On the Size of Some Trees Embedded in R d

Transcription

1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE On the Size of Some Trees Embee in R Pero Machao Manhães e Castro Olivier Devillers inria , version - 8 Jan N 779 Janvier apport e recherche ISSN ISRN INRIA/RR FR+ENG

2 inria , version - 8 Jan

3 On the Size of Some Trees Embee in R Pero Machao Manhães e Castro, Olivier Devillers Thème : Computational Geometry, Geometric Graphs Équipe-Projet Geometrica Rapport e recherche n 779 Janvier 3 pages inria , version - 8 Jan Abstract: This paper extens the result of Steele [6, 5] on the worst-case length of the Eucliean minimum spanning tree EM ST an the Eucliean minimum insertion tree EMIT of a set of n points S R. More precisely, we show that, if the weight w of an ege e is its Eucliean length to the power of α, the following quantities e EMST w(e an e EMIT w(e are both worst-case O(n α/, where is the imension an α, < α <, is the weight. Also, we analyze an compare the value of e T w(e for some trees T embee in R which are of interest in (but not limite to the point location problem []. Key-wors: Computational Geometry, Geometric Graphs This work is partially supporte by ANR Project Triangles an Région PACA. Centre e recherche INRIA Sophia Antipolis Méiterranée 4, route es Lucioles, BP 93, 69 Sophia Antipolis Ceex Téléphone : Télécopie :

4 Sûr la Taille e Quelques Arbres Plongées ans R Résumé : Ce papier éten le résultat e Steele [6, 5] sûr la taille au pire es cas e la plus petite arbre e couverture minimal Eucliienne et l arbre insertion minimal un ensemble e n points S R. Plus précisément, nous émontrons que si le pois w une arête e est sa longueur Eucliienne à la puissance α, les quantités suivantes e EMST w(e et e EMIT w(e valent au pire es cas O(n α/, où est la imension et α, < α <, est le poi. Nous éterminons et comparons aussi la valeur e e T w(e pour es arbres T plongées ans R, qui sont intèrêt au problème e la localisation es points []. Mots-clés : Géométrie Algorithmique, Graphes Géométriques inria , version - 8 Jan

5 On the Size of Some Trees Embee in R 3 Introuction inria , version - 8 Jan This paper extens the result of Steele [6, 5] on the worst-case length of the Eucliean minimum spanning tree EM ST an the Eucliean minimum insertion tree EMIT of a set of n points S R. More precisely, we show that, if the weight w of an ege e is its Eucliean length to the power α, the following quantities e EMST w(e an e EMIT w(e are both worst-case O(n α/, where is the imension an α, < α <, is the weight. Also, we analyze an compare the value of e T w(e for some trees T embee in R which are of interest in (but not limite to the point location problem []. Let S {p i, i n} be a set of points in R an G (V, E be the complete graph such that the vertex v i V is embee on the point p i S; the ege e ij E linking two vertices v i an v j is weighte by p i p j α, its Eucliean length to the power of α. G is usually referre to as the geometric graph of S. We will enote the sum of the weight of the eges of G by G α (or G if α. We will also refer to G α as the weighte-length of G. Consier the following trees: (i A star is a tree having one vertex that is linke to all others (see Figure a. (ii A path is a tree having all vertices of egree but two with egree (see Figure b. (iii Among all the trees spanning S, a tree with the minimal length is calle an Eucliean minimum spanning tree of S an enote EMST (S (see Figure c. (iv Consier that an orering is given by a permutation σ, vertices are inserte in the orer v σ(, v σ(,..., v σ(n. We buil incrementally a spanning tree T i for S i {p σ(j S, i j} with T {v σ( }, T i T i {v σ(i v σ(j } an a fixe k, with k < n, such that v σ(i v σ(j has the shortest length for any max (, i k j < i. This tree is calle the Eucliean minimum k-insertion tree, an will be enote by EMIT k (S (see Figure e; when k n, we will write EMIT (S (see Figure. EMIT (S epens on σ an for some permutations it coincies with EM ST (S. It is noteworthy that both the combinatorics of EMST (S an of EMIT (S are invariant to α (since f(λ λ α, is monotonically increasing for positive α an λ. What changes is the sum of the weights associate with the eges. Also, EMST assumes that the position of all the points in S is available (static whereas EM IT clearly oes not (ynamic. p 4 p 3 c p 7 p 6 p 5 p p p 8 (a Star p 4 p 3 p 5 p 6 p 7 p p p 8 (b Path p 4 p 3 p 5 p 6 p 7 p p p 8 (c EMST p 4 p 3 p 5 p 6 p 7 p p p 8 ( EMIT p 4 p 3 p 5 p 6 p 7 p p p 8 (e EMIT Figure : Trees embee in the plane. Steele proves [4] that if p i are i.i.. ranom variables with compact support, then EMST (S α O(n α/ with probability. For the extreme case of α, Alous an Steele [] show that EMST (S O( if the variables RR n 779

