Minimum Depth, Linear Size, and Fan-Out Two

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 log2 n logic gates, size 4n log2 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 log2 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. In this paper we introduce the first family of adders with an asymptotically optimum depth of log2 n+ o(log2 n), linear size O(n), and a fan-out bound of two.