具有渐近最小深度、线性尺寸和扇出2的二进制加法器电路
Binary Adder Circuits of Asymptotically Minimum Depth, Linear Size, and Fan-Out Two
摘要：
我们考虑构造快速和小型二进制加法器电路的问题。在广泛使用的加法器中，Kogge-Stone加法器通常被认为是最快的，因为它计算两个$n$位数（其中$n$是2的幂）的进位位，具有$2\log_2 n$逻辑门的深度，大小$4 n\log_2 n$，并且所有扇出都以2为界。避免了两个以上的扇出，因为它们导致插入中继器以重新为信号供电和物理实现中的额外深度。然而，Kogge-Stone加法器的深度界限偏离$\log_2 n$的下限两倍。这个界限是由Brent和Krapchenko在两个独立的构造中渐近实现的。Brent的构造既没有给出扇出的界限，也没有给出大小，而Krapchenko的加法器具有线性大小，但可以具有线性扇出。当扇出界限为2时，两种构造都不能达到小于$2 \log_2 n$的深度。在进一步的方法中，Brent和Kung提出了一种加法器，其具有线性大小和扇出两个，但深度是Kogge-Stone加法器的两倍。这些结果是33-43岁，并没有实质性的理论改进，自那时以来。在本文中，我们将所有以前的加法器电路的个别优势集成到一个新的全加器的家庭，第一次提高$2\log_2 n$的深度范围，同时保持扇出界的两个。我们的加法器实现了一个渐进的最佳逻辑门深度为$\log_2 n + o(\log_2 n)$和线性大小为$\mathcal {O}(n)$。
英文摘要：
We consider the problem of constructing fast and small binary adder circuits. Among widely-used adders, the Kogge-Stone adder is often considered the fastest, because it computes the carry bits for two $n$-bit numbers (where $n$ is a power of two) with a depth of $2\log_2 n$ logic gates, size $4 n\log_2 n$, and all fan-outs bounded by two. Fan-outs of more than two are avoided, because they lead to the insertion of repeaters for repowering the signal and additional depth in the physical implementation. However, the depth bound of the Kogge-Stone adder is off by a factor of two from the lower bound of $\log_2 n$. This bound is achieved asymptotically in two separate constructions by Brent and Krapchenko. Brent's construction gives neither a bound on the fan-out nor the size, while Krapchenko's adder has linear size, but can have up to linear fan-out. With a fan-out bound of two, neither construction achieves a depth of less than $2 \log_2 n$. In a further approach, Brent and Kung proposed an adder with linear size and fan-out two, but twice the depth of the Kogge-Stone adder. These results are 33-43 years old and no substantial theoretical improvement for has been made since then. In this paper we integrate the individual advantages of all previous adder circuits into a new family of full adders, the first to improve on the depth bound of $2\log_2 n$ while maintaining a fan-out bound of two. Our adders achieve an asymptotically optimum logic gate depth of $\log_2 n + o(\log_2 n)$ and linear size $\mathcal {O}(n)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
