POJ 3522 Slim Span (Kruskal ＋枚举 边权差最小的生成树)

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, …, v_n} and E is a set of undirected edges {e₁, e₂, …, e_m}. 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 T_a 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 T_a is 4. The slimnesses of spanning trees T_b, T_c and T_d 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.

The input consists of multiple datasets, followed by a line containing two zeros separated by a space. Each dataset has the following format.

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. a_k and b_k (k = 1, …, m) are positive integers less than or equal to n, which represent the two vertices v_ak and v_bk connected by the kth edge e_k. w_k is a positive integer less than or equal to 10000, which indicates the weight of e_k. 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).

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.

#include <cstdio>
#include <algorithm>
#include <cstring>
using namespace std;
int const INF = 0xffffff;
int const MAX = 105;
int n, m, ans;
int fa[MAX], rank[MAX];
bool flag, ok;
struct Edge
{
    int u, v, w;
}e[6000];

bool cmp(Edge a, Edge b)
{
    return a.w < b.w;
}

void UFset()
{
    for(int i = 0; i <= n; i++)
        fa[i] = i;
    memset(rank, 0, sizeof(rank));
}

int Find(int x)
{
    return x == fa[x] ? x : fa[x] = Find(fa[x]);
}

void Union(int a, int b)
{
    int r1 = Find(a);
    int r2 = Find(b);
    if(r1 == r2)
        return;
    if(rank[r1] > rank[r2])
        fa[r2] = r1;
    else
    {
        fa[r1] = r2;
        if(rank[r1] == rank[r2])
            rank[r1]++;
    }
}

int Kruskal(int v0)
{
    int num = 0, mi = INF, ma = -1;
    UFset();
    for(int i = v0; i < m; i++)
    {
        int u = e[i].u;
        int v = e[i].v;
        if(Find(u) != Find(v))
        {
            mi = min(mi, e[i].w); //记录最大边权
            ma = max(ma, e[i].w); //记录最小边权
            Union(u, v); 
            num++;
        }
        if(num == n - 1) //够造成一颗生成树
        {
            ok = true;   //ok表示存在一颗树
            flag = true; //flag表示当前情况下存在一棵树
            break;
        }
    }
    return ma - mi;
}

int main()
{
    while(scanf("%d %d", &n, &m) != EOF && (m + n))
    {
        ok = false;
        for(int i = 0; i < m; i++)
            scanf("%d %d %d", &e[i].u, &e[i].v, &e[i].w);
        sort(e, e + m, cmp);
        int x, y, ans = INF;
        for(int i = 0; i < m; i++) //枚举
        {
            flag = false;
            int tmp = Kruskal(i);
            if(flag)
                ans = min(ans, tmp);
        }
        printf("%d\n", !ok ? -1 : ans);
    }
}