有多少条线段将在一个平面内与水平线相交？ 找到这个的最有效方法是什么

Question

I'm trying to solve this Cyberchef challenge :

You are given N points in a plane (numbered 1 through N); for each valid i, the i-th point is Pi=(i,Ai). There are N−1 line segments between them (numbered 1 through N−1); for each valid i, the i-th line segment is formed by connecting the points Pi and Pi+1.

You are given Q horizontal line segments . Each query horizontal line segment is denoted by two points, from a point (x1,y) to a point (x2,y) (where it stops and does not propagate further). For Every Horizontal line segment, you have to find the number of line segments from those (N-1 line segments) it collides with on the way.

So this is the problem where:

• 2<=N<= 100000
• 1<=Q<= 100000

Can anyone teach me what approach will be best and most effective in terms of time complexity?

Answer 1

So you have the only polyline and want to check many queries for intersection with horizontal segments.

It is worth to build some kind of binary space partitioning (BSP) over that polyline. In this case you can check for intersection in O(log(n)) time, so overall time is O(nlogn)+O(qlogn) (building and checking)

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If yout polyline fortunately is convex polygon (without one closing edge), things are quite simpler - just sort edges by Y-coordinate and check only those intersecting query Y (binary search, log(n) per query). In general case (non-convex) your polyline might look like WMWMWMW and you have to check too many edges.

Answer 2

I used kd tree to solve this problem. Also dont ask questions that are on-going challenges.

Answer 3

所以扫描线算法是你基本上需要知道的，以解决这个问题。