Ô C2 − − 2 Most O(N ) [10], Where C2 =(1 Log2(1 + 1 Β ))/2 C1 −

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 uv2 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<ithen attach pjpi+1 to node pipi+1 and mark all leaf nodes pmpm+1 Figure 1: The relations between fade factors of β- for j +1≤ m ≤ i. If all descendant leaf nodes skeletons. of an internal node have marks, then also mark it. Delete all nodes with mark. Case 2. β ∈ [0, 1]. In this case, we have 1 = x2 + ∈ Ô 1 (e) If p S1,sayp = pj and it breaks multiple 2 − − 2 ≥ x2 2x1x2 cos θ. Let cos α = 1 β .Fromθ +α π, edges, choose the edge with the minimum in- we have dex prpr+1 and the maximum index psps+1.If j<rthen attach p p to p p and mark 1 ≥ x2 + x2 − 2x x cos(π − α)=x2 + x2 +2x x cos α j s+1 j j+1 1 2 1 2 1 2 1 2 all leaf nodes between p p and p p .If (2) j s+1 s s+1 j>s+1 then attach pspj to psps+1 and mark all leaf nodes between psps+1 and pj−1pj.If 2.2 Construction of the fade factor tree r +1 <j<sthen attach prpj to prpr+1 Our analysis of the spanning ratio is based on a con- and attach pjps+1 to psps+1, then mark all cept called fade factor tree, which intuitively records nodes between prpr+1 and psps+1.Ifanin- the edge-breaking sequence for a pair of points u and ternal node’s all descendant leaf nodes have v.

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