Theory of Computing ------------------- Title : Time Bounds for Streaming Problems Authors : Raphael Clifford, Markus Jalsenius, and Benjamin Sach Volume : 15 Number : 2 Pages : 1-31 URL : https://theoryofcomputing.org/articles/v015a002 Abstract -------- We give tight cell-probe bounds for the time to compute convolution, multiplication and Hamming distance in a stream. The cell probe model is a particularly strong computational model and subsumes, for example, the popular word RAM model. * We first consider online convolution where the task is to compute the inner product between a fixed $n$-dimensional vector and a vector of the $n$ most recent values from a stream. One symbol of the stream arrives at a time and then each output symbol must be computed before the next input symbol arrives. * Next we show bounds for online multiplication of two $n$-digit numbers where one of the multiplicands is known in advance and the other arrives one digit at a time, starting from the lower-order end. When a digit arrives, the task is to compute a single new digit from the product before the next digit arrives. * Finally we look at the online Hamming distance problem where the Hamming distance is computed instead of the inner product. For each of these three problems, we give a lower bound of $\Omega((\delta/w)\log n)$ time on average per output symbol, where $\delta$ is the number of bits needed to represent an input symbol and $w$ is the cell or word size. We argue that these bounds are in fact tight within the cell probe model. -------------- Preliminary versions of these results first appeared in the Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP), 2011, and the Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2013.