Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908945026646016 |
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| author | Asadi, Vahid R. Cleve, Richard |
| author_facet | Asadi, Vahid R. Cleve, Richard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02606 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling Asadi, Vahid R. Cleve, Richard Computational Complexity Data Structures and Algorithms 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. |
| title | Polynomial-Time Almost Log-Space Tree Evaluation by Catalytic Pebbling |
| topic | Computational Complexity Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.02606 |