![Ô C2 − − 2 Most O(N ) [10], Where C2 =(1 Log2(1 + 1 Β ))/2 C1 −](http://data.docslib.org/img/fc55669c0e70b1cb2146111b20bc36b4-1.webp)
CCCG 2003, Halifax, Nova Scotia, August 11–13, 2003
The spanning ratio of β-Skeletons
Weizhao Wang∗ Xiang-Yang Li∗ Kousha Moaveninejad∗ Yu Wang∗ Wen-Zhan Song∗
Abstract disconnected, so we restrict our attention to the case that 0 ≤ β ≤ 2. As β approaches 0, N(u, v, β)ap- In this paper we study the spanning ratio of the β- proximates the line segment connecting u and v.Thus, skeletons for β ∈ [0, 2]. Both our upper-bounds and except in degenerate situations (three or more points co- lower-bounds improve the previously best known results linear), all point pairs are β-neighbors under this scheme [10, 12]. for β sufficiently small, which means that we can find a β to make the β-skeleton of S a complete graph.
1 Introduction For β ∈ [0, 1], the spanning ratio of β-skeletons is at Ô c2 − − 2 most O(n ) [10], where c2 =(1 log2(1 + 1 β ))/2 c1 −
Proximity graphs [1, 2, 3, 4] have been used exten- and at least Ω(n ) [12], where c1 =1 log5(3 +
Ô sively in various fields including pattern recognition, 2+2 1 − β2). For some special β-skeletons such as GIS( Geographic Information System), computer vi- Gabriel graph (GG) [1, 13, 10] (β = 1) and the Rela- sion, and neural network [5, 3]. The spanning ra- tive Neighborhood Graph (RNG) [4, 14, 3,√ 15] (β = 2), tios of the proximity graphs are of great interest to Bose et al. [10] gave a bound which is Θ( n) and Θ(n) many applications. For example, several results showed respectively. Since the spanning ratio increases over β that Delaunay Triangulation has a constant bounded ∈ π for β [1, 2], the spanning ratio of the beta-skeletons spanning ratio, which is at least [6] and at most 1 2 for β for β ∈ [1, 2] is at least Ω(n 2 ) and at most O(n), 2π/(3 cos(π/6)) 2.42[2]. which is also the best known result till now. As one of the proximity graphs, β-skeletons have been The contribution of this paper is: We first prove that, studied extensively in [8, 9, 10, 11]. Our main concern for β ∈ [1, 2], the β-skeletons have spanning ratio at in this paper is about the spanning ratio (or dilation) − γ − 1 most (n 1) , where γ =1 2 log2(µ2 + 1), µ2 = of the β-skeletons. Given a set S of n points in a two − 2 β . We then show that the Gabriel Graph has exact dimensional plane, two points u and v are β-neighbors β √ in S if N(u, v, β) contains no point other than u or v spanning ratio n − 1 and the Relative Neighborhood in S in its interior 1. The most common definition of Graph has exact spanning ratio n − 1. The spanning N(u, v, β) is so-called Lune-Based Neighborhoods,which ratio of β-skeletons for β ∈ [0, 1] is at most (n − 1)γ , 1 − 1 Ô − 2 is defined as follows. where γ = 2 2 log2(µ1 +1), µ1 = 1 β . Finally, we Case 1: β ≥ 1. N(u, v, β) is the intersection of the construct a point set whose β-skeleton, for β ∈ [0, 1], has β uv c3 1 − 1 µ1+1 two circles of radius 2 centered at the points p1 = spanning ratio n , where c3 = 2 2 log2(1 + 2 ), − β β β − β which improves the previously best known lower bound (1 2 )u + 2 v and p2 = 2 u +(1 2 )v, respectively. Case 