103075: [Atcoder]ABC307 F - Virus 2

Problem Statement

There are $N$ rooms numbered $1$, $2$, $\ldots$, $N$, each with one person living in it, and $M$ corridors connecting two different rooms. The $i$-th corridor connects room $U_i$ and room $V_i$ with a length of $W_i$.

One day (we call this day $0$), the $K$ people living in rooms $A_1, A_2, \ldots, A_K$ got (newly) infected with a virus. Furthermore, on the $i$-th of the following $D$ days $(1\leq i\leq D)$, the infection spread as follows.

People who were infected at the end of the night of day $(i-1)$ remained infected at the end of the night of day $i$.
For those who were not infected, they were newly infected if and only if they were living in a room within a distance of $X_i$ from at least one room where an infected person was living at the end of the night of day $(i-1)$. Here, the distance between rooms $P$ and $Q$ is defined as the minimum possible sum of the lengths of the corridors when moving from room $P$ to room $Q$ using only corridors. If it is impossible to move from room $P$ to room $Q$ using only corridors, the distance is set to $10^{100}$.

For each $i$ ($1\leq i\leq N$), print the day on which the person living in room $i$ was newly infected. If they were not infected by the end of the night of day $D$, print $-1$.

Constraints

• $1 \leq N\leq 3\times 10^5$
• $0 \leq M\leq 3\times 10^5$
• $1 \leq U_i < V_i\leq N$
• All $(U_i,V_i)$ are different.
• $1\leq W_i\leq 10^9$
• $1 \leq K\leq N$
• $1\leq A_1<A_2<\cdots<A_K\leq N$
• $1 \leq D\leq 3\times 10^5$
• $1\leq X_i\leq 10^9$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$
$U_1$ $V_1$ $W_1$
$U_2$ $V_2$ $W_2$
$\vdots$
$U_M$ $V_M$ $W_M$
$K$
$A_1$ $A_2$ $\ldots$ $A_K$
$D$
$X_1$ $X_2$ $\ldots$ $X_D$

Output

Print $N$ lines.
The $i$-th line $(1\leq i\leq N)$ should contain the day on which the person living in room $i$ was newly infected.

Sample Input 1

4 4
1 2 2
2 3 1
2 4 3
3 4 2
1
1
2
3 3

Sample Output 1

0
1
1
2

The infection spreads as follows.

• On the night of day $0$, the person living in room $1$ gets infected.
• The distances between room $1$ and rooms $2,3,4$ are $2,3,5$, respectively. Thus, since $X_1=3$, the people living in rooms $2$ and $3$ are newly infected on the night of day $1$.
• The distance between rooms $3$ and $4$ is $2$. Thus, since $X_2=3$, the person living in room $4$ also gets infected on the night of day $2$.

Therefore, the people living in rooms $1,2,3,4$ were newly infected on days $0,1,1,2$, respectively, so print $0,1,1,2$ in this order on separate lines.

Sample Input 2

7 7
1 2 2
2 3 3
3 4 1
4 5 1
5 6 3
3 7 1
4 7 1
2
1 6
2
2 3

Sample Output 2

0
1
2
-1
2
0
-1

Sample Input 3

5 1
1 2 5
2
1 3
3
3 7 5

Sample Output 3

0
2
0
-1
-1

Note that it is not always possible to move between any two rooms using only corridors.