Path and Subsequence

AtCoder

IDarc150_c

Time2000ms

Memory256MB

Difficulty—

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

Samples

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

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`.

Input #2

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.

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

Output #3

Yes

API Response (JSON)

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  "problem": {
    "name": "Path and Subsequence",
    "description": {
      "content": "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 intege",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc150_c"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $V_i$.\nAdditionally, we are given an intege...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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