2: 0 ≤ β ≤ 1. N(u, v, β) is the intersection of [12]. D(uv) the two circles of radius 2β passing through both u and v. 2 Upper Bound of Spanning Ratio Here uv is the Euclidian distance between u and v. The β-skeleton of a point set S is the set of edges joining Consider a geometry graph G =(V,E)overn points V . β-neighbors in S. When β = 1, the closed N(u, v, β) For each pair of points (u, v), the length of the shortest corresponds exactly to the Gabriel neighborhood of u path connecting u and v measured by Euclidean dis- and v. When β = 2, the open N(u, v, β) is the rela- tance is denoted by DG(u, v), while the direct Euclidean tive neighborhood of u and v.Asβ approaches ∞,the distance is uv. The spanning ratio (also dilation ra- neighborhood of u and v approximates the infinite strip tio or length stretch factor) of the graph G is defined by formed by translating the line segment (u, v) normal to DG(u,v) ψ(G) = maxu,v∈G uv . If the graph G is not con- itself. Notice when β>2theβ-skeleton graph can be nected, then ψ(G) is infinity, so it is reasonable to focus ∗Department of Computer Science, Illinois Institute of on connected graphs only. Technology, Chicago, IL. Email {wangwei4, wangyu1, moavkoo, songwen}@iit.edu, [email protected]. 1There are two possible interpretations of the interior: one 2.1 Fade Factor of β-skeletons includes the boundary which is called Closed Neighbor and the other excludes the boundary which is called Open Neighbor.We Our analysis of the upper bound of the spanning ra- always consider closed neighbor here. tio of β-skeletons rely on our definition of fade fac-
1 15th Canadian Conference on Computational Geometry, 2003
tor of β-skeletons, which is defined as follows. Given 1. Construct the root node corresponding to uv. a 2-dimensional point set S and its β-skeleton G(β), choose any pair of points u, v ∈ S.Ifuv ∈ G(β), there 2. If there is no point inside N(u, v, β) then stop. Oth- ∈ must exist a point w ∈ S other than u, v in N(u, v, β). erwise, assume a point u0 N(u, v, β). We put We say that the point w breaks edge (u, v) and define edge uu0 as uv’s left child and edge u0v as uv’s right child and label these two branches with their x = uw , x = vw as the two fade factors of uv 1 uv 2 uv fade factor x and x respectively. The leaf nodes by w. We then study the property of fade factors of 1 2 uu , u v form path uu v. β-skeletons, illustrated by Figure 1. 0 0 0 Case 1: β ∈ [1, 2]. In this case, w must lie in the 3. If we already have a binary tree with k leaf nodes shaded area N(u, v, β). For symmetry, we assume that p p , p p , ···, p − p , where p = u, v = p .Let ≥ 2 0 1 1 2 k 1 k 0 k wu wv , In triangles wup1 and wvp1,we S = {p ,p , ··· ,p }. For every point p ∈ S,we 2 2 2 − 1 0 1 k have uw = up1 + wp1 2 up1 wp1 cos α test if p breaks edge p p . We consider five cases and vw2 = vp 2 +wp 2 −2vp wp cos(π−α). i i+1 1 1 1 1 here. Consequently,
