Hardness of Approximate Nearest Neighbor Search under L-infinity
aa r X i v : . [ c s . CC ] N ov Hardness of Approximate Nearest Neighbor Searchunder L-infinity
Young Kun Ko ∗ Min Jae Song † November 13, 2020
Abstract
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 specialcase of the Shortest Vector Problem (SVP), which captures many provably hard instances ofSVP, 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., FOCS2017), our reduction implies that finding a (1 + ε )-approximate nearest neighbor under ℓ ∞ withpolynomial preprocessing requires near-linear query time, unless the Strong Exponential TimeHypothesis (SETH) is false. This complements the results of Rubinstein (STOC 2018), whoshowed hardness of ANN under ℓ , ℓ , 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 showsa conditional separation between ANN under the ℓ /ℓ norm and the ℓ ∞ norm since there aresublinear time algorithms achieving better than 3-approximation for the ℓ and ℓ norm. Lastly,we show that the approximation factor of 3 is a barrier for any naive gadget reduction from theOrthogonal Vectors problem. ∗ Courant Institute of Mathematical Sciences, New York University. Email: [email protected] . † Courant Institute of Mathematical Sciences, New York University. Email: [email protected] . Researchsupported by the Simons Collaboration on Algorithms and Geometry and by the National Science Foundation (NSF)under Grant No. CCF-1814524. Introduction
Nearest Neighbor Search is formally defined as the following problem: Given a set of points P = { p , . . . , p N } ⊂ R d , and a query point t ∈ R d , find a point p ∈ P that is closest to t . This is afundamental algorithmic problem that has numerous applications in machine learning, computervision, and databases (See [SDI06] and references therein for applications).A naive solution is to enumerate all points in p ∈ P , compare the distance with t ∈ R d ,and return p that attains the minimum. This naturally gives a linear time ( O ( N ) where N = | P | )algorithm assuming one can compute the distance between two points in O (1)-time. Unfortunately,for a very large N , which is the usual setting, this is costly. Ideally, we would like to return thesolution without looking at every point in P . Another naive solution on the other extreme is toenumerate all possible query points t and, for each t , store its nearest neighbor in P in a look-uptable. That way, we can handle each query t using only O (1) query time. However, this wouldrequire almost exponential (in dimension d ) preprocessing time as one needs to enumerate throughall possible t and compute the corresponding nearest neighbor. Furthermore, this requires a hugespace overhead since one needs to store all the corresponding solutions for each t .A natural question is, can we obtain efficient preprocessing time (say polynomial in N and d )and efficient query time (sublinear in N ) simultaneously. When d = O (1), such algorithms areknown. If d = 1, for instance, one can construct a binary search tree with P and determine theclosest point to any query t ∈ R in O (log N ) time. For arbitrary d , the Voronoi diagram gives asolution using N O ( d ) space with ( d + log N ) O (1) query time, but it is not at all clear if this canbe further improved without paying an exponential price in d (this is typically called the curse ofdimensionality).Instead, if we are satisfied with an approximate nearest neighbor rather than an exact nearestneighbor, i.e., find ˜ p ∈ P such that d ( t, ˜ p ) ≤ γ · min p ∈ P d ( t, p ), we can do much better. We refer tosuch a problem as γ -approximate nearest neighbor search (or γ -ANN for short). Previous workshave shown that for many interesting metrics, we can indeed achieve improvements. For variousdistance metrics such as ℓ , ℓ , and edit distance, upper and lower bounds are well known. For ℓ (Hamming or Manhattan) and ℓ (Euclidean), one can use dimensionality reduction, localitysensitive hashing, or recently developed data-dependent hashing techniques [AR15] to improveupon the naive linear scan. Under these norms, current state-of-the-art techniques for worst-casedata achieve γ -approximation using O ( dN ρ ) preprocessing time and O ( dN ρ ) query time where ρ = γ − . Furthermore, these are known to be optimal for hashing based techniques [And+17a].We refer the reader to the survey by Andoni et al. [AIR18] for further information.Even if one uses non-hashing based techniques for these norms, one cannot hope to indefinitelyimprove the query time’s dependence on ε for (1 + ε )-approximation unless one refutes the StrongExponential Time Hypothesis (SETH). More precisely, for any δ >
0, there exists a constant ε = ε ( δ ) > ε )-ANN under ℓ , ℓ , and edit distance requires N − δ query time withpolynomial preprocessing time unless SETH is false [Rub18]. Yet, unlike ℓ or ℓ , ℓ ∞ remains amystery as we have not seen any improvement on either the upper or lower bound front for thepast decade [Ind01; ACP08; PTW10]. In Section 1.1, we elaborate on the current status of ANNunder ℓ ∞ . Moreover, we give a brief overview of the Shortest Vector Problem ( SVP ), a fundamentallattice problem central to post-quantum cryptography, and its connection to ANN.
