 Let $c$ be a finite closed path in a tessellation ${\mathcal{T}}$ of the plane $\mathbb  R^0$ into unit squares so that edges of $c$ are edges of ${\mathcal{T}}$. Then a minimal number $m_0(c)$ such that there is a finite sequence of [elementary homotopies]{} of type 0-0, which turns $c$ into a single point and which contains $m_0(c)$ [elementary homotopies]{} of type 0, can be computed in deterministic space $O((\log |c|)^0)$ or in deterministic time $O( |c|^0\log |c|)$, where $|c|$ is the length of $c$.  Furthermore, such a sequence of [elementary homotopies]{} of type 0-0, which turns $c$ into a single point and which contains $m_0(c)$ [elementary homotopies]{} of type 0, can also be computed in deterministic space $O((\log |c|)^0)$ or in deterministic time $O( |c|^0\log |c|)$