On the Complexity of Closest Pair via Polar-Pair of Point-Sets

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $\Delta$ is called the sphericity of $G$, and if the collection of spheres are non-overlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the geometric representation of a complete bipartite graph $K_{n,n}$ in various $L^p$ metrics and connect it to the complexity of the closest pair problem. We prove the following results. 1. We show that contact-dimension and sphericity of $K_{n,n}$ in the $L^0$-metric is $n$, and that the contact-dimension and sphericity of $K_{n,n}$ in $L^p$-metrics with $p>2$ is $\Theta(\log n)$. 2. We give alternate proofs for the lower bound on the sphericity and the upper bound on the contact-dimension of $K_{n,n}$ in the $L^2$-metric using spectral analysis. Our upper and lower bounds are slightly weaker than that in [Frankl&Maehara, DCG'88] and [Maehara, DCG'91], respectively. However, our proofs are arguably simpler than previous works including [Bilu&Linial, JCTB'05], which also uses spectral analysis. Using the above results, we are able to show quadratic-time hardness for the closest pair problem in the $L^p$-metric with $d=\omega(\log n)$ and $p>2$ under the Orthogonal Vectors Hypothesis. The hardness result is obtained by a reduction from the bichromatic variant of the closest pair problem in the Hamming metric whose hardness was recently established in [Alman&Williams, FOCS'15]. In addition, we prove subquadratic-time hardness for the closest pair (and thus bichromatic closest pair) problem in the $L^\infty$ metric, which does not follow from [Alman&Williams, FOCS'15].


Introduction
This paper studies the geometric representation of a bipartite complete graph in L p -metrics and connects it to the complexity of the closest pair problem. Given a point-set P in a d-dimensional L p -metric, an α-distance graph is a graph G = (V, E) with a vertex set V = P and an edge set E = {ab : a − b p ≤ α; a, b ∈ P ; a = b}.
In other words, points in P are centers of spheres of radius α, and G has an edge ab if and only if the spheres centered at a and b intersect. The sphericity of a graph G in an L p -metric, denoted by sph p (G), is the smallest dimension d such that G is isomorphic to some α-distance graph in a d-dimensional L p -metric, for some constant α > 0. The sphericity of a graph in the L ∞ -metric is known as cubicity. A notion closely related to sphericity is contact-dimension, which is defined in the same manner except that the spheres representing G must be non-overlapping. To be precise, an α-contact graph G = (V, E) of a point-set P is an α-distance graph of P such that every edge ab of G has the same distance (i.e., a − b p = α). Thus, G has the vertex set V = P and has an edge set E such that ∀ab ∈ E, a − b p = α and ∀ab ∈ E, a − b p > α.
The contact-dimension of a graph G in the L p -metric, denoted by cd p (G), is the smallest dimension such that G is isomorphic to a contact-graph in the d-dimensional L p -metric. We will use distance and contact graphs to means 1-distance and 1-contact graphs.
We are interested in determining the sphericity and the contact-dimension of the biclique K n,n in various L p -metrics. For notational convenience, we denote sph p (K n,n ) by bsph(∆), the biclique sphericity of a metric ∆, and denote cd p (K n,n ) by bcd(∆), the biclique contact-dimension of ∆. We call a pair of point-sets (A, B) polar if it is the partition of a contact graph isomorphic to K n,n . More precisely, a pair of point-sets (A, B) is polar in an L p -metric if there exists a constant α > 0 such that every inner-pair a, a ′ ∈ A (resp., b, b ′ ∈ B) has L p -distance greater than α while every crossing-pair a ∈ A, b ∈ B has L p -distance exactly α.
Next, we connect the biclique contact-dimension to the complexity of the closest pair problem (Closest Pair), which asks to find a closest pair of points from a given collection P of m points in a d-dimensional metric space. Specifically, we tie the complexity of the Closest Pair to its bichromatic variant, namely, the bichromatic closest pair problem (BCP), in which a point in the collection P is colored by either red or blue, and the goal is to find a closest pair of red-blue points. Alman and Williams [AW15] showed that BCP admits no subquadratic-time algorithm for high dimensions under the Orthogonal Vectors Hypothesis. Hence, it is a good candidate to base the complexity of Closest Pair on.
It is not hard to see that BCP is at least as hard as Closest Pair since we can apply an algorithm for BCP to solve Closest Pair with the same asymptotic running time. However, it is not clear whether the other direction is true. We will give a simple reduction from BCP to Closest Pair using a polar-pair of point-sets. First, take a polar-pair (A, B), each with cardinality n = m/2, in the L p -metric. Next, pair up vectors in A and B to red and blue points, respectively, and then attach a vertex a ∈ A (resp., b ∈ B) to its matching red (resp., blue) point. This reduction increases the distances between every pair of points, but by the definition of the polar-pair, this process has more affect on the distances of the monochromatic (i.e., red-red or blue-blue) pairs than that of bichromatic pairs, and the reduction, in fact, has no effect on the order of crossing-pair distances at all. Consequently, by scaling the vectors in A and B appropriately, the reduction gives an instance of Closest Pair whose closest pair of points must be a bichromatic pair. Thus, we have that BCP and Closest Pair in the L p metric are computationally equivalent for dimensions d = Ω(bcd(L p )).
Nevertheless, the hardness of BCP based on the Orthogonal Vectors Hypothesis need not be carried over to Closest Pair as the polar-pair (A, B) might have large dimension. Thus, we need an efficient construction of a polar-pair with n o(1) -dimension to show that BCP and Closest Pair in the considered L p -metric are equivalent in the quadratic-time complexity.

