Faster Carry Bit Computation for Adder Circuits with Prescribed Arrival Times

Abstract: We consider the fundamental problem of constructing fast circuits for the carry bit computation in binary addition. Up to a small additive constant, the carry bit computation reduces to computing an \aop, i.e., a formula of type $t_0 \land (t_1 \lor (t_2 \land ( \dots t_{m-1}) \dots )$ or $t_0 \lor (t_1 \land (t_2 \lor ( \dots t_{m-1}) \dots )$. We present an algorithm that computes the fastest known Boolean circuit for an \aop~ with given arrival times $a(t_0), \dotsc, a(t_{m-1})$ for the input signals. Our objective function is delay, a natural generalization of depth with respect to arrival times. The maximum delay of the circuit we compute is $\log_2 W + \log_2 \log_2 m + \log_2 \log_2 \log_2 m + 4.3$, where $W := \sum_{i = 0}^{m-1} 2^{a(t_i)}$. Note that $\lceil \log_2 W \rceil$ is a lower bound on the delay of any circuit depending on inputs $t_0, \dotsc, t_{m-1}$ with prescribed arrival times. Our method yields the fastest circuits for \aop s, carry bit computation and adders in terms of delay known so far.