Slim Span
| Time Limit: 5000MS | Memory Limit: 65536K | |
| Total Submissions: 7933 | Accepted: 4227 |
Description
Given an undirected weighted graph G, you should find one of spanning trees specified as follows.
The graph G is an ordered pair (V, E), where V is a set of vertices {v1, v2, …, vn} and E is a set of undirected edges {e1, e2, …, em}. Each edge e ∈ E has its weight w(e).
A spanning tree T is a tree (a connected subgraph without cycles) which connects all the n vertices with n ? 1 edges. The slimness of a spanning tree T is defined as the difference between the largest weight and the smallest weight among the n ? 1 edges of T.
Figure 5: A graph G and the weights of the edges
For example, a graph G in Figure 5(a) has four vertices {v1, v2, v3, v4} and five undirected edges {e1, e2, e3, e4, e5}. The weights of the edges are w(e1) = 3, w(e2) = 5, w(e3) = 6, w(e4) = 6, w(e5) = 7 as shown in Figure 5(b).
Figure 6: Examples of the spanning trees of G
There are several spanning trees for G. Four of them are depicted in Figure 6(a)~(d). The spanning tree Ta
in Figure 6(a) has three edges whose weights are 3, 6 and 7. The
largest weight is 7 and the smallest weight is 3 so that the slimness of
the tree Ta is 4. The slimnesses of spanning trees Tb, Tc and Td
shown in Figure 6(b), (c) and (d) are 3, 2 and 1, respectively. You can
easily see the slimness of any other spanning tree is greater than or
equal to 1, thus the spanning tree Td in Figure 6(d) is one of the
slimmest spanning trees whose slimness is 1.
Your job is to write a program that computes the smallest slimness.
Input
The
input consists of multiple datasets, followed by a line containing two
zeros separated by a space. Each dataset has the following format.
| n | m | |
| a1 | b1 | w1 |
| ? | ||
| am | bm | wm |
Every
input item in a dataset is a non-negative integer. Items in a line are
separated by a space. n is the number of the vertices and m the number
of the edges. You can assume 2 ≤ n ≤ 100 and 0 ≤ m ≤ n(n ? 1)/2. ak and bk (k = 1, …, m) are positive integers less than or equal to n, which represent the two vertices vak and vbk connected by the kth edge ek. wk is a positive integer less than or equal to 10000, which indicates the weight of ek. You can assume that the graph G = (V, E)
is simple, that is, there are no self-loops (that connect the same
vertex) nor parallel edges (that are two or more edges whose both ends
are the same two vertices).
Output
For
each dataset, if the graph has spanning trees, the smallest slimness
among them should be printed. Otherwise, ?1 should be printed. An output
should not contain extra characters.
Sample Input
4 5 1 2 3 1 3 5 1 4 6 2 4 6 3 4 7 4 6 1 2 10 1 3 100 1 4 90 2 3 20 2 4 80 3 4 40 2 1 1 2 1 3 0 3 1 1 2 1 3 3 1 2 2 2 3 5 1 3 6 5 10 1 2 110 1 3 120 1 4 130 1 5 120 2 3 110 2 4 120 2 5 130 3 4 120 3 5 110 4 5 120 5 10 1 2 9384 1 3 887 1 4 2778 1 5 6916 2 3 7794 2 4 8336 2 5 5387 3 4 493 3 5 6650 4 5 1422 5 8 1 2 1 2 3 100 3 4 100 4 5 100 1 5 50 2 5 50 3 5 50 4 1 150 0 0
Sample Output
1 20 0 -1 -1 1 0 1686 50
1 #include <iostream>
2 #include <cstring>
3 #include <cstdio>
4 #include <algorithm>
5
6 using namespace std;
7
8 const int maxn=105;
9 const int maxm=10005;
10 const int inf=1e9;
11
12 int par[maxn],Rank[maxn];
13
14 int n,m;
15
16 void init(){
17 for(int i=1;i<=n;i++){
18 par[i]=i;
19 Rank[i]=0;
20 }
21 }
22
23 int Find(int x){
24 if(x==par[x]) return x;
25 return par[x]=Find(par[x]);
26 }
27
28 void unite(int x,int y){
29 x=Find(x);
30 y=Find(y);
31 if(x==y) return;
32 if(Rank[x]<Rank[y]){
33 par[x]=y;
34 }
35 else{
36 par[y]=x;
37 if(Rank[x]==Rank[y]) Rank[x]++;
38 }
39 }
40
41 bool same(int x,int y){
42 return Find(x)==Find(y);
43 }
44
45 struct edge{
46 int s,t,v;
47 }E[maxm];
48
49 int ne;
50
51 int cmp(const edge&a,const edge&b){
52 return a.v<b.v;
53 }
54
55 void add_edge(int s,int t,int v){
56 E[ne].s=s;
57 E[ne].t=t;
58 E[ne++].v=v;
59 }
60
61 int kruskal(int s){
62 int mxe=-1;
63 int tot=0;
64 for(int i=s;i<ne;i++){
65 edge &e=E[i];
66 if(!same(e.s,e.t)){
67 unite(e.s,e.t);
68 tot++;
69 mxe=e.v;
70 }
71 }
72 if(tot!=n-1) return -1;
73 return mxe-E[s].v;
74 }
75
76 int main(){
77 while(scanf("%d %d",&n,&m)==2&&n){
78 ne=0;
79 for(int i=0;i<m;i++){
80 int s,t,v;
81 scanf("%d %d %d",&s,&t,&v);
82 add_edge(s,t,v);
83 add_edge(t,s,v);
84 }
85 sort(E,E+ne,cmp);
86 int res=inf;
87 for(int i=0;i<ne;i++){
88 init();
89 int tmp=kruskal(i);
90 if(tmp!=-1) res=min(res,tmp);
91 }
92 if(res==inf) printf("-1\n",res);
93 else printf("%d\n",res);
94 }
95 return 0;
96 }