Our Results and Contributions
Our main results are lower and upper bounds on the biclique-sphericity for the following metric spaces: L 0 ,L 1 ,L 2 and L p with p > 2.
Theorem 1. The following are upper and lower bounds of biclique-sphericity for the L p metric.
Note that bsph(∆) ≤ bcd(∆) for any metric ∆. Thus, it suffices to prove a lower bound for bsph(∆) and prove an upper bound for bcd(∆). In (1), (2), (3), and (5), we give bounds on sphericity and contact dimension for the respective metrics. For the L 2 -metric, we give alternate proofs for linear lower and upper bounds for bsph(L 2 ) and bcd(L 2 ) respectively. While our lower bounds are slightly weaker than the best known bounds [FM88,Mae91], our proofs are much shorter (the original proofs take more than ten pages while our proofs only take a couple of pages). Our proofs use spectral analysis similar to that in [BL05], but most arguments require no heavy machinery and thus are arguably simpler than the previous works [FM88,Mae91,BL05]. Additionally, while our upper bound in (4) is weaker than the one in [Mae85], our construction yields a more structured embedding: every pair of points not corresponding to an edge are also equidistant.
Next, we establish the following hardness result for Closest Pair under the Orthogonal Vectors Hypothesis (OVH) [Wil05] (a weaker assumption than the Strong Exponential Time Hypothesis  [IP01, IPZ01, CIP09]).
Corollary 2. For the L p metric with p > 2 and the L ∞ metric, there is a linear-time reduction from the bichromatic closest pair problem in the d-dimensional L p -metric (resp., L ∞ -metric) to the closest pair problem in the (d + O(log n))-dimension L p -metric (resp., L ∞ -metric).
Recently, Alman and Williams [AW15] showed that, for any ε > 0, p ≥ 0 and d = ω(log n), BCP in the d-dimensional L p -metric admits no (n 2−ε )-time algorithm unless the OVH is false. Thus, Corollary 2 implies the subquadratic-time hardness of Closest Pair in L p -metric with p > 2 in our case.
Corollary 3. For any ε > 0, p > 2 and d = ω(log n), the closest pair problem in the d-dimensional L p -metric admits no (n 2−ε )-time algorithm unless the Orthogonal Vectors Hypothesis is false.
We remark that the hardness of BCP (and thus Closest Pair) in the L ∞ -metric does not follow from [AW15]. Nonetheless, we prove the quadratic-time hardness of Closest Pair directly in Theorem 4.
Theorem 4. For any ε > 0 and d = ω(log n), the closest pair problem in the d-dimensional L ∞ -metric admits no (n 2−ε )-time algorithm unless the Orthogonal Vectors Hypothesis is false.
Additionally, for independent interest, we prove the quadratic-time hardness for a generalization of Closest Pair, namely, the set closest pair problem (Set Closest Pair) in which we are given a collection of point-sets (instead of points), and the goal is to find a closest pair of point-sets (two sets that minimizes the set-distances). Note that the distance between two sets U and V is the minimum distance over all pairs u ∈ U and v ∈ V , i.e., the distance of the closest pair between the sets. This computation seems to be inherent in many problems of machine learning, and in similarity searches in particular. Thus, the hardness result for Set Closest Pair might be of interest to the machine learning community. To be precise, we prove the following hardness result: Theorem 5. For any ε > 0, p > 1, and d = ω(log n), the set closest pair problem in the ddimensional L p -metric admits no (n 2−ε )-time algorithm unless the Orthogonal Vectors Hypothesis is false.
We remark that unlike Closest Pair, we could prove conditional hardness result for Set Closest Pair in the Euclidean metric.

Related Works
While our paper studies sphericity and contact-dimension of the complete bipartite graph, determining the contact-dimension of a complete graph in L p -metrics has also been extensively studied in the notion of equilateral dimension. To be precise, the equilateral dimension of a metric ∆ which is the maximum number of equidistant points that can be packed in ∆. An interesting connection is in the case of the L 1 -metric, for which we are unable to establish a strong lower bound for bsph(L 1 ). The equilateral dimension of L 1 is known to be at least 2d, and this bound is believed to be tight [Guy83]. This is a notorious open problem known as Kusner's conjecture, which is confirmed for d = 2, 3, 4 [BCL98, KLS00], and the best upper bound for d ≥ 5 is O(d log d) by Alon and Pavel [AP03]. If Kusner's conjecture is true for all d, then sph 1 (K n ) = n/2.
The complexity of Closest Pair has been a subject of study for many decades. There have been a series of developments on Closest Pair in the Euclidean space (see, e.g., [Ben80, HNS88, KM95, SH75, BS76]), which culminates in a deterministic O(2 O(d) n log n)-time algorithm [BS76] and a randomized O(2 O(d) n)-time algorithm [Rab76,KM95]. For low (i.e., constant) dimensions, these algorithms are tight as the matching lower bound of Ω(n log n) was shown by Ben-Or [Ben83] and Yao [Yao91] for the algebraic decision tree model, thus settling the complexity of Closest Pair in low dimensions. For high dimension (i.e., d = ω(log n)), there is a huge gap between the upper and lower bounds. There is no known algorithm that runs in time significantly better than a trivial O(n 2 d)-time algorithm for general d except for the case that d ≥ Ω(n), wherein there are subcubic algorithms in L 1 and L ∞ metrics [GS16,ILLP04]. Nevertheless, the best known lower bound for Closest Pair was Ω(max{n log n, nd}) [Ben83,Yao91] (note that nd is the input size). There were no better lower bounds for any L p -metrics prior to our work.

Organization
This paper is organized as follows. In Section 3, we prove lower and upper bounds on bsph(L 2 ) and bcd(L 2 ) (Eq. 4 in Theorem 1). In Section 4, we discuss the case of the L 1 -metric. Although we are unable to prove either a strong lower or upper bound for this case, we do make progress towards proving lower bound for bsph(L 1 ) and bcd(L 1 ). In Section 5, we prove tight bounds for bsph(L 0 ) and bcd(L 0 ) (Eq. 1 in Theorem 1). In Section 6, we prove logarithmic bounds on bsph(L p ) and bcd(L p ), for p > 2 (Eq. 5 in Theorem 1). In Section 7, we use the algorithmic construction of upper bounds in Section 6 to prove subquadratic hardness for Closest Pair and Set Closest Pair. Finally, in Section 8, we conclude our paper by giving open problems and highlighting some directions for future research.