Approximate nearest neighbors (ANN) in ℓ ∞ . Whether or not we can achieve a bettertime-space tradeoff for ℓ ∞ is not just a purely intellectual endeavor, but a practical one as well. For2nstance, if the coordinates are heterogeneous, adding up different coordinates may not make senseas it would be “comparing apples to oranges”. To circumvent this difficulty, one can convert eachcoordinate to a rank space and use the maximum rank difference as the distance measure [Fag96;Fag99]. Another motivation for studying ℓ ∞ comes from the fact that it can be used as a targetspace for embedding any general normed spaces. (e.g., one can embed any metric space into ℓ ∞ in O ( cN /c log N ) dimensions with 2 c − ℓ ∞ can be used as a black box to give efficient algorithms for other norms, as achieved in[And+17b]. Therefore, if one can design a better algorithm for ANN under ℓ ∞ , algorithms forother norms may see improvements as well.From a technical perspective, ℓ ∞ stands unique compared to other ℓ p norms. While therehas been fruitful algorithmic progress for other ℓ p norms by using various techniques such as di-mensionality reduction [JL84; Kle97], locality sensitive hashing [AI08] and data-dependent hash-ing [And+17a], ℓ ∞ remains resistant to these techniques. The seemingly unorthodox upper bounddevised by Indyk [Ind01] almost two decades ago remains state-of-the-art. This algorithm achieves O (log ρ log d )-approximation with O ( d poly log N ) query time with O ( dN ρ poly log N ) preprocessingtime for any ρ >
1. Surprisingly, this regime is known to be tight under some restricted computationmodel (decision tree model) [ACP08; PTW10; BK18].However, for a general computational model such as word-RAM, it is a major open problemwhether the tradeoff given by Indyk’s algorithm is optimal. Even if we allow a polynomial yetsublinear query time, it is open whether we can achieve a constant factor approximation withpolynomial preprocessing time (and no exponential dependence on log d ). Shortest vector problem (SVP).
SVP is a fundamental problem in lattice-based cryptography.For instance, the average-case hardness of learning with errors (LWE) [Reg09], which serves asthe basis of many post-quantum cryptography proposals [Ala+20], is based on the worst-casehardness of approximating
SVP up to polynomial factors in the ℓ norm. For finite ℓ p norms, NP-hardness of exact SVP p , the Shortest Vector Problem in the ℓ p norm, was shown by [Ajt98] usingrandomized reductions. This hardness result has been improved to hardness of constant factorapproximations [Mic01; Kho05], and 2 (log n ) − ε ( almost polynomial) factor approximations [Kho05;HR07] under the assumption that NP problems do not have randomized quasi-polynomial timealgorithms. Since approximating SVP up to factors larger than √ n is in NP ∩ coNP , it is unlikelythat NP-hardness can be shown for this regime [AR05]. For further information on SVP , we referthe reader to [MG02] and [Kho10].In this work, we consider the special case of finding short vectors in the ℓ ∞ norm, which wedenote by SVP ∞ . This problem is interesting because of its relevance to practical lattice-basedcryptosystems, in which the key size has to be as small as possible for efficiency while being largeenough to rule out attacks from any conceivable adversaries. For instance, the practical securityof Dilithium, a recent lattice-based signature scheme by [Duc+18], is based on the intractability ofsolving SVP ∞ . NP-hardness of exact SVP ∞ was shown by [Boa81], and [Din02] showed hardnessof approximation up to factor n c/ log log n . Recently, [BGSD17] proved that for any constant ε > γ ε ∈ (1 ,
2) such that
SVP ∞ does not have a 2 (1 − ε ) n time algorithm achieving γ ε -approximation assuming SETH. We note that the best known provable algorithm for γ -approximate SVP ∞ runs in time 3 n · (cid:16) γγ − (cid:17) n [AM18]. Here we remark that the only known technique for word-RAM lower bounds uses the cell-probe model [Yao79], inwhich computation is given for free and one only gets charged for information access. Proving any super-logarithmicquery time lower bound for the cell-probe model (under any polynomial preprocessing time) would lead to a break-through in circuit complexity [DGW19].
