There might be a really clever hash you could build, but it's actually a somewhat complex problem. Finding an overlapping range between two entities is easy... finding it for n-entities is a bit more difficult particularly if you want to do it efficiently. I can imagine a couple of answers.. you can build a lookup table (which makes sense if the time resolution is low enough) and then find gaps in it.

It's actually a pretty good interview question if your going to ask a whiteboard question. It has lots of answers depending on requirements. You can get into parallelizing it for speed. It's more of a practical problem than most and you can implement it basically any language.

Try having actual time ranges, not just slots, and you need to find, say, a 30m contiguous block of time.