Ex - Walk

AtCoder

IDabc317_h

Time8000ms

Memory256MB

Difficulty—

We have a directed graph with $N$ vertices numbered $1$ to $N$. The graph has no multi-edges but can have self-loops. Also, every edge in the graph satisfies the following condition. * If the edge goes from vertex $s$ to vertex $t$, then $s$ and $t$ satisfy at least one of $0 \leq t - s\leq 2$ and $t = 1$. The presence or absence of an edge in the graph is represented by sequences $A, B, C, D$, each of length $N$. Each element of $A, B, C, D$ has the following meaning. (Let $A_n$ denote the $n$\-th element of $A$; the same applies to $B_n, C_n, D_n$.) * $A_n$ is $1$ if there is an edge from vertex $n$ to vertex $n$, and $0$ otherwise. * $B_n$ is $1$ if there is an edge from vertex $n$ to vertex $n+1$, and $0$ otherwise. (Here, $B_N = 0$.) * $C_n$ is $1$ if there is an edge from vertex $n$ to vertex $n+2$, and $0$ otherwise. (Here, $C_{N-1} = C_N = 0$.) * $D_n$ is $1$ if there is an edge from vertex $n$ to vertex $1$, and $0$ otherwise. (Here, $D_1 = A_1$.) In the given graph, find the number, modulo $998244353$, of walks with $K$ edges starting at vertex $1$ and ending at vertex $N$. Here, a walk with $K$ edges starting at vertex $1$ and ending at vertex $N$ is a sequence of vertices $v_0 = 1, v_1, \dots, v_K = N$ such that for each $i$ $(0 \leq i \lt K)$ there is an edge from vertex $v_i$ to vertex $v_{i + 1}$. Two walks are distinguished when they differ as sequences. ## Constraints * $2 \leq N \leq 5 \times 10^4$ * $1 \leq K \leq 5 \times 10^5$ * $A_i, B_i, C_i, D_i \in \lbrace 0, 1 \rbrace$ * $A_1 = D_1$ * $B_N = C_{N-1} = C_N = 0$ ## Input The input is given from Standard Input in the following format: $N$ $K$ $A_1$ $A_2$ $\dots$ $A_N$ $B_1$ $B_2$ $\dots$ $B_N$ $C_1$ $C_2$ $\dots$ $C_N$ $D_1$ $D_2$ $\dots$ $D_N$ [samples]

Samples

Input #1

Output #1

6

The following figure shows the graph.
![image](https://img.atcoder.jp/abc317/2106e1b4faaa87d208ed3e3a275cda1b.jpg)
The following six walks satisfy the conditions.

*   $1, 1, 1, 3$
*   $1, 1, 2, 3$
*   $1, 1, 3, 3$
*   $1, 2, 3, 3$
*   $1, 3, 1, 3$
*   $1, 3, 3, 3$

Input #2

Output #2

Input #3

10 500000
0 1 0 1 0 0 0 0 1 1
1 1 1 0 1 1 1 0 1 0
0 0 1 1 0 0 1 1 0 0
0 1 1 1 1 1 0 1 1 0

Output #3

866263864

API Response (JSON)

{
  "problem": {
    "name": "Ex - Walk",
    "description": {
      "content": "We have a directed graph with $N$ vertices numbered $1$ to $N$. The graph has no multi-edges but can have self-loops. Also, every edge in the graph satisfies the following condition. *   If the edge ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 8000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc317_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a directed graph with $N$ vertices numbered $1$ to $N$. The graph has no multi-edges but can have self-loops. Also, every edge in the graph satisfies the following condition.\n\n*   If the edge ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments