E - Come Back Quickly

Score : $500$ points

Problem Statement

In the Republic of AtCoder, there are $N$ towns numbered $1$ through $N$ and $M$ roads numbered $1$ through $M$.
Road $i$ is a one-way road from Town $A_i$ to Town $B_i$, and it takes $C_i$ minutes to go through. It is possible that $A_i = B_i$, and there may be multiple roads connecting the same pair of towns.
Takahashi is thinking about taking a walk in the country. We will call a walk valid when it goes through one or more roads and returns to the town it starts at.
For each town, determine whether there is a valid walk that starts at that town. Additionally, if the answer is yes, find the minimum time such a walk requires.

Constraints

• $1 \le N \le 2000$
• $1 \le M \le 2000$
• $1 \le A_i \le N$
• $1 \le B_i \le N$
• $1 \le C_i \le 10^5$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$ $C_1$

$A_2$ $B_2$ $C_2$

$A_3$ $B_3$ $C_3$

$\hspace{25pt} \vdots$

$A_M$ $B_M$ $C_M$

Output

Print $N$ lines. The $i$-th line $(1 \le i \le N)$ should contain the following:

• if there is a valid walk that starts at Town $i$, the minimum time required by such a walk;
• otherwise, -1.

4 4
1 2 5
2 3 10
3 1 15
4 3 20

30
30
30
-1

By Roads $1, 2, 3$, Towns $1, 2, 3$ forms a ring that takes $30$ minutes to go around.
From Town $4$, we can go to Towns $1, 2, 3$, but then we cannot return to Town $4$.

4 6
1 2 5
1 3 10
2 4 5
3 4 10
4 1 10
1 1 10

10
20
30
20

There may be a road such that $A_i = B_i$.
Here, we can use just Road $6$ to depart from Town $1$ and return to that town.

4 7
1 2 10
2 3 30
1 4 15
3 4 25
3 4 20
4 3 20
4 3 30

-1
-1
40
40

Note that there may be multiple roads connecting the same pair of towns.