Hardness of Approximate Nearest Neighbor Search under L-infinity
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We show conditional hardness of Approximate Nearest Neighbor Search (ANN) under the ℓ_∞ norm with two simple reductions. Our first reduction shows that hardness of a special case of the Shortest Vector Problem (SVP), which captures many provably hard instances of SVP, implies a lower bound for ANN with polynomial preprocessing time under the same norm. Combined with a recent quantitative hardness result on SVP under ℓ_∞ (Bennett et al., FOCS 2017), our reduction implies that finding a (1+ε)-approximate nearest neighbor under ℓ_∞ with polynomial preprocessing requires near-linear query time, unless the Strong Exponential Time Hypothesis (SETH) is false. This complements the results of Rubinstein (STOC 2018), who showed hardness of ANN under ℓ_1, ℓ_2, and edit distance. Further improving the approximation factor for hardness, we show that, assuming SETH, near-linear query time is required for any approximation factor less than 3 under ℓ_∞. This shows a conditional separation between ANN under the ℓ_1/ ℓ_2 norm and the ℓ_∞ norm since there are sublinear time algorithms achieving better than 3-approximation for the ℓ_1 and ℓ_2 norm. Lastly, we show that the approximation factor of 3 is a barrier for any naive gadget reduction from the Orthogonal Vectors problem.
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