2 2 2 2 2 2 (a) If p doesn’t break any pipi+1 then continue to uw − up1 − wp1 uw − vp1 − wp1 + =0 try other points in S. up1 vp1 2 2 2 (b) If p ∈ S − S and p breaks a single edge x1 x2 1 wp1 2 1 1 ⇒ + = + ≤ . 2 − β β 2 uv 2 β(2 − β) 2 − β pipi+1 then similar to step (2), attach pip as the left child and ppi+1 as the right child of ≤ 2−β ≤ edge p p . Suppose that 0 µ2 = β 1. We have the relation i i+1 ∈ ≥ between the fade factors when β [1, 2] and x1 x2, (c) If p ∈ S − S1 and it breaks multiple edges, choose such broken edge prpr+1 with the min- x2 + µ x2 ≤ 1 (1) 1 2 2 imum index r and psps+1 with the maximum index s. Attach node prp to node prpr+1 and node pps+1 to node psps+1 in the tree. Mark p 1 all leaf nodes between prp and pps+1. If all de- w α scendant leaf nodes of an internal node have x1 x 2 marks, then also mark it. Delete all nodes α π−α u v with marks. p1 p2
(d) If p ∈ S ,sayp = p ,and it breaks single edge 1 j A C pipi+1.Ifj>i+ 1 then attach pipj to node θ x 1 x 2 p p , and mark all leaf nodes p p for B i i+1 m m+1 ≤ ≤ − (a) β ∈ [1, 2] (b) β ∈ [0, 1] i+1 m j 1. If j
2 CCCG 2003, Halifax, Nova Scotia, August 11–13, 2003
from left-to-right, we get a sequence of edges 1. If U(β,nr +1)≥ µ2U(β,nl + 1), we have
uE0,B1E1, ··· ,Bl−1El−1,Blv. Ô ψ(G) ≤ (U(β,nl +1)+U(β,nr + 1))/ 1+µ2
1 Ô γ 1−1+ log (1+µ2) Observation 1 Observations of fade factor tree. ≤ (nl + nr) · 2 2 2 / 1+µ2 ≤ (n − 1)γ = U(β,n) 1. For every 0 ≤ i ≤ l − 1, we have Ei = Bi+1, so the sequence can be written as u0u1,u1u2, ··· ,ul−1ul. 2. If U(β,nr +1)≤ µ2U(β,nl + 1), we have ψ(G) ≤ (u0 = u, ul = v). 2 1 2 U(β,nl +1) + U(β,nr +1) .Letf(x)= µ2 2. l ≤ n − 1,wheren is the number of total points in x2γ + 1 (n − x − 1)2γ . Differentiating f(x), we µ2 S. − − get f (x)=2γ[x2γ 1 − 1 (n − x − 1)2γ 1]. Since µ2 1/2 ≤ γ ≤ 1, f(x) reaches its minimum at 3. u0u1,u1u2, ··· ,ul−1ul corresponds to a simple n−1 ≥ point x0 = 1 , increases when x x0, path connecting u and v in the β-skeleton. 2γ−1 1+µ2 and decreases when 0 ≤ x ≤ x . Notice that We can show that the above algorithm terminates. 0 U(β,nr +1)≤ µ2U(β,nl + 1), which implies that For detail of the proof, see the full version of the paper. n−1 ≤ ≤ − xl = (1/γ) x n 2=xr.Itiseasytoshow µ2 +1 n−1 n−1 ≤ x − γ ≤ 2.3 Upper bound when β ∈ [1, 2] that 1/γ 0 = 1/(2γ 1) when 1and 1+µ2 1+µ2 µ2 ≤ 1. Previously, Bose et al. [10] gave an upper bound O(n) for β-skeletons when β ∈ [1, 2] from the fact that, for a Consequently, U(β,nl +1)x1 + U(β,nr +1)x2 reaches its maximum at point xl or xr = n − 2. point set S,theβ1-skeleton belongs to the β2-skeleton We can show that it reaches the maximum at point when β1 ≥ β2. They use the upper bound of the RNG (β = 2) as upper bound for β ∈ [1, 2]. We improve it to xl.Thus,
1 Ô γ ≤ · − γ γ γ U(β,n)=(n − 1) , ψ(G) 1+µ2 (n 1) /(µ2 +1) .
1 { − 1 } − Notice (µ γ +1)γ strictly increases over [0, 1] for γ. where γ = max 1 2 log2(µ2 +1),g(µ2) =1 2 1 2−β Thus 2 log2(µ2 + 1), µ2 = β ,andg(µ2) is the solution to
1 1 Ô g(µ2) 2g(µ2) ≤ · − γ γ γ the equation (µ2 +1) =1+µ2 (See appendix ψ(G) 1+µ2 (n 1) /(µ2 +1) 1 about the details of g(µ2)andγ). Ô ≤ · − γ g(µ2) g(µ2) γ Before presenting our proof, we list some simple re- 1+µ2 (n 1) /(µ2 +1) = n sults. If x1 ≥ x2 and subject to the constraint of (1), then for a, b ≥ 0, The upper bound we proved so far could be a loose
