## Abstract

We revisit the k-mismatch problem in the streaming model on a pattern of length m and a streaming text of length n, both over a size-f alphabet. The current state-of-the-art algorithm for the streaming k-mismatch problem, by Clifford et al. [SODA 2019], uses∼O(k) space and∼O ôp k fworst-case time per character. The space complexity is known to be (unconditionally) optimal, and the worst-case time per character matches a conditional lower bound. However, there is a gap between the total time cost of the algorithm, which is∼O(n p k), and the fastest known offline algorithm, which costs∼O ô n + min ô pnk m,n time. Moreover, it is not known whether improvements over the∼O(n p k) total time are possible when using more than O(k) space. We address these gaps by designing a randomized streaming algorithm for the k-mismatch problem that, given an integer parameter k s m, uses∼O(s) space and costs∼O ô n + min ô nk2 m , pnk s nm s total time. For s = m, the total runtime becomes∼O ô n + min ô pnk m, fn f, which matches the time cost of the fastest offline algorithm. Moreover, the worst-case time cost per character is still∼O ôp k ff. 2012 ACM Subject Classification Theory of computation ! Pattern matching.

Original language | English |
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Title of host publication | 31st Annual Symposium on Combinatorial Pattern Matching, CPM 2020 |

Editors | Inge Li Gortz, Oren Weimann |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771498 |

DOIs | |

State | Published - 1 Jun 2020 |

Event | 31st Annual Symposium on Combinatorial Pattern Matching, CPM 2020 - Copenhagen, Denmark Duration: 17 Jun 2020 → 19 Jun 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 161 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st Annual Symposium on Combinatorial Pattern Matching, CPM 2020 |
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Country/Territory | Denmark |

City | Copenhagen |

Period | 17/06/20 → 19/06/20 |

### Bibliographical note

Publisher Copyright:© 2020 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

### Funding

Funding This work was supported in part by ISF grants no. 1278/16 and 1926/19, by a BSF grant no. 2018364, and by an ERC grant MPM under the EU’s Horizon 2020 Research and Innovation Programme (grant no. 683064).

Funders | Funder number |
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Horizon 2020 Framework Programme | 683064 |

European Commission | |

United States-Israel Binational Science Foundation | 2018364 |

Israel Science Foundation | 1926/19, 1278/16 |

## Keywords

- Hamming distance
- K-mismatch
- Streaming pattern matching