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 ${\log_{α_k} N \pm O(1)}$, where $α_k$ 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 $k=\infty$.

翻译：本文证明了在扇出限制为$k$的情况下，具有$N$个输入的最优前缀电路（即零亏量电路）的最小深度为${\log_{α_k} N \pm O(1)}$，其中$α_k$是多项式${2+x+ x^2+\ldots + x^{k-2}-x^k}$的唯一正根。该界限在$k=2$和$k=\infty$的情形下已为已知结果。