Problem
I have a case where I generate axis aligned rectangles that may or may not overlap each other - they often will. Ultimately, I need a set of rectangles that do not overlap, but cover (at least) the same regions. I'm looking for algorithms to accomplish this efficiently online (at run-time) in a real-time setting. I have some ideas, but they may be naive or brutal and I'd like to avoid reinventing the wheel if this is a well studied and resolved problem.
I'm having difficulty thinking of other applications or metaphors that might clue me in to a name. Is there a specific name for this kind of problem/algorithmic solution? Kind of like, say, the "four color theorem" in trying to color maps with no adjacent regions of the same color. "Four Color Theorem" being, in a sense, the algorithm with "coloring regions of a map such that no adjacent region share a color" being the problem instance.
Context
The specific application is to generate rectangular height fields for collision purposes around objects, but objects in close proximity will cause overlapping height fields, which will cause collision artifacts. Preserving existing rectangles and favoring those with a larger area is preferable to minimize the amount of memory that needs to be moved/copied into a new rectangle height field.
Example Images
