2D range overlap (window query)

Question

[tuom.larsen]

Hello!

I'm in search for an algorithm/data structure which would enable efficient "window query" over a set of rectangles (2D). That is, given many (possibly overlapping) rectangles report all the ones overlapped by a given query rectangle. In 1D, this is known as interval intersection or range overlap. I found two very interesting posts here:

1. Fast Interval Intersection Methodologies
2. What Is The Quickest Algorithm For Range Overlap?

So I was wondering that perhaps someone could have an idea on how to approach this problem. I know about R-trees but perhaps there are some additional suitable data structures?

Thank you in advance!

[Devon Ryan]

You'll want to post this to stackoverflow, which has a more CS-centric community. For what it's worth, my go to structure would also be an R-tree (or variant thereof) for this sort of task.

[tuom.larsen]

Thanks a lot for the reply!

Unfortunately, I cannot think of a way how to make the "intersection" efficient, i.e. better than O(n). What I mean by that is consider the following example:

▢▢▢▢▢▢▢▢▢▢▢▢▢▢▢▢▢▢

and if the window overlaps just one of the rectangles I first would have to report everything along the "x" axis only to discard most of the result in the "y query".

Answer 1

[Sean Davis]

If you really have just rectangles, then perhaps two 1-D indices are all that is needed? An intersection with the rectangle will mean intersection with both the x-extent and the y-extent, each of which can be represented by a 1-D index. There are multiple implementations of very performant 1-D indexing in various languages.

Edit: as Devon points out, an R-tree is one of the best approaches to these types of problems. http://en.wikipedia.org/wiki/R-tree