Tight Cell-Probe Bounds for Online Integer Multiplication and Convolution

TL;DR

This paper establishes tight cell-probe bounds for online integer multiplication and convolution, showing optimal bounds on the number of memory probes needed per output in the online setting.

Contribution

It provides the first tight bounds for online multiplication and convolution in the cell-probe model, extending understanding of their computational complexity.

Findings

Theta((d/w)*log n) bounds on average per output digit for multiplication

Theta((d/w)*log n) bounds per new number in convolution

Bounds hold under randomisation and amortisation

Abstract

We show tight 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 Theta((d/w)*log n) bound on average per output digit for this problem where 2^d 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 a Theta((d/w)*log n) bound for the number of probes 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.