Score : $600$ points

问题描述

我们有 $N$ 个在 $xy$ 平面上的多边形。
这些多边形的所有边都与 $x$ 轴或 $y$ 轴平行，并且每个内角为 $90$ 或 $270$ 度。所有这些多边形都是简单的。
第 $i$ 个多边形有 $M_i$ 个顶点，其中第 $j$ 个顶点坐标为 $(x_{i, j}, y_{i, j})$。
该多边形的边是由第 $j$ 个顶点和第 $(j+1)$ 个顶点连接而成的线段。（假设 $(M_i+1)$-th 顶点是第 $1$ 个顶点。）

一个多边形是简单的，当...

对于它的任何两条不相邻的边，它们之间都不会相交（交叉或接触）。

您将获得 $Q$ 个查询。对于每个 $i = 1, 2, \dots, Q$，第 $i$ 个查询如下所示。

• 在这 $N$ 个多边形中，有多少个多边形包含点 $(X_i + 0.5, Y_i + 0.5)$ 在其内部？

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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:

$N$

$M_1$

$x_{1, 1}$ $y_{1, 1}$ $x_{1, 2}$ $y_{1, 2}$ $\dots$ $x_{1, M_1}$ $y_{1, M_1}$

$M_2$

$x_{2, 1}$ $y_{2, 1}$ $x_{2, 2}$ $y_{2, 2}$ $\dots$ $x_{2, M_2}$ $y_{2, M_2}$

$\vdots$

$M_N$

$x_{N, 1}$ $y_{N, 1}$ $x_{N, 2}$ $y_{N, 2}$ $\dots$ $x_{N, M_N}$ $y_{N, M_N}$

$Q$

$X_1$ $Y_1$

$X_2$ $Y_2$

$\vdots$

$X_Q$ $Y_Q$

Output

Print $Q$ lines.
The $i$-th line should contain the answer to the $i$-th query.

Sample 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

Sample Output 1

0
2
1

Note that different polygons may cross or touch each other.

Sample 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

Sample Output 2

0
2
1
1

Although the polygons are simple, they may not be convex.

update @ 2024/3/10 09:26:31