We consider the fundamental problem of constructing fast and small circuits for binary addition. We propose a new algorithm with running time $\mathcal O(n \log_2 n)$ for constructing linear-size $n$-bit adder circuits with a significantly better depth guarantee compared to previous approaches: Our circuits have a depth of at most $\log_2 n + \log_2 \log_2 n + \log_2 \log_2 \log_2 n + \text{const}$, improving upon the previously best circuits by [12] with a depth of at most $\log_2 n + 8 \sqrt{\log_2 n} + 6 \log_2 \log_2 n + \text{const}$. Hence, we decrease the gap to the lower bound of $\log_2 n + \log_2 \log_2 n + \text{const}$ by [5] significantly from $\mathcal O (\sqrt{\log_2 n})$ to $\mathcal O(\log_2 \log_2 \log_2 n)$. Our core routine is a new algorithm for the construction of a circuit for a single carry bit, or, more generally, for an And-Or path, i.e., a Boolean function of type $t_0 \lor ( t_1 \land (t_2 \lor ( \dots t_{m-1}) \dots ))$. We compute linear-size And-Or path circuits with a depth of at most $\log_2 m + \log_2 \log_2 m + 0.65$ in time $\mathcal O(m \log_2 m)$. These are the first And-Or path circuits known that, up to an additive constant, match the lower bound by [5] and at the same time have a linear size. The previously fastest And-Or path circuits are only by an additive constant worse in depth, but have a much higher size in the order of $\mathcal O (m \log_2 m)$.

翻译：我们考虑构造快速且紧凑的二进制加法电路这一基础性问题。我们提出了一种运行时间为$\mathcal O(n \log_2 n)$的新算法，用于构造线性规模的$n$位加法器电路，其深度性能相比先前方法有显著提升：我们的电路深度至多为$\log_2 n + \log_2 \log_2 n + \log_2 \log_2 \log_2 n + \text{const}$，优于文献[12]中深度至多为$\log_2 n + 8 \sqrt{\log_2 n} + 6 \log_2 \log_2 n + \text{const}$的最优电路。因此，我们将与文献[5]中$\log_2 n + \log_2 \log_2 n + \text{const}$下界的差距从$\mathcal O(\sqrt{\log_2 n})$显著减小至$\mathcal O(\log_2 \log_2 \log_2 n)$。我们的核心程序是一种用于构造单进位位电路（更一般地，用于与或路径电路）的新算法，其中与或路径是形如$t_0 \lor ( t_1 \land (t_2 \lor ( \dots t_{m-1}) \dots ))$的布尔函数。我们在时间$\mathcal O(m \log_2 m)$内计算出深度至多为$\log_2 m + \log_2 \log_2 m + 0.65$的线性规模与或路径电路。这是已知首个在加法常数范围内匹配文献[5]下界且同时保持线性规模的与或路径电路。此前最快的与或路径电路在深度上仅相差加法常数，但其规模高达$\mathcal O(m \log_2 m)$量级。