SVP has already been explored in previousworks [BJG15; Bec+16; Laa15], but the focus of these works was on speeding up heuristic sievingalgorithms for
SVP using nearest neighbor search. More precisely, LSH-based ANN algorithms havebeen used as a subprocedure in sieving algorithms to solve
SVP with better time-space tradeoffsunder heuristic assumptions. In this work, we turn this relationship around and prove a near-linearquery time lower bound for γ -ANN with polynomial preprocessing time using the aforementioned(conditional) lower bound on SVP ∞ [BGSD17]. We give two separate reductions that show conditional hardness of ANN with polynomial prepro-cessing time under the ℓ ∞ norm. Our result demonstrates a separation between the ℓ /ℓ normand the ℓ ∞ norm. That is, while algorithms with polynomial preprocessing and sublinear querytime for (1 + ε )-ANN, where ε > d = O (log N ), exist for the ℓ and ℓ norm [IM98; Val15; AIR18], our second reduction(Corollary 1.4) shows that we cannot expect to have sublinear algorithms for γ -ANN under the ℓ ∞ norm for any γ < γ - SVP { , } p , which is the Shortest Vector Problem under the ℓ p norm restricted to lattice vectorswith { , } -coefficients in the given basis (See Section 2 for a formal definition), into a hardnessresult (Corollary 1.2) for γ -ANN under the ℓ p norm. While this reduction leads to a weaker hard-ness result compared to Corollary 1.4, it establishes a simple connection between the two canonicalproblems which could potentially lead to fine-grained hardness results that are not based on SETH(See Section 1.3). We remark that an analogous reduction can be shown for a special case of theClosest Vector Problem ( CVP ), which is the problem of computing the distance from some targetpoint to the lattice (See Remark 3.4).
Theorem 1.1 (Informal) . For any ε > , γ > and ≤ p ≤ ∞ , if there is no algorithm whichsolves γ - SVP { , } p in (1 − ε ) n time, where n is the rank of the given lattice, then there exists a δ = δ ( ε ) > such that γ -Approximate Nearest Neighbor under the ℓ p norm cannot be solved withpolynomial preprocessing and N − δ query time. The key observation that connects this rather generic translation between
SVP { , } and ANNto a quantitative hardness result on ANN under the ℓ ∞ norm is that k -SAT reduces to this specialsubcase of SVP ∞ (with a small gap, in fact). The quantitative hardness of SVP { , }∞ based onSETH [BGSD17] (See Theorem 2.8) implies the hardness of ANN under the ℓ ∞ norm. Corollary 1.2 (Informal) . Assuming SETH, for any δ > , there exists a constant γ ∈ (1 , suchthat γ -ANN under the ℓ ∞ norm cannot be solved with polynomial preprocessing and N − δ querytime. One caveat of Corollary 1.2 is that the precise relationship between the approximation factorand query time lower bound is not known (See Remark 2.9). Moreover, the approximation factorfor which hardness can be shown is less than 2. A natural question is whether hardness canbe shown for larger approximation factors. Further strengthening the hardness result based onSETH, we show that a modification of the reduction from online partial matching by Indyk [Ind01]demonstrates hardness with a larger gap of γ = 3 for Bichromatic Closest Pair under ℓ ∞ . Theorem 1.3 (Informal) . Assuming SETH, for any δ > and γ < , there exists a constant c > such that γ -approximate Bichromatic Closest Pair on instances with dimension d = c log N underthe ℓ ∞ norm cannot be solved in N − δ time. γ -approximate Bichromatic Closest Pair implies a lower bound for γ -ANN with polynomial preprocessing time [WW18], Theorem 1.3 implies the following corollary. Corollary 1.4 (Informal) . Assuming