License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.95
URN: urn:nbn:de:0030-drops-181472
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18147/
Oshman, Rotem ;
Roth, Tal
The Communication Complexity of Set Intersection Under Product Distributions
Abstract
We consider a multiparty setting where k parties have private inputs X_1,…,X_k ⊆ [n] and wish to compute the intersection ⋂_{? =1}^k X_? of their sets, using as little communication as possible. This task generalizes the well-known problem of set disjointness, where the parties are required only to determine whether the intersection is empty or not. In the worst-case, it is known that the communication complexity of finding the intersection is the same as that of solving set disjointness, regardless of the size of the intersection: the cost of both problems is Ω(n log k + k) bits in the shared blackboard model, and Ω (nk) bits in the coordinator model.
In this work we consider a realistic setting where the parties' inputs are independent of one another, that is, the input is drawn from a product distribution. We show that this makes finding the intersection significantly easier than in the worst-case: only Θ̃((n^{1-1/k} (H(S) + 1)^{1/k}) + k) bits of communication are required, where {H}(S) is the Shannon entropy of the intersection S. We also show that the parties do not need to know the exact underlying input distribution; if we are given in advance O(n^{1/k}) samples from the underlying distribution μ, we can learn enough about μ to allow us to compute the intersection of an input drawn from μ using expected communication Θ̃((n^{1-1/k}?[|S|]^{1/k}) + k), where |S| is the size of the intersection.
BibTeX - Entry
@InProceedings{oshman_et_al:LIPIcs.ICALP.2023.95,
author = {Oshman, Rotem and Roth, Tal},
title = {{The Communication Complexity of Set Intersection Under Product Distributions}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {95:1--95:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-278-5},
ISSN = {1868-8969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18147},
URN = {urn:nbn:de:0030-drops-181472},
doi = {10.4230/LIPIcs.ICALP.2023.95},
annote = {Keywords: Communication complexity, intersection, set disjointness}
}
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Keywords: |
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Communication complexity, intersection, set disjointness |
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Collection: |
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50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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05.07.2023 |
