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 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 opens a new line of attack on putting TreeEval in logspace, and 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）中的树评估问题（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解决树评估问题：一方面，J. Cook与Mertz（STOC 2024）提出了运行于超对数空间$O(\log n\log\log n)$和超多项式时间$n^{O(\log\log n)}$的TreeEval算法；另一方面，通过将TreeEval简化为电路评估并结合Buhrman等人（STOC 2014）的结果，可得到一种在自由空间为对数级$O(\log n)$、运行时间为多项式但催化空间为多项式级的催化TreeEval算法。我们证明了后一结果可进一步优化：我们提出了一种催化TreeEval算法，其具有对数级$O(\log n)$自由空间、多项式运行时间以及亚多项式级$2^{\log^εn}$催化空间（对任意$ε>0$）。该结果为将TreeEval纳入对数空间开辟了新路径，并通过Williams（STOC 2025）的归约直接改进了时间对催化空间的模拟效果。我们的催化TreeEval算法灵感来源于匹配向量族与私有信息检索的关联，而（均匀）匹配向量族的构造改进将直接推动本算法的优化。