We introduce the notion of nested distance desert automata as a joint generalization of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in 22O(n) space whether the language accepted by an n-state non-deterministic automaton is of a star height less than a given integer h (concerning rational expressions with union, concatenation and iteration), which is the first ever complexity bound for the star height problem.
