What kinds of indexing structures are used instead, and how do they differ from B+trees? Do you have examples of which relational databases have replaced B+tree indexes?
The property being optimized for, relative to B+trees, is extreme compactness of representation. In the pantheon of possible indexing algorithms, B+trees are pretty far on the bloated end of the spectrum in terms of the ratio between data space and index space. All indexes have a scaling performance cliff due to the index structure filling up and eventually overflowing available cache, crowding out the data and forcing page faults for almost every index write. In B+tree indexes this happens relatively early and often.
Radically improving index compactness is achieved by loosening design constraints on B+trees: the indexes represent a partial order which only converges on a total order at the limit and the search structure is unbalanced. In the abstract these appear slightly less efficient but it enables the use of selectivity-maximizing succinct representations of the key space that can get pretty close to the information theoretic limits. Scalability gains result from the radical reduction in cache footprint when represented this way.
Optimal compressive indexes are not computable (being equivalent to AI), so the efficient approximation strategies people come up with tend to be diverse, colorful, and sometimes impractical. Tangentially, some flavors have excellent write performance. It is not a trivial algorithm problem but there are a few design families that generalize well to real databases engines. I wouldn't describe this as a fully solved problem but many ordinary cases are covered.
There isn't much incentive to design a relational database engine that can use these types of indexes, since the types of workloads and data models that recommend them usually aren't relational. Someone could, there just isn't much incentive. It is more de rigueur for graph, spatiotemporal, and some types of analytical databases, where there is no other practical option if scalability matters at all.
It's definitely possible, and can make a lot of sense -- MyRocks/RocksDB for MySQL seems like an interesting and well designed system to me. It is fairly natural to compare MyRocks to InnoDB, since they're both MySQL storage engines. That kind of comparison is usually far more useful than an abstract comparison that ignores the practicalities of concurrency control and recovery.
The fact that MyRocks doesn't use B+Trees seems like half the story. Less than half, even. The really important difference between MyRocks and InnoDB is that MyRocks uses log-structured storage (one LSM tree for everything), while InnoDB uses a traditional write-ahead log with checkpoints, and with logical UNDO. There are multiple dimensions to optimize here, not just a single dimension. Focusing only on time/speed is much too reductive. In fact, Facebook themselves have said that they didn't set out to improve performance as such by adopting MyRocks. The actual goal was price/performance, particularly better write amplification and space amplification.
Databases designed to query large amounts of data tend to rely on partition key to allow parallel workloads (i.e. map reduce). And this partition data is stored like hash index.
