POJ 3522--slim Span —————— minimum spanning tree, maximum edge and minimum edge

Slim Span

Time Limit: 5000MS		Memory Limit: 65536K
Total Submissions: 7102		Accepted: 3761

Description

Given an undirected weighted graph G, you should find one of spanning trees specified as follows.

The graph 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 have 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 GAnd the weights of the edges

For example, a graph G in Figure 5 (a) have four vertices {v1, v2, v3, v4} An D 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 is several spanning trees for G. Four of them is depicted in Figure 6 (a) ~ (d). The Spanning Tree Ta in Figure 6 (a) had three edges whose weights are 3, 6 and 7. The largest weight is 7 and the smallest weight are 3 so, 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) is 3, 2 and 1 , respectively. You can easily see the slimness of any other spanning tree are greater than or equal to 1, thus the spanning tree Td in Fig Ure 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 and 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 is 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) is positive integers less than or equal to n, which R Epresent the vertices vak and VBK connected by the k-th edge ek. wk is a positive an integer less than or equal to 10000, which indicates the weight of ek. You can assume that the graph G = (V, E) is easy, that's, there is no self-loops (tha T connect the same vertex) nor parallel edges (that is, or more edges whose both ends is the same-vertices).

Output

For each dataset, if the graph has spanning trees, the smallest slimness among them should is printed. Otherwise,−1 should be printed. An output should not contain extra characters.

Sample Input

4 51 2 31 3 51 4 62 4 63 4 74 61 2 101 3 1001 4 902 3 202 4 803 4 402 11 2 13 03 11 2 13 31 2 22 3 51 3 65 101 2 1101 3 12 01 4 1301 5 1202 3 1102 4 1202 5 1303 4 1203 5 1104 5 1205 101 2 93841 3 8871 4 27781 5 69162 3 77942 4 83362 5 53873 4 49 33 5 66504 5 14225 81 2 12 3 1003 4 1004 5 1001 5 502 5 503 5 504 1 1500 0

Sample Output

1200-1-110168650

Source

Japan 2007 Main topic: let you ask for a spanning tree, the maximum value of the tree edge is the smallest difference from the minimum value. The idea of solving problems: in fact, it is kruskal to find the minimum spanning tree. Violent enumeration of different minimum edges.

#include <stdio.h> #include <algorithm> #include <string.h> #include <iostream>using namespace    std;const int MAXN = 110;const int maxe = 11010;struct edge{int from,to,dist,idx; Edge () {} edge (int _from,int _to,int _dist,int _idx): From (_from), to (_to), Dist (_dist), idx (_IDX) {}}edges[maxe];struct set{int Pa,rela;} Sets[maxn];int Ans[maxn];bool cmp (Edge A,edge b) {return a.dist < b.dist;}    void init (int n) {for (int i = 0; I <= N; i++) {SETS[I].PA = i;    }}int Find (int x) {if (x = = SETS[X].PA) {return x;    } int tmp = SETS[X].PA;    SETS[X].PA = Find (TMP); return SETS[X].PA;}    int main () {int n, m;        while (scanf ("%d%d", &n,&m)!=eof&& (n+m)) {init (n);        int a,b,c;            for (int i = 0; i < m; i++) {scanf ("%d%d%d", &a,&b,&c);        Edges[i] = Edge (a,b,c,i);        } sort (edges,edges+m,cmp);        int pos = 0, cnt = 0;           for (int i = 0; i < m; i++) { Edge & e = edges[i];            int Rootx, Rooty;            Rootx = Find (E.from);            Rooty = Find (e.to);            if (Rootx = = Rooty) {continue;            } cnt++;            SETS[ROOTY].PA = Rootx;        pos = i;            } if (cnt! = n-1) {puts ("-1");        Continue        } int ans = edges[pos].dist-edges[0].dist;            for (int j = 1; J <= M-n + 1; j + +) {cnt = 0;            for (int i = 0; I <= N; i++) {SETS[I].PA = i;                } for (int i = j; i < m; i++) {Edge & e = edges[i];                int Rootx, Rooty;                Rootx = Find (E.from);                Rooty = Find (e.to);                if (Rootx = = Rooty) {continue;                } SETS[ROOTY].PA = Rootx;                cnt++;            pos = i;            } if (CNT < n-1) {break;             }else{   int tmp = edges[pos].dist-edges[j].dist;            ans = min (ans,tmp);    }} printf ("%d\n", ans); } return 0;}

　　

POJ 3522--slim Span —————— minimum spanning tree, maximum edge and minimum edge