Score : $500$ points

问题描述

令 $N$ 为一个正整数。我们有一个 $(2N+1) \times (2N+1)$ 的网格，其中行编号从0到$2N$，列编号也从0到$2N$。用 $(i,j)$ 表示第 $i$ 行和第 $j$ 列的方格。

我们有一个白色兵（pawn），它最初位于 $(0, N)$。另外，我们有 $M$ 个黑色兵，其中第 $i$ 个位于 $(X_i, Y_i)$。

当白色兵位于 $(i, j)$ 时，你可以执行以下操作之一来移动它：

• 如果满足 $0\leq i\leq 2N-1$ 和 $0 \leq j\leq 2N$，且 $(i+1,j)$ 不包含 黑色兵，则将白色兵移动到 $(i+1, j)$。
• 如果满足 $0\leq i\leq 2N-1$ 和 $0 \leq j\leq 2N-1$，且 $(i+1,j+1)$ 包含 黑色兵，则将白色兵移动到 $(i+1,j+1)$。
• 如果满足 $0\leq i\leq 2N-1$ 和 $1 \leq j\leq 2N$，且 $(i+1,j-1)$ 包含 黑色兵，则将白色兵移动到 $(i+1,j-1)$。

你不能移动黑色兵。

找出通过重复这些操作可以使白色兵到达 $(2N, Y)$ 的 $Y$ 值的数量。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

Let $N$ be a positive integer. We have a $(2N+1)\times (2N+1)$ grid where rows are numbered $0$ through $2N$ and columns are also numbered $0$ through $2N$. Let $(i,j)$ denote the square at Row $i$ and Column $j$.

We have one white pawn, which is initially at $(0, N)$. Also, we have $M$ black pawns, the $i$-th of which is at $(X_i, Y_i)$.

When the white pawn is at $(i, j)$, you can do one of the following operations to move it:

• If $0\leq i\leq 2N-1$, $0 \leq j\leq 2N$ hold and $(i+1,j)$ does not contain a black pawn, move the white pawn to $(i+1, j)$.
• If $0\leq i\leq 2N-1$, $0 \leq j\leq 2N-1$ hold and $(i+1,j+1)$ does contain a black pawn, move the white pawn to $(i+1,j+1)$.
• If $0\leq i\leq 2N-1$, $1 \leq j\leq 2N$ hold and $(i+1,j-1)$ does contain a black pawn, move the white pawn to $(i+1,j-1)$.

You cannot move the black pawns.

Find the number of integers $Y$ such that it is possible to have the white pawn at $(2N, Y)$ by repeating these operations.

Constraints

• $1 \leq N \leq 10^9$
• $0 \leq M \leq 2\times 10^5$
• $1 \leq X_i \leq 2N$
• $0 \leq Y_i \leq 2N$
• $(X_i, Y_i) \neq (X_j, Y_j)$ $(i \neq j)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$X_1$ $Y_1$

$:$

$X_M$ $Y_M$

Output

Print the answer.

Sample Input 1

2 4
1 1
1 2
2 0
4 2

Sample Output 1

3

We can move the white pawn to $(4,0)$, $(4,1)$, and $(4,2)$, as follows:

• $(0,2)\to (1,1)\to (2,1)\to (3,1)\to (4,2)$
• $(0,2)\to (1,1)\to (2,1)\to (3,1)\to (4,1)$
• $(0,2)\to (1,1)\to (2,0)\to (3,0)\to (4,0)$

On the other hand, we cannot move it to $(4,3)$ or $(4,4)$. Thus, we should print $3$.

Sample Input 2

1 1
1 1

Sample Output 2

0

We cannot move the white pawn from $(0,1)$.

update @ 2024/3/10 09:16:24