Optimal-depth circuits for prefix computation and addition

Abstract

Addition and prefix computation are among the most fundamental problems in arithmetic and algebraic computation. In this paper, we present efficient circuits for performing prefix computation and addition with small depth and size and flexible fan-in (i.e., the maximum fan-in can be selected as a small constant or a larger constant/nonconstant number). In particular, we show that any prefix operation of n inputs can be computed using a circuit of fan-in k, depth log/sub k/n+o(log/sub k/n)+O(1), gate complexity O(n), and edge complexity O(n log/sup d-1**...*d-1/n), for any constant integer d. We show that the sum of two n-bit numbers can be found using an AND-OR circuit of fan-in k, depth log/sub k/n+o(log/sub k/n)+O(1), and edge complexity O(n(log/sup d-1**...*d-1/n)/sup 2/), for any constant integer d. In particular, the depths of our circuits for prefix computation and addition are optimal within a factor of 1+o(1), for any fan-in k=n/sup o(1)/.