SETH, for any δ > and γ < , there exists a constant c > such that γ -ANN on instances with dimension d = c log N under the ℓ ∞ norm cannot besolved with polynomial preprocessing and N − δ query time. Furthermore, we show that any naive gadget reduction from Orthogonal Vectors (Definition 2.2),which essentially captures most known reduction techniques from SETH, to γ -approximate Bichro-matic Closest Pair cannot show hardness for any approximation factor larger than 3 (Theorem 3.9).This essentially follows from the triangle inequality. We note that this limitation was mentionedin [Rub18] as well. Hence, showing any hardness with an approximation factor larger than 3 wouldrequire a novel technique bypassing the naive gadget reduction techniques. Therefore, our hardnessresult for ANN under ℓ ∞ is tight in this sense.We note that the result of [Rub18], who showed hardness under the ℓ norm, already impliesthe hardness of ANN under the ℓ ∞ norm since one can embed ℓ into a subspace of ℓ ∞ with lowdistortion [Ind03]. This reduction is similar to that of [RR06], who showed that ℓ is, in a certainsense, the “easiest norm” for lattice problems. However, hardness under the ℓ ∞ norm through thisreduction only holds for d = poly log N (as opposed to d = O (log N ) in our results) because ofthe embedding’s dimension blow-up. Moreover, our reductions are simpler and do not require thedistributed PCP [ARW17] machinery used in [Rub18]. Our Corollary 1.4 is also stronger in thesense that it holds for larger (and known) approximation factors. The approximation factor forhardness in [Rub18] is exponentially small in δ and depends on an unspecified relation between thereduction parameters due to SETH. Our result shows that the hardness of ANN under ℓ ∞ for approximation factor 3 is the best onecan hope for with current reduction techniques. The difficulty of proving hardness beyond 3-approximation was hinted in [Rub18], but no norm which explicitly instantiates this barrier wasknown previously.An obvious next direction is determining whether hardness of γ -ANN under ℓ ∞ for γ > γ -ANN and γ - SVP { , } suggests a direction for overcoming this technical barrier. Insteadof basing hardness on SETH, consider the following conjecture on SVP { , }∞ . Conjecture 1.5.
For any ε ∈ (0 , / , there exists γ = ω (1) such that there is no (1 − ε ) n timealgorithm for γ - SVP { , }∞ . It is straightforward to see that Lemma 3.1 and 3.2 (formal version of Theorem 1.1) togetherwith Conjecture 1.5 would imply hardness of ω (1)-ANN under ℓ ∞ . We remark that almost allinstances arising in NP-hardness proofs of γ - SVP ∞ are indeed captured by SVP { , }∞ [MG02; Din02;BGSD17]. One potential direction is to prove Conjecture 1.5 assuming SETH. However, this would thenovercome the barrier of 3-approximation for ANN, suggesting that such a result would be difficult to [Din02] generates a lattice in which coefficients of the short lattice vector take values in {− , , } instead of { , } . Still, there are only 3 n many candidates for the short vector, so our reduction applies to this case as well withminor modifications. SVP { , }∞ is itself a natural problem which encapsulates thedifficulty of canonical problems such as k -SAT and Subset Sum. Additional motivation for thisdirection is given by [BGSD17; Agg+21], who remark that the concrete lower bound for general SVP ∞ may in fact be 3 n since the kissing number in the ℓ ∞ norm is 3 n −
1. Since it is not atall clear whether a 3 n lower bound for SVP ∞ can be deduced from SETH, it may make sense todirectly assume the quantitative hardness of SVP ∞ or SVP { , }∞ . Acknowledgments.
