2026-07-23T08:48:11Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:11195 2024-03-06T10:47:41Z ddc:004 open_access

Hardness of Bichromatic Closest Pair with Jaccard Similarity Pagh, Rasmus Stausholm, Nina Mesing Thorup, Mikkel fine-grained complexity set similarity search bichromatic closest pair jaccard similarity Consider collections A and B of red and blue sets, respectively. Bichromatic Closest Pair is the problem of finding a pair from A x B that has similarity higher than a given threshold according to some similarity measure. Our focus here is the classic Jaccard similarity |a cap b|/|a cup b| for (a,b) in A x B. We consider the approximate version of the problem where we are given thresholds j_1 > j_2 and wish to return a pair from A x B that has Jaccard similarity higher than j_2 if there exists a pair in A x B with Jaccard similarity at least j_1. The classic locality sensitive hashing (LSH) algorithm of Indyk and Motwani (STOC '98), instantiated with the MinHash LSH function of Broder et al., solves this problem in Õ(n^(2-delta)) time if j_1 >= j_2^(1-delta). In particular, for delta=Omega(1), the approximation ratio j_1/j_2 = 1/j_2^delta increases polynomially in 1/j_2. In this paper we give a corresponding hardness result. Assuming the Orthogonal Vectors Conjecture (OVC), we show that there cannot be a general solution that solves the Bichromatic Closest Pair problem in O(n^(2-Omega(1))) time for j_1/j_2 = 1/j_2^o(1). Specifically, assuming OVC, we prove that for any delta>0 there exists an epsilon>0 such that Bichromatic Closest Pair with Jaccard similarity requires time Omega(n^(2-delta)) for any choice of thresholds j_2 < j_1 < 1-delta, that satisfy j_1 <= j_2^(1-epsilon). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Rasmus Pagh and Nina Mesing Stausholm and Mikkel Thorup 2019 Is Part Of LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2019.74 urn:nbn:de:0030-drops-111951 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.74 eng https://creativecommons.org/licenses/by/3.0/legalcode