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arXiv · 2512.23657

A note on the depth of optimal fanout-bounded prefix circuits

Igor S. Sergeev
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Abstract

It is shown that the minimal depth of an optimal prefix circuit (i.e., a zero-deficiency circuit) on $N$ inputs with fanout bounded by $k$ is ${\log_{\alpha_k} N \pm O(1)}$, where $\alpha_k$ is the unique positive root of the polynomial ${2+x+ x^2+\ldots + x^{k-2}-x^k}$. This bound was previously known in the cases $k=2$ and $k=\infty$.

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Igor S. Sergeev. 2025-12-29. A note on the depth of optimal fanout-bounded prefix circuits. https://arxiv.org/abs/2512.23657

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