Preliminaries
We use the following standard terminologies and notations. Below we define below L p -norms, which are distance measures in the metric spaces considered in this paper.
Distance Measures. For any vector x ∈ R d , the L p -norm of x is denoted by i.e., the number of non-zero coordinates of x. These norms define distance measures of R d . The distance of two points x and y w.r.t. the L p -norm is, thus, ||x − y|| p . The distances measures that are well studied in literature are the Hamming distance L 0 -norm, the Rectilinear distance L 1 -norm, the Euclidean distance L 2 -norm, the Chebyshev distance (a.k.a, Maximum-norm) L ∞ -norm. Given two sets of points A, B ⊆ R d , the L p -distance between the two sets is defined to be d p (A, B) = min a∈A,b∈B a − b p .
Problems. We define Closest Pair as a decision problem in a usual manner. In particular, in Closest Pair, we are given a real number α and a collection of points P ⊆ Σ d in a d-dimensional L p -metric, for some finite set Σ, and the goal is to determine if there exists a pair of distinct points a, b ∈ P such that a − b p ≤ α.
For BCP, we are given a real α, and we specify the two color classes (the collections of red and blue points) by A and B, and the goal is to determine if there exists a pair of points a ∈ A and b ∈ B such that a − b p ≤ α.
A generalization of Closest Pair that we study is the set closest pair problem (Set Closest Pair).
In Set Closest Pair, we are given a real α and a collection (possible, multiset) X of N subsets of points in a d-dimensional metric space, and the goal is to determine if there exists a pair of distinct Note that the search variant of the above problems are at least as hard as the decision variants described above, and thus any lower bounds on the above problems also applies to their search version.

Fine-Grained Complexity and Conditional Hardness.
Conditional hardness is the current trend in proving running-time lower bounds for polynomial-time solvable problems. This has now developed into the area of Fine-Grained Complexity. Please see, e.g., [Wil15,Wil16] and references therein.
The Orthogonal Vectors Hypothesis (OVH) is a popular complexity theoretic assumption in fine-grained complexity to base the hardness of a problem on. OVH states that in the Word RAM model with O(log n) bit words, any algorithm requires n 2−o(1) time in expectation to determine whether collections of vectors U, W ⊆ {0, 1} d , each with n/2 vectors, where d = ω(log n), contains an orthogonal pair u ∈ U and w ∈ W (i.e., d i=1 u i · w i = 0). Another popular conjecture is the Strong Exponential-Time Hypothesis for SAT (SETH), which states that, for any ε > 0, there exists an integer k, such that k-SAT on n variables cannot be solved in O(2 (1−ε)n )-time. Williams showed that SETH implies OVH [Wil05].

Geometric Representation of Biclique in L 2
In this section, we prove a lower bound on bsph(L 2 ) and an upper bound on bcd(L 2 ).
Theorem 6. For every n, d ∈ N, and any two sets A, B ⊆ R d , each of cardinality n, suppose the following holds for some non-negative real numbers a and b with a > b.

For every u in
Then the dimension d must be at least n−3 2 .
Proof. Let |A| = |B| = n be arbitrary two sets of vectors in R d that satisfy the above conditions. We will show that d ≥ n−3 2 . First, we scale all the vectors in A ∪ B so that the vector with the largest ℓ 2 -norm in A ∪ B has ℓ 2 -norm that is equal to 1 (by this scaling, the parameters a, b are scaled as well by, say, s. For brevity, we will write a for a/s and similarly for b.). We modify A and B in two steps as follows. We add one new coordinate to all of the vectors with value K ≫ 1 (to be determined exactly later) and obtain A 1 , B 1 ⊆ R d+1 . Note that each element in the new set of vectors A 1 and B 1 has ℓ 2 -norm roughly equal to K. More specifically, the square of the ℓ 2 -norm is bounded between K 2 and K 2 + 1 and the vector with the largest ℓ 2 -norm in A 1 ∪ B 1 has ℓ 2 -norm that is equal to √ K 2 + 1.
By adding to the last coordinate of each vector u in A 1 ∪ B 1 a positive value c u smaller than 1/K, we can impose that all the vectors are with ℓ 2 -norm that is equal to √ K 2 + 1. To see this, note that if we have a vector u 1 in A 1 ∪ B 1 that has ℓ 2 -norm that is equal to K (namely, as small as possible), then by setting c u 1 to satisfy we get that the ℓ 2 -norm of u 1 is √ K 2 + 1. Any c u 1 that solves Equation 6 is smaller than 1/K and by assuming that u 1 has a larger ℓ 2 -norm we would have get better bounds on c u 1 .
Let A ′ 1 ∪ B ′ 1 be the set of vectors that was obtained by adding c u 's as described above. Let u, v be vectors in A 1 ∪ B 1 and let u ′ , v ′ be the corresponding vectors in A ′ 1 ∪ B ′ 1 . By definition, the following holds: Hence, by choosing K to satisfy 1/K 2 ≤ a 2 −b 2 2 it follows that A ′ 1 ∪ B ′ 1 satisfies the conditions of the theorem with a ′ = a and b ′ = b 2 + a 2 −b 2 2 < a ′ . Again, for brevity, we refer to a ′ as a and b ′ as b.
. . , a n be the vectors from A ′ 1 , and b 1 , b 2 , . . . , b n be the vectors from B ′ 1 . Consider the following matrix in R 2(d+1)×2n : M = a 1 , a 2 , . . . , a n b 1 , b 2 , . . . , b n b 1 , b 2 , . . . , b n a 1 , a 2 , . . . , a n (7) Define the set A 2 to be the first n column vectors of M and define B 2 to be the last n column vectors of M . Note that A 2 ∪ B 2 ⊆ R 2(d+1) and satisfies the conditions of the theorem with a 2 = 2a 1 > 2b 1 = b 2 . Consider the inner product matrix M T M ∈ R 2n×2n written in a block matrix form as follows: where M 1,1 , M 1,2 , M 2,1 , M 2,2 ∈ R n×n and c is such that the matrix 3. M 1,2 = M 2,1 . This follows since M 1,2 = M T 2,1 = M 2,1 . Here the first equality follows since M T M is a symmetric matrix, and the last equality follows by item 1.
In the rest of the proof, we analyze some of the eigenvectors of M T M , for this we consider the matrix M 1,1 − M 1,2 . Since both M 1,1 and M 1,2 are symmetric, we have that M 1,1 − M 1,2 is symmetric as well and has real eigenvalues. Moreover, by the conditions of the theorem, it holds that M 1,1 − M 1,2 is strictly negative (i.e., all the entries of the matrix are negative). This follows because all the vectors have the same ℓ 2 -norm. Let x 1 , x 2 , . . . , x n be the eigenvectors of M 1,1 − M 1,2 with eigenvalues λ 1 , λ 2 , . . . , λ n . By the Perron-Frobenius Theorem it follows that λ 1 is strictly smaller than λ 2 , λ 3 , . . . , λ n .
Let x i ∈ R n be an eigenvector of M 1,1 − M 1,2 with eigenvalue λ i . Then the following holds.
Hence the vectors with eigenvalues λ 1 , λ 2 , . . . , λ n . The operation of adding cI 2n×2n to Note that M T M is a positive semidefinite matrix. Hence, λ 1 + c, λ 2 + c, . . . , λ n + c ≥ 0. More specifically, λ 1 + c ≥ 0 and λ 2 + c, . . . , λ n + c > 0 (since λ 1 < λ 2 , λ 3 . . . , λ n ). It follows that M T M has at least n − 1 positive eigenvalues. Hence, the rank of M T M is at least n − 1. By standard linear algebra arguments, it holds that the rank of M is at least the rank of M T M , and the rank of M is at most 2 (d + 1). That is, Therefore, d ≥ (n − 3)/2, and the theorem follows.

