To improve her mathematical knowledge, Bessie has been taking a graph theory course and finds herself stumped by the following problem. Please help her!
You are given a connected, undirected graph with vertices labeled 1…N and edges labeled 1…M(2≤N≤2⋅105, N−1≤M≤4⋅105). For each vertex v in the graph, the following process is conducted:
Let S={v}and h=0.
While |S|<N,
1.Out of all edges with exactly one endpoint in S, let e be the edge with the minimum label.
2.Add the endpoint of e not in S to S.
3.Set h=10h+e.
Return h ( mod 109+7).
Determine all the return values of this process.
INPUT FORMAT (pipe stdin):
The first line contains N and M .Then follow M lines, the eth containing the endpoints (ae,be) of the eth edge (1≤ae<be≤N). It is guaranteed that these edges form a connected graph, and at most one edge connects each pair of vertices.
OUTPUT FORMAT (pipe stdout):
Output N lines, where the ith line should contain the return value of the process starting at vertex i.
SAMPLE INPUT:
3 2
1 2
2 3
SAMPLE OUTPUT:
12
12
21
SAMPLE INPUT:
5 6
1 2
3 4
2 4
2 3
2 5
1 5
SAMPLE OUTPUT:
1325
1325
2315
2315
5132
Consider starting at i=3. First, we choose edge 2, after which S={3,4} and h=2.Second, we choose edge 3, after which S={2,3,4}and h=23. Third, we choose edge 1, after which S={1,2,3,4} and h=231. Finally, we choose edge 5, after which S={1,2,3,4,5}and h=2315. The answer for i=3 is therefore 2315.
SAMPLE INPUT:
15 14
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8 9
9 10
10 11
11 12
12 13
13 14
14 15
SAMPLE OUTPUT:
678925929
678925929
678862929
678787329
678709839
678632097
178554320
218476543
321398766
431520989
542453212
653475435
764507558
875540761
986574081
Make sure to output the answers modulo 109+7.
SCORING:
Input 4: N,M≤2000
Inputs 5-6: N≤2000
Inputs 7-10: N≤10000
Inputs 11-14: ae+1=be
for all e
Inputs 15-23: No additional constraints.
Problem credits: Benjamin Qi
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