000 secs, 3752 KB] Test 5: TEST OK [0.054 secs, 3752 KB] Test 6: TEST OK [0.032 secs, 3752 KB] Test 7: TEST OK [0.054 secs, 3752 KB] Test 8: TEST OK [0.054 secs, 3752 KB] Test 9: TEST OK [0.043 secs, 3752 KB]All tests OK.
Your program ('cowtour') produced all correct answers! This is yoursubmission #2 for this problem. Congratulations!
Here are the test data inputs:
------- test 1 ---- 8 10 10 15 10 20 10 15 15 20 15 30 15 25 10 30 10 01000000 10111000 01001000 01001000 01110000 00000010 00000101 00000010 ------- test 2 ---- 2 0 0 1 0 00 00 ------- test 3 ---- 4 0 0 1 0 0 20 1 20 0100 1000 0001 0010 ------- test 4 ---- 30 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10 0 11 0 12 0 13 0 14 0 0 100 1 100 2 100 3 100 4 100 5 100 6 100 7 100 8 100 9 100 10 100 11 100 12 100 13 100 14 100 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 111111111111111000000000000000 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 000000000000000111111111111111 ------- test 5 ---- 148 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 10 0 11 0 12 0 13 0 14 0 15 0 16 0 17 0 18 0 19 0 20 0 21 0 22 0 23 0 24 0 25 0 26 0 27 0 28 0 29 0 30 0 31 0 32 0 33 0 34 0 35 0 36 0 37 0 38 0 39 0 40 0 41 0 42 0 43 0 44 0 45 0 46 0 47 0 48 0 49 0 50 0 51 0 52 0 53 0 54 0 55 0 56 0 57 0 58 0 59 0 60 0 61 0 62 0 63 0 64 0 65 0 66 0 67 0 68 0 69 0 70 0 71 0 72 0 73 0 0 100 1 100 2 100 3 100 4 100 5 100 6 100 7 100 8 100 9 100 10 100 11 100 12 100 13 100 14 100 15 100 16 100 17 100 18 100 19 100 20 100 21 100 22 100 23 100 24 100 25 100 26 100 27 100 28 100 29 100 30 100 31 100 32 100 33 100 34 100 35 100 36 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0 160 0 170 0 180 0 190 0 200 0 210 0 220 0 230 0 240 0 250 0 260 0 270 0 280 0 290 0 300 0 310 0 320 0 330 0 340 0 350 0 360 0 370 0 380 0 390 0 400 0 410 0 420 0 430 0 440 0 450 0 460 0 470 0 480 0 490 0 500 0 510 0 520 0 530 0 540 0 550 0 560 0 570 0 580 0 590 0 600 0 610 0 620 0 630 0 640 0 650 0 660 0 670 0 680 0 690 0 700 0 710 0 720 0 730 0 740 0 750 0 760 0 770 0 780 0 790 0 800 0 810 0 820 0 830 0 840 0 850 0 860 0 870 0 880 0 890 0 900 0 910 0 920 0 930 0 940 0 950 0 960 0 970 0 980 0 990 0 1000 0 1010 0 1020 0 1030 0 1040 0 1050 0 1060 0 1070 0 1080 0 1090 0 1100 0 1110 0 1120 0 1130 0 1140 0 1150 0 1160 0 1170 0 1180 0 1190 0 1200 0 1210 0 1220 0 1230 0 1240 0 1250 0 1260 0 1270 0 1280 0 1290 0 1300 0 1310 0 1320 0 1330 0 1340 0 1350 0 1360 0 1370 0 1380 0 1390 0 1400 0 1410 0 1420 0 1430 0 1440 0 1450 0 1460 0 1470 0 1480 0 1481 0 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