Square

AtCoder

IDcaddi2018_d

Time2000ms

Memory256MB

Difficulty—

Takahashi has an $N \times N$ grid. The square at the $i$\-th row and the $j$\-th column of the grid is denoted by $(i,j)$. Particularly, the top-left square of the grid is $(1,1)$, and the bottom-right square is $(N,N)$. An integer, $0$ or $1$, is written on $M$ of the squares in the Takahashi's grid. Three integers $a_i,b_i$ and $c_i$ describe the $i$\-th of those squares with integers written on them: the integer $c_i$ is written on the square $(a_i,b_i)$. Takahashi decides to write an integer, $0$ or $1$, on each of the remaining squares so that the condition below is satisfied. Find the number of such ways to write integers, modulo $998244353$. * For all $1\leq i < j\leq N$, there are even number of $1$s in the square region whose top-left square is $(i,i)$ and whose bottom-right square is $(j,j)$. ## Constraints * $2 \leq N \leq 10^5$ * $0 \leq M \leq min(5 \times 10^4,N^2)$ * $1 \leq a_i,b_i \leq N(1\leq i\leq M)$ * $0 \leq c_i \leq 1(1\leq i\leq M)$ * If $i \neq j$, then $(a_i,b_i) \neq (a_j,b_j)$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $a_1$ $b_1$ $c_1$ $:$ $a_M$ $b_M$ $c_M$ [samples]

Samples

Input #1

Output #1

8

For example, the following ways to write integers satisfy the condition:

101   111
011   111
000   011

Input #2

Output #2

Input #3

Output #3

Input #4

Output #4

Input #5

100000 0

Output #5

342016343

API Response (JSON)

{
  "problem": {
    "name": "Square",
    "description": {
      "content": "Takahashi has an $N \\times N$ grid. The square at the $i$\\-th row and the $j$\\-th column of the grid is denoted by $(i,j)$. Particularly, the top-left square of the grid is $(1,1)$, and the bottom-rig",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "caddi2018_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Takahashi has an $N \\times N$ grid. The square at the $i$\\-th row and the $j$\\-th column of the grid is denoted by $(i,j)$. Particularly, the top-left square of the grid is $(1,1)$, and the bottom-rig...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments