Catalytic Tree Evaluation From Matching Vectors

2026-02-15

AI Summary

This work addresses the efficient solvability of the Tree Evaluation problem (TreeEval) in the catalytic computation model. By integrating matching vector families and private information retrieval techniques, and leveraging a reduction from trees to Boolean circuits, the authors devise a novel catalytic algorithm. This algorithm operates in polynomial time using only $O(\log n)$ work space while reducing the required catalytic space to subpolynomial size $2^{\log^\varepsilon n}$ for any $\varepsilon > 0$, substantially improving upon prior methods that necessitated polynomial catalytic space. Notably, TreeEval becomes the first natural problem known to be efficiently solvable with logarithmic work space and catalytic space below $n^{1-\varepsilon}$, despite not being known to reside in standard logspace, thereby advancing the understanding of trade-offs between time and catalytic space in computational simulations.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

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 (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 $2^{\log^\epsilon n}$ catalytic space (for any $\epsilon>0$). Our result gives the first natural problem known to be solvable with logarithmic free space and even $n^{1-\epsilon}$ 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.

Problem

• catalytic computing
• tree evaluation
• space complexity
• matching vector families
• logspace

Innovation

• catalytic computing
• tree evaluation
• matching vector families
• subpolynomial catalytic space
• logspace