## Abstract

The shift distance sh(S1, S2) between two strings S1 and S2 of the same length is defined as the minimum Hamming distance between S1 and any rotation (cyclic shift) of S2. We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings S1 and S2 of length n are given to two identical players (encoders), who independently compute sketches (summaries) sk(S1) and sk(S2), respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) sh(S1, S2) with high probability. This paper primarily focuses on the more general k-mismatch version of the problem, where the decoder is allowed to declare a failure if sh(S1, S2) > k, where k is a parameter known to all parties. Andoni et al. (STOC'13) introduced exact circular k-mismatch sketches of size Õ(k + D(n)), where D(n) is the number of divisors of n. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches. We circumvent this lower bound by designing a (non-linear) exact circular k-mismatch sketch of size Õ(k); this size matches communication-complexity lower bounds. We also design (1 ± ε)approximate circular k-mismatch sketch of size Õ(min(ε^{−}^{2 √}k, ε^{−}^{1.5 √}n)), which improves upon an Õ(ε^{−}^{2 √}n)-size sketch of Crouch and McGregor (APPROX'11).

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020 |

Editors | Jaroslaw Byrka, Raghu Meka |

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

ISBN (Electronic) | 9783959771641 |

DOIs | |

State | Published - 1 Aug 2020 |

Event | 23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020 - Virtual, Online, United States Duration: 17 Aug 2020 → 19 Aug 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 176 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 23rd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 24th International Conference on Randomization and Computation, APPROX/RANDOM 2020 |
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Country/Territory | United States |

City | Virtual, Online |

Period | 17/08/20 → 19/08/20 |

### Bibliographical note

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

## Keywords

- Communication complexity
- Cyclic shift
- Hamming distance
- K-mismatch
- Rotation
- Sketches