You'd use mutable state, either in the form of java collections, or atoms and family.
Even Haskell, where the State monad itself is actually immutable, provides an escape hatch into mutable state for when it's really, truly required. Just if you do use it when it's not, great shame shall be visited on you and your kin. By which I mean someone in #haskell will very politely point out how you could have done it purely.
Right, using a mutable structure is the easy way. But is there a way that would maintain the advantages of immutability? (eg memory-efficient and easy undo)
Easy undo can be achieved just as easily by reversing operations, in many cases. No need to keep two copies of a potentially large structure when you could just keep the diff and reverse apply it. Oddly, I think typically you would do both. Keep a few large snapshots with small diffs between stages. That is digressing, though.
As for memory-efficiency... not sure how keeping many potentially large copies is more efficient than just modifying a single copy.
Now, I can agree that in many cases it is nice that it prevents you from worrying about race conditions across threads. Though, I'm also not convinced that immutable things are any better than traditional locking strategies. Is that race truly won?
One doesn't need to keep large copies, since immutability allows common substructures to be shared. (That said, it still seems rather unlikely that an immutable implementation will use less memory.)
That's not a property of immutability. That's a property of certain specialized data structures, see "Purely Functional Data Structures" for a start, implemented in clojure.
Even Haskell, where the State monad itself is actually immutable, provides an escape hatch into mutable state for when it's really, truly required. Just if you do use it when it's not, great shame shall be visited on you and your kin. By which I mean someone in #haskell will very politely point out how you could have done it purely.