Almost All Even Yao-Yao Graphs Are Spanners

Transcription

1 Almost All Even Yao-Yao Graphs Are Spanners Jian Li 1 and Wei Zhan 2 1 Institte for Interdisciplinary Information Sciences, Tsingha University, Beijing, China lijian83@mail.tsingha.ed.cn 2 Institte for Interdisciplinary Information Sciences, Tsingha University, Beijing, China zhanw-13@mails.tsingha.ed.cn Abstract It is an open problem whether Yao-Yao graphs YY k (also known as sparse-yao graphs) are all spanners when the integer parameter k is large enogh. In this paper we show that, for any integer k 42, the Yao-Yao graph YY 2k is a t k -spanner, with stretch factor t k = O(k 1 ) when k tends to infinity. Or reslt generalizes the best known reslt which asserts that all YY 6k are spanners for k 6 [Baer and Damian, SODA 13]. Or proof is also somewhat simpler ACM Sbject Classification F.2.2 Nonnmerical Algorithms and Problems, I.3.5 Comptational Geometry and Object Modeling Keywords and phrases Yao-Yao graph, geometric spanner, crved trapezoid Digital Object Identifier /LIPIcs.ESA Introdction Let P be a set of points on the Eclidean plane R 2. A simple ndirected graph G = (V, E) with vertex set V = P is called a geometric t-spanner, if for any pair of vertices (, v) there is a path 1... s v in G sch that the total length of the path s v is at most t times the Eclidean distance v, where t is a constant, called the stretch factor. The concept of geometric spanners was first proposed in [7], and can be considered as a special case of general graph spanners [21] if we consider the complete graph with edge lengths defined by the Eclidean metric. Yao graph was first introdced by Andrew Yao in his seminal work on high-dimensional Eclidean minimm spanning trees [23]. Let C (γ 1, γ 2 ) be the cone with apex and consisting of the rays with polar angles in [γ 1, γ 2 ). Given a positive integer parameter k, the constrction of Yao graph Y k is described in the following process. Initially Y k is an empty graph. For each point : For each j = 0,..., k 1: Let C = C (2jπ/k, 2(j + 1)π/k); Select v C P sch that v is the shortest; Add edge v into Y k. JL and WZ are spported in part by the National Basic Research Program of China grants 2015CB358700, 2011CBA00300, 2011CBA00301, and the National NSFC grants , Jian Li and Wei Zhan; licensed nder Creative Commons License CC-BY 24th Annal Eropean Symposim on Algorithms (ESA 2016). Editors: Piotr Sankowski and Christos Zaroliagis; Article No. 62; pp. 62:1 62:13 Leibniz International Proceedings in Informatics Schloss Dagsthl Leibniz-Zentrm für Informatik, Dagsthl Pblishing, Germany

