CF160D Edges in MST

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• 151提交
• 题目来源
• 评测方式 RemoteJudge
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题意翻译

题目大意:

给一个带权的无向图,保证没有自环和重边. 由于最小生成树不唯一,因此你需要确定每一条边是以下三种情况哪一个 1.一定在所有MST上 2.可能在某个MST上 3.一定不可能在任何MST上 输入

第一行给出n,m表示点数和边数. 数据范围见原题面 之后m行,每行给出ai,bi,wi 表示一个边的两个端点和边的权值.保证没有自环与重边. 输出

你需要输出m行,每行依次表示输入的第几条边,如果是情况1,输出 "any"(不包含引号);如果是情况2,输出 "at least one"(不包含引号);如果是情况3,输出 "none"(不包含引号).

题目描述

You are given a connected weighted undirected graph without any loops and multiple edges.

Let us remind you that a graph's spanning tree is defined as an acyclic connected subgraph of the given graph that includes all of the graph's vertexes. The weight of a tree is defined as the sum of weights of the edges that the given tree contains. The minimum spanning tree (MST) of a graph is defined as the graph's spanning tree having the minimum possible weight. For any connected graph obviously exists the minimum spanning tree, but in the general case, a graph's minimum spanning tree is not unique.

Your task is to determine the following for each edge of the given graph: whether it is either included in any MST, or included at least in one MST, or not included in any MST.

输入输出格式

输入格式：

The first line contains two integers $n$ and $m$ ( $2<=n<=10^{5}$ , ) — the number of the graph's vertexes and edges, correspondingly. Then follow $m$ lines, each of them contains three integers — the description of the graph's edges as " $a_{i}$ $b_{i}$ $w_{i}$ " ( $1<=a_{i},b_{i}<=n,1<=w_{i}<=10^{6},a_{i}≠b_{i}$ ), where $a_{i}$ and $b_{i}$ are the numbers of vertexes connected by the $i$ -th edge, $w_{i}$ is the edge's weight. It is guaranteed that the graph is connected and doesn't contain loops or multiple edges.

输出格式：

Print $m$ lines — the answers for all edges. If the $i$ -th edge is included in any MST, print "any"; if the $i$ -th edge is included at least in one MST, print "at least one"; if the $i$ -th edge isn't included in any MST, print "none". Print the answers for the edges in the order in which the edges are specified in the input.

输入输出样例

输入样例#1： 复制
4 5
1 2 101
1 3 100
2 3 2
2 4 2
3 4 1

输出样例#1： 复制
none
any
at least one
at least one
any

输入样例#2： 复制
3 3
1 2 1
2 3 1
1 3 2

输出样例#2： 复制
any
any
none

输入样例#3： 复制
3 3
1 2 1
2 3 1
1 3 1

输出样例#3： 复制
at least one
at least one
at least one


说明

In the second sample the MST is unique for the given graph: it contains two first edges.

In the third sample any two edges form the MST for the given graph. That means that each edge is included at least in one MST.

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