ALT1 is the least modal logic containing the formula characterizing deterministic frames. The unification problem in ALT1 is to determine, given a formula A(x1,...,xn), whether there exists formulas B1, ..., Bn such that A(B1,...,Bn) is in ALT1. In this paper, we show that the unification problem in ALT1 is in PSPACE. We also show that there exists an ALT1-unifiable formula that has no minimal complete set of unifiers. Finally, we study sub-Boolean variants of the unification problem in ALT1.