6 4 P. e Castro & O. Devillers inria , version - 8 Jan above are evenly istribute in the unit cube. Without any epenence on probabilistic hypotheses, Steele proves [6] that the complexity of EM ST (S is boune by O(n / in the worst case. Finally, the asymptotic length of EMIT (S is shown to be the same as the one of EMST (S [5]. In other wors, EMIT (S O(n /. This result is surprising because it means that a priori knowlege of S oes not affect the asymptotic length of trees following the greey strategy of an EMST. This fact has application in the ynamic point location problem []. This work is presente as follow: First, in Section, we exten the result of Steele [5] stating that EMIT an EMST have the same asymptotical behavior to the case of EMIT α an EMST α for < α < as well. Secon, in Section 3, we obtain the expecte weighte-length of some stars of interest insie the unit ball. Then in Section 4 we obtain some ratios between the expecte weighte-length of such stars an ranom paths in the unit ball. Finally, in Section 5 we obtain bouns on the expecte weighte-length of EMIT k. Weighte Eucliean Minimum Insertion Tree We exten the result of Steele [5] for EMIT α (an consequently EMST α with the following theorem: Theorem. Let S be a sequence of n points in [, ], then EMST (S α EMIT (S α γ,α n α/, with an < α <. Where, γ,α + 4 / ( α (/α. The proof of Theorem follows exactly the same line as Steele [5], an starts with the two lemmas below. Given a fixe sequence S {p, p,..., p n } of points in R, then we can buil a spanning tree for S by sequentially joining x i to the tree forme by {p, p,..., p i } for < i n. Let w i R be efine as follow: w i min p i p j α, ( j<i then w i is the minimal cost of joining p i to a vertex of a spanning tree of {p, p,..., p i }. Now, we have that EMIT (S α <i n w i. Lemma. If {p, p,..., p n } [, ] an w i min p i p j α, for < i n, j<i an < α <, then for any < λ < we have w /α i 8 /. ( λ w i< α λ Proof. Let C {i : λ w i < α λ} an for each i C let B i be a ball of raius r i 4 w/α i with center p i. We will argue by contraiction that B i B j for all i < j. If B i B j, then the bouns r i α λ an r j α λ gives us p i p j 4 (w i /α + w j /α < λ /α. (3 But, by efinition of w j we have p i p j α w j for all i < j, which implies p i p j w /α j for all i < j; an, by the lower boun on the summans in Eq.( INRIA

