从最近一条线的一组点中找到点的最快算法是什么？

Question

I have:
- a set of points of known size (in my case, only 6 points)
- a line characterized by x = s + t * r, where x, s and r are 3D vectors

I need to find the point closest to the given line. The actual distance does not matter to me.

I had a look at several different questions that seem related (including this one) and know how to solve this on paper from my highschool math classes. But I cannot find a solution without calculating every distance, and I am sure there has to be a better/faster way. Performance is absolutely crucial in my application.

One more thing: All numbers are integers (coordinates of points and elements of s and r vectors). Again, for performance reasons I would like to keep the floating-point math to a minimum.

Answer 1

You have to process every point at least once to know their distance. Unless you want to repeat the process many times with different lines, simply computing the distance of every point is unavoidable. So the algorithm has to be O(n).

Since you don't care about the actual distance, we can make some simplification to the point-distance computation. The exact distance is computed by ( source ):

d^2 = |r⨯(p-s)|^2 / |r|^2

where ⨯ is the cross product and |r|^2 is the squared length of vector r . Since |r|^2 is constant for all points, we can omit it from the distance computation without changing result:

d^2 = |r⨯(p-s)|^2

Compare the approximated square distances and keep the minimum. The advantage of this formula is that you can do everything with integers since you mentioned that all coordinates are integers.

Answer 2

I'm afraid you can't get away with computing less than 6 distances (if you could, at least one point would be left out -- including the nearest one).

See if it makes sense to preprocess: Is the line fixed and the points vary? Consider rotating coordinates to make the line horizontal.

As there are few points, it is doubtful that this is your bottleneck. Measure where the hot spots are, redesign algorithms/data representation, spice up compiler optimization, compile to assembly and bum that. Strictly in that order.

Jon Bentley's "Writing Efficient Programs" (sadly long out of print) and "Programming Pearls" (2nd edition) are full of advise on practical programming.