2 62:2 Almost All Even Yao-Yao Graphs Are Spanners In the above process, ties are broken in an arbitrary bt consistent manner. The above process is sally referred to as a Yao step. Note that the edges v we added are directed edges (the directions are sefl in the constrction of Yao-Yao graphs). Besides the seflness in constrcting minimm spanning trees, Yao graphs are also sparse graphs with srprisingly nice spanning properties. It is known that Yao graph Y k has a stretch factor (1 2 sin(π/k)) 1 when k 6 (see e.g., [18] and [5]). Later, the spanning properties of Y 4, Y 6 and Y 5 are proved in a series of work [10, 11, 2]. De to the spanning properties and the simplicity of the constrction, they have important applications in networking and wireless commnication; we refer to [20] and [16] for more details. One may notice that a Yao graph may not have a bonded degree. This is a serios drawback in certain wireless networking applications, since a wireless node can only commnicate with a bonded nmber of neighbors. The isse was initially realized by Li et al. [18]. To address the isse, they proposed a modified constrction with two Yao steps: the first Yao step prodces a Yao graph Y k, and the second step, called the reverse Yao step, eliminates a sbset of edges of Y k to ensre the maximm degree is bonded. The reverse Yao step can be described by the following procedre: Initially YY k is an empty graph. For each point : For each j = 0,..., k 1: C = C (2jπ/k, 2(j + 1)π/k); Select v C P, v Y k sch that v is the shortest; Add edge v into YY k. The reslting graph, YY k, is named as Yao-Yao graph or Sparse-Yao graph in the literatre. The node degrees in YY k are clearly pper-bonded by 2k. It has long been conjectred that YY k are also geometric spanners when k is larger than some constant threshold [18, 16, 3]: Conjectre 1. There exists a constant k 0 sch that for any integer k > k 0, YY k is a geometric spanner. In sharp contrast to Yao graphs, or knowledge abot the spanning properties of Yao-Yao graphs is still qite limited. Li et al. [18] proved that YY k is connected for k > 6, and provided extensive experimental evidence sggesting YY k are indeed spanners for larger k. Jia et al. [14] and Damian [9] showed that YY k is spanner in certain special cases (the nderlying point set satisfies certain restrictions). It is also known that for small constant k s, none of YY k with 2 k 6 admits a constant stretch factor [2, 10, 13]. Recently, a sbstantial progress was made by Baer and Damian [3], who showed that YY 6k are spanners with stretch factor for all integer k 6 and the factor drops to 4.75 when k 8. In fact, a closer examination of the proofs in their paper actally implies an asymptotic stretch factor of 2 + O(k 1 ). None of YY k with other k vales have been proved or disproved to be spanners. We note here that some of the aforementioned work [18, 9] focsed on UDGs (nit disk graphs). Bt their argments can be easily translated to general planar point sets as well. 1 1 Sppose a spanning property holds for any UDG. For a general set of points, we first scale it so that its diameter is less than 1. We can see that the UDG defined over the scaled set is a complete graph. So the spanning property also holds for the set of points.

3 J. Li and W. Zhan 62:3 In this paper, we improve the crrent knowledge of Yao-Yao graphs and make a step towards the resoltion of Conjectre 1, by showing that almost all Yao-Yao graphs with even k are geometric spanners for k large enogh. Formally, or reslts is as follows: Theorem 2. For any k 42, YY 2k is a t k -spanner, where t k = O(k 1 ). Or Techniqe Or proof contains two major steps. 1. (Section 3) We first introdce two classes of intermediate graphs, called overlapping Yao graphs (denoted as OY k ) and trapezoidal Yao graphs (denoted as TY k ). The constrction of overlapping Yao graphs is similar to that of Yao graphs, except that the cones incident on a vertex overlap with each other. We can easily show that OY k is a geometric spanner. The definition of trapezoidal Yao graphs is also similar to Yao graphs, bt takes advantage of a shape called crved trapezoid (defined in Section 2). We can show that OY k is a sbgraph of TY k, implying TY k is a geometric spanner as well. 2. (Section 4) In the second step, we show that YY 2k spans TY 2k (i.e., for any, v P, the shortest -v path in YY 2k is at most a constant times longer than the shortest -v path in TY 2k ). Or proof makes crcial se of the properties of crved trapezoids. Roghly speaking, crved trapezoids are more flexible than triangles which were sed in [3]), which is the main reason for the improvement from YY 6k to more general YY 2k. Related Works Replacing the Eclidean distance by a power κ with a constant κ 2 leads to the definition of power spanners. Since the power of length models the energy consmed in wireless transmissions, power spanners have important implications in wireless networking applications. In this setting, Yao-Yao graphs YY k have been proved to be power spanners for any κ 2 when k > 6 [14, 22]. It is clear that when a graph is a geometric spanner, it mst also be a power spanner (See [17, Lemma 1]), however the reverse does not hold. We also note that neither Yao graphs nor Yao-Yao graphs can be garanteed to be planar graphs [22, 15], whereas Delanay trianglation provides another type of spanner which is planar bt withot bonded degree [12, 4]. In order to facilitate network design in certain applications, some previos work [6, 19] made se of both Yao graphs and Delanay graphs to prodce degree-bonded and planar geometric spanners. The paper is organized as follows. We introdce some standard notations and sefl tools, inclding the shape of crved trapezoid, in Section 2. We introdce overlapping Yao graphs, trapezoidal Yao graphs, and prove their spanning properties in Section 3. We prove or main reslt in Section 4. Finally, we conclde with some ftre work in Section 5. 2 Preliminaries P is the nderlying set of points in R 2. D(a, ρ) denotes an open disk centered at point a with radis ρ. The bondary and closre of a region R are denoted by R and R, respectively. Let S(a, ρ) = D(a, ρ) be the circle centered at a with radis ρ. The length of shortest -v path in a graph G is denoted by d G (, v). On R 2, a point with coordinates (x, y) is denoted by (x, y). Let o(0, 0) be the origin of R 2. The positive direction of x-axis is fixed as the polar axis throghot the constrction and analysis. For a point a R 2, we se x a to denote its x-coordinate and y a its y-coordinate. We se ϕ(v) to represent the polar angle of vector v. The angle comptations are all nder the modlo of 2π, and angle sbtraction is regarded as the E S A

