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Matcom Online Grader
Faculty of Mathematics and Computer Science of University of Havana
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You are given a convex polygon with $n$ vertices (no three of them lie on the same line) and a triangulation for that polygon. You should answer $q$ queries of the type $x_1, y_1, x_2, y_2$. For each query you should output the minimum number of diagonals you need to cross when walking from point ($x_1, y_1$) to point ($x_2, y_2$) without stepping out of the polygon.
Image is related to sample input
- If we are at point $H$ inside $\triangle AFG$ and want to get to point $I$ in $\triangle BCD$, best route implies crossing all 4 given diagonals ($\overline{AF}$, $\overline{AE}$, $\overline{BE}$, $\overline{BD}$).
- If we are at point $J$ inside $\triangle ABE$ and want to get to point $K$ in $\triangle AEF$, since points are in adjacent triangles, it is sufficient to cross the diagonal $\overline{AE}$.
- If we are at point $L$ inside $\triangle BCD$ and want to get to point $I$ in $\triangle BCD$, since both points are inside the same triangle, the amount of diagonals to cross is 0.
Input
First line of input contains an integer $n$ ($3 \leq n \leq 10^5$) describing the number of vertices of the polygon. $N$ lines follow, the $i$-th of them with two integers $x_i, y_i$ ($-10^9 \leq x_i, y_i \leq 10^9$), the coordinates of the $i$-th vertex of the polygon, given in counter-clockwise order.
Next $n-3$ lines describes the triangulation, giving the list of diagonals that make up the triangulation. Each of those lines contains two integers $a_i, b_i$ ($1 \leq a_i, b_i \leq n$), indicating there is a diagonal from the $a_i$-th vertex to the $b_i$-th vertex, according to the order in which vertices were given in the input. It is guaranteed that the given list of diagonals make up a valid triangulation for the polygon.
Next line gives you the integer $q$ ($1 \leq q \leq 10^5$), the number of queries to process. And the final $q$ lines contain four integers $x_1, y_1, x_2, y_2$ describing the query. It is guaranteed that both points are inside the polygon and do not lie on a given diagonal nor on a side of the polygon.
Output
For each query output a line with answer for that query.
Sample test(s)
Input
7
2 1
6 1
10 2
10 5
9 6
4 6
1 4
1 6
1 5
2 4
2 5
3
2 3 9 3
5 2 6 5
8 2 9 3
Output
4
1
0
Input
4
100 100
200 100
200 200
100 200
1 3
3
150 140 150 160
190 140 190 160
110 140 110 160
Output
1
0
0