7 On the Size of Some Trees Embee in R 5 we have λ w j, which means λ /α w /α j, so we also see p i p j λ /α. Since p i p j λ /α contraicts Eq.(3, we have B i B j. Now, since all of the balls B i are isjoint an containe in a sphere with raius, the sum of their volumes is boune by the volume of the sphere of raius. Thus, if ω enotes the volume of the unit ball in R, we have the boun ω w /α i 4 ω / from which Eq.( follows. i C Lemma ( 3. Let Ψ be a positive an non-increasing function on the interval, ], then for any < a < b, with an < α <, a w i b w i (/α+ Ψ(w i b α α 8 / a/ α Ψ(λλ. (4 inria , version - 8 Jan Proof. By Lemma we have for any < λ <, w /α i I(λ w i < α λ 8 /, where a w i<b I(λ w i < α λ I( α w i λ < w i is the inicator function. If we multiply by Ψ(λ an integrate over [ a, b], we α fin wi b (/α w i Ψ(λλ 8 / Ψ(λλ. (5 w i/ α a/ α a w i b Since Ψ is non-increasing, the integran on the left-han sie of Eq.(5 is boune from below by Ψ(w i, so Ψ(w i w i ( w i α w i/ Ψ(λλ, an Eq.(4 α follows from Eq(5. Proof of Theorem. Divie the set {w, w 3,..., w n } in two sets R {w i : w i n α/ } an R {w i : w i > n α/ }. We have the trivial boun [ w i w i n n α/ n α/. Now, let [a, b] n α/, ], w i R w i n α/ Ψ(λ λ /α in Eq.(4, then we have: w i n α/ w i ( α α 8 / (/α α n α/ (/α /. An hence, n w i i w i R w i w i R w i + 4 / ( α (/α n α/. Now, we have that w i R w i ( 4 / + ( α n α/, (/α from which Theorem follows. The constant is γ,α + 4 / ( α (/α. RR n 779

8 6 P. e Castro & O. Devillers 3 Two Stars Embee in R Assume a set of points S {p i, i n}, evenly istribute insie the unit ball B. Let c be a point in B, an E( cp α, p B be the expecte value of the istance between c an a point insie the unit ball to the power of α, with α >, as n. The following theorem istinguish two stars of particular interest. Theorem 4. The shortest an largest expecte weighte-length stars insie the ball are respectively: the star centere at O, the center of the ball; an the star centere at Ω, a point on the bounary of the ball, enote by H. inria , version - 8 Jan We have that lim n i n p i c α E( cp α, p B. n In other wors, lim n E( cp α, p B is the weighte-length of an ege of the star centere in c as n. Now, let us turn the attention to the so-calle weighte Steiner star of S, the star with the smallest weighte-length we can imagine of S. Consier that the weighte Steiner star of the first n points is centere at c n, then we have n p i c n α i n p i c α, (6 for any n > an c B. Let A n (c n i p i c α /n. The proof of Theorem 4 is ivie into several lemmas. First, we show that A n (c n converges. Then we show that there is a point c B such that A n (c an A n (c n converge to the same limit. Finally, we argue that c must be the center of the ball, the point O. Lemma 5. Let A n (c n i p i c α /n an c n be the center of the weighte Steiner star of the first n points A n (c n. Then A n (c n converges. i Proof. As points are evenly istribute in B, c n is a non-constant sequence with probability, an thus we can assume that < A n (c n < α for any n > an. We have from Eq.(6 A n+ (c n+ A n+ (c n n p i c n α i n n i p i c n α n + + p n+ c n α. n + p n+ c n α n + But p n+ c n, an thus A n+ (c n+ A n (c n + α /n, which means A n+ (c that lim n+ n A n (c. Analogously, we have, from the fact that p n+ n c n+ an Eq.(6, A n (c n ( n+ n An+ (c n+ α n. This means that A n+ (c lim n+ A n+ (c n A n (c, an thus lim n+ n n A n (c. As < A n (c n < α for n any n > an, A n (c n converges. INRIA

