 Both the diagram problem and the minimal diagram problem for group presentation can be solved in deterministic space $O((\log|W|)^0)$ or in deterministic time $O(|W|^0\log|W|)$.  Furthermore, let $W$ be a word such that $W \overset {{\mathcal{G}}_0} =0$ and let $$\tau({\Delta}) = (\tau_0({\Delta}), \ldots, \tau_{s_\tau}({\Delta}))$$ be a tuple of integers, where the absolute value $| \tau_i({\Delta}) |$ of each $\tau_i({\Delta})$ represents the number of certain vertices or faces in a disk diagram ${\Delta}$ over such that ${\varphi}({\partial}{\Delta}) \equiv W$. Then, in deterministic space $O( (\log |W|)^0 )$, one can algorithmically construct such a minimal diagram ${\Delta}$ which is also smallest relative to the tuple $\tau({\Delta})$ (the tuples are ordered lexicographically).  We 