An Improved Overlap Argument for On-line Multiplication

Abstract

A lower bound of cN1ogN is proved for the mean time complexity of an on-line multitape Turing machine performing the multiplication of N-digit binary integers. For a more general class of machines the corresponding bound is cN1ogN. These bounds compare favorably with know upper bounds of the form cN(1ogN) k, and for some classes the upper and lower bounds coincide. The proofs are based on the "overlap" argument due to Cook and Aanderaa.