2026-07-21T19:38:40Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18147 2024-03-06T11:01:09Z ddc:004 open_access

The Communication Complexity of Set Intersection Under Product Distributions Oshman, Rotem Roth, Tal Communication complexity intersection set disjointness We consider a multiparty setting where k parties have private inputs X_1,…,X_k ⊆ [n] and wish to compute the intersection ⋂_{𝓁 =1}^k X_𝓁 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}𝔼[|S|]^{1/k}) + k), where |S| is the size of the intersection. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Rotem Oshman and Tal Roth 2023 Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2023.95 urn:nbn:de:0030-drops-181472 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.95 eng https://creativecommons.org/licenses/by/4.0/legalcode