问题
Suppose I have n line segments in general position. How can I quickly count, for each of my n segments, how many of the other n-1 it intersects?
I can do this naively in O(n2) time. I can find all intersections using a fairly straightforward sweep line algorithm (Bentley-Ottmann) in O((n + k) log n) time, where k is the number of such intersections, and then aggregate the intersections I found into a bunch of counts.
I don't need to find the intersections, though; I just want to know how many there are. I don't see how to modify the sweep line algorithm to be faster since it needs to reorder two things in a tree for every intersection, and I can't think of any other techniques that don't suffer from the same problem.
I'm also interested in hearing how to count how many total intersections there are.
回答1:
I have a hard time believing that you can do better than Bentley Ottman in the general case. You can simplify the computation a bit if you don't care where the line segments intersect, but I don't see how you could count crossings without finding them.
In essence, Bentley Ottman is a way to simplify the search space for intersections. There are other ways, which might work for particular arrangements of line segments, but unless there is some predictable geometric relationship between your segments, you're not going to be able to better than first using some clever way of finding candidate intersections combined with an individual verification of each candidate.
Unless your problem domain has some specific features which might make provide exploitable substructure, I'd think your best bet for speed would be to adapt Bentley Ottman (or some similar sweeping algorithm) for parallel execution. (Clipping the line segments into bands is a simple one. Figuring out an optimal set of bands would be interesting, too.) Of course, that's a practical rather than an academic exercise; the parallel algorithm might well end up doing more work in total; it just exploits hardware to do the work in (a constant factor) less time.
来源：https://stackoverflow.com/questions/14009029/is-there-an-efficient-way-to-count-the-number-of-intersections-among-a-given-set