We thank Noah Stephens-Davidowitz and Oded Regev for helpful com-ments.
We present two reductions that show hardness of γ -ANN. Our first reduction reduces a special caseof the Shortest Vector Problem to γ -ANN. Our second reduction reduces the Subset Query problemto γ -ANN. Both reductions involve an intermediate problem, referred to as the Bichromatic ClosestPair problem in [Rub18]. In this section, we present formal definitions of these problems and theirlower bounds assuming SETH. Definition 2.1 (Strong Exponential Time Hypothesis [IP99]) . For any ε > , there exists k = k ( ε ) such that k -SAT on n variables cannot be solved in (1 − ε ) n time. Orthogonal Vectors (OV) is an intermediate problem commonly used to show fine-grained hard-ness results in P [Wil05; Wil07]. Note that SETH implies the Orthogonal Vectors Conjecture, whichstates that Orthogonal Vectors cannot be solved in subquadratic time. For recent developmentsin fine-grained complexity assuming the Orthogonal Vectors Conjecture, we refer the reader to thesurvey [Wil17] and references therein. Definition 2.2 (Orthogonal Vectors) . Given two sets
A, B ⊂ { , } d , decide whether there existsa pair ( a, b ) ∈ A × B such that h a, b i = 0 . Conjecture 2.3 (Orthogonal Vectors Conjecture) . For every δ > , there exists a constant c = c ( δ ) such that given two sets A, B ⊂ { , } d each of cardinality N , deciding if there is a pair ( a, b ) ∈ A × B such that h a, b i = 0 cannot be solved in O ( N − δ ) time on instances with d = c log N . Definition 2.4 (Approximate Nearest Neighbor Search) . The γ -Approximate Nearest NeighborSearch problem ( γ -ANN) under the ℓ p norm is defined as follows. Let A ⊂ R d be a set of N vectors. Given numbers γ > , r > , and a vector x ∈ R d , distinguish between the following twocases after preprocessing A : • (YES) There exists a ∈ A such that with k a − x k p ≤ r . • (NO) For all a ∈ A , k a − x k p ≥ γr . Definition 2.5 (Approximate Bichromatic Closest Pair) . The γ -approximate Bichromatic ClosestPair problem under the ℓ p norm is defined as follows. Let A, B ⊂ R d each be a set of size N . Givennumbers γ > and r > , distinguish between the following two cases: (YES) There exists a ∈ A and b ∈ B with k a − b k p ≤ r . • (NO) For all a ∈ A and b ∈ B , k a − b k p ≥ γr . A standard technique [WW18] shows that any γ -ANN algorithm with sublinear query time andpolynomial preprocessing time implies a subquadratic algorithm for γ -approximate BichromaticClosest Pair (See Lemma 3.1). A lattice L is defined as the set of all integer combinations of linearly independent basis vectors b , . . . , b n ∈ R d , i.e., L = L ( b , . . . , b n ) = ( n X i =1 α i b i | α i ∈ Z ) . We denote the rank of L by n and the ambient dimension by d . Definition 2.6 (Shortest Vector Problem) . The γ -approximate Shortest Vector Problem under the ℓ p norm ( γ - SVP p ) is defined as follows. Let L be a d -dimensional lattice of rank n given in the formof a basis B = ( b , . . . , b n ) , where B ⊂ R d . Given numbers γ > and r > , distinguish betweenthe following two cases: • (YES) There exists a non-zero vector ~α ∈ Z n such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 α i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ r . • (NO) For any non-zero ~α ∈ Z n , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 α i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≥ γr . Definition 2.7 (Shortest Vector Problem with Binary Restriction) . The γ -approximate ShortestVector Problem with Binary Restriction under the ℓ p norm ( γ - SVP { , } p ) is defined as follows. Let L be a d -dimensional lattice of rank n given in the form of a basis B = ( b , . . . , b n ) , where B ⊂ R d .Given numbers γ > and r > , distinguish between the following two cases: • (YES) There exists a non-zero vector ~α ∈ { , } n such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 α i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ r . • (NO) For any non-zero ~α ∈ Z n , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 α i b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≥ γr . Theorem 2.8 ([BGSD17, Corollary 6.7]) . Assuming SETH, for any ε > , there exists γ = γ ( ε ) such that γ - SVP { , }∞ cannot be solved in (1 − ε ) n time, where n is the rank of the given lattice. Remark 2.9.
