POJ 1679 The Unique MST(次小生成树&Kruskal)

最新推荐文章于 2021-07-26 17:51:50 发布

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#MST #Kruskal #POJ

POJ 同时被 3 个专栏收录

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MST (最小生成树)

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本文探讨了如何判断一个无向图的最小生成树是否唯一，通过Kruskal算法求解最小生成树，并在此基础上进行特定操作来验证唯一性。

The Unique MST

http://poj.org/problem?id=1679

Description

Given a connected undirected graph, tell if its minimum spanning tree is unique.

Definition 1 (Spanning Tree): Consider a connected, undirected graph G = (V, E). A spanning tree of G is a subgraph of G, say T = (V', E'), with the following properties:
1. V' = V.
2. T is connected and acyclic.

Definition 2 (Minimum Spanning Tree): Consider an edge-weighted, connected, undirected graph G = (V, E). The minimum spanning tree T = (V, E') of G is the spanning tree that has the smallest total cost. The total cost of T means the sum of the weights on all the edges in E'.

Input

The first line contains a single integer t (1 <= t <= 20), the number of test cases. Each case represents a graph. It begins with a line containing two integers n and m (1 <= n <= 100), the number of nodes and edges. Each of the following m lines contains a triple (xi, yi, wi), indicating that xi and yi are connected by an edge with weight = wi. For any two nodes, there is at most one edge connecting them.

Output

For each input, if the MST is unique, print the total cost of it, or otherwise print the string 'Not Unique!'.

Sample Input

Sample Output

3
Not Unique!

思路：

先求一个 MST，再从原图G中删掉从有向树（以找的次序为方向）的每个点出发的边（每次删掉一条边），对删掉一条边的新图G'求MST与原有比较即可。

完整代码：

/*16ms,224KB*/

#include<cstdio>
#include<cstring>
#include<algorithm>
using namespace std;
const int Max_V = 102;
const int Max_E = 4952;

struct EDGE
{
	int u, v, w;
	bool ischoosed;
} edge[Max_E];

int path[Max_V], V, E, bj[Max_V + 2];

bool cmp(const EDGE&a, const EDGE&b)
{
	return a.w < b.w;
}

int search_path(int x)
{
	if (x != path[x])
		path[x] = search_path(path[x]);
	return path[x];
}

int Kruskal(int start)
{
	int count = 0, sum = 0, x, y;
	for (int i = 1; i <= V; i++)
		path[i] = i;
	for (int i = start; i < E && count < V - 1; i++)
	{
		x = search_path(edge[i].u);
		y = search_path(edge[i].v);
		if (x == y)
			continue;
		bj[count++] = i;//??????????????????????
		path[x] = path[y];//修正
		sum += edge[i].w;
	}
	return count < V - 1 ? 0 : sum;
}

int Seckruskal()
{
	int count = 0, sum = 0, x, y;
	for (int i = 1; i <= V; i++)
		path[i] = i;
	for (int i = 0; i < E && count < V - 1; i++)
	{
		x = search_path(edge[i].u);
		y = search_path(edge[i].v);
		if (x == y || edge[i].ischoosed == true)//把这条边删掉，看有没有另外的MST
			continue;
		count++;
		path[x] = path[y];
		sum += edge[i].w;
	}
	return count < V - 1 ? 0 : sum;
}

int main()
{
	int T, i, temp, minMST;
	bool flag;
	scanf("%d", &T);
	while (T--)
	{
		memset(bj, 0, sizeof(bj));
		memset(path, 0, sizeof(path));
		memset(edge, 0, sizeof(edge));
		flag = false;
		scanf("%d%d", &V, &E);
		for (i = 0; i < E; i++)
			scanf("%d%d%d", &edge[i].u, &edge[i].v, &edge[i].w);
		sort(edge, edge + E, cmp);
		///////////////////////////////////////////////////////
		minMST = Kruskal(0);
		for (i = 0; i < V - 1; i++)
		{
			edge[bj[i]].ischoosed = true;
			temp = Seckruskal();
			edge[bj[i]].ischoosed = false;
			if (temp == minMST && temp)
			{
				flag = true;
				break;
			}
		}
		if (flag)
			printf("Not Unique!\n");
		else
			printf("%d\n", minMST);
	}
	return 0;
}