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F - Teleporter Setting

Score : $500$ points

Problem Statement

There are $N$ towns numbered Town $1$, Town $2$, $\ldots$, Town $N$.
There are also $M$ Teleporters, each of which connects two towns bidirectionally so that a person can travel from one to the other in one minute.

The $i$-th Teleporter connects Town $U_i$ and Town $V_i$ bidirectionally. However, for some of the Teleporters, one of the towns it connects is undetermined; $U_i=0$ means that one of the towns the $i$-th Teleporter connects is Town $V_i$, but the other end is undetermined.

For $i=1,2,\ldots,N$, answer the following question.

When the Teleporters with undetermined ends are all determined to be connected to Town $i$, how many minutes is required at minimum to travel from Town $1$ to Town $N$? If it is impossible to travel from Towns $1$ to $N$ using Teleporters only, print $-1$ instead.

Constraints

• $2 \leq N \leq 3\times 10^5$
• $1\leq M\leq 3\times 10^5$
• $0\leq U_i<V_i\leq N$
• If $i \neq j$, then $(U_i,V_i)\neq (U_j,V_j)$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$U_1$ $V_1$

$U_2$ $V_2$

$\vdots$

$U_M$ $V_M$

Output

Print $N$ integers, with spaces in between. The $k$-th integer should be the answer to the question when $i=k$.

3 2
0 2
1 2

-1 -1 2

When the Teleporters with an undetermined end are all determined to be connected to Town $1$,
the $1$-st and the $2$-nd Teleporters both connect Towns $1$ and $2$. Then, it is impossible to travel from Town $1$ to Town $3$.

When the Teleporters with an undetermined end are all determined to be connected to Town $2$,
the $1$-st Teleporter connects Town $2$ and itself, and the $2$-nd one connects Towns $1$ and $2$. Again, it is impossible to travel from Town $1$ to Town $3$.

When the Teleporters with an undetermined end are all determined to be connected to Town $3$,
the $1$-st Teleporter connects Town $3$ and Town $2$, and the $2$-nd one connects Towns $1$ and $2$. In this case, we can travel from Town $1$ to Town $3$ in two minutes.

• Use the $2$-nd Teleporter to travel from Town $1$ to Town $2$.
• Use the $1$-st Teleporter to travel from Town $2$ to Town $3$.

Therefore, $-1,-1$, and $2$ should be printed in this order.

Note that, depending on which town the Teleporters with an undetermined end are connected to, there may be a Teleporter that connects a town and itself, or multiple Teleporters that connect the same pair of towns.

5 5
1 2
1 3
3 4
4 5
0 2

3 3 3 3 2