The precise form of the approximation factor γ in Theorem 2.8 is γ = 1 + 2 / ( k − ,where k = k ( ε ) comes from the hard k - SAT instance from SETH. Since the dependence k = k ( ε ) isnot known, so is the dependence of γ on ε . .4 Subset Query Problems Definition 2.10 (Subset Query) . The Subset Query problem is defined as follows. Given N sets S , . . . , S N ⊂ [ d ] and a query Q ⊂ [ d ] , distinguish between the following two cases: • (YES) There exists i ∈ [ N ] such that Q ⊆ S i . • (NO) For all i ∈ [ N ] , Q S i . Definition 2.11 (Bichromatic Subset Query) . The Bichromatic Subset Query problem is definedas follows. Given two collections of sets S , . . . , S N ⊂ [ d ] and T , . . . , T N ⊂ [ d ] , distinguish betweenthe following two cases: • (YES) There exists i, j ∈ [ N ] such that T i ⊆ S j . • (NO) For all i, j ∈ [ N ] , T i S j . It is well-known that Bichromatic Subset Query is SETH-hard (as it is equivalent to OrthogonalVectors) due to the following celebrated result of Williams [Wil05].
Theorem 2.12 ([Wil05, Theorem 5.1]) . Assuming SETH, for any δ > , there exists a constant c = c ( δ ) such that Bichromatic Subset Query cannot be solved in N − δ time on instances with d = c log N . We give two simple reductions that imply the hardness of γ -ANN assuming SETH. To this end,we first present a reduction from Bichromatic Closest Pair to ANN. Then, we reduce SVP { , }∞ andBichromatic Subset Query to Bichromatic Closest Pair, thereby showing the conditional quantita-tive hardness of γ -ANN. Furthermore, we show that the hard approximation factor of 3, which ourreduction from Bichromatic Subset Query achieves, is the best possible for any “natural” reductionfrom the Orthogonal Vectors problem (See Theorem 3.9). Since most known SETH-based hard-ness results are obtained via a reduction from Orthogonal Vectors, our result implies that showinghardness for approximation factor larger than 3 would require new techniques. A standard reduction from [WW18] shows that a sublinear algorithm for γ -ANN with polynomialpreprocessing time implies a subquadratic algorithm for γ -approximate Bichromatic Closest Pair.We include a short proof of this fact for completeness. Lemma 3.1.
Given numbers δ > and C > , if there exists an algorithm for γ -ANN under the ℓ p norm with preprocessing time N C and query time N − δ , where N denotes the number of datapoints, then there exists a N − Ω ( δC − ) -time algorithm for γ -approximate Bichromatic Closest Pairunder ℓ p .Proof. Consider the following algorithm for γ -approximate Bichromatic Closest Pair under the ℓ p norm for two sets A = { p , . . . , p N } ⊂ R d and B = { q , . . . , q N } ⊂ R d . • Divide A into N/ℓ batches, each of size ℓ . • Preprocess each batch separately, constructing
N/ℓ many data structures. • For each query point q j , query each data structure created by the preprocessing step.8he total time used for preprocessing is Nℓ · ( ℓ ) C = N · ℓ C − . For each query q j , one needs to query N/ℓ data structures. Hence, the total query time spent on q , . . . , q N is N · Nℓ · ℓ − δ = N ℓ δ . Let δ ′ > δ ′ − δ ′ < δC − and choose ℓ such that N δ ′ /δ < ℓ < N − δ ′ C − . This gives a O ( N − δ ′ )-time algorithm for γ -approximate Bichromatic Closest Pair. SVP { , } p Hardness implies Hardness of ANN under ℓ p In this section, we show that a fast algorithm for γ -approximate Bichromatic Closest Pair under ℓ p implies a fast algorithm for γ - SVP { , } p . Combined with Lemma 3.1, this shows that a lowerbound for γ - SVP { , } p implies a lower bound for γ -ANN under ℓ p . Assuming SETH, lower boundsfor γ - SVP { , }∞ are known (Theorem 2.8). Hence, our reduction implies hardness of γ -ANN underthe ℓ ∞ norm. Lemma 3.2.