4 62:4 Almost All Even Yao-Yao Graphs Are Spanners O γ γ O λo w o(0, 0) O + O α β v Figre 1 Left: Affine transformations. Right: Proof for Lemma 3. conterclockwise difference. A cone between polar angles γ 1 < γ 2 with apex at the origin is denoted as C(γ 1, γ 2 ) = { ϕ(o) [γ 1, γ 2 )}. It is also necessary to introdce some notations of standard affine transformations on a geometric object O in the plane: (Dilation) If O is niformly scaled by factor λ with the origin as the center, the reslt is denoted by λo = {λz z O}. (Translation) If O is translated so that the original point goes to point, the reslt is denoted by + O = { + z z O}. (Rotation) If O rotated an angle γ conterclockwise with respect to the origin, the reslt is denoted by O γ. (Reflection) If O is reflected throgh the x-axis, the reslt is denoted by O. By the above notations, we can denote the cone with apex by +C(γ 1, γ 2 ) (abbreviated as C (γ 1, γ 2 )). Geometric Ineqalities In order to attain a constant stretch factor for Yao graphs and their variants, one often needs to bond a certain geometric ratio. Here we present a simple yet general lemma for this prpose. Let the open strip on the segment v be the collection of points Lemma 3. Sppose τ 1 is a constant. Let w be a point with wv, wv [0, π/2) (see Figre 1). When τ vw < v, the following statements abot the ratio w /( v τ vw ) hold: 1. If w is restricted within a compact segment of an arc centered at, the ratio is maximized when vw is the largest; 2. If w is restricted within a compact segment of a ray from v, the ratio is maximized when vw is the largest; 3. If w is restricted within a compact segment of a ray originated from, the maximm ratio is achieved when w makes w the largest or smallest; Proof. The first statement is straightforward since both v and w are fixed. Now, we show the last two properties. Let α = wv and β = wv. By the law of sines and the fact that sin(α + β) = sin α cos β + cos α sin β, we can see that w v τ vw = sin β sin(α + β) τ sin(α) = 1 cos α sin α (τ cos β)/ sin β. (1)

