## Abstract

Proving super-logarithmic data structure lower bounds in the static group model has been a fundamental challenge in computational geometry since the early 80's. We prove a polynomial (n{Omega(1)}) lower bound for an explicit range counting problem of n{3} convex polygons in mathbb{R}{2} (each with n{tilde{O}(1)} facets/semialgebraic-complexity), against linear storage arithmetic data structures in the group model. Our construction and analysis are based on a combination of techniques in Diophantine approximation, pseudorandomness, and compressed sensing-in particular, on the existence and partial derandomization of optimal binary compressed sensing matrices in the polynomial sparsity regime (k=n{1-delta}). As a byproduct, this establishes a (logarithmic) separation between compressed sensing matrices and the stronger RIP property.

Original language | American English |
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Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |

Publisher | IEEE Computer Society |

Pages | 740-751 |

Number of pages | 12 |

ISBN (Electronic) | 9781728196213 |

DOIs | |

State | Published - Nov 2020 |

Externally published | Yes |

Event | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States Duration: 16 Nov 2020 → 19 Nov 2020 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2020-November |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
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Country/Territory | United States |

City | Virtual, Durham |

Period | 16/11/20 → 19/11/20 |

### Bibliographical note

Publisher Copyright:© 2020 IEEE.

## Keywords

- compressed sensing
- computational geometry
- data structures
- pseudorandomness