My 2c, the choice of algorithm is quite complex... It is data dependent and also depends on your read/write ratio, latency and accuracy needs, memory/cpu, and many other factors. If I were you I'd look for a versatile solution more than "the best" algorithm cos that choice will likely change... just sayin'...
+1 to sibling comment about not choosing the SOTA just cuz SOTA (if you're in the market for an ANN algo). Almost all the ones there are pretty darn fast these days and many (e.g. hnsw) are implemented out of the box by libraries like nmslib.
Also, most search use cases have requirements besides just the "get_k_nearest" functionality too (scoring-level filtering, business logic built into scoring unrelated to similarity, disparate query patterns that don't fit nicely into a single similarity space), so unless you only need to retrieve the K most-similar documents it might not be a good solution.