bound, and usually the β-skeletons cannot reach this 2 2 upper bound. But at some extreme cases, we can show max{ax1 + bx2} = a + b /µ2 ;ifb ≤ µ2a (3) x1,x2 that the upper bounds are indeed tight. When β =1,theβ-skeleton is the Gabriel Graph. Ô 2−β 1 max{ax1 + bx2} =(b + a)/ µ2 +1; ifb ≥ µ2a (4) Then µ2 = =1.Itiseasytoverifyg(1) = . x ,x β 2 1 2 Thus, Now we prove our upper bound by induction on n. 1 1 1 γ = max{1 − log (2), } = When n = 3, there are only three points u, v and w, 2 2 2 2 and suppose that uv is longest edge. If w doesn’t break In the following section, we construct an example such 1 uv, then ψ(G)=1≤ U(β). Otherwise, the relation of that GG has spanning ratio (n − 1) 2 . Consequently, we the fade factors from (1) implies have
− 1 1 2 log2(1+µ2) Theorem 1 The spanning ratio of Gabriel Graph is ex- ψ(G)=x1 + x2 ≤ 2 = U(β,3). 1 actly U(1,n)=(n − 1) 2 . ≤ Suppose for all k 3 15th Canadian Conference on Computational Geometry, 2003 An−1 An An−1 2.4 Upper bound when β ∈ [0, 1] β α β αn−2 Ô An−2 An−3 2 2 2 α The fade factors satisfy x + x +2x1x2 1 − β ≤ 1 α α Ô1 2 2 C n−2 when β<1. Let µ1 = 1 − β here. For symmetry, Õ √1 assume that x1 ≥ x2.Thus,0≤ x2 ≤ . 2+2 1−β2 If x1 ≥ x2 and subject to the constraint (2), then for A5 α a>0, b>0, 4 A4 α3 A3 A5 β α A4 α2 A2 β Ô α1 2 2 A1 A0 a + b − 2abµ1 α α α max{ax + bx } = if b ≥ aµ (5) A 1 2 1 A3 α 2 x1,x2 2 β − C 4 1 µ1 β C 3 C 2 α α α C 1 A1 A0 max{ax1 + bx2} = aµ1 if b When β ∈ [0, 1], we prove that the spanning ratio is Figure 2: Point sets that achieve the upper bounds of at most the spanning ratio. − 1 log2(1+µ1) U(β,n)=(n − 1) 2 , 1 We prove this bound similar to the case β ∈ [1, 2]. When 2. Let α = ∠A − A A . Then sin α = √ k k 1 k k+1 k n−k k = 3, it is easy to verify the correctness of the bound. and ∠αk ≤ ∠αk+1. For every 1 ≤ k ≤ n − 2, Suppose when k 4 CCCG 2003, Halifax, Nova Scotia, August 11–13, 2003 In this paper, we give a different construction that Table 1: Lower and upper bounds for spanning ratio achieves a better lower bound. Suppose that α = Ô of β-skeletons. Here the constants used are c =1− arccos( 1 − β2), and θ = π − α. Then for any n = √ 1 1 − 1 1 − k k log5(3 + 2+2µ1), c2 = 2 2 log2(1 + µ1), c3 = 2 2 +1, let P (β,k) be a path of 2 segment (defined along 1 µ1+1 − 1 our construction). Figure 3 illustrates our construction 2 log2(1 + 2 ), and c4 =1 2 log2(µ2 + 1). And Ô 2 of this β-skeleton for n points, which is described as µ1 = 1 − β and µ2 =(2− β)/β. ∈ ∈ follows. β (0, 1) β √=1 β √(1, 2) β =2 c OldLower Ω(n 1 ) Ω(√n) Ω( n) Ω(n) c2 1. If k = 1, construct a triangle ABC such that OldUpper O(n ) √O( n) O(n) O(n) π−θ π+θ √ ∠ ∠ ∠ c3 − − ABC = ACB = 4 ,so BAC = 2 . Then OurLower Ω(n ) √n 1 Ω( n) n 1 P (β,1) is segments BAC. Call segment BC the OurUpper O(nc2 ) n − 1 O(nc4 ) n − 1 supporting segment of P (β,1). 