We examine the fundamental problem of constructing depth-optimum circuits for binary addition. More precisely, as in literature, we consider the following problem: Given auxiliary inputs $t_0, \dotsc, t_{m-1}$, so-called generate and propagate signals, construct a depth-optimum circuit over the basis {AND2, OR2} computing all $n$ carry bits of an $n$-bit adder, where $m=2n-1$. In fact, carry bits are AND-OR paths, i.e., Boolean functions of the form $t_0 \lor ( t_1 \land (t_2 \lor ( \dots t_{m-1}) \dots ))$. Classical approaches construct so-called prefix circuits which do not achieve a competitive depth. For instance, the popular construction by Kogge and Stone is only a $2$-approximation. A lower bound on the depth of any prefix circuit is $1.44 \log_2 m$ + const, while recent non-prefix circuits have a depth of $\log_2 m$ + $\log_2 \log_2 m$ + const. However, it is unknown whether any of these polynomial-time approaches achieves the optimum depth for all $m$. We present a new exponential-time algorithm solving the problem optimally. The previously best exact algorithm with a running time of $\mathcal O(2.45^m)$ is viable only for $m \leq 29$. Our algorithm is significantly faster: We achieve a running time of $\mathcal O(2.02^m)$ and apply sophisticated pruning strategies to improve practical running times dramatically. This allows us to compute optimum circuits for all $m \leq 64$. Combining these computational results with new theoretical insights, we derive the optimum depths of $2^k$-bit adder circuits for all $k \leq 13$, previously known only for $k \leq 4$. In fact, we solve a more general problem occurring in VLSI design: $delay$ optimization of a $generalization$ of AND-OR paths where AND and OR do not necessarily alternate. Our algorithm arises from our new structure theorem which characterizes delay-optimum generalized AND-OR path circuits.

翻译：我们检查了为二进制构建深度- 优化电路的根本问题。 更准确地说, 正如在文献中, 我们考虑了以下问题: 在辅助投入$t_ 0,\dostsc, t\\m-1}$, 所谓的生成和传播信号, 在基础 {AND2, OR2} 建立深度- 优化电路, 计算所有美元都包含一个 美元- 比特添加器的位元。 事实上, 传输比特( 美元=2n-1美元 ) 和 或路径。 也就是说, 以美元( 美元) 的深度计算, 即: 美元- 美元( diot_ 0\ dosc, t_\\ land ( t_ 2\ hl) 创建并传播信号。 经典方法构建了所谓的普通电路路, 无法达到竞争深度。 例如, Kogge 和 Stone 的流行构造只是2美元- 和 方程式。 在任何新的电路的深度中, 以美元- 美元- 美元 美元 美元 美元- 美元- 美元- 运行算算算算算算算算算算算算算一个多少 。