## Abstract

We study k-party number-in-hand set disjointness in the simultaneous message-passing model, and show that even if each element i ∈ [n] is guaranteed to either belong to all k parties or to at most O(1) parties in expectation (and to at most O(log n) parties with high probability), then Ω(n min(log 1/δ, log k)/k) communication is required by any δ-error communication protocol for this problem (assuming k = Ω(log n)). We use the strong promise of our lower bound, together with a recent characterization of turnstile streaming algorithms as linear sketches, to obtain new lower bounds for the well-studied problem in data streams of approximating the frequency moments. We obtain a space lower bound of Ω(n^{1−2/p}ε^{−2} logM log 1/δ) bits for any algorithm giving a (1 + ε)-approximation to the p-th moment (Formula Presented.) of an n-dimensional vector x ∈ {±M}^{n} with probability 1 − δ, for any (Formula Presented.). Our lower bound improves upon a prior Ω(n^{1−2/p}ε^{−2} log M) lower bound which did not capture the dependence on δ, and our bound is optimal whenever ε ≤ 1/poly(log n). This is the first example of a lower bound in data streams which uses a characterization in terms of linear sketches to obtain stronger lower bounds than obtainable via the one-way communication model; indeed, our set disjointness lower bound provably cannot hold in the one-way model.

Original language | American English |
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Title of host publication | Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings |

Editors | Magnus M. Halldorsson, Naoki Kobayashi, Bettina Speckmann, Kazuo Iwama |

Publisher | Springer Verlag |

Pages | 1082-1093 |

Number of pages | 12 |

ISBN (Print) | 9783662476710 |

DOIs | |

State | Published - 2015 |

Externally published | Yes |

Event | 42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 - Kyoto, Japan Duration: 6 Jul 2015 → 10 Jul 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9134 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 |
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Country/Territory | Japan |

City | Kyoto |

Period | 6/07/15 → 10/07/15 |

### Bibliographical note

Funding Information:O. Weinstein and D.P. Woodruff—Research supported by a Simons Fellowship in Theoretical Computer Science and NSF Award CCF-1215990.

Publisher Copyright:

© Springer-Verlag Berlin Heidelberg 2015.

## Keywords

- Frequency moments
- Information complexity
- Information theory
- Multiparty communication complexity