The Communication Complexity of Set Intersection Under Product Distributions

Rotem Oshman*, Tal Roth*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We consider a multiparty setting where k parties have private inputs X1, . . ., Xk ⊆ [n] and wish to compute the intersectionTkℓ=1 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((n1−1/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(n1/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((n1−1/k E[|S|]1/k) + k), where |S| is the size of the intersection.

Original languageEnglish
Title of host publication50th International Colloquium on Automata, Languages, and Programming, ICALP 2023
EditorsKousha Etessami, Uriel Feige, Gabriele Puppis
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772785
DOIs
StatePublished - Jul 2023
Event50th International Colloquium on Automata, Languages, and Programming, ICALP 2023 - Paderborn, Germany
Duration: 10 Jul 202314 Jul 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume261
ISSN (Print)1868-8969

Conference

Conference50th International Colloquium on Automata, Languages, and Programming, ICALP 2023
Country/TerritoryGermany
CityPaderborn
Period10/07/2314/07/23

Keywords

  • Communication complexity
  • intersection
  • set disjointness

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