If there exists a f ( N ) -time algorithm that solves γ -approximate Bichromatic ClosestPair under the ℓ p norm (where ≤ p ≤ ∞ ), then there exists an algorithm for γ - SVP { , } p withthe same ambient dimension that runs in O (cid:0) f (2 n/ ) (cid:1) time, where n denotes the rank of the inputlattice.Proof. Consider an instance of γ - SVP { , } p with a lattice basis B ⊂ R d of size n . Divide B into B and B , each of size n/
2, and consider the following sets of vectors. A := (X i α i b i | b i ∈ B , ~α ∈ { , } n/ ) A := (X i α i b i | b i ∈ B , ~α ∈ { , } n/ , ~α = ~ ) B := ( − X i α i b i | b i ∈ B , ~α ∈ { , } n/ ) B := ( − X i α i b i | b i ∈ B , ~α ∈ { , } n/ , ~α = ~ ) . Let A be an f ( N )-time algorithm for γ -approximate Bichromatic Closest Pair. We run A twotimes with the following inputs, A ( A , B ), A ( A , B ) and return the OR of the outputs. Since | A | , | B | ≤ | A | , | B | ≤ N = 2 n/ , then the runtime is O ( f ( N )) = O ( f (2 n/ )). It remains to showthe correctness of our reduction. Claim 3.3 (Correctness) . Let L { , } = { P ni =1 α i b i | b i ∈ B , ~α ∈ { , } n } and A − B = { a − b | a ∈ A, b ∈ B } . Then ( A − B ) ∪ ( A − B ) = L { , } \ { ~ } . roof. First, notice that A − B = L { , } . It follows that A − B = L { , } \ A since B = B \ { ~ } and B = B ∪ B is a basis. Similarly, A − B = L { , } \ B . Hence,( A − B ) ∪ ( A − B ) = ( L { , } \ A ) ∪ ( L { , } \ B )= L { , } \ ( A ∩ B )= L { , } \ { ~ } . From Claim 3.3, we know that if there exists a vector in A − B or A − B with ℓ p -norm less than r , then there exists a vector in L { , } with ℓ p -norm less than r . If all vectors in ( A − B ) ∪ ( A − B )have ℓ p -norm greater than γr , then all non-zero vectors in L { , } must have ℓ p -norm greater than γr which concludes our proof. Remark 3.4.
A similar reduction can be shown for γ - CVP { , } p , which is the CVP analogue of γ - SVP { , } p . To see this, take B ′ = B + t in the proof of Lemma 3.2, where t ∈ R d is a target pointwhich is guaranteed to be close to a { , } -coefficient lattice vector. Then, run the BichromaticClosest Pair algorithm on the input ( A , B ′ ) . Combining Lemma 3.1 and Lemma 3.2, we obtain our main theorem which connects hardnessof γ - SVP { , } and hardness of γ -ANN. Theorem 3.5.
For any δ > and C > , if there is an algorithm for γ -ANN under the ℓ p normwith N C preprocessing time and N − δ query time, then there is a − Ω ( δC − )) n -time algorithm for γ - SVP { , } p with the same ambient dimension, where n denotes the rank of the input lattice.Proof. By Lemma 3.1, we get a N − δ ′ -time algorithm for γ -approximate Bichromatic Closest Pairwhere δ ′ = Ω (cid:16) δC − (cid:17) from the algorithm for γ -ANN. By Lemma 3.2, this in turn gives a 2( − δ ) n -timealgorithm for γ - SVP { , } p .Note that the SETH-hard instance of γ - SVP { , }∞ in Theorem 2.8 has ambient dimension d = O ( n ) by the Sparsification Lemma [IPZ01]. Hence, we obtain the following lower bound for γ -ANNunder ℓ ∞ . Corollary 3.6.
