On stars and Steiner stars

Transcription

1 On stars and Steiner stars Adrian Dumitrescu Csaba D. Tóth Guangwu Xu March 9, 9 Abstract A Steiner star for a set P of n points in R d connects an arbitrary point in R d to all points of P, while a star connects one of the points in P to the remaining n points of P. All connections are realized by straight line segments. Let the length of a graph be the total Euclidean length of its edges. Fekete and Meijer showed that the minimum star is at most times longer than the minimum Steiner star for any finite point configuration in R d. The supremum of the ratio between the two lengths, over all finite point configurations in R d, is called the star Steiner ratio in R d. It is conjectured that this ratio is / = in the plane and /3 = in three dimensions. Here we give upper bounds of.363 in the plane, and.3833 in 3-space. These estimates yield improved upper bounds on the maximum ratios between the minimum star and the maximum matching in two and three dimensions. We also verify that the conjectured bound / in the plane holds in two special cases. Our method exploits the connection with the classical problem of estimating the maximum sum of pairwise distances among n points on the unit sphere, first studied by László Fejes Tóth. It is quite general and yields the first non-trivial estimates below on the star Steiner ratios in arbitrary dimensions. We show, however, that the star Steiner ratio in R d tends to as d goes to infinity. As it turns out, our estimates are related to the classical infinite Wallis product: = n= Introduction ( n n = Let P be a set of n points in R d. A Steiner star for P connects an arbitrary point in R d to all points of P, while a star connects one of the points in P to the remaining n points of P. All connections are realized by straight line segments. The study of minimum Steiner stars and minimum stars is motivated by applications in facility location and computational statistics [7, 8, 9, ]. The Weber point (or Weber center, also known as the Fermat-Torricelli point or Euclidean median, is the point of the space that minimizes the sum of distances to n given points in R d. It is known that even in the plane, the Weber point cannot be computed exactly, already for n 5 [5, ]. For n = 3 and, resp., Torricelli and Fagnano gave algebraic solutions. The Weber point can however be approximated with arbitrary precision [8, 9], based on Weiszfeld s algorithm [9]. The problem of finding such a point can be asked in any metric space. The Fermat-Torricelli problem has been also studied in Minkowski spaces [8]. More information on the problem can be found in []. Preliminary versions of this article appeared in the Proceedings of the 9th and th ACM-SIAM Symposia on Discrete Algorithms [, 3]. Department of Computer Science, University of Wisconsin-Milwaukee, WI 53-78, USA. ad@cs.uwm.edu Supported in part by NSF CAREER grant CCF-88. Department of Mathematics, University of Calgary, AB, Canada TN N. cdtoth@ucalgary.ca Supported in part by NSERC grant RGPIN Cs. D. Tóth s research was conducted at the Massachusetts Institute of Technology. Department of Computer Science, University of Wisconsin-Milwaukee, WI 53-78, USA, gxuuwm@uwm.edu Supported in part by National 973 Project of China (No. 7CB8793.

