Catalytic Tree Evaluation From Matching Vectors

TL;DR

This paper introduces a new catalytic algorithm for tree evaluation that operates in logarithmic free space and subpolynomial catalytic space, advancing the understanding of space complexity in the catalytic-computing model.

Contribution

It improves existing algorithms by achieving logarithmic free space and subpolynomial catalytic space for TreeEval, opening new avenues for space-efficient computation in the catalytic model.

Findings

Achieves logarithmic free space for TreeEval.

Uses subpolynomial catalytic space for the first time in this context.

Connects tree evaluation complexity to matching-vector families and private information retrieval.

Abstract

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 $2^{\log^\epsilon n}$ catalytic space (for any $\epsilon > 0$). 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.