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

Vahid R. Asadi, Richard Cleve

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.

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

Abstract

The Tree Evaluation Problem () is a computational problem originally proposed as a candidate to prove a separation between complexity classes and . Recently, this problem has gained significant attention after Cook and Mertz (STOC 2024) showed that can be solved using bits of space. Their algorithm, despite getting very close to showing , 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 . For any , our algorithm solves in time while using space. Furthermore, our algorithm has the additional property that it requires only 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.

Paper Structure

This paper contains 16 sections, 2 theorems, 10 equations, 4 figures.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. Overview of construction in terms of circuit diagrams
  4. Formal analysis in terms of algorithms
  5. Input:
  6. The recursive step (input node $u\in \{0,1\}^{\leq r}$):
  7. Input:
  8. The recursive step (input subtree ${t}$ where ${t}\in\{0,\dots,k-1\}^{\leq w}$):
  9. Analysis of the algorithm
  10. Proof of correctness.
  11. Runtime analysis.
  12. Space analysis.
  13. Conclusion
  14. Miscellaneous comments
  15. Low-degree extension.
  16. ...and 1 more sections

Key Result

Theorem 1

For any $\varepsilon > 0$, we have that $\mathsf{TreeEval}$ can be solved in time $n^{1/\varepsilon+O(1)}$, while using $O(\log n)$ bits of free space and $\widetilde{O}(\log^{1+\varepsilon} n)$ bits of catalytic space. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: A circuit corresponding to \ref{['lem:formula']}, illustrated for $p=5$. A simple pattern of gates is repeated $m = p-1$ times. The $\oplus$ and $\ominus$ symbols denote addition and subtraction (modulo $p$). For any initial values of $\tau_1,\tau_2,\dots,\tau_{m} \in \mathbb{Z}_p$, the first $m-1$ registers incur no net change and the output of the last register is $\tau_m + x_1x_2\cdots x_{m-1} \bmod p$. The gate labeled $\omega$, multiplies by $\omega_m$, a fixed primitive $m$-th root of unity in $\mathbb{Z}_p$. The gate acting on $m$ registers is an arithmetic generalization of a Toffoli gate, that maps each $(a_1,\dots,a_{m-1},a_{m})\in \mathbb{Z}_p^m$ to $(a_1,\dots,a_{m-1},a_{m}+a_1a_2\dots a_{m-1} \bmod p)$.
  • Figure 2: Condensed circuit notation, where the wires carry arrays consisting of $\ell$ registers, illustrated for $\ell=3$. The input to $F$ is $2\ell$ elements of $\mathbb{Z}_p$ and the output is $\ell$ elements of $\mathbb{Z}_p$.
  • Figure 3: Depiction of two levels of the Cook-Mertz recursive tree-evaluation algorithm to compute $G(X_0,X_1,X_2,X_3) = F(F_0(X_0,X_1),F_1(X_2,X_3))$. Each dashed-line box labelled "$m$ times" is shorthand for $m$ sequential repetitions of its contents.
  • Figure 4: Computing $G(X_0,X_1,X_2,X_3) = F(F_0(X_0,X_1),F_1(X_2,X_3))$ by the catalytic pebbling method (the $r=2$ case of a subtree of depth $r$). The first four (in general, $2^r$) registers store the catalytic pebbles.

Theorems & Definitions (5)

  • Theorem 1: Main Theorem
  • Definition 2: Tree evaluation problem
  • Remark 3
  • Lemma 4: CM23
  • proof : Proof of \ref{['thm:main']}