Upper Bound on the Biclique Contact-Dimension
Now we show that bcd(L 2 ) ≤ 2n − 1. Our proof is constructive and yields a linear-time algorithm that constructs a polar-pair (A, B) with strong property: the distance of every inner pair a, a ′ ∈ A (resp., b, b ′ ∈ B) is the same.
Theorem 7. For every n ∈ N, there exists two sets A, B ⊆ R 2n−1 each of cardinality n such that the following holds:

For every u in
Moreover, there exists a deterministic algorithm that outputs A and B and runs in linear-time w.r.t. the size of the output (which is Θ(n 2 )).
Proof. Let G = K n,n be the complete bipartite graph. Consider the adjacency matrix A G of the graph G and note that n · I + A G is a positive semi-definite matrix (this follows since the smallest eigenvalue of G is −n). Let U be the symmetric matrix n · I + A G and W be a matrix with 2n columns that satisfies W T W = U . One can check that the rank of W in this case is 2n − 1. We construct A and B as follows. The first n columns of W are the vectors of A and the last n columns of W are the vectors of B. If both u, v ∈ A (and similarly for B) then we note the following: On the other hand, if u ∈ A and v ∈ B, then we have the following: The following claim shows that we can construct W , and consequently, A and B efficiently (note that, in general, factoring a positive semi-definite matrix takes O n 3 ).
Claim 8. The matrix W can be constructed in O n 2 time.
Proof. Let 1 be the all one column vector and 1 P be a column vector with values 1 on the coordinates of A and −1 on the coordinates of B. Note that A G is a rank 2 matrix because, One can check that 1 and 1 P are eigenvectors of A G with eigenvalues n and −n, respectively. The other eigenvectors are orthogonal to 1 and 1 P , and their associated eigenvalues are 0 (because A G has rank 2).
Recall that the Hadamard vectors H 1 , H 2 , . . . , H n of dimension n is a collection of n vectors in {1, −1} n with the property that every two vectors are orthogonal and H 1 is the all one vector. There exist well known recursive constructions of the Hadamard vectors (that are linear in the output size) when n is a power of 2. In the case that it is not a power of 2, it suffices to work with the smallest power of 2 which is greater than n.
Consider the following matrix: We note that the column vectors v 1 , v 2 , . . . , v 2n of V are the eigenvectors of A G . Furthermore, U has the same eigenvectors (with n added to the eigenvalues). So, if v 1 , v 2 , . . . , v 2n are the normalized eigenvectors of U with positive eigenvalues λ 1 , λ 2 , . . . , λ 2n−1 and λ 2n = 0, then we have The last matrix multiplication can be done in O n 2 since diag The proof follows by appropriately scaling the vectors in A and B.

