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Given is a set S of n axis-aligned cubes. The task is to find the volume of the union of all cubes in S. This means that every volume-overlap of 2 or more cubes is only counted once. The set specifically contains all the coordinates of all the cubes.

I have found several papers regarding the subject, presenting algorithms to complete the task. This paper for example generalizes the problem to any dimension d rather than the trivial d=3, and to boxes rather than cubes. This paper as well as a few others solve the problem in time O(n^1.5) or slightly better. Another paper which looks promising and is specific to 3d-cubes is this one which solves the task in O(n^4/3 log n). But the papers seem rather complex, at least for me, and I cannot follow them clearly.

Is there any relatively simple pseudocode or procedure that I can follow to implement this idea? I am looking for a set of instructions, what exactly to do with the cubes. Any implementation will be also excellent. And O(n^2) or even O(n^3) are fine.

Currently, my approach was to compute the total volume, i.e the sum of all volumes of all the cubes, and then compute the overlap of every 2 cubes, and reduce it from the total volume. The problem is that every such overlap may (or may not) be of a different pair of cubes, meaning an overlap can be for example shared by 5 cubes. In this approach the overlap will be counted 10 times rather just once. So I was thinking maybe an inclusion-exclusion principle may prove itself useful, but I don't know exactly how it may be implemented specifically. Computing every overlap (naively) is already O(n^2), but doable. Is there any better way to compute all such overlaps? Anyways, I don't assume this is a useful approach, to continue along these lines.