## Abstract

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of t ≥ ω(log^{2} n) on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space (s = (1 + ε)n), would already imply a semi-explicit (P^{NP}) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial (t ≥ n^{δ}) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime (s = n + o(n)), we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the “inner" and “outer" dimensions of a matrix (Paturi and Pudlák, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest.

Original language | American English |
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Title of host publication | STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Moses Charikar, Edith Cohen |

Publisher | Association for Computing Machinery |

Pages | 967-978 |

Number of pages | 12 |

ISBN (Electronic) | 9781450367059 |

DOIs | |

State | Published - 23 Jun 2019 |

Externally published | Yes |

Event | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States Duration: 23 Jun 2019 → 26 Jun 2019 |

### Publication series

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

### Conference

Conference | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 |
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Country/Territory | United States |

City | Phoenix |

Period | 23/06/19 → 26/06/19 |

### Bibliographical note

Publisher Copyright:© 2019 Association for Computing Machinery.

## Keywords

- Circuit lower bound
- Codes
- Data structures
- Rigidity