查找边缘是否位于一组不相交的矩形内

Question

I'm using a triangulation library to compute the Constrained Delaunay Triangulation of a set of rectangles within some large boundary. The algorithm returns all the edges, but also adds edges inside of the rectangles that define the constraints. I want to be able to find if an edge lies inside of a rectangle in O(1) time.

Here's a more general description of the problem I want to solve. Given a set of nonoverlapping rectangles (the borders of the rectangles may touch) and an edge e with endpoints (x1,y1) and (x2, y2), find in O(1) time if e lies within any of the rectangles (including the border).

Also let me know of any data structures I can use for speedups! I'm also implementing this in java so I have easy access to hash sets, maps and all those nice data structures.

Answer 1

Since the rectangles are completely enclosed, the inside of each rectangle will simply be the CDT of the rectangle itself -- which is to say, two triangles, meeting along a diagonal of the rectangle. So you can just insert all rectangles' diagonals (remember, two possible diagonals per rectangle) into a hashtable, and check which edges exactly match those endpoints.

Answer 2

It is possible to break up the area that all of the rectangles cover into a N by M grid of boxes. By labeling each box with the rectangle it is in or the rectangles it overlaps. It is possible to obtain O(1) queries, with O(N*M) pre-processing.

However, in order for it to work, the grid has to created based on an algorithm that allows for calculating which box a point lies in in O(1). It also requires that the number of rectangles a box overlaps be very small (ideally no more than 2 or 3) as otherwise the average query time could be O(log N) or worst. This means that the number of boxes can get very large.