9 On the Size of Some Trees Embee in R 7 Now, we nee the following easy lemma. Lemma 6. Assume a, b R, a, b, for each α > there is a function f α (b such that (a + b α a α + f α (b an lim b f α (b. Proof. For α, we have f α (b b α, as (a + b α a α + b α. For α an α N, we have f α (b α b, as (a + b α a α + b α i ( α a α i b i a α + α b. i Let, {x} signify x x, then for α > an α / N, we have f α (b α b {α} + α b, as inria , version - 8 Jan (a + b α (a + b α (a + b {α} ( a α + α b (a {α} + b {α} a α + α b {α} + α b. Lemma 7. There is a point c B such that A n (c an A n (c n converge to the same limit. Proof. If a topological space X is compact, then every infinite subset of X has an accumulation point. Assume c is an accumulation point of the sequence {c i } i,,...,, then we have a subsequence of inices ζ : N N such that {c ζ(i } i,,..., converges to c. Because of the triangulation inequality an by irect application of Lemma 6, for any i >, we have that A ζ(i (c A ζ(i (c ζ(i + f α( cc ζ(i. As cc ζ(i converges to, then f α( cc ζ(i also converges to (see Lemma 6. An thus lim A ζ(i (c lim A ζ(i (c ζ(i. Therefore, as A i i n(c n converges, lim A ζ(i(c ζ(i lim A n(c n an lim A n(c lim A ζ(i (c lim A i n n i n(c n. n Proof of Theorem 4. By symmetry, lim A n(o lim A n(c for any c B, n n an from Eq.(6 an Lemma 7, we have that lim A n(o lim A n(c n n n lim A n(c lim A n(o. Therefore, at the limit, we have that the length of n n the weighte Steiner star is equivalent to the length of the star centere at O. With analogous arguments, we have that the largest weighte-star an a star centere at the bounary of the ball have equivalent length. Denote the shortest an largest expecte weighte-length stars insie the ball by S an H respectively. Let E( Op α, p B an E( Ωp α, p B be the expecte value of an ege of S an H respectively, then as n the size of S an H are given accoringly by n E( Op α, p B an n E( Ωp α, p B. We analyze in the sequel the values of E( Op α, p B an E( Ωp α, p B. RR n 779

10 8 P. e Castro & O. Devillers Theorem 8. When points are uniformly i.i. in a ball, the expecte size E( Op α, p B of an ege of the star centere at the center of the unit ball, with positive α, is given by: (. + α Proof. Let B l be a ball with raius l centere at the origin, we have E( Op α, p B l α P rob(p B l+l \ B l l +α l + α, l α (V (l/ll V ( where V (l is the volume of a ball of raius l (an V (l/l is its area. inria , version - 8 Jan Theorem 9. When points are uniformly i.i. in a ball, the expecte size E( Ωp α, p B of an ege of the star centere at the bounary of the unit ball, with positive α, is given by: +α ( + α + α B ( +, + + α B ( +,, where B(x, y λx ( λ y λ is the so-calle Beta function. The computation of the average is more involve than in Theorem 8, an we split the computation into several lemmas. Lemma. Consier the spherical cap H h forme by crossing a ball B R with raius R centere at the origin, with the plane x R h. Denote h the height of the cap. The volume of H h is the volume of the intersection between the half-space x R h an B R. This volume is given by: R π Γ( + arccos ( R h R sin (λλ. (7 Proof. The volume V (r of a ball with raius r in imension is given by r π /Γ( +. Each cross-section x R h + δ, δ h is a ( - imensional ball. If we integrate all of those balls along the x axis, we have R R h V ( R t t. Eq.(7 follows from replacing t by λ R cos (t. Lemma. Let Ω be a point on the bounary of the unit ball B unit, an P H (l P rob( Ωp l ; p B unit be the cumulative istribution function of istances between an uniformly istribute ranom point insie B unit an Ω, then ( arccos ( l / arccos (l/ P H (l B ( +, sin (λλ + l sin (λλ, where B(x, y λx ( λ y λ is the Beta function. Proof. If we enote B l the ball of raius l centere in Ω, the esire probability is clearly volume(b l B unit /volume(b unit. B l B unit is the union of two spherical caps limite by the plane x l / which can be compute using Lemma. INRIA

