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Inverted index is designed to be fast for searching but not that easy for updating. So far as I know incremental index is used to update inverted index data.
So what are the algorithms or main concepts of real-time searching, how did they do that?
Real-time search can still use an inverted index. You have to maintain this index in a form that can be both efficiently updated and queried, of course, which doesn’t come free – but it’s certainly doable. Another common technique is to have multiple indexes – one master index (updated frequently), and some overlay (or patch) indices that only record things that have changed since the last master index.