POJ 3522 Slim Span

Original title address: http://poj.org/problem?id=3522

Slim Span

Time Limit: 5000MS		Memory Limit: 65536K
Total Submissions: 7041		Accepted: 3732

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 Weig HT 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) have four vertices {v1,v2, v3, v4} and F iveundirected 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 Depictedin Figure 6 (a) ~ (d). The Spanning Tree Ta in Figure 6 (a) Hasthree edges whose weights is 3, 6 and 7. The largest weight is 7 and thesmallest weight are 3 so, the slimness of the tree Ta is4. The slimnesses of spanning trees Tb, Tcand 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 isgreater than or equal to 1, thus the spanning tree Td in Figu Re 6 (d) is one ofthe 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 twozeros separated by a space. Each dataset has the following format.

n	M
a 1	b 1	w 1
am	BM p>	wm

Every input item in a dataset is a non-negative integer. Items in a lineare separated by a space. n is the number of the vertices and m the number ofthe edges. You can assume 2≤ n ≤100 and 0≤ m ≤ n(n− 1)/2. AKand BK (k = 1, ..., m) is positive integers lessthan or equal to n, which re Present the vertices vakand VBK connected by the k-th edge ek. wk is a positive an integer less than or equal to 10000, whichindicates the weight of ek. You can assume that the graph G= (V, E) was simple, which is, there was no self-loops (that Connect the same vertex) nor parallel edges (that is, or more edges Whoseboth ends is the same).

Output

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

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

7 ·

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

Test instructions

The spanning tree with the smallest edge and minimum margin difference, output-1 if no spanning tree exists.

Ideas:

Enumerate the minimum edges with the Kruskal algorithm

1#include <cstdio>2#include <algorithm>3 using namespacestd;4 Const intN = the, M = the;5 structSIDE6 {7     int  from;8     intto ;9     intdistance;Ten }edge[m]; One intCnt[n]; A BOOLcmpConstSIDE A,ConstSIDE B) { -     returnA.distance <b.distance; - } the intFind (intx) { -     returnCNT[X] = = x?X:find (cnt[x]); - } - intMainvoid) + { -     intN, M, A, B, MIN, OK; +      while(SCANF ("%d%d", &n, &m), n+m) A     { at          for(intI=0; i<m; ++i) -scanf" %d%d%d", &edge[i]. from, &edge[i].to, &edge[i].distance); -Sort (Edge, edge+m, CMP); -MIN =0x3f3f3f3f; -OK =0; -          for(intstart=0; start<=m-n+1; ++start) in         { -              for(intI=1; i<=n; ++i) toCnt[i] =i; +             intI, j =0; -              for(I=start; i<m && j<n-1; ++i) the             { *A = Find (Edge[i]. from); $b =Find (edge[i].to);Panax Notoginseng                 if(A! =b) -CNT[B] = A, + +J; the             } +             if(J = = n1) Amin = min (min, edge[i-1].distance-edge[start].distance), OK =1; the         } +printf"%d\n"Ok? MIN:-1); -     } $}

POJ 3522 Slim Span