2 Ratio Lower bound Old upper bound New upper bound ρ : L(S min /L(SS min = = ρ 3 : L(S min /L(SS min 3 = = ρ : L(S min /L(SS min 5 = = ρ 5 : L(S min /L(SS min 35 = = ρ : L(S min /L(SS min... = η : L(S min /L(M max 3 = η 3 : L(S min /L(M max = Table : Lower and upper bounds on star Steiner ratios ρ,ρ 3,ρ,ρ 5,ρ, and matching ratios η,η 3 for some small values of d. Those marked with are new. We now introduce some definitions and notations, following [6]. Consider a finite point set P in R d. Let S min denote a minimum star, SS min denote a minimum Steiner star, and M max denote a maximum matching (for an even number of points. Let ρ d (P denote the maximum ratio between the lengths of the minimum star and the minimum Steiner star for P. Let ρ d (n be the supremum of ρ d (P over all point sets of size n in R d. Finally, let ρ d be the supremum of ρ d (n over all positive integers n. Obviously ρ d (n ρ d, for each n. Fekete and Meijer [6] were the first to study the star Steiner ratio. They proved that ρ d holds for any dimension d. It is conjectured that ρ = / =.73..., and ρ 3 = /3 = , which are the limit ratios for a uniform mass distribution on a circle, and a sphere, respectively (in these cases the Weber point is the center of the circle, or sphere [6]. For a given finite even-size point set P in R d, let η d (P denote the maximum ratio between the lengths of the minimum star and that of a maximum matching of P. Let η d (n be the supremum of η d (P over all point sets of size n in R d (n even. Finally, let η d be the supremum of η d (n over all positive even integers n. Obviously η d (n η d, for each even n. By using their bounds on ρ d, Fekete and Meijer also established nontrivial bounds on η d in two and three dimensions (d =,3. In this paper we make further progress and obtain better bounds on the ratios ρ d and η d, by exploiting the connection with the classical problem of estimating the maximum sum of pairwise distances among n points on the unit sphere, first studied by László Fejes Tóth [5]. We first prove that ρ.363, and ρ We also show that the conjectured bound / in the plane holds in two special cases, corresponding to the lower bound construction. Based on the above estimates on ρ and ρ 3, we then further obtain improved estimates on η and η 3, using the method developed by Fekete and Meijer [6]. Our improvements are summarized in Table. Finally, our method yields the first non-trivial estimates below on the star Steiner ratios in arbitrary dimensions. Among others, we show that the upper bound on the star Steiner ratio given by Fekete and Meijer is in fact a very good approximation for higher dimensions d. Thus in this sense the problem in the plane is the most interesting one, and the gap between the upper and the lower bound on the star Steiner ratio is the largest. Among n points, q,q,...,q n, on the unit sphere S d in R d, the sum of pairwise Euclidean distances is i<j q iq j. Let G(d,n denote the maximum of i<j q iq j over all n-element point sets {q,q,...,q n } S d. As in [, 3, ], define the constant of uniform density for the sphere in R d, c d, as the average distance from a point of S d to all other points of S d. That is, c d = S pq dq d, for d S dq d for any point p S d. For example, c =, c = /, and c 3 = /3 (more details in Section 3. The

3 sequence c d, d, is increasing and our Theorem shows that lim d c d =. The connection with this problem will be evident in the next section. Björck and others [6] have shown that G(d,n c d G(d, n n and lim n n = c d. ( László Fejes Tóth [5] gave a nice closed formula for G(,n for every n : G(,n = n tan n = n 6 + O ( n. ( The exact determination of G(d, n for d 3 is considered to be a difficult geometric discrepancy problem [3, pp. 98]. For d = 3, Alexander [, ] has shown that Stars in R d 3 n n / < G(3,n < 3 n. (3 A geometric graph G in R d is a pair (V,E where V is a finite set of points in R d, and E is a set of segments (edges connecting points in V. The length of G, denoted L(G, is the sum of the Euclidean lengths of all edges in G. Let P = {p,...,p n } be a set of n points in R d. Let SS min be a minimal Steiner star for P, and assume that its center c is not an element of P. As noted in [6], the minimality of the Steiner star implies that the sum of the unit vectors rooted at c and oriented to the points vanishes, that is, n =. cp i For completeness, we include here the brief argument. Fix an arbitrary orthogonal coordinate system in R d. Assume that p i = (p i,p i,...,p id, for i =,,...,n. The length of the star S x centered at an arbitrary point x = (x,x,...,x d is n n L(S x = xp i = d (x j p ij. If c = (c,c,...,c d is the Weber point, then all partial derivatives of L(S x vanish at x = c: L(S x x j = x=c n j= cp i c j p ij d j= (c j p ij =, for j =,,...,d. ( Our setup is as follows. Refer to Figure for an example in the plane. We may assume that the Weber point is not in P, since otherwise the star Steiner ratio is. Choose the coordinate system such that the Weber point is the origin o and the closest point in P to o is p = (,,...,, hence L(SS min = ( + δn, for some δ. For i =,,...,n, let q i be the intersection point of line segment op i and the unit sphere S d in R d. Then oq i = op i op i, that is, oq i is the unit vector corresponding to op i. Let a i = op i, b i = p q i, and a i = p p i, for i =,,...,n. We have L(SS min = n a i. Finally, let α ij [,] be the angle between the positive x j -axis and line segment op i. Henceforth ( can be rewritten in the more convenient form n cos α ij =, for j =,,...,d. (5 3

