Ex - Trespassing Takahashi

Score : $600$ points

Problem Statement

There are $N$ points numbered $1$ through $N$, and $M$ roads. The $i$-th ($1 \leq i \leq M$) road connects Point $a_i$ and Point $b_i$ bidirectionally and requires $c_i$ minutes to pass through. One can travel from any point to any other point using some number of roads. There is a house on Points $1,\ldots, K$.

For $i=1,\ldots,Q$, solve the following problem.

Takahashi is currently at the house at Point $x_i$ and wants to travel to the house at Point $y_i$.
Once $t_i$ minutes have passed since his last sleep, he cannot continue moving anymore.
He can get sleep only at a point with a house, but he may do so any number of times.
If he can travel from Point $x_i$ to Point $y_i$, print Yes; otherwise, print No.

Constraints

• $2 \leq K \leq N \leq 2 \times 10^5$
• $N-1 \leq M \leq \min (2 \times 10^5, \frac{N(N-1)}{2})$
• $1 \leq a_i \lt b_i \leq N$
• If $i \neq j$, then $(a_i,b_i) \neq (a_j,b_j)$.
• $1 \leq c_i \leq 10^9$
• One can travel from any point to any other point using some number of roads.
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq x_i \lt y_i \leq K$
• $1 \leq t_1 \leq \ldots \leq t_Q \leq 10^{15}$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

$a_1$ $b_1$ $c_1$

$\vdots$

$a_M$ $b_M$ $c_M$

$Q$

$x_1$ $y_1$ $t_1$

$\vdots$

$x_Q$ $y_Q$ $t_Q$

Output

Print $Q$ lines. The $i$-th line should contain the answer for the $i$-th problem.

6 6 3
1 4 1
4 6 4
2 5 2
3 5 3
5 6 5
1 2 15
3
2 3 4
2 3 5
1 3 12

No
Yes
Yes

In the $3$-rd problem, it takes no less than $13$ minutes from Point $1$ to reach Point $3$ directly. However, he can first travel to Point $2$ in $12$ minutes, get sleep in the house there, and then travel to Point $3$. Thus, the answer is Yes.