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 2069278600 1430582609 1571696081
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. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. We show how to compute any symmetric Boolean function on n variables over any field (as well as the integers) with a probabilistic polynomial of degree O(a#x221A;n log(1 a#x03B5;))) and error at most a#x03B5;. The degree dependence on n and a#x03B5; is optimal, matching a lower bound of Razborov (1987) and Smolensky (1987) for the MAJORITY function. The proof is constructive: a low-degree polynomial can be efficiently sampled from the distribution. This polynomial construction is combined with other algebraic ideas to give the first sub quadratic time algorithm for computing a (worst-case) batch of Hamming distances in super logarithmic dimensions, exactly. To illustrate, let c(n): N -> N. Suppose we are given a database D of n vectors in 0, 1 (c(n) log n) and a collection of n query vectors Q in the same dimension. For all u in Q, we wish to compute a v in D with minimum Hamming distance from u. We solve this problem in n(2-1 O(c(n) log2 c(n))) randomized time. Hence, the problem is in "truly sub quadratic" time for O(log n) dimensions, and in sub quadratic time for d = o((log2 n) (log log n)2). We apply the algorithm to computing pairs with maximum inner product, closest pair in l1 for vectors with bounded integer entries, and pairs with maximum Jaccard coefficients. Guruswami and Sinop give a @math approximation guarantee for the non-uniform Sparsest Cut problem by solving @math -level Lasserre semidefinite constraints, provided that the generalized eigenvalues of the Laplacians of the cost and demand graphs satisfy a certain spectral condition, namely, @math . Their key idea is a rounding technique that first maps a vector-valued solution to @math using appropriately scaled projections onto Lasserre vectors. In this paper, we show that similar projections and analysis can be obtained using only @math triangle inequality constraints. This results in a @math approximation guarantee for the non-uniform Sparsest Cut problem by adding only @math triangle inequality constraints to the usual semidefinite program, provided that the same spectral condition, @math , holds.