Covering an ellipsoid with equal balls


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1 Journal of Combinatorial Theory, Series A 113 (2006) Covering an ellipsoid with equal balls Ilya Dumer College of Engineering, University of California, Riverside, CA 92521, USA Received 1 October 2005 Available online 13 June 2006 Abstract The thinnest coverings of ellipsoids are studied in the Euclidean spaces of an arbitrary dimension n. Given any ellipsoid, our goal is to find the minimum number of unit balls needed to cover this ellipsoid. A tight asymptotic bound on the logarithm of this number is obtained Elsevier Inc. All rights reserved. Keywords: Ellipsoid; Euclidean space; Spherical covering; Unit ball 1. Introduction 1.1. Ellipsoids and coverings Consider the ball Bε n(y) of radius ε centered at some point y = (y 1,...,y n ) in an n dimensional Euclidean space R n : { Bε n def (y) = x R n n (x i y i ) 2 ε }. 2 For any subset A R n, a subset M ε (A) R n is called its εcovering if A is contained in the union of the balls of radius ε centered at points y M ε (A): A Bε n (y). y M ε (A) address: /$ see front matter 2006 Elsevier Inc. All rights reserved. doi: /j.jcta
2 1668 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) The socalled εentropy [1] ℵ ε (A) of a set A is defined as ℵ ε (A) = log min M ε (A), M ε (A) where minimum is taken over all coverings and the logarithm is base e. Below we study the εentropy of an arbitrary ellipsoid { } def = x R n n x 2 i 1, (1) E n a a 2 i where a = (a 1,...,a n ) is a vector with n positive symbols. Without loss of generality, we assume that 0 <a 1 a 2 a n. By linear transformation of R n, we can always replace a subset A and its covering M ε (A) using the unit balls B1 n (y) on the rescaled subset A/ε. More generally, we can use different scaling factors b i for different axis x i. Thus, the following three problems are equivalent: (1) covering an ellipsoid Ea n with unit balls; (2) covering an ellipsoid Ea n with balls of radius ε; (3) covering an ellipsoid Ea n with (smaller) ellipsoids { } Eb n def (y) = x R n n (x i y i ) 2 1. b 2 i Due to this equivalence, we will consider coverings with unit balls and remove the subscript ε from our notation. Our main goal is to find the asymptotic (unit) entropy ℵ(Ea n ) as a function of n and a. Here we consider the subsets of ellipsoids such that ℵ(Ea n) Coverings of the balls Optimal coverings have been long studied for an Euclidean ball Bρ n = Bn ρ (0). Various bounds on its minimum covering size are obtained in papers [2,3]. In particular, it follows from these papers that for any n 1 and ρ 1, ℵ ( Bρ) n n log ρ + c log(n + 1). (2) Here and in the sequel, c and c i denote some universal constants. We also mention the Few Coxeter Rogers lower bound ℵ ( Bρ) n n log ρ + c0 if ρ>n. For more details, we refer to the monographs [4,5], and survey [6], which give a detailed account of the subject along with an extensive bibliography. Coverings of other sets have also been studied for general convex bodies (see [7] and references therein). 2. Prior and present results Note that ℵ(E 1 a ) = log a for n = 1. Thus, we assume that n 2. Given some θ (0, 1/2), we decompose the set of positions N ={1,...,n} into the three consecutive subsets:
3 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) J θ,0 = { j:0<aj 2 1 θ} ; J θ,1 = { j:1 θ<aj 2 1} ; J ={j: a j > 1}. Let μ = μ θ = J θ,1 + J, m= J. (3) We also assume that m 1, since otherwise Ea n is covered with the single ball Bn. Given an ellipsoid Ea n, consider the geometric mean of the m largest halfaxes and the sum of their logarithms ρ def = a 1/m j, (4) j J K def = n j=n m+1 log a j = m log ρ. (5) Note that the ball Bρ m has the same volume as the mdimensional subellipsoid { def = x R m x 2 } j 1 E m a a 2 j J j spanned over the m largest axes of the original ellipsoid Ea n. We begin with a lower bound. Theorem 1 (Generalized packing bound [7]). The entropy of any ellipsoid Ea n satisfies inequality ℵ ( Ea n ) K. (6) The following theorem is valid for all ellipsoids and directly follows from the more general results [7] for convex bodies. It is also a reformulation of Theorem 3 of [8]. Theorem 2. For any θ (0, 1/2), the entropy of an ellipsoid Ea n satisfies inequality ℵ ( Ea n ) K + μθ log(3/θ). (7) From now on, consider the sets of ellipsoids Ea n with K, so that subellipsoids Em a have growing size. According to Theorem 2, asymptotic equality ℵ ( Ea n ) = K + o(k) (8) holds if there exists θ (0, 1/2) such that K/μ θ. (9) Note, however, that condition (9) is very restrictive and holds only for the sets of expanding ellipsoids, such that ρ. In particular, it fails on a ball Bρ n of any given radius ρ>1. The following asymptotic bound of [8] removes this drawback. Theorem 3. Asymptotic equality (8) holds for the ellipsoids Ea n provided that ( ) log a n K log ρ = o. log n (10)
