Teleporter Setting

AtCoder
IDabc257_f
Time2000ms
Memory256MB
Difficulty
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: $N$ $M$ $U_1$ $V_1$ $U_2$ $V_2$ $\vdots$ $U_M$ $V_M$ [samples]
Samples
Input #1
3 2
0 2
1 2
Output #1
\-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.
Input #2
5 5
1 2
1 3
3 4
4 5
0 2
Output #2
3 3 3 3 2
API Response (JSON)
{
  "problem": {
    "name": "Teleporter Setting",
    "description": {
      "content": "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 ot",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc257_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ towns numbered Town $1$, Town $2$, $\\ldots$, Town $N$.  \nThere are also $M$ _Teleporters_, each of which connects two towns bidirectionally so that a person can travel from one to the ot...",
      "is_translate": false,
      "language": "English"
    }
  ]
}
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