103775: [Atcoder]ABC377 F - Avoid Queen 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 in row $i$
• Placed in column $j$
• Placed on any square $(a,b)\ (1\leq a\leq N,1\leq b\leq N)$ where $i+j=a+b$
• Placed on any square $(a,b)\ (1\leq a\leq N,1\leq b\leq N)$ where $i-j=a-b$

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

• $1\leq N\leq10^9$
• $1\leq M\leq10^3$
• $1\leq a_k\leq N,1\leq b_k\leq N\ (1\leq k\leq M)$
• $(a_k,b_k)\neq(a_l,b_l)\ (1\leq k\lt l\leq M)$
• 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

2

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

Therefore, you can place your piece on only two squares: squares $(6,6)$ and $(7,7)$.

Sample Input 2

1000000000 1
1 1

Sample Output 2

999999997000000002

Out of $10^{18}$ squares, the squares that cannot be used are: squares in row $1$, squares in column $1$, and squares $(1,1)$, $(2,2)$, $\ldots$, $(10^9,10^9)$, totaling $3\times10^9-2$ squares.

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

77