4 1670 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Note that condition (10) implies that K log n, (11) in which case the volume of the largest subellipsoid Ea m exceeds any polynomial in n. Our main goal is to obtain asymptotic equality (8) for a broader class of ellipsoids. Firstly, we shall refine condition (10) so that the largest coefficient a n will depend on m and ρ only. Secondly, we show that all positions of the subset J θ,0 are insignificant for the entropy ℵ(Ea n ), which allows us to replace parameter n in (11) with parameter μ θ. Finally, we show that asymptotic bounds only slightly depend on θ and can be extended to the case of θ 0. These results are summarized as follows. Theorem 4. Consider the set of ellipsoids Ea n that satisfy restriction ( ) log a n K log ρ = o (12) log m for any m 2. Then for any parameter θ (0, 1/2), any ellipsoid Ea n has entropy ℵ ( Ea n ) K + o(k) + c1 log(μ θ + 1) + c 2 log(1/θ), K, (13) where c 1 and c 2 are universal constants. By taking θ = e K/log K, we obtain the following corollary. Corollary 5. Consider the set of ellipsoids Ea n that satisfy condition (12). Let m = m (K) = { a j : aj 2 1 e K/log K}. Then ellipsoids Ea n satisfy asymptotic equality (8) if K. log m (14) Note that new conditions (12) and (14) loosen former conditions (10) and (11). In particular, restriction (12) holds whenever the longest halfaxis a n is a polynomial ρ s of increasing degree s, as long as s = o(m log ρ/log m). Also, (14) admits any ellipsoid whose size grows faster than a polynomial in m. Finally, this number m includes only those m halfaxes a j, which either exceed 1 or are arbitrarily close to 1 (within an exponentially declining margin e K/o(K) for any function o(k) ). On the other hand, the following lemma shows that there exist ellipsoids, whose entropy (13) is dominated by the term log μ θ. Lemma 6. Consider an ellipsoid E n a with halfaxes a 1 = =a n 1 = 1 and any a n > 1. No n unit balls can cover E n a. 3. Proofs Proof of Theorem 4. The proof includes three main steps. In the first step, any ellipsoid Ea n will be enclosed into a finite number of subsets DR n, each of which is a direct product of the balls (of lesser dimensions). In the second step, we design a covering for each DR n. In the third step, we obtain a universal upper bound on the entropy ℵ(Ea n ) and optimize its asymptotic parameters.
5 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Step 1. Divide the set N of n positions into some number t + 1 of consecutive intervals J 0 = J θ,0, J 1 = J θ,1, J 2,...,J t, where t m + 1. Below we use notation J i =[n i + 1,n i+1 ] for any interval J i and denote its length s i = n i+1 n i. Here n 0 = 0 and n t+1 = n. For any θ, m, and μ = μ θ, we also use the parameter z = 2(μ + 1) 2 /θ 2 and define t + 1 approximation grids P i, where P 0 : pl 0 = e lθ/2, l = 0,..., θ 2 log z, P 1 : pl 1 = e l(log z)/μ, l = 0,...,μ, P i : p l = e l(log z)/m, l = 0,...,m, i = 2,...,t. Thus, all t + 1 grids have the same range [1/z, 1], and the last t 1 grids are identical. Without loss of generality, we assume that θ 2 log z is an integer. These grids are used as follows. For any point x Ea n, we take any interval J i and consider the subvector x Ji = (x j j J i ), i = 0,...,t, of length s i. Then we define the vector R = R(x) = (r 0,...,r t ) with symbols (15) r i = j J i x 2 j /a2 j, i = 0,...,t. By the definition of an ellipsoid E n a, r i 1. i=0 Each symbol r i is then rounded off to the two closest (but not necessarily different) points r i and r i on the grid P i taken as follows: { ri = r i = 1/z if r i 1/z, (17) r i r i r i, r i, r i P i if r i > 1/z. Finally, for any vector R, we use its approximation R = ( r 0,..., r t ). These vectors R form the set {R} of size N P 0 P 1 P i t 1 = 2(m + 1) t 1 (μ + 1)(log z)/θ. (18) Now, for any i, define the ball { B s i } ρ i = x Ji xj 2 ρ2 i, ρi 2 = a2 n i+1 r i, j J i of dimension s i and radius ρ i. Then we consider the direct products DR n D n R def = t i=0 { B s i ρ i = x R n j J i of all t + 1 balls: (16) xj 2 } an 2 r i,i= 0,...,t. (19) i+1
