題目：
Problem B: Audiophobia Consider yourself lucky! Consider yourself lucky to be still breathing and having fun participating in this contest. But we apprehend that many of your descendants may not have this luxury. For, as you know, we are the dwellers of one of the most polluted cities on earth. Pollution is everywhere, both in the environment and in society and our lack of consciousness is simply aggravating the situation.
However, for the time being, we will consider only one type of pollution - the sound pollution. The loudness or intensity level of sound is usually measured in decibels and sound having intensity level 130 decibels or higher is considered painful. The intensity level of normal conversation is 6065 decibels and that of heavy traffic is 7080 decibels.
Consider the following city map where the edges refer to streets and the nodes refer to crossings. The integer on each edge is the average intensity level of sound (in decibels) in the corresponding street.
To get from crossing A to crossing G you may follow the following path: ACFG. In that case you must be capable of tolerating sound intensity as high as 140 decibels. For the paths ABEG, ABDG and ACFDG you must tolerate respectively 90, 120 and 80 decibels of sound intensity. There are other paths, too. However, it is clear that ACFDG is the most comfortable path since it does not demand you to tolerate more than 80 decibels.
In this problem, given a city map you are required to determine the minimum sound intensity level you must be able to tolerate in order to get from a given crossing to another.
Input
The input may contain multiple test cases.
The first line of each test case contains three integers , and where C indicates the number of crossings (crossings are numbered using distinct integers ranging from 1 to C), S represents the number of streets and Q is the number of queries.
Each of the next S lines contains three integers: c1, c2 and d indicating that the average sound intensity level on the street connecting the crossings c1 and c2 ( ) is d decibels.
Each of the next Q lines contains two integers c1 and c2 ( ) asking for the minimum sound intensity level you must be able to tolerate in order to get from crossing c1 to crossing c2.
The input will terminate with three zeros form C, S and Q.
Output
For each test case in the input first output the test case number (starting from 1) as shown in the sample output. Then for each query in the input print a line giving the minimum sound intensity level (in decibels) you must be able to tolerate in order to get from the first to the second crossing in the query. If there exists no path between them just print the line ``no path".
Print a blank line between two consecutive test cases.
Sample Input
7 9 3
1 2 50
1 3 60
2 4 120
2 5 90
3 6 50
4 6 80
4 7 70
5 7 40
6 7 140
1 7
2 6
6 2
7 6 3
1 2 50
1 3 60
2 4 120
3 6 50
4 6 80
5 7 40
7 5
1 7
2 4
0 0 0
Sample Output
Case #1
80
60
60
Case #2
40
no path
80
題目大意：
從a點到b點， 找到一條路徑，使得這條路徑上的所有噪音中最大的值是所有路徑中最小的， 這個噪音值便是要求的。
分析與總結：
用floyd是找出所有路徑中長度最小的，只需要稍微變形一下，便可求得答案。
除了用這個方法，還可以用Kruskal算法，按照Kruskal算法的步驟，一條邊一條邊的加入樹中，每加入一次，就判斷所有要問的起點和終點是否已經連通， 一旦有連通的，那麼那條路徑中的最大值便是當前加入的這條邊的權值，因為加入的邊是按照從小到大順序加入的。
代碼：
1.Kruskal
[cpp]
#include<cstdio>
#include<cstring>
#include<algorithm>
#define N 1005
using namespace std;
int n,m,Q, f[N],rank[N],ans[N*10];
bool vis[N*10];
struct Edge{
int u,v,val;
friend bool operator<(const Edge&a,const Edge&b){
return a.val < b.val;
}
}arr[N];
struct Query{
int id, u, v;
}q[N*10];
inline void init(){
for(int i=0; i<N; ++i)
f[i]=i,rank[i]=0;
}
int find(int x){
int i,j=x;
while(j!=f[j]) j=f[j];
while(x!=j){
i=f[x]; f[x]=j; x=i;
}
return j;
}
bool Union(int x, int y){
int a=find(x), b=find(y);
if(a==b)return false;
if(rank[a]>rank[b])
f[b]=a;
else{
if(rank[a]==rank[b])
++rank[b];
f[a]=b;
}
return true;
}
int main(){
int a,b,c,cas=1;
while(~scanf("%d%d%d",&n,&m,&Q)){
if(!n&&!m&&!Q) break;
for(int i=0; i<m; ++i){
scanf("%d%d%d",&a,&b,&c);
arr[i].u=a, arr[i].v=b, arr[i].val=c;
}
for(int i=0; i<Q; ++i){
scanf("%d%d",&a,&b);
q[i].id=i, q[i].u=a, q[i].v=b;
}
init();
memset(vis, 0, sizeof(vis));
memset(ans, -1, sizeof(ans));
sort(arr,arr+m);
for(int i=0; i<m; ++i){
if(Union(arr[i].u,arr[i].v)){
for(int j=0; j<Q; ++j)if(!vis[j]){
int a=find(q[j].u), b=find(q[j].v);
if(a==b){
vis[j] = true;
ans[j] = arr[i].val;
}
}
}
}
if(cas!=1) printf("\n");
printf("Case #%d\n", cas++);
for(int i=0; i<Q; ++i){
if(ans[i]==-1) printf("no path\n");
else printf("%d\n", ans[i]);
}
}
return 0;
}
2.Floyd
[cpp]
#include<cstdio>
#include<cstring>
#include<algorithm>
const int N = 105;
const int INF = 1000000000;
using namespace std;
int d[N][N], n, m, Q;
inline void read_graph(){
for(int i=1; i<N; ++i){
d[i][i] = INF;
for(int j=i+1; j<N; ++j)
d[i][j]=d[j][i]=INF;
}
int a,b,c;
for(int i=0; i<m; ++i){
scanf("%d%d%d",&a,&b,&c);
d[a][b]=d[b][a]=c;
}
}
inline void Floyd(){
for(int k=1; k<=n; ++k){
for(int i=1; i<=n; ++i){
for(int j=1; j<=n; ++j){
int tmp = max(d[i][k], d[k][j]);
d[i][j] = min(d[i][j], tmp);
}
}
}
}
inline void output(){
int u,v;
for(int i=0; i<Q; ++i){
scanf("%d%d",&u,&v);
if(d[u][v]!=INF) printf("%d\n",d[u][v]);
else printf("no path\n");
}
}
int main(){
int a,b,c,cas=1;
while(~scanf("%d%d%d",&n,&m,&Q)){
if(!n&&!m&&!Q) break;
read_graph();
Floyd();
if(cas!=1) puts("");
printf("Case #%d\n",cas++);
output();
}
return 0;
}
