Faster Linear-Size And-Or Path and Adder Circuits

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 $\mathcal O (m \log_2 m)$.