11 On the Size of Some Trees Embee in R 9 Proof of Theorem 9. The theorem follows from: E( Ωp α, p B l α l α P H(ll ( l l 4 B ( +, + l arccos (l/ sin (λλ B ( +, l inria , version - 8 Jan l +α ( l 4 B ( +, l + l +α arccos (l/ sin (λλ B ( +, l The right part of the expression above correspons exactly to the expecte value of l α where l is the length of a ranom segment etermine by two evenly istribute points in the unit ball [3, 7]. Its value is given by: l +α arccos (l/ sin (λλ ( B ( +, l +α + α The left part of the expression can be obtaine as follows: Finally, we have l +α ( l 4 B ( +, B ( +, + + α B ( +,. l +α y ( +α y y B ( +, +α z + α ( z z B ( +, +α B ( +, + + α B ( +,. E( Ωp α, p B +α ( + α + α B ( +, + + α B ( +,. 4 Some Ratios We may ask now what is the value of the ratio ρ(, α between E( Ωp α, p B an E( Op α, p B. It is an easy exercise to verify that ρ(, α α. In Corollary, we compute lim ρ(, α. Corollary. The ratio ρ(, α E( Ωp α, p B/E( Op α, p B when is given by α/. Proof. Computing ρ(, α with Theorems 8 an 9 gives: ( ( + α B ρ(, α +α +, + + α B ( +, (8 RR n 779

12 P. e Castro & O. Devillers Using the Stirling s ientities: B(a, b π aa b b, a, b, (9 (a + b a+b inria , version - 8 Jan we have: { ( + α lim ρ(, α lim +α +α π ( lim ( π + + α B(a, b Γ(ba b, a b >, ( α/ lim { ( B + + α, + } + α/ e α/4 e α/4 α/ ( B ( +, + + α + + α ( + + α ( + + ( + + α +α ( + + α + ( + + α +α If we consier a tree which is a ranom path in the unit ball, then the average size of its ege is given by the expecte weighte-length of a ranom segment etermine by two evenly istribute points in the unit ball. This is given by: +α ( + α B ( +, + + α } B ( +, ( The reaer may refer to Tu an Fischbach [7] for a proof. From Theorem 8, Corollary an Eq.( we obtain the following corollary: Corollary 3. The ratio between the weighte-length of a ranom path in the unit ball an the weighe-length of a star centere at the center of the unit ball is α +α/ when an α/ when. From Theorem 9 an Eq.( we obtain the following corollary: Corollary 4. The ratio between the weighte-length of a star centere at the bounary of the unit ball an the weighte-length of a ranom path in the unit ball is given by +α. 5 Weighte Eucliean Minimum k-insertion Tree for Ranom Points Now, we will compute a boun on the expecte weighte-length of an ege of EMIT k α for points evenly istribute in the unit ball. Theorem 5. When points are uniformly i.i. in a ball, the expecte length E(length of an ege of EMIT k α, with positive α, verifies: ( α ( B k +, α ( E(length α α ( B k +, α, ( where B(x, y λx ( λ y λ is the Beta function. INRIA

13 On the Size of Some Trees Embee in R First, we will evaluate the weighte-istance between the origin an the closest amongst k points {p, p,..., p k } evenly istribute in the unit ball. This provies the lower-boun. Then, we will fin an upper-boun on the weighte-istance between any point insie the ball an the closest amongst k points {p, p,..., p k } evenly istribute in the unit ball. Lemma 6. Let c be a point insie the unit ball, P rob ( cp l P c (l be the probability that the istance between a point p B an c is less or equal to l, an P c,k (l P rob (min( cp j j k l be the cumulative istribution function of the minimum istance among k points following a uniformly i.i. insie the unit ball an c, then P c,k (l ( P c (l k. Proof. inria , version - 8 Jan P c,k (l P rob (min( cp j j k l P rob ( cp j > l, j k P rob ( cp > l k ( P c (l k. A irect consequence of Lemma 6 is the following corollary. Corollary 7. Let P B,k (l P rob (min( Op j j k l be the cumulative istribution function of the minimum istance among k points following a uniformly i.i. insie the unit ball, an the center of the unit ball, then P B,k (l ( l k. Lemma 8. The expecte value E (min( Op j α j k of the minimum weighteistance among k points following a uniformly i.i. insie the unit ball an the center of the unit ball, with positive α, is given by E (min( Op j α j k ( α B ( k +, α. Proof. Using Corollary 7, we have: E (min( Op j α j k k ( α B l α P B,k(ll k λ α/ ( λ k λ kb ( k +, α. l +α ( l k l ( k, + α Now, we shall obtain the upper-boun, which is more involve. First we will obtain a general expression for the expecte value of min( cp j α j k. Assume δ(c + Oc in what follows. RR n 779

