### 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 Ω((δ/w)logn) time on average per output symbol, 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 bounds are in fact tight within the cell probe model.

Original language | English |
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Article number | 2 |

Pages (from-to) | 1-31 |

Number of pages | 31 |

Journal | Theory of Computing |

Volume | 15 |

DOIs | |

Publication status | Published - 7 Sep 2019 |

### Keywords

- Cell probe
- Lower bounds
- Online algorithms
- Streaming algorithms

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## Cite this

*Theory of Computing*,

*15*, 1-31. [2]. https://doi.org/10.4086/toc.2019.v015a002