2. If k ≥ 1, first construct P (β,1) = BAC. Then This theorem can be enhanced such that we can con- construct two P (β,k−1), scale the supporting seg- struct examples for any integer n, but with a small con- ments to length AB, and align their supporting stant degradation of the spanning ratio. For Gabriel segments to AB and BC respectively. Notice there Graph, from previous result by Eppstein [12], we get a are two possible ways to place P (β,k−1), we should spanning ratio of Ω(nc)for0.077 5 15th Canadian Conference on Computational Geometry, 2003 ∈ { − 1 Theoretical Computer Science, vol. 262, no. 1, pp. 459– which means for µ2 [0, 1] γ = max 1 2 log2(µ2 + 471, 2001. } − 1 1),g(µ2) =1 2 log2(µ2 + 1). See full version of the [6] P.L. Chew, “There is a planar graph as good as the paper for arithmetic proof. complete graph,” in Proc. of the 2nd Symp. on Compt. 2 0.5 f(u) Previous low boundary 1.9 1+u Geometry, 1986, pp. 169–177. 0.45 Our low boundary Our up boundary 1.8 0.4 [7] D.P. Dobkin, S.J. Friedman, and K.J. Supowit, “De- 1.7 0.35 1.6 0.3 launay graphs are almost as good as complete graphs,” 1.5 0.25 Discrete Compt. Geometry, 1990. 1.4 0.2 1.3 0.15 [8] N. Amenta, M. Bern, and D. Eppstein, “The crust and 1.2 0.1 1.1 0.05 1 0 the β-skeleton: Combinatorial curve reconstruction,” 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Graphical models and image processing: GMIP,vol.60, (a) g(µ2) is unnecessary. (b) lower bounds for β ∈ [0, 1]. no. 2, pp. 125–135, 1998. ∈ [9] F. P. Preparata and M. I. Shamos, Computational Ge- Figure 4: (a) The upper bound for β [1, 2] can be ometry: an Introduction, Springer-Verlag, 1985. simplified. (b) Our lower bound for β ∈ [0, 1] is strictly better than previous result. [10] P. Bose, L. Devroye, W. Evans, and D. Kirkpatrick, “On the spanning ratio of gabriel graphs and beta- skeletons,” in Proceedings of the Latin American The- oretical Infocomatics (LATIN), 2002. [11] Xiang-Yang Li, Peng-Jun Wan, and Yu Wang, “Power efficient and sparse spanner for wireless ad hoc net- works,” in IEEE ICCCN, 2001. [12] David Eppstein, “Beta-skeletons have unbounded dila- tion,” Tech. Rep. ICS-TR-96-15, University of Califor- nia, Irvine, 1996. [13] D. W. Matula and R. R. Sokal, “Properties of gabriel graphs relevant to geographical variation research and the clustering of points in the plane,” Geographical Analysis, , no. 12, pp. 205–222, 1984. [14] J. Katajainen, “The region approach for computing relative neighborhood graphs in the lp metric,” Com- puting, vol. 40, pp. 147–161, 1988. [15] J.W. Jaromczyk and G.T. Toussaint, “Relative neigh- borhood graphs and their relatives,” Proceedings of IEEE, vol. 80, no. 9, pp. 1502–1517, 1992. 5 Appendix In subsection 2.3, we show that the β-skeletons, β ∈ [1, 2], have spanning ratio at most (n − 1)γ , where γ = { − 1 } − ∈ max 1 2 log2(µ2 +1),g(µ2) , µ2 =(2 β)/β (0, 1]. − 1 ≥ We then show that 1 2 log2(µ2 +1) g(µ2). 1/x 2x ∈ Let f(x)=(µ2 +1) . For any µ2 [0, 1], it is 1/x 2x ≥ easy to verify that both µ2 +1 and a (a 1) are increasing on [0, 1], here a is a fixed constant. Thus, f(x) increases over [0, 1]. With f(0) = 1 ≤ 1+µ2 1 2 ≥ g(µ2) and f(1) = (1 + µ2) 1+µ2, the equation (µ2 + 2g(µ2) 1) =1+µ2 has exactly one solution g(µ2)over [0, 1]. In fact, any solution to the inequality below is an 1 g(µ2) 2g(µ2) ≥ upper bound (µ2 +1) 1+µ2. − 1 Now we compare the the value of 1 2 log2(µ2 +1) and g(µ2), which is equivalent to compare the value of 1 − 1 1 log2(1+µ2) − 2 2 log2(1+µ2) f(µ2)=(µ2 +1) and 1 + µ2 for µ2 ∈ [0, 1]. Figure 4(a) shows that f(µ2) ≥ 1+µ2, 6