The Possibility of Representing Biclique in L 1
In this section, we discuss the case of the L 1 -metric, which is the only case that we are unable to prove neither strong lower bound nor linear upper bound. A weak lower bound bsph(L 1 ) ≥ Ω(log n) follows from the proof for the L p -metric with p > 2 in Section 6.1, and a quadratic upper bound bcd(L 1 ) ≤ n 2 follows from the proof for the L 0 -metric in Section 6.2. However, we cannot prove any upper bound smaller than Ω(n 2 ) or any lower bound larger than of O(log n). There is a huge gap in this case. Hence, we study an average case relaxation of the question.
We show in Theorem 10 that there is no distribution whose expected distances simulate a polar-pair of point-sets in the L 1 -metric. Consequently, even though we could not prove the biclique sphericity lower bound for the L 1 -metric, we are able to refute an existence of an geometric representation with large gap for any dimension as shown in Corollary 11. (A similar result was shown in [DM94].) Definition 9 (L 1 -distribution). Let X, Y be two random variables taking values in R d . An L 1distribution is constructed by X, Y if the followings hold.
Lemma 10. For any two finite-supported random variables X, Y that are taking values in R d there is no L 1 -distribution.
Proof. Assume towards a contradiction that there exist two finite-supported random variables X, Y that are taking values in R d and that are satisfying Inequality 8 of Definition 9. Given a vector x ∈ R d , we denote by x (i) the value of the i-th coordinate of x. Hence the following inequalities hold, Thus for some i ⋆ ∈ [d] the following holds, Fix i ⋆ ∈ [d] satisfying the above inequality. For the sake of clarity, we assume that the random variables X, Y are taking values in R (i.e., projection on the i ⋆th coordinate). We can assume that the size of supp (X) ∪ supp (Y ) is greater than 1 because, if supp (X) ∪ supp (Y ) contains a single point then E [ x 1 − x 2 1 ] = 0, contradicting Inequality 10. Let supp (X) ∪ supp (Y ) contains t ≥ 2 points. We prove by induction on t, that there are no X, Y over R satisfying Inequality 10. The induction base case is when t = 2. By Inequality 10, there exists 3 pointsx 1 ,x 2 ,ỹ 1 in R such that, Since supp (X) ∪ supp (Y ) contains exactly two points thenx 1 ,x 2 ,ỹ 1 are supported by two distinct points in R. Hence, there are two cases, either that x 1 = x 2 (and y 1 = x 1 ), or that x 1 = x 2 (and eitherỹ 1 =x 1 orỹ 1 =x 2 ). It is easy to see that none of these cases satisfy Inequality 11, resulting in a contradiction.
We assume from induction hypothesis that there are no X, Y taking values over R satisfying Inequality 10 when the size of supp (X) ∪ supp (Y ) is equal to k ≥ 2. We consider the case when t = k + 1 ≥ 3. Sort the points in supp (X) ∪ supp (Y ) by their value and denote by s i the value of the i-th point of supp (X) ∪ supp (Y ). For the sake of simplicity we say that we change the value of s t−1 tos t−1 , where s t−2 ≤s t−1 ≤ s t , if after changing its value we change the values of (at least one of) X, Y toX,Ỹ in such a way that the value of the t − 1-th point (after sorting) of supp X ∪ supp Ỹ is equal tos t−1 (if s t−2 =s t−1 then the value of the (t − 2)-th point of supp X ∪ supp Ỹ is equal tos t−1 ). Define the function f : [s t−2 , s t ] → R as follows: whereX,Ỹ are obtained after changing s t−1 to x ∈ [s t−2 , s t ]. The crucial observation is that the function f is linear. Hence either f (s t−2 ) ≥ f (s t−1 ) or f (s t ) ≥ f (s t−1 ) and we can reduce the size of supp (X) ∪ supp (Y ) by 1. However, this contradicts our induction hypothesis.
The next lemma refutes the existence of a polar-pair of point-sets with large gap in any dimension.
Corollary 11 (No Polar-Pair of Point-Sets in L 1 with Large Gap). For any α > 0, there exist no subsets A, B ⊆ R d of n/2 vectors with d < n/2 such that • For any u, v both in A, or both in B, we have u − v 1 ≥ 1 1−2/n · α.
• For any u ∈ A and v ∈ B, u − v 1 < α.
Proof. Assume towards a contradiction that there exist a polar-pair of point-sets (A, B) in the L 1 -metric that satisfies the conditions above. We can create a distribution X and Y such that To see this, we create a uniform random variable X (resp., Y ) for the set A (resp., B). Now the expected distance of two independent copies of X (resp., Y ) is at least 1 1−2/n α · 1 − 1 n/2 = α, which follows because we may pick the same point twice. Since the expected distance of the crossing pair a ∈ A and ∈ B is less than α. This contradicts Theorem 10.
In stark contrast, a result similar to Theorem 10 but in the L 0 metric, i.e., Lemma 15 implies that there is no polar-pair in L 0 . Finally, one can show that biclique sphericity lower bound for L 2 implies that there is no polar-pair distribution for L 2 as well. This completes the picture of non-existence of polar-pair distributions for all the three metrics (i.e., L 2 , L 1 , and L 0 ).

Geometric Representation of Biclique in L 0
In this section, we prove a lower bound on bsph(L 0 ) and upper bounds on bcd(L 0 ). The proofs in this section are mostly combinatorial due to the behavior of the Hamming metric.
We start by providing real-to-binary reductions in Section 5. Then we proceed to prove the lower bound on bsph(L 0 ) in Section 5.1 and then the upper bounds on bcd(L 0 ) in Section 5.2.
Real to Binary Reductions First we prove the following lemma, which allows mapping from vectors in R d to zero-one vectors.
Lemma 12 ((Trivial) Real to Binary Reduction). Let S ⊆ R be a finite set of real numbers, and let P ⊆ S d . Then there exists a transformation φ : S d → {0, 1} d|S| such that, for any x, y ∈ P , Proof. First, we order element in S in arbitrary order and write it as S = {r 1 , r 2 , . . . , r |S| }. Next we define ψ : S → {0, 1} |S| so that the i-th coordinate of ψ(r i ) is 1 and others are zeroes. That is, Then we define φ(x) = (ψ(x 1 ), ψ(x 2 ), . . . , ψ(x d )). Clearly, ψ(r i ) − ψ(r j ) 0 = 1 if and only if r i = r j . Therefore, we conclude that for any x, y ∈ P , Next we present a randomized reduction which gives a better dimension bound for some parameters. We require the additive form of Chernoff's bound.
Lemma 14 (Randomized Real to Binary Reduction with Given Bar). Let β > 0 be any integer. Let P ⊆ R d be a set of d-dimensional points with cardinality n. Then there exists a function φ : P → {0, 1} dL , where L > 8β ln n, that transforms points in P to binary strings such that, for any points u, v, x, y ∈ P , Proof. We prove the claim by probabilistic method.
First, observe that, for each coordinate, there can be at most n different values of p i , for any vector p ∈ P . Thus, we may think that the i-th coordinate of any point in P is a letter from an alphabet set of size n, denoted by Σ i . Then we construct a function φ i : Σ i → {0, 1} L for each alphabet set Σ i by uniformly and independently at random setting each coordinate of φ i (a), for a ∈ Σ i , to be 0 and 1 with probability 1/2. That is, each letter a ∈ Σ i is mapped to a binary string b = φ i (a) ∈ {0, 1} L in such a way that b j = 1 with probability 1/2 0 otherwise Note that we need to construct a mapping for each coordinate independently. The mapping φ is then defined as φ(x) = (φ 1 (x 1 ), φ 2 (x 2 ), . . . , φ d (x d )).
Let us take any points u, v, x, y ∈ P such that α = u − v 0 > β ≥ x − y 0 . (Thus, α ≥ β + 1). Observe that α (resp., β) is the number of coordinates that the vectors u and v (resp., x and y) are different.
, then each bit of the blocks φ i (u) and φ(v) (i.e., φ i (u) j and φ i (v) j ) are different with probability 1/2, and these events are independent. Thus, By Chernoff's bound (Lemma 13), we have Next consider φ(x) − φ(y) 0 . By the same arguments as that of u and v, we conclude that Therefore, by taking union bound over all pairs, we conclude that there exists φ such that

