Consider collections @math and @math of red and blue sets, respectively. Bichromatic Closest Pair is the problem of finding a pair from @math that has similarity higher than a given threshold according to some similarity measure. Our focus here is the classic Jaccard similarity @math for @math . We consider the approximate version of the problem where we are given thresholds @math and wish to return a pair from @math that has Jaccard similarity higher than @math if there exists a pair in @math with Jaccard similarity at least @math . The classic locality sensitive hashing (LSH) algorithm of Indyk and Motwani (STOC '98), instantiated with the MinHash LSH function of , solves this problem in @math time if @math . In particular, for @math , the approximation ratio @math increases polynomially in @math . 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 @math time for @math . Specifically, assuming OVC, we prove that for any @math there exists an @math such that Bichromatic Closest Pair with Jaccard similarity requires time @math for any choice of thresholds @math , that satisfy @math .
The proof of Theorem will be based on a result by Rubinstein @cite_3 : Assuming the Orthogonal Vectors Conjecture, a @math -approximation to Bichromatic Closest Pair with Hamming, Edit or Euclidean distance requires time @math . The required approximation factor @math depends on @math , and tends to 1 as @math tends to zero. We translate this into an equivalent conditional lower bound for Jaccard similarity for certain constants @math and @math .
@cite_3
2963964051
We prove conditional near-quadratic running time lower bounds for approximate Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance. Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false, for every δ>0 there exists a constant e>0 such that computing a (1+e)-approximation to the Bichromatic Closest Pair requires Ω(n2−δ) time. In particular, this implies a near-linear query time for Approximate Nearest Neighbor search with polynomial preprocessing time. Our reduction uses the recently introduced Distributed PCP framework, but obtains improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG codes have been constructed in other settings before, but our construction is the first to yield new hardness results.