4 p i p j a i q j q i a i p i b i q i α i o p o p Figure : Left: estimating the length of a star centered at p = (, P. Right: estimating pairwise distances. Let S i be the star centered at p i, for i =,,...,n. By definition, we have L(S min = min L(S i. i n Using the local optimality condition (5, Fekete and Meijer [6] show that if one moves the center of SS min from the Weber point to a closest point of P, the sum of distances increases by a factor of at most, and this bound is best possible. It follows (see [6] for details that for any d, ρ d (n, thus ρ d. (6 From the opposite direction, by considering a uniform mass distribution on the unit sphere in R d, one has for any d : c d ρ d. (7 Our new argument in a nutshell is as follows. If δ is large, we consider the star centered at a point in P closest to the Weber point, as a good candidate for approximating the minimum star. If δ is small, we upper bound the length of the minimum star by the average of all n stars. In the end we balance the two estimates obtained. Applying the averaging argument (for small δ leads naturally to the problem of maximizing the sum of pairwise distances among n points on the unit sphere (or unit circle. Theorem Let d, and let c d denote the constant of uniform density for the sphere in R d. The star Steiner ratio in R d is bounded as follows: c d ρ d c d + c d. (8 Proof. Consider an n-element point set P in R d, and the setup above. It is enough to show that L(S min L(SS min cd +. c d

5 By the triangle inequality (see also Fig. (left, we have a i b i + a i, for i =,...,n. Hence n n n L(S = a i (b i + a i = b i + δn. (9 By Lemma in [6], the local optimality condition (5 implies n b i n. It follows that L(S ( + δn. Hence the star Steiner ratio is at most L(S L(SS min + δ + δ. ( Clearly, the sum of the lengths of the n stars (centered at each of the n points equals twice the sum of pairwise distances among the points. n By the triangle inequality (see also Fig. (right L(S i = i<j p i p j. p i p j p i q i + q i q j + q j p j = (a i + q i q j + (a j. By summing over all pairs i < j, and using the upper bound on G(d,n in (, we obtain n L(S i i<j n q i q j + (n i = (a q i q j + δ(n n (c d + δn. i<j The minimum of the n stars, S min, clearly satisfies: L(S min n L(S i, hence L(S min (c d + δn. ( n It follows that the star Steiner ratio is at most L(S min L(SS min c d + δ + δ. ( Inequalities ( and ( imply ( L(S min + δ L(SS min max min δ + δ, c d + δ. + δ Observe that +δ +δ is a monotonically decreasing function of δ, and c d+δ +δ is a monotonically increasing function of δ, for every < c d <. Their minimum is maximized for a value of δ satisfying +δ +δ = c d+δ +δ. The solution δ = c d balances the two upper estimates in ( and (, and it yields ρ d ρ d (n c d +. c d (3 This completes the proof of Theorem. By substituting the values of c d for d = and d = 3, one obtains: and ρ + <.363, 3 ρ 3 7 (6 3 <

