Lower Bounds for Online Integer Multiplication and Convolution in the Cell-Probe Model

Abstract

We show time lower bounds for both online integer multiplication and convolution in the cell-probe model with word size w. For the multiplication problem, one pair of digits, each from one of two n digit numbers that are to be multiplied, is given as input at step i. The online algorithm outputs a single new digit from the product of the numbers before step i + 1. We give a lower bound of $\Omega(\frac{\delta}{w} \log n)$ time on average per output digit for this problem where 2^δ is the maximum value of a digit. In the convolution problem, we are given a fixed vector V of length n and we consider a stream in which numbers arrive one at a time. We output the inner product of V and the vector that consists of the last n numbers of the stream. We show an $\Omega(\frac{\delta}{w}\log n)$ lower bound for the time required per new number in the stream. All the bounds presented hold under randomisation and amortisation. Multiplication and convolution are central problems in the study of algorithms which also have the widest range of practical applications.