We give new algorithms for tree evaluation (S. Cook et al. TOCT 2012) in the catalytic-computing model (Buhrman et al. STOC 2014). Two existing approaches aim to solve tree evaluation (TreeEval) in low space: on the one hand, J. Cook and Mertz (STOC 2024) give an algorithm for TreeEval running in super-logarithmic space $O(\log n\log\log n)$ and super-polynomial time $n^{O(\log\log n)}$. On the other hand, a simple reduction from TreeEval to circuit evaluation, combined with the result of Buhrman et al. (STOC 2014), gives a catalytic algorithm for TreeEval running in logarithmic $O(\log n)$ free space and polynomial time, but with polynomial catalytic space. We show that the latter result can be improved. We give a catalytic algorithm for TreeEval with logarithmic $O(\log n)$ free space, polynomial runtime, and subpolynomial $2^{\log^εn}$ catalytic space (for any $ε> 0$). Our result gives the first natural problem known to be solvable with logarithmic free space and even $n^{1-ε}$ catalytic space, that is not known to be in standard logspace even under assumptions. Our result immediately implies an improved simulation of time by catalytic space, by the reduction of Williams (STOC 2025). Our catalytic TreeEval algorithm is inspired by a connection to matching vector families and private information retrieval, and improved constructions of (uniform) matching vector families would imply improvements to our algorithm.

翻译：我们针对催化计算模型（Buhrman等人，STOC 2014）中的树求值问题（S. Cook等人，TOCT 2012）提出了新算法。现有两种方法旨在以低空间复杂度求解树求值问题：一方面，J. Cook与Mertz（STOC 2024）提出了在超对数空间$O(\log n\log\log n)$和超多项式时间$n^{O(\log\log n)}$内运行的树求值算法；另一方面，通过将树求值问题归约至电路求值问题，并结合Buhrman等人（STOC 2014）的研究成果，可得到一种在自由空间$O(\log n)$与多项式时间内运行的催化树求值算法，但其催化空间需求为多项式级别。本文证明后一结果存在改进空间：我们提出了一种催化树求值算法，其自由空间为$O(\log n)$，运行时间为多项式级别，且催化空间需求为亚多项式级别$2^{\log^εn}$（对任意$ε>0$成立）。该成果首次揭示了一个可在对数自由空间乃至$n^{1-ε}$催化空间内求解的自然问题，且该问题即使在假设条件下也未被证明属于标准对数空间。通过Williams（STOC 2025）的归约方法，我们的结果可直接推演出催化空间对时间复杂度的改进模拟。本催化树求值算法的设计灵感来源于匹配向量族与私有信息检索的关联性研究，若（均匀）匹配向量族的构造技术获得改进，将直接推动本算法的性能提升。