Lower Bound on the Biclique-Sphericity
Now we will show that bsph(L 0 ) ≥ n. Our proof requires the following lemma, which rules out a randomized algorithm that generates a polar-pair of point-sets.
Lemma 15 (No Distribution for L 0 ). For any α > β ≥ 0, regardless of dimension, there exist no distributions A and B of points in R d with finite supports such that Proof. We prove by contradiction. Assume to a contrary that such distributions exist. Then Let A and B be supports of A and B, respectively. Since A and B have finite supports, we may assume wlog that A and B are multi-sets of equal size. Moreover, by Lemma 12, we may further assume that vectors in A and B are binary vectors.
Observe that each coordinate of vectors in A and B contribute to the expectations independently. In particular, Eq. (12) can be written as where ρ A 0,i , ρ A 1,i , ρ B 0,i and ρ B 1,i are the probability that the i-th coordinate of x ∈ A (resp., y ∈ B) is 0 (resp., 1). Thus, to show a contradiction, it is sufficient to consider the coordinate which contributes the most to the summation in Eq. (13). The contribution of this coordinate to the summation is Since ρ A 0 + ρ A 1 = 1 and ρ B 0 + ρ B 1 = 1, the summation in Eq.(14) can be non-negative only if ρ A 0 = ρ B 0 and ρ A 1 = ρ B 1 . But, then this implies that the summation in Eq.(14) is zero. We have a contradiction since this coordinate contributes the most to the summation in Eq. (13) which we assume to be positive.
The next Lemma shows that bsph(L 0 ) ≥ n.
Theorem 16 (Lower Bound for L 0 with Arbitrary Alphabet). For any integers α > β ≥ 0 and n > 0, there exist no subsets A, B ⊆ R d of n vectors with d < n such that • For any a, a ′ ∈ A, a − a 0 ≥ α.
• For any a ∈ A and b ∈ B, a − b 0 < β.
Proof. Suppose for a contradiction that such subsets A and B exist with d < n. We build uniform distributions A and B by uniformly at random picking a vector in A and B, respectively. Then it is easy to see that the expected value of inner distance is The intra distance of B is similar. We know that α − β ≥ 1 because they are integers and so are ℓ 0 -distances. But, then if α < n, we would have distributions that contradict Lemma 15. Note that α and β are at most d (dimension). Therefore, we conclude that d ≥ n.

Upper Bound on the Biclique Contact-Dimension
Now we show that, for vectors in R d , bcd(L 0 ) ≤ n.
Theorem 17 (Upper Bound for L 0 with Arbitrary Alphabet). For any integer n > 0 and d = n, there exist subsets A, B ⊆ R d each with n vectors such that • For any a, a ′ ∈ A, a − a 0 = d.
• For any b, b ′ ∈ B, a − a 0 = d.
• For any a ∈ A and b ∈ B, a − b 0 = d − 1.
Proof. First we construct a set of vectors A. For i = 1, 2, . . . , n, we define the i-th vector a of A so that a is an all-i vector. That is, a = (i, i, . . . , i) Next we construct a set of vectors B. The first vector of B is (1, 2, . . . , n). Then the i + 1-th vector of B is the left rotation of the i-th vector. Thus, the i-th vector of B is b = (i, i + 1, . . . , n, 1, 2, . . . , i − 1).
It can be seen that the L 0 distance between any two vectors from the same set is d because all the coordinates are different. Any vectors from different set, say a ∈ A and b ∈ B, must have at least one common coordinate. Thus, their L 0 distance is d − 1. This proves the lemma.
Below is the upper bounds for zero-one vectors, which is a corollary of Theorem 17.
Corollary 18 (Upper Bound for L 0 with Binary Vectors). For any integer n > 0 and d = n 2 , there exist subsets A, B ⊆ R d each with n vectors such that • For any a, a ′ ∈ A, a − a 0 = n.
• For any b, b ′ ∈ B, a − a 0 = n.
• For any a ∈ A and b ∈ B, a − b 0 = n − 1.
Proof. We take the construction from Theorem 17 and denote the two sets by A ′ and B ′ and denote their dimension by d ′ = n.
We transform A ′ and B ′ to sets A and B by applying the transformation φ in Lemma 12. That is, Since the alphabet set in Lemma 12 is [n], we have a construction of A and B with dimension d = n 2 .
The corollary below follows directly from Theorem 17 and Lemma 12.
Corollary 19 (Upper Bound for ℓ 0 with Binary Vectors). For any integer n > 0 and d = n 2 , there exist subsets A, B ⊆ R d each with n vectors such that • For any a, a ′ ∈ A, a − a 0 = n.
• For any b, b ′ ∈ B, a − a 0 = n.
• For any a ∈ A and b ∈ B, a − b 0 = n − 1.
It is not hard to see that the following corollary holds. • For any a, a ′ ∈ A, a − a 0 = d.
• For any b, b ′ ∈ B, a − a 0 = d.
• For any a ∈ A and b ∈ B, a − b 0 = d − 1.
Proof Sketch. First, we follow the proof of Lemma 12. Here we can reduce the dimension by a factor of K because each coordinate can represent K values. The corollary then follows by following the proof of Corollary 18.
6 Geometric Representation of Biclique in L p for p > 2 In this section, we show the lower bound on bsph(L p ) and an upper bound on bcd(L p ) for p > 2. Both bounds are logarithmic. The latter upper bound is constructive and thus implies the subquadratic-time equivalent between Closest Pair and BCP.

Lower Bound on the Biclique Sphericity
Now we show the lower bound on the biclique sphericity of a complete bipartite graph in L p -metrics with p > 2. In fact, we prove the lower bound for the case of a star graph on n vertices, denoted by S n , and then use the fact that bsph(H) ≤ bsph(G) for all induced subgraph H of G (i.e., bsph(K n/2,n/2 , L p ) ≥ bsph(S n/2 , L p )).
In short, we show in Lemma 23 that O(log n) is the maximum number of L p -balls of radius 1/2 that we can pack in an L p -ball of radius 1 so that no two of them intersect or touch each other. This upper bounds, in turn, implies the lower bound on the dimension. We proceed the proof by volume arguments, which are commonly used in proving the minimum number of points in an ǫ-net that are sufficient to cover all the points in a sphere.
Definition 21 (ǫ-net). The unit L p -ball in R d centered at o is denoted by For brevity, we write B L d p , o as B L d p . Let (X, d) be a metric space and let S be a subset of X and ǫ be a constant greater than 0. A subset N ǫ of X is called an ǫ-net of S under d if for every point x ∈ S it holds for some point y ∈ N ǫ that d (x, y) ≤ ǫ.
The following lemma is well known in literature, and we provide below a proof using essentially the same arguments as in [Ver10] for the sake of completeness. Proof. Let us fix ǫ > 0 and choose N ǫ of maximal cardinality such that x − y p > ǫ for all x = y both in N ǫ . We claim that N ǫ is an ǫ-net of the B L d p . Otherwise, there would exist an x ∈ B L d p that is at least ǫ-far from all points in N ǫ . Thus, N ǫ ∪ {x} contradicts the maximality of N ǫ . After establishing that N ǫ is an ǫ-net, we note that by the triangle inequality, we have that the balls of radii ǫ/2 centered at the points in N ǫ are disjoint. On the other hand, by the triangle inequality all such balls lie in (1 + ǫ/2) B L d p . Comparing the volumes gives us that vol ǫ 2 B L d p · |N ǫ | ≤ vol 1 + ǫ 2 B L d p . Since for all r ≥ 0, we have that vol r · B L d p = r d · vol B L d p , we conclude the following: Theorem 23. For every N, d ∈ N, and any two sets A, B ⊆ R d , each of cardinality N , suppose the following holds for some non-negative real numbers a and b with a > b.
1. For every u and v both in A we have that u − v p > a.
2. For every u and v both in B we have that u − v p > a.

