Time Bounds for Streaming Problems

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 output 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 the each output must be computed before the next symbols arrives. • Next we show bounds for online multiplication where the stream consists of pairs of digits, one from each of two n digit numbers that are to be multiplied. One pair arrives at a time and the task is to output a single new digit from the product before the next pair of digits arrives. • Finally we look at the online Hamming distance problem where the Hamming distance is outputted instead of the inner product. For each of these three problems, we give a lower bound of Ω ( δ w log n ) time on average per output, where δ is the number of bits needed to represent an input symbol and w is the cell or word size. We argue that these bound are in fact tight within the cell probe model.