Stair-like Grid

AtCoder

IDabc357_g

Time6000ms

Memory256MB

Difficulty—

There is a special grid with $N$ rows. ($N$ is even.) The $i$\-th row from the top has $\left \lceil \frac{i}{2} \right \rceil \times 2$ cells from the left end. For example, when $N = 6$, the grid looks like the following: ![image](https://img.atcoder.jp/abc357/00fb5a3d30c86d7f23b529a62eb586b6.jpg) Let $(i, j)$ denote the cell at the $i$\-th row from the top and $j$\-th column from the left. Each cell is either an empty cell or a wall cell. There are $M$ wall cells, and the $i$\-th wall cell is $(a_i, b_i)$. Here, $(1, 1)$ and $(N, N)$ are empty. Starting from $(1, 1)$, how many ways are there to reach $(N, N)$ by repeatedly moving right or down to an adjacent empty cell? Find the count modulo $998244353$. ## Constraints * $2 \leq N \leq 2.5 \times 10^5$ * $N$ is even. * $0 \leq M \leq 50$ * $1 \leq a_i \leq N$ * $1 \leq b_i \leq \left \lceil \frac{a_i}{2} \right \rceil \times 2$ * $(a_i, b_i) \neq (1, 1)$ and $(a_i, b_i) \neq (N, N)$. * $(a_i, b_i) \neq (a_j, b_j)$ if $i \neq j$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $a_1$ $b_1$ $a_2$ $b_2$ $\vdots$ $a_M$ $b_M$ [samples]

Samples

Input #1

4 2
2 1
4 2

Output #1

2

There are two paths that satisfy the conditions of the problem:

*   $(1, 1) \to (1, 2) \to (2, 2) \to (3, 2) \to (3, 3) \to (3, 4) \to (4, 4)$
*   $(1, 1) \to (1, 2) \to (2, 2) \to (3, 2) \to (3, 3) \to (4, 3) \to (4, 4)$

Input #2

Output #2

Input #3

Output #3

619611437

Input #4

100000 10
552 24
4817 255
7800 954
23347 9307
28028 17652
39207 11859
48670 22013
74678 53158
75345 45891
88455 4693

Output #4

175892766

API Response (JSON)

{
  "problem": {
    "name": "Stair-like Grid",
    "description": {
      "content": "There is a special grid with $N$ rows. ($N$ is even.) The $i$\\-th row from the top has $\\left \\lceil \\frac{i}{2} \\right \\rceil \\times 2$ cells from the left end.   For example, when $N = 6$, the grid ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 6000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc357_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a special grid with $N$ rows. ($N$ is even.) The $i$\\-th row from the top has $\\left \\lceil \\frac{i}{2} \\right \\rceil \\times 2$ cells from the left end.  \nFor example, when $N = 6$, the grid ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments