Rectilinear Polygons

AtCoder

IDabc211_f

Time2000ms

Memory256MB

Difficulty—

We have $N$ polygons on the $xy$\-plane. Every side of these polygons is parallel to the $x$\- or $y$\-axis, and every interior angle is $90$ or $270$ degrees. All of these polygons are simple. The $i$\-th polygon has $M_i$ corners, the $j$\-th of which is $(x_{i, j}, y_{i, j})$. The sides of this polygon are segments connecting the $j$\-th and $(j+1)$\-th corners. (Assume that $(M_i+1)$\-th corner is the $1$\-st corner.) A polygon is simple when...for any two of its sides that are not adjacent, they do not intersect (cross or touch) each other. You are given $Q$ queries. For each $i = 1, 2, \dots, Q$, the $i$\-th query is as follows. * Among the $N$ polygons, how many have the point $(X_i + 0.5, Y_i + 0.5)$ inside them? ## Constraints * $1 \leq N \leq 10^5$ * $4 \leq M_i \leq 10^5$ * Each $M_i$ is even. * $\sum_i M_i \leq 4 \times 10^5$ * $0 \leq x_{i, j}, y_{i, j} \leq 10^5$ * $(x_{i, j}, y_{i, j}) \neq (x_{i, k}, y_{i, k})$ if $j \neq k$. * $x_{i, j} = x_{i, j+1}$ for $j = 1, 3, \dots M_i-1$. * $y_{i, j} = y_{i, j+1}$ for $j = 2, 4, \dots M_i$. (Assume $y_{i, M_i +1} = y_{i, 1}$.) * The given polygons are simple. * $1 \leq Q \leq 10^5$ * $0 \leq X_i, Y_i \lt 10^5$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M_1$ $x_{1, 1}$ $y_{1, 1}$ $x_{1, 2}$ $y_{1, 2}$ $\dots$ $x_{1, M_1}$ $y_{1, M_1}$ $M_2$ $x_{2, 1}$ $y_{2, 1}$ $x_{2, 2}$ $y_{2, 2}$ $\dots$ $x_{2, M_2}$ $y_{2, M_2}$ $\vdots$ $M_N$ $x_{N, 1}$ $y_{N, 1}$ $x_{N, 2}$ $y_{N, 2}$ $\dots$ $x_{N, M_N}$ $y_{N, M_N}$ $Q$ $X_1$ $Y_1$ $X_2$ $Y_2$ $\vdots$ $X_Q$ $Y_Q$ [samples]

Samples

Input #1

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

Output #1

0
2
1

![image](https://img.atcoder.jp/ghi/5fccf008dddd93f10ebfc7f13d04a0e0.png)  
Note that different polygons may cross or touch each other.

Input #2

2
4
12 3 12 5 0 5 0 3
12
1 1 1 9 10 9 10 0 4 0 4 6 6 6 6 2 8 2 8 7 2 7 2 1
4
2 6
4 4
6 3
1 8

Output #2

0
2
1
1

![image](https://img.atcoder.jp/ghi/1c97f791a2aadcf5637b1f10736fb820.png)  
Although the polygons are simple, they may not be convex.

API Response (JSON)

{
  "problem": {
    "name": "Rectilinear Polygons",
    "description": {
      "content": "We have $N$ polygons on the $xy$\\-plane.   Every side of these polygons is parallel to the $x$\\- or $y$\\-axis, and every interior angle is $90$ or $270$ degrees. All of these polygons are simple.   Th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc211_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have $N$ polygons on the $xy$\\-plane.  \nEvery side of these polygons is parallel to the $x$\\- or $y$\\-axis, and every interior angle is $90$ or $270$ degrees. All of these polygons are simple.  \nTh...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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