 There is a deterministic algorithm that, for given word $W$ over the alphabet ${\mathcal{A}}^{\pm 0}$ such that $W \overset{{\mathcal{F}}({\mathcal{A}})}{=} 0$, where ${\mathcal{F}}({\mathcal{A}}) = \langle {\mathcal{A}}\  \|  \  \varnothing \rangle$ is the free group over ${\mathcal{A}}$, constructs a pattern of cancellations of letters in $W$ that result in the empty word and the algorithm operates in space $O( (\log |W|)^0 )$.  Furthermore, let ${\Delta}$ be a disk diagram over ${\mathcal{F}}({\mathcal{A}}) $ that corresponds to a pattern of cancellations of letters in $W$, i.e., ${\varphi}({\partial}{\Delta}) \equiv W$, 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 vertices in ${\Delta}$ of certain degree. Then, also in deterministic space $O( (\log |W|)^0 )$, one can algorithmically construct such a diagram ${\Delta}$ which is smallest relative to the tuple $\tau({\Delta})$.  