6 . Minimum star to maximum matching ratios in two and three dimensions First consider d =. Fekete and Meijer [6] showed that L(SS min 3 L(M max. By our Theorem, L(S min + L(SS min. Combining the two upper bounds yields the following new upper bound on η. Corollary The maximum ratio η between the minimum star and the maximum matching in the plane is less than That is, for any finite planar point set L(S min L(M max The best known lower bound for this ratio, /3, is given in [6]. It is obtained by placing n/3 points on each vertex of an equilateral triangle (for n divisible by 3. Note that this ratio can be approached arbitrarily close using distinct points. Let now d = 3. Fekete and Meijer [6] showed that L(SS min L(M max. By our Theorem, L(S min 7 (6 3 L(SS min. Combining the two yields the following new upper bound on η 3. Corollary The maximum ratio η 3 between the minimum star and the maximum matching in R 3 is less than.956. That is, for any finite point set in R 3 L(S min L(M max 7 (6 3 = 7 (8 3 <.956. The best known lower bound for this ratio, 3/, is given in [6]. It is obtained by placing n/ points on each vertex of a regular tetrahedron (for n divisible by. Again, this ratio can be approached arbitrarily close using distinct points. 3 Asymptotic results and approximations Theorem Let c d be the constant of uniform density for the sphere in R d, for d. Then lim c d = lim ρ d =. d d The following closed formula approximations hold for d 3: e (d 3 ρ d e 5(d + e 5(d Proof. In order to establish the limits, we start by computing the constant of uniform density c d. Recall that c d equals the average distance from a point on the unit sphere in R d to all the other points on the same sphere. It is easy to verify that c d is given by the following integral formula: c d = / sin d (α sin α dα /, for. ( sin d (α dα 6.

7 Some initial values are c =, c = =.73..., c 3 = 3 = , c = 6 5 = , c 5 = 8 35 = In order to establish a recurrence on c d, define Some initial values are a ij = / sin i α cos j α dα, i,j. Expanding sin α yields then a =, a = a =, a =, a = a =. c d = a d,d a d,d. Recall that integration by parts leads to the well-known recurrence relations for a ij, for i,j : a ij = / sin i α cos j α dα = sini α cos j+ α i + j = sini+ α cos j α i + j / / + i i + j + j i + j / / Plugging these in the formula for c d immediately gives a recurrence for c d. For any d : d d+ d c d+ = d a d,d d d d d a d,d Recall at this point the infinite Wallis product [7]: Let = W n = k= ( k k n ( k k= k ( = d d c d = + d c d. = , and Z n = k=n+ ( k k sin i α cos j α dα sin i α cos j α dα. denote the partial finite and respectively partial infinite Wallis products, so that W n Z n = /, for every n. Our recurrence for c d yields that c d is an increasing sequence satisfying also c d+ c d+ = c c W d, for d. (5 Since c d is bounded, it converges to some limit c. The value of c can be obtained by solving the equation c = c c =., 7

8 We thus have lim d c d =. Since c d ρ d, we also have lim d ρ d =. From Equation (5, we also get that for d 3, c c W d c d c c W d or c c W d c d c c W d. (6 Observe that k=n+ k = k=n+ ( k = k + (n +. Standard inequalities e x/5 + x e x for x [,/3] now imply that for each n and Z n = Z n = k=n+ k=n+ Since W n = (//Z n, we have and consequently (6 gives ( + ( + k=n+ k ep k e P 5 k=n+ e (n+ W n e k = e (n+, k = e 5(n+. 5(n+, e (d 3 c d e 5(d, or equivalently e (d 3 c d e 5(d. Taking into account (8 and substituting the above upper bound on c d, we get the final estimate The proof of Theorem is now complete. e (d 3 ρ d e 5(d + e 5(d, for d 3. Verification of Conjecture in two special cases We show that the bound c = in R is best possible in two special cases, corresponding to the lower /n bound construction. Specifically, we show that the star Steiner ratio is at most tan(/n < (i for any finite point set on a circle centered at the Weber point in R, and (ii for any finite point set in the plane where the angles from the Weber center to the n points are uniformly distributed (that is, α i = i/n, for i =,,...,n. The two tight bounds in the plane essentially follow from the closed formula ( for G(,n due to László Fejes Tóth [5] and mentioned in the Introduction. We also give suitable generalizations of these two planar results for points in R d. Here we have chosen x/5 to simplify the resulting expressions. 8

