White Pawn

AtCoder

IDabc203_e

Time2000ms

Memory256MB

Difficulty—

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: $N$ $M$ $X_1$ $Y_1$ $:$ $X_M$ $Y_M$ [samples]

Samples

Input #1

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 $3$.

Input #2

1 1
1 1

Output #2

0

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

API Response (JSON)

{
  "problem": {
    "name": "White Pawn",
    "description": {
      "content": "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 $i$ an",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc203_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $i$ an...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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