Edges in MST

2000ms

262144K

Description:

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.

Input:

The first line contains two integers n and m (2 ≤ n ≤ 10⁵, $\text{[math]}$ ) — 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 ≤ 10⁶, 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.

Output:

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.

Sample Input:

Sample Output:

none
any
at least one
at least one
any

Sample Input:

Sample Output:

any
any
none

Sample Input:

Sample Output:

at least one
at least one
at least one

Note:

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.

Informação

Provedor Codeforces

Código CF160D