Jeff Dean did not invent map/reduce
The correct name is really MapShuffleReduce and the most important step is the shuffle, because it's a distributed sort. Of course, they didn't invent distributed sort, but combining these three concepts together in a distributed fashion and running it as a production system is really what was important.
The mappers, shufflers, and reducers are all independent of each other, reading and writing from the filesystem, and managed by a coordinator. There's nothing like MPI, other than the use of the Stubby RPC system, which sort of resembles MPI but has completely different distributed communication semantics.