Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling

The Tree Evaluation Problem ($\mathsf{T}\mathsf{r}\mathsf{e}\mathsf{e}\mathsf{E}\mathsf{v}\mathsf{a}\mathsf{l}$) is a computational problem originally proposed as a candidate to prove a separation between complexity classes $\mathsf{P}$ and $\mathsf{L}$. Recently, this problem has gained significant attention after Cook and Mertz (STOC 2024) showed that $\mathsf{T}\mathsf{r}\mathsf{e}\mathsf{e}\mathsf{E}\mathsf{v}\mathsf{a}\mathsf{l}$ can be solved using $O(\log n\log \log n)$ bits of space. Their algorithm, despite getting very close to showing $\mathsf{T}\mathsf{r}\mathsf{e}\mathsf{e}\mathsf{E}\mathsf{v}\mathsf{a}\mathsf{l}\in \mathsf{L}$, falls short, and in particular, it does not run in polynomial time. In this work, we present the first polynomial-time, almost logarithmic-space algorithm for $\mathsf{T}\mathsf{r}\mathsf{e}\mathsf{e}\mathsf{E}\mathsf{v}\mathsf{a}\mathsf{l}$. For any $\epsilon >0$, our algorithm solves $\mathsf{T}\mathsf{r}\mathsf{e}\mathsf{e}\mathsf{E}\mathsf{v}\mathsf{a}\mathsf{l}$ in time $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n)$ while using $O({\log }^{1+\epsilon }n)$ space. Furthermore, our algorithm has the additional property that it requires only $O(\log n)$ bits of free space, and the rest can be catalytic space. Our approach is to trade off some (catalytic) space usage for a reduction in time complexity.