9 Theorem 3 Let P = {p,p,...,p n } be set of n (not necessarily distinct points in R d such that their Weber point is the origin o, and a closest point to o in P is p, at unit distance from o. For i =,,...,n, let q i be the intersection point of line segment op i and the unit sphere S d in R d ; in particular, q = p. If the star over Q = {q,q,...,q n } centered at q has length n i= q oq i n G(d,n, then the star Steiner ratio of P is at most G(d,n c n d. Proof. Let a i = op i, for i =,,...,n, as in our general setup in Section. We have L(SS min = n a i = ( + δn = n + δn, for some δ. By the triangle inequality, p p i p q i + q i p i = q q i + (a i. Therefore, the length of the star over P centered at p is n n p p i n q q i + (a i G(d,n n + δn. It is easy to see that G(d,n n, or equivalently, n G(d,n n. Using this and the previous inequality, we deduce that the star Steiner ratio in this special case is: G(d,n/n + δn n + δn G(d,n n c d. Corollary 3 (i The star Steiner ratio for a set of n points in R d that lie on a sphere centered at the Weber center is at most G(d,n c n d. (ii In particular, the star Steiner ratio for a set of n points in the plane that lie on a circle centered at the Weber center is at most G(, n n = < = c. n tan n Proof. (i We may assume that the points in P lie on a unit sphere S d centered at the origin. The total length of the n stars for P is at most G(d,n, by the definition of G(d,n. By using the first inequality in (, there is a point p P such that the length of the star centered at p is at most n G(d,n. An application of Theorem 3 with p = p completes the proof. (ii By the closed formula (, we have G(,n = n tan, and the bound follows as in part (i. n Corollary The star Steiner ratio for a set P = {p,p,...,p n } of n points in the plane where point p i is visible from the origin under angle i/n, for i =,,...,n, is at most n tan n <. Proof. We may assume w.l.o.g. that p = (, is a closest point in P to the origin o. By equation (5, the Weber center of P is the origin o. For i =,,...,n, let q i be the intersection point of line segment op i and the unit circle S in R ; in particular, q = p. Note that the point set Q is symmetric under rotation by /n about the origin. It follows that every star of Q has the same length. The total length of all stars of Q is at most G(,n, by the definition of G(,d. Hence, the length of the star of Q centered at q is at most n G(,n. An application of Theorem 3 completes the proof. The earlier proof of this result in [] had an error. 9

10 5 Conclusion We think that the sharper upper bound on the star Steiner ratio in the plane in the two special cases (in Corollary 3 and Corollary always holds, so we venture here a slightly stronger version of the conjecture proposed by Fekete and Meijer [6]: Conjecture The star Steiner ratio for n points in the plane is ρ (n = n tan n Fekete and Meijer [6] conjectured that ρ = / and ρ 3 = /3, where these values coincide with c and c 3, respectively. We extend their conjecture to all dimensions d : Conjecture The star Steiner ratio in R d equals the constant of uniform density for the sphere in R d, that is, ρ d = c d for every d.. References [] R. Alexander: On the sum of distances between n points on a sphere, Acta Mathematica Academiae Scientiarum Hungaricae 3 (97, 3 8. [] R. Alexander: On the sum of distances between n points on a sphere. II, Acta Mathematica Academiae Scientiarum Hungaricae 9 (977, [3] R. Alexander, J. Beck, and W. Chen: Geometric discrepancy theory and uniform distribution, in J. Goodman and J. O Rourke (editors, Handbook of Discrete and Computational Geometry, pages 79 3, Chapman & Hall, second edition,. [] R. Alexander and K. Stolarsky: Extremal problems of distance geometry related to energy integrals, Transactions of the American Mathematical Society 93 (97, 3. [5] C. Bajaj: The algebraic degree of geometric optimization problems, Discrete & Computational Geometry 3 (988, [6] G. Björck: Distributions of positive mass which maximize a certain generalized energy integral, Arkiv för Matematik 3 (956, [7] V. Boltyanski, H. Martini, and V. Soltan: Geometric Methods and Optimization Problems, Kluwer Acad. Publ., 999. [8] P. Bose, A. Maheshwari, and P. Morin: Fast approximations for sums of distances, clustering and the Fermat-Weber problem, Computational Geometry: Theory and Applications (3, [9] R. Chandrasekaran and A. Tamir: Algebraic optimization: The Fermat-Weber problem, Mathematical Programming 6 (99, 9. [] E. J. Cockayne and Z. A. Melzak: Euclidean constructibility in graph-minimization problems, Mathematical Magazine (969, 6 8.

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