## Abstract

We revisit a fundamental problem in string matching: given a pattern of length m and a text of length n, both over an alphabet of size σ, compute the Hamming distance (i.e., the number of mismatches) between the pattern and the text at every location. Several randomized (1+ϵ)-approximation algorithms have been proposed in the literature (e.g., by Karloff (Inf. Proc. Lett., 1993), Indyk (FOCS 1998), and Kopelowitz and Porat (SOSA 2018)), with running time of the form O(ϵ^{-O(1)}nlognlogm), all using fast Fourier transform (FFT). We describe a simple randomized (1+ϵ)-approximation algorithm that is faster and does not need FFT. Combining our approach with additional ideas leads to numerous new results (all Monte-Carlo randomized) in different settings: (1) We design the first truly linear-time approximation algorithm for constant ; the running time is O(ϵ^{-2}n). In fact, the time bound can be made slightly sublinear in n if the alphabet size σ is small (by using bit packing tricks). (2) We apply our approximation algorithms to design a faster exact algorithm computing all Hamming distances up to a threshold k; its runtime of O(n + min(nkglogm/gm,nk^{2}/m)) improves upon previous results by logarithmic factors and is linear for k≤ gm. (3) We alternatively design approximation algorithms with better ϵ-dependence, by using fast rectangular matrix multiplication. In fact, the time bound is O(n polylog n) when the pattern is sufficiently long, i.e., m≥ ϵ^{-c} for a specific constant c. Previous algorithms with the best ϵ-dependence require O(ϵ^{-1}n polylog n) time. (4) When k is not too small, we design a truly sublinear-time algorithm to find all locations with Hamming distance approximately (up to a constant factor) less than k, in time O((n/k^{ω(1)}+occ)n^{o(1)}) time, where occ is the output size. The algorithm leads to a property tester for pattern matching that costs O((^{-1/3}n^{2/3} + ^{-1}n/m) n) time and, with high probability, returns true if an exact match exists and false if the Hamming distance is more than δm at every location. (5) We design a streaming algorithm that approximately computes the Hamming distance for all locations with the distance approximately less than k, using O(ϵ^{-2}gk n) space. Previously, streaming algorithms were known for the exact problem with O(k n) space (which is tight up to the polylogn factor) or for the approximate problem with O(ϵ^{-O(1)}gmpolylogn) space. For the special case of k=m, we improve the space usage to O(ϵ^{-1.5}gmpolylogn).

Original language | English |
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Title of host publication | STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy |

Publisher | Association for Computing Machinery |

Pages | 643-656 |

Number of pages | 14 |

ISBN (Electronic) | 9781450369794 |

DOIs | |

State | Published - 8 Jun 2020 |

Event | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States Duration: 22 Jun 2020 → 26 Jun 2020 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 |
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Country/Territory | United States |

City | Chicago |

Period | 22/06/20 → 26/06/20 |

### Bibliographical note

Publisher Copyright:© 2020 ACM.

### Funding

∗This work was supported in part by ISF grants no. 1278/16 and 1926/19, by a BSF grant 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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Bloom's Syndrome Foundation | 2018364 |

Horizon 2020 Framework Programme | 683064 |

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

European Commission |

## Keywords

- Hamming distance
- Pattern matching
- Property testing
- Sampling
- Streaming
- Sublinear