15,472 research outputs found

    Lower Bounds for Oblivious Near-Neighbor Search

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    We prove an Ω(dlgn/(lglgn)2)\Omega(d \lg n/ (\lg\lg n)^2) lower bound on the dynamic cell-probe complexity of statistically oblivious\mathit{oblivious} approximate-near-neighbor search (ANN\mathsf{ANN}) over the dd-dimensional Hamming cube. For the natural setting of d=Θ(logn)d = \Theta(\log n), our result implies an Ω~(lg2n)\tilde{\Omega}(\lg^2 n) lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for ANN\mathsf{ANN}. This is the first super-logarithmic unconditional\mathit{unconditional} lower bound for ANN\mathsf{ANN} against general (non black-box) data structures. We also show that any oblivious static\mathit{static} data structure for decomposable search problems (like ANN\mathsf{ANN}) can be obliviously dynamized with O(logn)O(\log n) overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page

    Tradeoffs for nearest neighbors on the sphere

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    We consider tradeoffs between the query and update complexities for the (approximate) nearest neighbor problem on the sphere, extending the recent spherical filters to sparse regimes and generalizing the scheme and analysis to account for different tradeoffs. In a nutshell, for the sparse regime the tradeoff between the query complexity nρqn^{\rho_q} and update complexity nρun^{\rho_u} for data sets of size nn is given by the following equation in terms of the approximation factor cc and the exponents ρq\rho_q and ρu\rho_u: c2ρq+(c21)ρu=2c21.c^2\sqrt{\rho_q}+(c^2-1)\sqrt{\rho_u}=\sqrt{2c^2-1}. For small c=1+ϵc=1+\epsilon, minimizing the time for updates leads to a linear space complexity at the cost of a query time complexity n14ϵ2n^{1-4\epsilon^2}. Balancing the query and update costs leads to optimal complexities n1/(2c21)n^{1/(2c^2-1)}, matching bounds from [Andoni-Razenshteyn, 2015] and [Dubiner, IEEE-TIT'10] and matching the asymptotic complexities of [Andoni-Razenshteyn, STOC'15] and [Andoni-Indyk-Laarhoven-Razenshteyn-Schmidt, NIPS'15]. A subpolynomial query time complexity no(1)n^{o(1)} can be achieved at the cost of a space complexity of the order n1/(4ϵ2)n^{1/(4\epsilon^2)}, matching the bound nΩ(1/ϵ2)n^{\Omega(1/\epsilon^2)} of [Andoni-Indyk-Patrascu, FOCS'06] and [Panigrahy-Talwar-Wieder, FOCS'10] and improving upon results of [Indyk-Motwani, STOC'98] and [Kushilevitz-Ostrovsky-Rabani, STOC'98]. For large cc, minimizing the update complexity results in a query complexity of n2/c2+O(1/c4)n^{2/c^2+O(1/c^4)}, improving upon the related exponent for large cc of [Kapralov, PODS'15] by a factor 22, and matching the bound nΩ(1/c2)n^{\Omega(1/c^2)} of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal complexities n1/(2c21)n^{1/(2c^2-1)}, while a minimum query time complexity can be achieved with update complexity n2/c2+O(1/c4)n^{2/c^2+O(1/c^4)}, improving upon the previous best exponents of Kapralov by a factor 22.Comment: 16 pages, 1 table, 2 figures. Mostly subsumed by arXiv:1608.03580 [cs.DS] (along with arXiv:1605.02701 [cs.DS]

    Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors

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    [See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the cc-Approximate Near Neighbor Search problem. For the dd-dimensional Euclidean space and nn-point datasets, we develop a data structure with space n1+ρu+o(1)+O(dn)n^{1 + \rho_u + o(1)} + O(dn) and query time nρq+o(1)+dno(1)n^{\rho_q + o(1)} + d n^{o(1)} for every ρu,ρq0\rho_u, \rho_q \geq 0 such that: \begin{equation} c^2 \sqrt{\rho_q} + (c^2 - 1) \sqrt{\rho_u} = \sqrt{2c^2 - 1}. \end{equation} This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor c>1c > 1, improving upon [Kapralov, PODS 2015]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni, Razenshteyn, STOC 2015]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole above trade-off in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower bounds for one and two probes that match the above trade-off for ρq=0\rho_q = 0, improving upon the best known lower bounds from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.Comment: 62 pages, 5 figures; a merger of arXiv:1511.07527 [cs.DS] and arXiv:1605.02701 [cs.DS], which subsumes both of the preprints. New version contains more elaborated proofs and fixed some typo

    Lower Bounds on Time-Space Trade-Offs for Approximate Near Neighbors

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    We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In articular, our lower bound matches the upper bound recently shown in [Laarhoven 2015] for the random instance on a Euclidean sphere (which we show in fact extends to the entire space Rd\mathbb{R}^d using the techniques from [Andoni, Razenshteyn 2015]). We also show tight, unconditional cell-probe lower bounds for one and two probes, improving upon the best known bounds from [Panigrahy, Talwar, Wieder 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than for one probe. To show the result for two probes, we establish and exploit a connection to locally-decodable codes.Comment: 47 pages, 2 figures; v2: substantially revised introduction, lots of small corrections; subsumed by arXiv:1608.03580 [cs.DS] (along with arXiv:1511.07527 [cs.DS]

    The random link approximation for the Euclidean traveling salesman problem

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    The traveling salesman problem (TSP) consists of finding the length of the shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where the cities are distributed randomly and independently in a d-dimensional unit hypercube. Working with periodic boundary conditions and inspired by a remarkable universality in the kth nearest neighbor distribution, we find for the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive analytical predictions for these quantities using the random link approximation, where the lengths between cities are taken as independent random variables. From the ``cavity'' equations developed by Krauth, Mezard and Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3, numerical results show that the random link approximation is a good one, with a discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we argue that the approximation is exact up to O(1/d^2) and give a conjecture for beta_E(d), in terms of a power series in 1/d, specifying both leading and subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte

    Probabilistic Polynomials and Hamming Nearest Neighbors

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    We show how to compute any symmetric Boolean function on nn variables over any field (as well as the integers) with a probabilistic polynomial of degree O(nlog(1/ϵ))O(\sqrt{n \log(1/\epsilon)}) and error at most ϵ\epsilon. The degree dependence on nn and ϵ\epsilon 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 subquadratic time algorithm for computing a (worst-case) batch of Hamming distances in superlogarithmic dimensions, exactly. To illustrate, let c(n):NNc(n) : \mathbb{N} \rightarrow \mathbb{N}. Suppose we are given a database DD of nn vectors in {0,1}c(n)logn\{0,1\}^{c(n) \log n} and a collection of nn query vectors QQ in the same dimension. For all uQu \in Q, we wish to compute a vDv \in D with minimum Hamming distance from uu. We solve this problem in n21/O(c(n)log2c(n))n^{2-1/O(c(n) \log^2 c(n))} randomized time. Hence, the problem is in "truly subquadratic" time for O(logn)O(\log n) dimensions, and in subquadratic time for d=o((log2n)/(loglogn)2)d = o((\log^2 n)/(\log \log n)^2). We apply the algorithm to computing pairs with maximum inner product, closest pair in 1\ell_1 for vectors with bounded integer entries, and pairs with maximum Jaccard coefficients.Comment: 16 pages. To appear in 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015
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