4,215 research outputs found

    Hardness of Approximate Nearest Neighbor Search

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    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\delta>0 there exists a constant ϵ>0\epsilon>0 such that computing a (1+ϵ)(1+\epsilon)-approximation to the Bichromatic Closest Pair requires n2δn^{2-\delta} time. In particular, this implies a near-linear query time for Approximate Nearest Neighbor search with polynomial preprocessing time. Our reduction uses the Distributed PCP framework of [ARW'17], but obtains improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG codes have been constructed in other settings before [BKKMS'16, BCGRS'17], but our construction is the first to yield new hardness results

    Distributed PCP Theorems for Hardness of Approximation in P

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    We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment x{0,1}nx \in \{0,1\}^n to a CNF formula φ\varphi is shared between two parties, where Alice knows x1,,xn/2x_1, \dots, x_{n/2}, Bob knows xn/2+1,,xnx_{n/2+1},\dots,x_n, and both parties know φ\varphi. The goal is to have Alice and Bob jointly write a PCP that xx satisfies φ\varphi, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of xx. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of 2(logn)1o(1)2^{(\log n)^{1-o(1)}}; only (1+o(1))(1+o(1))-factor lower bounds (under SETH) were known before

    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

    SANNS: Scaling Up Secure Approximate k-Nearest Neighbors Search

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    The kk-Nearest Neighbor Search (kk-NNS) is the backbone of several cloud-based services such as recommender systems, face recognition, and database search on text and images. In these services, the client sends the query to the cloud server and receives the response in which case the query and response are revealed to the service provider. Such data disclosures are unacceptable in several scenarios due to the sensitivity of data and/or privacy laws. In this paper, we introduce SANNS, a system for secure kk-NNS that keeps client's query and the search result confidential. SANNS comprises two protocols: an optimized linear scan and a protocol based on a novel sublinear time clustering-based algorithm. We prove the security of both protocols in the standard semi-honest model. The protocols are built upon several state-of-the-art cryptographic primitives such as lattice-based additively homomorphic encryption, distributed oblivious RAM, and garbled circuits. We provide several contributions to each of these primitives which are applicable to other secure computation tasks. Both of our protocols rely on a new circuit for the approximate top-kk selection from nn numbers that is built from O(n+k2)O(n + k^2) comparators. We have implemented our proposed system and performed extensive experimental results on four datasets in two different computation environments, demonstrating more than 1831×18-31\times faster response time compared to optimally implemented protocols from the prior work. Moreover, SANNS is the first work that scales to the database of 10 million entries, pushing the limit by more than two orders of magnitude.Comment: 18 pages, to appear at USENIX Security Symposium 202
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