platinum_A_Graph_Problem (USACO December_Contest)
Time limit: 2 seconds | Memory limit: 256 MB | Task type: batch
Algorithms: data structures, spanning tree, graph theory, union-find

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 $1\dots N$ and
edges labeled $1\dots M$ ($2\le N\le 2\cdot 10^5$, $N-1\le M\le 4\cdot 10^5$).
For each vertex $v$ in the graph, the following process is conducted:
​
Let $S=\{v\}$ and $h=0$.
While $|S|<N$, 
​
Out of all edges with exactly one endpoint in $S$, let $e$ be the edge with
the minimum label.
Add the endpoint of $e$ not in $S$ to $S$.
Set $h=10h+e$.
​
Return $h\pmod{10^9+7}$.
​
Determine all the return values of this process.
​
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains $N$ and $M$.
​
Then follow $M$ lines, the $e$th containing the endpoints $(a_e,b_e)$ of the 
$e$th edge ($1\le a_e<b_e\le N$). It is guaranteed that these edges form a
connected graph, and at most one edge connects each pair of vertices.
​
OUTPUT FORMAT (print output to the terminal / stdout):
Output $N$ lines, where the $i$th 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 $10^9+7$.
SCORING:
Input 4: $N,M\le 2000$
Inputs 5-6: $N\le 2000$
Inputs 7-10: $N\le 10000$
Inputs 11-14: $a_e+1=b_e$ for all $e$
Inputs 15-23: No additional constraints.
Problem credits: Benjamin Qi

Starter code:

Grader code:

Setup script:

Evaluation script: