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arXiv · 2509.10188

Constant Time with Minimal Preprocessing, a Robust and Extensive Complexity Class

Étienne Grandjean·Louis Jachiet
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Abstract

In this paper, we study the class $\mathtt{cstPP}$ of operations $\mathtt{op}: \mathbb{N}^k\to\mathbb{N}$, of any fixed arity $k\ge 1$, satisfying the following property: for each fixed integer $d\ge 1$, there exists an algorithm for a RAM machine which, for any input integer $N\ge 2$, - pre-computes some tables in $O(N)$ time, - then reads $k$ operands $x_1,\ldots,x_k 1$, or conversely, is reduced to $N^{\varepsilon}$, for any positive $\varepsilon<1$ (provided the set of primitive operation includes $+$, $\mathtt{div}$ and $\mathtt{mod}$). To complete the picture, we demonstrate that the $\mathtt{cstPP}$ class degenerates if the preprocessing time reduces to $N^{o(1)}$.

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Étienne Grandjean, Louis Jachiet. 2025-09-12. Constant Time with Minimal Preprocessing, a Robust and Extensive Complexity Class. https://arxiv.org/abs/2509.10188

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