101683: [AtCoder]ABC168 D - .. (Double Dots)

Problem Statement

There is a cave.

The cave has $N$ rooms and $M$ passages. The rooms are numbered $1$ to $N$, and the passages are numbered $1$ to $M$. Passage $i$ connects Room $A_i$ and Room $B_i$ bidirectionally. One can travel between any two rooms by traversing passages. Room $1$ is a special room with an entrance from the outside.

It is dark in the cave, so we have decided to place a signpost in each room except Room $1$. The signpost in each room will point to one of the rooms directly connected to that room with a passage.

Since it is dangerous in the cave, our objective is to satisfy the condition below for each room except Room $1$.

• If you start in that room and repeatedly move to the room indicated by the signpost in the room you are in, you will reach Room $1$ after traversing the minimum number of passages possible.

Determine whether there is a way to place signposts satisfying our objective, and print one such way if it exists.

Constraints

• All values in input are integers.
• $2 \leq N \leq 10^5$
• $1 \leq M \leq 2 \times 10^5$
• $1 \leq A_i, B_i \leq N\ (1 \leq i \leq M)$
• $A_i \neq B_i\ (1 \leq i \leq M)$
• One can travel between any two rooms by traversing passages.

Input

Input is given from Standard Input in the following format:

$N$ $M$
$A_1$ $B_1$
$:$
$A_M$ $B_M$

Output

If there is no way to place signposts satisfying the objective, print No.

Otherwise, print $N$ lines. The first line should contain Yes, and the $i$-th line $(2 \leq i \leq N)$ should contain the integer representing the room indicated by the signpost in Room $i$.

Sample Input 1

4 4
1 2
2 3
3 4
4 2

Sample Output 1

Yes
1
2
2

If we place the signposts as described in the sample output, the following happens:

• Starting in Room $2$, you will reach Room $1$ after traversing one passage: $(2) \to 1$. This is the minimum number of passages possible.
• Starting in Room $3$, you will reach Room $1$ after traversing two passages: $(3) \to 2 \to 1$. This is the minimum number of passages possible.
• Starting in Room $4$, you will reach Room $1$ after traversing two passages: $(4) \to 2 \to 1$. This is the minimum number of passages possible.

Thus, the objective is satisfied.

Sample Input 2

6 9
3 4
6 1
2 4
5 3
4 6
1 5
6 2
4 5
5 6

Sample Output 2

Yes
6
5
5
1
1

If there are multiple solutions, any of them will be accepted.