6 1672 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Lemma 7. The original ellipsoid E n a is contained in the union of the sets Dn R : E n a {R} D n R. (20) Proof. For any point x Ea n, consider the corresponding vectors R(x) and R. Recall that a ni+1 = max{a j,j J i }. Then xj 2 /a2 n i+1 xj 2 /a2 j = r i r i, j J i j J i and x Ji B s i ρ i for all i. Thus, by considering all possible vectors R, we find a subset DR n that covers any point x Ea n, and (20) holds. Step 2. Given any vector R, our next goal is to cover each subset DR n defined in (19) with unit balls. In doing so, we cover each ball B s i ρ i,i= 0,...,t,with the balls B s i e i of some radius e i. Then the direct product DR n of the balls is completely covered by the direct product of their coverings. Given any vector R, we choose the covering radii { ei 2 def a 2 n1 r 0 if i = 0, = ε i = r i (1 2μ 1 ) if i 1. (21) The following lemma shows that the direct product of the covering balls is contained in the unit ball: t i=0 B s i e i B n. Lemma 8. For any vector R, vector E = (ε 0,...,ε t ) satisfies restrictions ε i 1. i=0 (22) Proof. First, consider the intervals J i,i= 0,...,t,on which r i 1/z. Since t μ, ε i 1/z t + 1 θ 2 z μ + 1. (23) i: r i 1/z i: r i 1/z Next we proceed with r i > 1/z. For r 0 > 1/z, we use the fact that a 2 n 1 1 θ in definition (21). Also, e θ/2 1 + θ for θ [0, 1]. Then ε 0 r 0 (1 θ) r 0 (1 + θ)(1 θ)= r 0 ( 1 θ 2 ). (24) Also, definition (21) shows that ( ε i r i 1 1 2μ 1 i t r i >1/z Thus, (22) follows from (23) (25): ) ( (1 r 0 ) 1 1 ). (25) 2μ
7 i=0 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) ( ε i r 0 1 θ 2 ) ( + (1 r 0 ) 1 1 ) + θ 2 2μ μ + 1 { max 1 θ 2, 1 1 } + θ μ μ + 1 Step 3. Our goal is to estimate the entropy ℵ ( ( Ea n ) ) ℵ DR n max ℵ( D n ) R + log N. (26) R R Consider any set DR n. According to (21), the first ball Bs 0 ρ 0 is entirely covered by the ball of the same radius e 0. Also, for the remaining t balls B s i ρ i, the universal bound (2) gives the following estimates: ℵ ( DR n ) ( ) = ℵ ei B s i ρ i s i log ρ i + c log(s i + 1). (27) e i Note that log(1 α) 2α for any α [0, 1/2]. Thus for all i 1, definitions (15), (17), and (21) give inequalities log r i log r i + log r i log z ε i r i ε i m + 1 μ, log r 1 = log r 1 + log r 1 log z ε 1 r 1 ε 1 μ + 1 μ. These two inequalities are used as follows. For i = 1, the interval J 1 has length s 1 = μ m. Then s 1 log ρ 1 = s 1 e 1 2 log a2 n 1 r 1 s 1 ε 1 2 log r 1 log z + 1 ε (28) Similarly, for any i 2, we obtain a uniform estimates for all vectors R: s i log ρ i s i e i 2 log a2 n i+1 r i ( log z s i log a ni+1 + ε i m + 1 ) s i μ 2 s i log a ni+1 + log z (29) Now note that log(s i + 1) log(μ + 1) + (t 1) log(m + 1). (30) Thus, estimates (26) (30) give the universal bound ℵ ( Ea n ) s i log a ni+1 + c log(s i + 1) + log z + log N + 1 s i log a ni+1 + c 1 log(μ + 1) + c 2 log θ + C(t 1) log(m + 1). (31)
8 1674 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) For m = 1, we have t 2, and bound (13) readily follows from (31). For m 2 and K, let η = η(k) be a positive function such that ( ) lim η = 0, log a n ηk K log ρ = o. (32) log m Obviously, our original condition (12) can be replaced with (32) if function η approaches 0 slowly enough. We then choose the intervals J i of length m log m s i s =, i = 2,...,t 1, ηk s t s. Then ηk t 1 = m/s, log m (t 1) log(m + 1) 2ηK + log(m + 1). (33) Finally, note that restriction (32) can be rewritten as ( ηk 2 ) log a n = o. m log m Then the first term in (31) gives n s i log a ni+1 log a j + (s i 1) log a n i+1 a n K + (s 1) log a ni +1 a n m+1 j=n m = K + o(k). The latter bound combined with (31) and (33) gives our main estimate (13), and the proof of Theorem 4 is completed. Proof of Lemma 6. Assume that n unit balls cover Ea n. Then the centers of the balls belong to some hyperplane H in R n. First, suppose that H does not contain the longest axis x n. Consider the orthogonal line OA H from the origin O that crosses the surface of Ea n at some point A. Since a n > 1, this point A is located at the distance d(a,h) > 1 and is not covered by any unit ball. Secondly, let H contain the axis x n. We take any positive parameter λ< a n 1 (34) a n + 1 and consider L n + 1 points O i = ( 0,...,0,λ i), i = 1,...,L, on the axis x n. Next, we consider the L lines O i A i H, all orthogonal to H. Here the points A i belong to the surface of Ea n. For each A i, let D i H be the center of its covering ball B(D i ). By the definitions of the ellipsoid Ea n and the unit ball B(D i), the Euclidean distance d(x,y) satisfies the following: d 2 (O, O i )/an 2 + d2 (O i,a i ) = 1, d 2 (O i,d i ) + d 2 (O i,a i ) 1.
