What is the best way to turn a Map[A, Set[B]] into a Map[B, Set[A]]?

For example, how do I turn a

Map(1 -> Set("a", "b"),
    2 -> Set("b", "c"),
    3 -> Set("c", "d"))

into a

Map("a" -> Set(1),
    "b" -> Set(1, 2),
    "c" -> Set(2, 3),
    "d" -> Set(3))

(I'm using immutable collections only here. And my real problem has nothing to do with strings or integers. :)

scala collections dictionary
Edit
Report