Assuming SETH, for any δ > and C > , there exists a constant γ = γ ( δ, C ) ∈ (1 , such that γ -ANN under ℓ ∞ (where d = O (log N ) ) cannot be solved with N C preprocessingand N − δ query time. ℓ ∞ In this section, we show that solving γ -ANN for γ < γ -approximate Bichromatic Closest Pair under ℓ ∞ is as hardas Bichromatic Subset Query for any γ < Lemma 3.7.
For any γ < , if there exists a f ( N ) -time algorithm for γ -approximate BichromaticClosest Pair under the ℓ ∞ norm, then there exists a f ( N ) + O ( dN ) -time algorithm for BichromaticSubset Query. roof. Define functions f and g as f ( x ) = ( x = 0 if x = 1 g ( x ) = ( if x = 01 if x = 1For a set S ⊂ [ d ], consider its corresponding vector χ S ∈ { , } d where χ S,j = 1 iff j ∈ S . Let F : { , } d → { , } d and G : { , } d → { , } d be defined as F ( T ) = ( f ( χ T, ) , . . . , f ( χ T,d )) G ( S ) = ( g ( χ S, ) , . . . , g ( χ S,d )) . Define D ( S, T ) := k G ( S ) − F ( T ) k ∞ . If T ⊆ S , then D ( S, T ) = , and D ( S, T ) = 1 other-wise. Then, an instance of Bichromatic Subset Query, S , . . . , S N ⊂ [ d ] and T , . . . , T N ⊂ [ d ],can be reduced to a γ -approximate Bichromatic Closest Pair instance with G ( S ) , . . . , G ( S N ) and F ( T ) , . . . , F ( T N ) for γ <
3. If there exists a pair G ( S i ) , F ( T j ) with D ( S i , T j ) ≤ / T j ⊆ S i .If no such pair exists, we know that for any i, j ∈ [ N ], T j S i .Combining Lemma 3.1, Lemma 3.7 and Theorem 2.12, we get the following theorem. Theorem 3.8.
Assuming SETH, for any δ > , C > , and γ < , there exists c = c ( δ, C ) suchthat γ -ANN under the ℓ ∞ norm cannot be solved with N C preprocessing and N − δ query time oninstances with d = c log N .Proof. Suppose there exists some δ > , C >
1, and γ < γ -ANN under the ℓ ∞ norm with N C preprocessing time and N − δ query time for all d = O (log N ).Then, by Lemma 3.1, there is a N − δ ′ -time algorithm for γ -approximate Bichromatic Closest Pairunder ℓ ∞ with δ ′ = Ω (cid:16) δC − (cid:17) that works for all d = O (log N ). By Lemma 3.7, such an algorithmfor Bichromatic Closest Pair implies a O ( N − δ ′ )-time algorithm for Bichromatic Subset Queryapplying to all d = O (log N ). Such an algorithm for Bichromatic Subset Query cannot exist unlessSETH is false (Theorem 2.12). We show that the hard approximation factor of 3 in Theorem 3.8 is, in a certain sense, tight.More precisely, we show that any “natural” reduction from Orthogonal Vectors (which is how mostSETH-based hardness results are obtained) to γ -ANN under any metric cannot show hardness for γ > Theorem 3.9.
Let d > be a positive integer and ( M , D ) be a metric space. Let F : { , } d → M and G : { , } d → M be embeddings such that for any two strings S, T ∈ { , } d , • If DISJ ( S, T ) = 1 , then D ( F ( S ) , G ( T )) ≤ r . • If DISJ ( S, T ) = 0 , then D ( F ( S ) , G ( T )) ≥ γr .Then, it must be the case that γ ≤ . roof. For the sake of contradiction, suppose that there exists such F and G with γ >
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