Theory of Computing
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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
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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.
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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.