## Abstract

This paper proves the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new approach and use it to prove a Ω (lg
^{1.5} n) lower bound on the operational time of a wide range of boolean data structure problems, most notably, on the query time of dynamic range counting over F
_{2} (Pǎtraşcu, 2007). Proving an (lg n) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Pǎtraşcu’s obituary (Thorup, 2013). This result also implies the first (lg n) lower bound for the classical 2D range counting problem, one of the most fundamental data structure problems in computational geometry and spatial databases. We derive similar lower bounds for boolean versions of dynamic polynomial evaluation and 2D rectangle stabbing, and for the (non-boolean) problems of range selection and range median. Our technical centerpiece is a new way of “weakly" simulating dynamic data structures using efficient one-way communication protocols with small advantage over random guessing. This simulation involves a surprising excursion to low-degree (Chebyshev) polynomials which May be of independent interest, and offers an entirely new algorithmic angle on the “cell sampling" method of Panigrahy et al. (2010).

Original language | English |
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Title of host publication | STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Monika Henzinger, David Kempe, Ilias Diakonikolas |

Number of pages | 12 |

Place of publication | New York, NY, USA |

Publisher | Association for Computing Machinery |

Publication date | 20 Jun 2018 |

Pages | 978-989 |

ISBN (Print) | 978-1-4503-5559-9 |

ISBN (Electronic) | 9781450355599 |

DOIs | |

Publication status | Published - 20 Jun 2018 |

Event | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018 |

### Conference

Conference | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 |
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Country/Territory | United States |

City | Los Angeles |

Period | 25/06/2018 → 29/06/2018 |

Sponsor | ACM Special Interest Group on Algorithms and Computation Theory (SIGACT) |

Series | A C M. S I G A C T - S I G M O D - S I G A R T Symposium on Principles of Database Systems. Proceedings |
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Number | 2018 |

ISSN | 1055-6338 |

## Keywords

- Cell probe complexity
- Data structures
- Dynamic problems
- Lower bounds
- Range searching
- CELL PROBE COMPLEXITY
- Dynamic Problems
- Data Structures
- Range Searching
- Cell Probe Complexity
- Lower Bounds