9 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) Thus, d(o i,d i ) d(o,o i )/a n = λ i /a n. Then for any two indices i<j, d(o i,d i ) + d(o j,d j ) λi + λ j λ i 1 + λ < 2λi a n a n a n + 1, in accordance with restriction (34). On the other hand, d(o i,o j ) = λ i λ j λ i (1 λ) > Now we see that 2λi a n + 1. d(o i,d i ) + d(o j,d j )<d(o i,o j ), (35) and no two points A i and A j can have the same center D = D i = D j, by contradiction to (35). Thus, no L 1 separate balls with centers on the same hyperplane H can cover all L points A i (despite the fact that these points have vanishing distance if n or a n 1). This contradiction shows that the centers D i may not belong to the same hyperplane and there are more than n centers needed for complete covering. Note that similar arguments can also be extended to the case when some coefficients a i are less than 1. In particular, Lemma 6 can be verified for any a n > 1 and any parameter c (1,a n ) if we take an ellipsoid En a, where a1 2 = =a2 n 1 1 λ2n+1 (c 1) 2an 2, λ= a n c a n + c Concluding remarks Replacing an original ellipsoid with direct products of the balls was first used in [8]. Present design differs in the following aspects. Firstly, exponentially declining steps are now used instead of the uniform quantization of [8]. Secondly, different approximation grids are applied to different positions. Finally, for each vector R, our radii e i are specified directly. Instead of this, nonconvex optimization was performed in [8] to find the worst vectors R, which give the highest contribution to the entire entropy. Our design includes two parts. In the first, approximation, part, we increase and round off the original quantities r i employed to cover the axes of an ellipsoid. To minimize the overhead caused by this expansion, the approximation grids {P i } have to be stretched to a very low level 1/z. These grids should also have sufficient density (to yield small approximation errors) and sufficiently small size (to avoid prohibitively many subsets D n R ). Exponentially declining levels { r i} resolved these problems. In the second, covering, part, we have to compensate for the increase in radius e 0 employed on the first interval J θ,0. This problem is addressed by using a small multiplying step e θ/2 in the first grid P 0. To obtain sufficiently small approximation errors r i /ε i on the remaining intervals, we define the radii e i through the second set of approximation levels { r i }. In turn, this double approximation allowed us to meet the restrictions of Theorem 4 and Corollary 5. Acknowledgment The author thanks V. Prelov for helpful suggestions.
10 1676 I. Dumer / Journal of Combinatorial Theory, Series A 113 (2006) References [1] A.N. Kolmogorov, V.M. Tikhomirov, εentropy and εcapacity, Uspekhi Mat. Nauk 14 (1959) [2] C.A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963) [3] J.L. VergerGaugry, Covering a ball with smaller equal balls in R n, Discrete Comput. Geom. 33 (2005) [4] K. Böröczky Jr., Finite Packing and Covering, Cambridge Tracts in Math., vol. 154, Cambridge Univ. Press, Cambridge, [5] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, SpringerVerlag, New York, [6] G. Fejes Tóth, New results in the theory of packing and covering, in: P.M. Gruber, J.M. Wills (Eds.), Convexity and Its Applications, 1983, pp [7] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, Cambridge Univ. Press, Cambridge, [8] I. Dumer, V. Prelov, M. Pinsker, On the coverings of an ellipsoid in the Euclidean space, IEEE Trans. Inform. Theory 50 (10) (2004)
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