201502: [AtCoder]ARC150 C - Path and Subsequence

Problem Statement

We have a connected undirected graph $G$ with $N$ vertices and $M$ edges. The vertices are numbered $1$ to $N$. The $i$-th edge connects vertices $U_i$ and $V_i$.

Additionally, we are given an integer sequence of length $N$, $A=(A_1,\ A_2, \dots,\ A_N)$, and an integer sequence of length $K$, $B=(B_1,\ B_2,\ \dots,\ B_K)$.

Determine whether $G$, $A$, and $B$ satisfy the following condition.

• For every simple path from vertex $1$ to $N$ in $G$, $v=(v_1,\ v_2, \dots,\ v_k)\ (v_1=1,\ v_k=N)$, $B$ is a (not necessarily contiguous) subsequence of $(A_{v_1},\ A_{v_2},\ \dots,\ A_{v_k})$.

Constraints

• $2 \leq N \leq 10^5$
• $1 \leq K \leq N$
• $N-1 \leq M \leq 2 \times 10^5$
• $1 \leq U_i < V_i \leq N$
• $(U_i,\ V_i) \neq (U_j,\ V_j)$ if $i \neq j$.
• $1 \leq A_i,\ B_i \leq N$
• All values in the input are integers.
• The given graph $G$ is connected.

Input

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

$N$ $M$ $K$
$U_1$ $V_1$
$U_2$ $V_2$
$\vdots$
$U_M$ $V_M$
$A_1$ $A_2$ $\dots$ $A_N$
$B_1$ $B_2$ $\dots$ $B_K$

Output

If the condition is satisfied, print Yes; otherwise, print No.

Sample Input 1

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

Sample Output 1

Yes

There are two simple paths from vertex $1$ to vertex $6$: $(1,\ 2,\ 4,\ 6)$ and $(1,\ 3,\ 5,\ 6)$. The $(A_{v_1},\ A_{v_2},\ \dots,\ A_{v_k})$ corresponding to these paths are $(1,\ 2,\ 5,\ 6)$ and $(1,\ 4,\ 2,\ 6)$. Both of them have $B=(1,\ 2,\ 6)$ as a subsequence, so the answer is Yes.

Sample Input 2

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

Sample Output 2

No

For a simple path $(1,\ 2,\ 5)$ from vertex $1$ to vertex $5$, the $(A_{v_1},\ A_{v_2},\ \dots,\ A_{v_k})$ is $(1,\ 2,\ 2)$, which does not have $B=(1,\ 3,\ 2)$ as a subsequence.

Sample Input 3

10 20 3
5 6
5 10
5 7
3 5
3 7
2 6
3 8
4 5
5 8
7 10
1 6
1 9
4 6
1 2
1 4
6 7
4 8
2 5
3 10
6 9
2 5 8 5 1 5 1 1 5 10
2 5 1

Sample Output 3

Yes