Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling

TL;DR

This paper introduces the first polynomial-time algorithm for the Tree Evaluation Problem that uses nearly logarithmic space, leveraging catalytic space to optimize resource trade-offs.

Contribution

It presents a novel polynomial-time algorithm for TreeEval with almost log-space complexity, improving upon previous space bounds and introducing catalytic space techniques.

Findings

Achieves polynomial-time complexity for TreeEval.

Uses O(log^{1+ε} n) space for any ε>0.

Requires only O(log n) free space with additional catalytic space.

Abstract

The Tree Evaluation Problem ($\mathsf{TreeEval}$) is a computational problem originally proposed as a candidate to prove a separation between complexity classes $\mathsf{P}$ and $\mathsf{L}$. Recently, this problem has gained significant attention after Cook and Mertz (STOC 2024) showed that $\mathsf{TreeEval}$ can be solved using $O(\log n\log\log n)$ bits of space. Their algorithm, despite getting very close to showing $\mathsf{TreeEval} \in \mathsf{L}$, falls short, and in particular, it does not run in polynomial time. In this work, we present the first polynomial-time, almost logarithmic-space algorithm for $\mathsf{TreeEval}$. For any $\varepsilon>0$, our algorithm solves $\mathsf{TreeEval}$ in time $\mathrm{poly}(n)$ while using $O(\log^{1 +\varepsilon}n)$ space. Furthermore, our algorithm has the additional property that it requires only $O(\log n)$ bits of free space, and the rest can be catalytic space. Our approach is to trade off some (catalytic) space usage for a reduction in time complexity.