## Abstract

In 2010, Pǎtraşcu proposed a dynamic set-disjointness problem, known as the Multiphase problem, as a candidate for proving polynomial lower bounds on the operational time of dynamic data structures. He conjectured that any data structure for the Multiphase problem must make n{epsilon} cell-probes in either update or query phases, and showed that this would imply similar unconditional lower bounds on many important dynamic data structure problems. There has been almost no progress on this conjecture in the past decade since its introduction. We show an tilde{Omega}(sqrt{n}) cell-probe lower bound on the Multiphase problem for data structures with general (adaptive) updates, and queries with unbounded but'layered' adaptivity. This result captures all known set-intersection data structures and significantly strengthens previous Multiphase lower bounds, which only captured non-adaptive data structures. Our main technical result is a communication lower bound on a 4-party variant of Pǎtraşcu's Number-On-Forehead Multiphase game, using information complexity techniques. We then use this result to make progress on understanding the power of nonlinear gates in networks computing linear operators, a long-standing open problem in circuit complexity and network design: We show that any depth-d circuit that computes a random m times n linear operator x mapsto Ax using gates of degree k (width-k DNFs) must have Omega(m cdot n{1/2(d+k)}) wires. Finally, we show that a lower bound on Pǎtraşcu's original NOF game would imply a polynomial wire lower bound (n{1+ Omega(1/d)}) for circuits with arbitrary gates computing a random linear operator. This suggests that the NOF conjecture is much stronger than its data structure counterpart.

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 | 752-761 |

Number of pages | 10 |

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

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