103772: [Atcoder]ABC377 C - Avoid Knight Attack

Problem Statement

There is a grid of $N^2$ squares with $N$ rows and $N$ columns. Let $(i,j)$ denote the square at the $i$-th row from the top $(1\leq i\leq N)$ and $j$-th column from the left $(1\leq j\leq N)$.

Each square is either empty or has a piece placed on it. There are $M$ pieces placed on the grid, and the $k$-th $(1\leq k\leq M)$ piece is placed on square $(a_k,b_k)$.

You want to place your piece on an empty square in such a way that it cannot be captured by any of the existing pieces.

A piece placed on square $(i,j)$ can capture pieces that satisfy any of the following conditions:

• Placed on square $(i+2,j+1)$
• Placed on square $(i+1,j+2)$
• Placed on square $(i-1,j+2)$
• Placed on square $(i-2,j+1)$
• Placed on square $(i-2,j-1)$
• Placed on square $(i-1,j-2)$
• Placed on square $(i+1,j-2)$
• Placed on square $(i+2,j-1)$

Here, conditions involving non-existent squares are considered to never be satisfied.

For example, a piece placed on square $(4,4)$ can capture pieces placed on the squares shown in blue in the following figure:

How many squares can you place your piece on?

Constraints

• $1\leq N\leq10^9$
• $1\leq M\leq2\times10^5$
• $1\leq a_k\leq N,1\leq b_k\leq N\ (1\leq k\leq M)$
• $(a_k,b_k)\neq(a_l,b_l)\ (1\leq k\lt l\leq M)$
• All input values are integers.

Input

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

$N$ $M$
$a_1$ $b_1$
$a_2$ $b_2$
$\vdots$
$a_M$ $b_M$

Output

Print the number of empty squares where you can place your piece without it being captured by any existing pieces.

Sample Input 1

8 6
1 4
2 1
3 8
4 5
5 2
8 3

Sample Output 1

38

The existing pieces can capture pieces placed on the squares shown in blue in the following figure:

Therefore, you can place your piece on the remaining $38$ squares.

Sample Input 2

1000000000 1
1 1

Sample Output 2

999999999999999997

Out of $10^{18}$ squares, only $3$ squares cannot be used: squares $(1,1)$, $(2,3)$, and $(3,2)$.

Note that the answer may be $2^{32}$ or greater.

Sample Input 3

20 10
1 4
7 11
7 15
8 10
11 6
12 5
13 1
15 2
20 10
20 15

Sample Output 3

338

问题陈述

有一个由$N^2$个方格组成的网格，有$N$行和$N$列。 用$(i,j)$表示从顶部开始的第$i$行$(1\leq i\leq N)$和从左边开始的第$j$列$(1\leq j\leq N)$的方格。

每个方格要么是空的，要么上面放有一个棋子。 网格上有$M$个棋子，第$k$个$(1\leq k\leq M)$棋子放在方格$(a_k,b_k)$上。

你想要在一个空方格上放置你的棋子，以使得它不能被任何现有的棋子捕获。

放在方格$(i,j)$上的棋子可以捕获满足以下任一条件的棋子：

• 放在方格$(i+2,j+1)$上
• 放在方格$(i+1,j+2)$上
• 放在方格$(i-1,j+2)$上
• 放在方格$(i-2,j+1)$上
• 放在方格$(i-2,j-1)$上
• 放在方格$(i-1,j-2)$上
• 放在方格$(i+1,j-2)$上
• 放在方格$(i+2,j-1)$上

这里，涉及不存在的方格的条件被认为从不满足。

例如，放在方格$(4,4)$上的棋子可以捕获放在以下图中蓝色方格上的棋子：

你可以放置你的棋子的方格有多少个？

约束条件

• $1\leq N\leq10^9$
• $1\leq M\leq2\times10^5$
• $1\leq a_k\leq N,1\leq b_k\leq N\ (1\leq k\leq M)$
• $(a_k,b_k)\neq(a_l,b_l)\ (1\leq k\lt l\leq M)$
• 所有输入值都是整数。

输入

输入从标准输入以以下格式给出：

$N$ $M$
$a_1$ $b_1$
$a_2$ $b_2$
$\vdots$
$a_M$ $b_M$

输出

打印你可以放置棋子且不会被任何现有棋子捕获的空方格的数量。

示例输入 1

8 6
1 4
2 1
3 8
4 5
5 2
8 3

示例输出 1

38

现有的棋子可以捕获放在以下图中蓝色方格上的棋子：

因此，你可以将你的棋子放在剩余的$38$个方格上。

示例输入 2

1000000000 1
1 1

示例输出 2

999999999999999997

在$10^{18}$个方格中，只有