For every u in
Then the dimension d must be at least log 5 (N ).
Proof. Scale and translate the sets A, B in such a way that b = 1 and that 0 ∈ B. It follows that A ⊆ B L d p . By Lemma 22 we can fix an 1/2-net N 1/2 for B L d p of size 5 d . Note that for every x ∈ N 1/2 it holds that 1/2 · B L d p , x contains at most one point from A. Note also that N 1/2 covers B L d p . Thus |A| ≤ 5 d which implies that d ≥ log 5 (N ).

Upper Bound on the Biclique Contact-Dimension
We first give a simple randomized construction that gives a logarithmic upper bound on the biclique contact-dimension of L p . The construction is simple. We uniformly at random take a subset A of n vectors from {−1, 1} d/2 × {0} d/2 and a subset B of n vectors from {0} d/2 × {−1, 1} d/2 . Observe that, for any p > 2, the L p -distance of any pair of vectors a ∈ A and b ∈ B is exactly d while the expected distance between the inner pair a, a ′ ∈ A (resp., b, b ′ ∈ B) is strictly larger than d.
Thus, if we choose d to be sufficiently large, e.g., d ≥ 10 ln n, then we can show by using a standard concentration bound (e.g., Chernoff's bound) that the probability that the inner-pair distance is strictly larger than d is at least 1 − 1/n 3 . Applying the union bound over all inner-pairs, we have that the d-neighborhood graph of A ∪ B is a bipartite complete graph with high probability. Moreover, the distances between any crossing pairs a ∈ A and b ∈ B are the same for all pairs. This shows the upper bound for the contact-dimension of a biclique in the L p -metric for p > 2.
The above gives a simple proof of the upper bound on the biclique contact-dimension of the L p -metric. Moreover, it shows a randomized construction of the polar-pair in O(log n)-dimensional L p -metric, for p > 2, thus implying that Closest Pair and BCP are equivalent for these L pmetrics.
For a deterministic construction, the proof is more involved, and we show it using appropriate codes.
Proof. In literature, we note that for any constant δ > 0, there is an explicit binary code of (some) constant rate and relative distance at least 1 2 − δ and the entire code can be listed in quasilinear time with respect to the size of the code (see Appendix E.1.2.5 from [Gol08], or Justesen codes [Jus72]). Using such a code, for some constant α, we obtain C ⊂ {0, 1} αn of cardinality 2 n/2 , such that for every two x, y ∈ C, x and y differ on at least 1 2 − δ αn coordinates, for some constant δ ∈ 0, 1 4 − 1 2 p . Moreover, C can be computed in 2 n/2 n O(1) time. From C, we obtain C ′ ⊂ {−h, h} αn by replacing 0 with −h, and 1 with h in each of the coordinates in all the points of C. We construct the requisite P and Q as subsets of {−h, 0, h} 2αn . For every i ∈ [2 n/2 ], the i th point of P is given by the concatenation of the i th point of C ′ with 0 αn . Similarly, the i th point of Q is given by the concatenation of 0 αn with the i th point of C ′ (note the reversal in the order of the concatenation). In particular, points in P and Q are of the form (x i , 0) and ( 0, x i ), respectively, where x i is the i th point in C ′ and 0 is the zero-vector of length αn.
First, consider any two points in the same set, say a, a ′ ∈ P (respectively, b, b ′ ∈ Q). We have from the distance of C that on at least 1 2 − δ αn coordinates the two points differ by 2h, thus implying that their L p -distance is at least This proves the first two items of the Lemma. Next we prove the third item. Consider any two points from different sets, say a ∈ P and b ∈ Q. It is easy to see from the construction that a and b differ in every coordinate by exactly h. Thus, the L p -distance between any two points from different set is exactly

Connection to Closest Pair
In this section, we prove the quadratic-time hardness for Closest Pair in the L ∞ -metric and Set Closest Pair. Our reductions are from OV which we rephrase it as follows: Given a pair of collection of vectors U, W ⊆ {0, 1} d , the goal is to find a pair of vectors u ∈ U and w ∈ W such that (u i , w i ) ∈ {(0, 0), (0, 1), (1, 0)} for all i ∈ [d]. Throughout, we denote by n the total number of vectors in U and W .
We start by giving a base reduction, which will be used in the hardness of Closest Pair in the L ∞ -metric and Set Closest Pair.

Base Reduction
Let U, W ⊆ {0, 1} d be an instance of OV. We may assume that U and W have no duplicates. Otherwise, we may sort vectors in U (resp., W ) in lexicographic order and then sequentially remove duplicates; this preprocessing takes O(dn log n)-time.
We construct a pair of sets A, B ⊆ R d of BCP from U, W as follows. For each vector u ∈ U (resp., w ∈ W ), we create a point a ∈ A (resp., b ∈ B) such that Observe that, for any vectors a ∈ A and b ∈ B, |a j − b j | = 2 only if a j = b j = 1; otherwise, |a j − b j | = 1. It can be seen that a − b p = d if and only if their corresponding vectors u ∈ U and w ∈ W are orthogonal. Thus, this gives an alternate proof for the quadratic-time hardness of BCP under OVH.

