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Catalytic Tree Evaluation From Matching Vectors

Alexandra Henzinger, Edward Pyne, Seyoon Ragavan

TL;DR

The 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 the algorithm.

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 $2^{\log^εn}$ 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.

Catalytic Tree Evaluation From Matching Vectors

TL;DR

The 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 the algorithm.

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 and super-polynomial time . 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 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 free space, polynomial runtime, and subpolynomial catalytic space (for any ). 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.
Paper Structure (41 sections, 28 theorems, 19 equations)

This paper contains 41 sections, 28 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Implications.
  3. Overview
  4. The tree evaluation problem.
  5. Inspiration from cryptography.
  6. Connecting tree evaluation to private information retrieval.
  7. Which PIR scheme to use?
  8. Related Work
  9. Algorithms for Tree Evaluation.
  10. Time-Space Tradeoffs.
  11. Matching Vectors.
  12. Open Questions
  13. Preliminaries
  14. Notation.
  15. The Catalytic Space Model
  16. ...and 26 more sections

Key Result

Theorem 1.1

For every $\epsilon > 0$, $\mathsf{TreeEval}$ can be solved in $O(\log n)$ free space, $2^{O(\log^{\epsilon} n)}$ catalytic space, and $\mathsf{poly}(n)$ runtime. (See cor:eps for a parameterized statement.)

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4: Informal
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3: Representations of rings on the catalytic tape
  • Theorem 2.4: CM23
  • Theorem 2.5: BCKLS14
  • Remark 2.6
  • Lemma 2.7
  • ...and 48 more