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

TL;DR

This paper establishes the minimal depth of optimal fanout-bounded prefix circuits, showing it scales logarithmically with input size based on a specific polynomial root, extending known results for certain fanout bounds.

Contribution

It generalizes the known bounds for minimal depth of prefix circuits to all fanout bounds by deriving a new formula involving the polynomial root lpha_k.

Findings

Minimal depth is log_{lpha_k} N  O(1) for fanout ounded circuits.

The bound extends previous results for fanout 2 and infinite fanout.

The depth depends on the unique positive root of a specific polynomial.

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 $k=\infty$.