Quadratic-Time Hardness of Closest Pair in L ∞
Here we show that the reduction in Section 7.1 rules out both exact and 2-approximation algorithm for Closest Pair in L ∞ that runs in subquadratic-time. (unless OVH is false). That is, we prove Theorem 4, which follows from the theorem below.
Theorem 25. Assuming OVH, for any ε > 0 and d = ω(log n), there is no O n 2−ε -time algorithm that, given a point-set P ⊆ R d , distinguishes between the following two cases: • There exists a pair of vectors in P with L ∞ -distance one.
• Every pair of vectors in P has L ∞ -distance two.
In particular, it is OVH-hard to approximate Closest Pair in the L ∞ -metric to within a factor of two.
Proof. Consider the sets A and B constructed from an instance of the OV problem in Section 7.1.
First, observe that every inner pair has L ∞ -distance at least 2. To see this, consider an inner pair a, a ′ ∈ A. Since all inner pairs are distinct, they must have at least one different coordinate, say a j = a ′ j for some j ∈ {1, . . . , n/2}. Consequently, (a j , a ′ j ) ∈ {(0, 2), (2, 0)}, implying that the L ∞ -distance of a and a ′ is at least 2. The case of an inner pair b, b ′ ∈ B is similar. Thus, any pair of vectors with L ∞ -distance less than two must be a crossing pair a ∈ A, b ∈ B. Now suppose there is a pair of orthogonal vectors u * ∈ U, w * ∈ W , and let a * ∈ A and b * ∈ B be the corresponding vectors of u * and w * in the Closest Pair instance, respectively. Then we know from the construction that (a * j , b * j ) ∈ {(0, 1), (0, −1), (2, 1)} for all coordinates j ∈ [n]. Thus, the L ∞ -distance of a * and b * must be one.
Next suppose there is no orthogonal pair of vectors in (U, W ). Then every vectors u ∈ U and w ∈ W must have one coordinate, say j, such that u j = w j = 1. So, the corresponding vectors a and b (of u and w, respectively) must have a i = 2, b j = −1. This means that a and b have L ∞distance at least three. (Note that there might be an inner pair with L ∞ -distance two.) Therefore, we conclude that every pair of points in A ∪ B has L ∞ -distance at least two.

Quadratic-Time Hardness of Set Closest Pair
In this section, we consider Set Closest Pair in the L p -metric with p > 1. The naive algorithm for this problem tries all possible pairs of subsets a, b ∈ X and returns the smallest one. The running time of this trivial algorithm is thus O(dtn 2 ). There is no better algorithm in literature, and we show that it might not be possible to improve the bound unless OVH is false.
More precisely, we rule out subquadratic-time algorithm for Set Closest Pair by a reduction from OVH, i.e., we prove Theorem 5. For an overview, we take a hardness instance (A, B) of BCP constructed in the base reduction in Section 7.1. Then we attach a codeword to each vector (possibly with duplicates) in A and B. This can be seen as a set set version of polar-pair.
Our construction requires the following functions.
The functions f 1 , f 2,1 and f 2,2 are illustrated in Figure 1.
We will abuse notation and define f * : {0, 1} l → R 2l to be the function that maps a vector v with l coordinates to a vector with 2l coordinates by applying f * on each coordinate of v.

Constructing Set-Version of Polar-Pair.
Consider the code from the proof of Lemma 24 (namely for 0 < δ < 1 2 − 1 2 p . For some constant α > 0, there is a binary code with constant rate and relative distance at least 1 2 − δ. The code can be explicitly constructed in quasilinear-time with respect to the size of the code). Let q = O(log n) be the dimension of the mentioned code. We obtain a collection of codewords C ⊆ {0, 1} αq of cardinality 2 · 2 q such that, for every two x, y ∈ C, x and y differ on at least 1 2 − δ αq coordinates. Next we construct the sets P and Q so that P = {{f 1 (c)} : c ∈ C} and Q = {{f 2,1 (c), f 2,2 (c)} : c ∈ C}.
We use the following lemma which is a variant of Theorem 24.
Lemma 26. The following properties hold for Q, P.
3. For all X ∈ P, Y ∈ Q it holds that dist p (X, Y ) = (αq) 1/p . Moreover, there exists a deterministic algorithm that outputs P and Q in time O(2 q n O(1) ).
Here the last equality follows by the definition of d, f 1 , f 2,1 , f 2,2 .
• Case 1: X, Y ∈ A. Let a, a ′ ∈ A be vectors corresponding to X and Y , respectively. Then by the construction of A and Lemma 26, we have that The last inequality follows because M ≫ d 2 and 2 p (1/2 − δ) > 1.
• Case 2: X, Y ∈ B. This case is exactly the same as the previous case, and we have that dist p (X, Y ) > (M + d) 1/p .
• Case 3: X ∈ A, Y ∈ B. Let a ∈ A and b ∈ B be points corresponding to X and Y , respectively. Then by the construction of X and Y , we have that Thus, in any cases, the L p -distance between X and Y is strictly greater than (M + d) 1/p . This concludes the claim.

Conclusion and Discussion
We have studied the sphericity and contact dimension of the complete bipartite graph in various metrics. We have proved new bounds on these measures for some metrics and given alternate proofs of the old bounds for the Euclidean metric. However, biclique sphericity and biclique contact dimension in the L 1 -metric remains poorly understood as we are unable to show any strong upper or lower bounds. However, we believe that both L 1 and L 2 metrices have linear upper and lower bounds. To be precise, we raise the following conjecture: Conjecture 28 (L 1 -Biclique Sphericity Conjecture).
In addition, we have showed conditional lower bounds for Closest Pair the L p -metric, for all p ∈ R >2 ∪ {∞}, by using polar-pairs of point-sets. However, it is unlikely that our techniques could be extended to to popular metrics L 2 , L 1 , and L 0 . An open question is thus whether there exists an alternative technique to derive a lower bound from OVH to Closest Pair for these metrics. The answer might be on the positive side, i.e., there might exist an algorithm that performs well in the L 2 -metric because there are more tools available, e.g., Johnson-Lindenstrauss' dimension reduction. Thus, it is possible that there exists a strongly subquadratic-time algorithm in the L 2 -metric. This question is still mysterious and remains a long standing open problem.