14 P. e Castro & O. Devillers Lemma 9. The expecte value E (min( cp j α j k of the minimum weighteistance among k points following a uniformly i.i. insie the unit ball an c, with positive α, is given by E (min( cp j α j k δ(c αl α ( P c (l k l. Proof. As P c ( an P c (δ(c, integration by parts gives us the following ientity: inria , version - 8 Jan δ(c δ(c α l α P c(lp c i (l δ(c αl α Pc(l i, i >. (3 i From Lemma 6, we also have the following expression for E (min( cp j α j k : E (min( cp j α j k δ(c k i Replacing Eq.(3 in Eq.(4 leas to: kl α P c(l( P c (l k ( i ( k i k ( k δ(c ( i kl α P i c(lpc(ll i i δ(c kl α P c(lp c(ll. i (4 k ( k δ(c ( i αl α P i c(ll i k ( k αl α ( i P i i c(ll i δ(c δ(c i αl α ( P c (l k l. Proof of Theorem 5. Lemma 8 gives us the lower-boun in Theorem 5. Now, if we take a function Ψ(l such that Ψ(l P c (l for l δ(c, it upperbouns the integral in Lemma 9. Take Ψ(l (l/δ(c, then we have: E (min( cp j α j k δ(c δ(c δ(c αl α ( P c (l k l αl α ( Ψ(l k l ( k αl α l δ(c l αδ(c α λ α ( λ k λ ( δ(c α α y (α/ ( y k y ( δ(c α α ( B k +, α. INRIA

15 On the Size of Some Trees Embee in R 3 An thus, δ(c (c on the bounary maximizes the value above. completes the proof. This Remark: The results obtaine here for evenly istribute points in the unit ball (expecte values, ratios, bouns so far are vali for any positive α. Then, we have that the expecte weighte-length of the i-th step s i of EMIT in the unit ball for evenly istribute point is i + B (i +, s i B (i +, i +. Evaluating for n steps leas to inria , version - 8 Jan Ω(log n n+ i n+ i EMIT i i O(log n. Unlike EMST of points evenly istribute insie the unit cube, which converges to a constant as n [], the expecte value of EMIT of points evenly istribute insie the unit ball is Θ(log n. With the same argument, for k > we have that the expecte EMIT k Θ ( log k + n k. References [] D. Alous an J. M. Steele. Asymptotics for eucliean minimal spanning trees on ranom points. Probab. Theory Relate Fiels, 9:47 58, 99. [] Pero Machao Manhães e Castro an Olivier Devillers. Self-Aapting Point Location. Technical report. [3] Luis A. Santaló. Integral Geometry an Geometric Probability. Aison- Wesley, 976. [4] J. M. Steele. Growth rates of eucliean minimal spanning trees with power weighte eges. AnnProb, 6: , 988. [5] J. M. Steele. Cost of sequential connection for points in space. Operations Research Letters, 8:37 4, 989. [6] J. M. Steele an T. L. Snyer. Worst-case growth rates of some classical problems of combinatorial optimization. SIAM J. Comput., 8:78 87, 989. [7] Shu-Ju Tu an Ephraim Fischbach. A new geometric probability technique for an n-imensional sphere an its applications to physics. arxiv.org Mathematical Physics,. RR n 779

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