5 J. Li and W. Zhan 62:5 T θ w T θ θ o(0, 0) p(1, 0) o a v b p R 2 Figre 2 Left: The crved trapezoid. The critical arc is shown in bold. Right: An illstration of Lemma 6 with its proof. One can see the ratio is maximized when α is maximized, which implies the second statement. Moreover, (τ cos β)/ sin β is a convex fnction of β for β (0, π/2). Hence, it is maximized when β is maximized or minimized, which implies the last statement. De to Lemma 3, to compte an pper bond for w /( v vw ) in a region, it sffices to consider a small nmber of extreme positions. In particlar, the following corollary is an immediately conseqence, which is an improvement of a classical reslt mentioned in [8]: Corollary 4. R = D(, v ) C (γ 1, γ 2 ) is a sector with apex and with v on its arc. Sppose α = max{ϕ(v) γ 1, γ 2 ϕ(v)} < π/3. Then, for all w R \ {}, it holds that w (1 v vw 2 sin α ) 1. 2 Proof. Take τ = 1 in Lemma 3. By considering the rays from and the arcs centered at, and applying the first and third cases in Lemma 3, we can see that the ratio can only be maximized when w is at the two corners of the sector with β = (π α)/2, or when w approaches with β = 0. The two pper bonds in eqality 1 in these cases are (1 2 sin(α/2)) 1 and (cos α) 1, and the former one is larger. Crved Trapezoid T θ To constrct the trapezoidal Yao Graphs TY k, we need to define an open shape, called crved trapezoid, as follows. Definition 5 (Crved Trapezoid T θ ). Fix two points o(0, 0) and p(1, 0). Given θ [π/4, π/3), we define the crved trapezoid T θ as: T θ = {(x, y) 0 < x < 1, 0 < y < sin θ, o < 1, p < 1}. Intitively, it is the convex hll of two sectors with apices at o and p respectively. We regard T θ as a shape attached to the origin o, and the closed arc not incident on o is called the critical arc. See Figre 2 for an example. We list some sefl properties of T θ here. It is straightforward to show that T θ is symmetric with respect to the vertical line x = 1 2. For any T θ it holds that 0 < ϕ(o) < π/2. If is E S A

6 62:6 Almost All Even Yao-Yao Graphs Are Spanners on the critical arc, it holds that 0 < ϕ(o) < θ. The following crcial lemma, illstrated in Figre 2, is also an immediate conseqence from the definition. Lemma 6. Denote R 2 as the lower half-plane {(x, y) y 0}. Consider any two points, v R 2 satisfying 0 < x < 1, ϕ(v) π < 1 π, and o, pv [ v, 1). We constrct 6 a similar shape T θ := + v (T θ ) ϕ(v). We have that T θ T θ is completely contained in lower half plane R 2. Proof. We first consider the extreme case when θ = π/3. Since, v R 2, if the two arcs of T θ do not intersect with the line y = 0, then T θ is flly contained in R 2 and the proof is done. Now sppose the two intersections are a, b, illstrated in the RHS of Figre 2. Notice that a = v o and vb = v pv. Combining with the condition that 0 < x < 1, we can see that both a and b lie in the segment op. For any point w T θ R2, it sffices to prove ow, pw < 1 so that w is in T θ. This is done by examining the perpendiclar bisector of aw and bw: since a > w, is in the pper half-plane of the bisector of aw, and so is p. Ths pw < pa op = 1. The same argments hold for ow < 1. The above shows that T θ T θ R 2 when θ = π/3, and if θ is smaller, we only need to show that the distance d(w, op) from w to line y = 0 is less than sin θ. This is done by the observation that d(w, ab) d(w, v) in T θ, and d(w, v) < v sin θ < sin θ. This completes the proof for Lemma 6. 3 Overlapping Yao Graphs and Trapezoidal Yao Graphs In this section we consider two variants of Yao graphs, overlapping Yao graphs and trapezoidal Yao graphs. k 2π Definition 7 (Overlapping Yao Graph OY k ). Let γ = 4 k. For every P and j = 0,..., k 1, select shortest v with v C (2jπ/k, 2jπ/k + γ). The chosen edges form the overlapping Yao graph OY k (P). The angle γ in the definition is actally the smallest mltiple of 2π/k which is no less than π/2. We note here that the term overlapping" comes from the fact that the cones with the same apex overlap with each other, while in the original Yao graphs they are disjoint. First we claim that OY k is a spanner when k is large. ( Lemma 8. If k > 24, then OY k is a τ k -spanner where τ k = 1 2 sin ( π k + π ) ) 1. 8 Proof. We prove d OY (, v) τ k v by indction on the length v for, v P. Notice that it is always possible to find j so that v C (2jπ/k + π/4, 2(j + 1)π/k + π/4). See Figre 3 for an illstration. This cone is contained in C (γ 1, γ 2 ), where γ 1 = 2jπ/k and γ 2 = 2jπ/k + γ, since k > 24 and ths γ π/2. If v is the shortest in C (γ 1, γ 2 ), then v OY k and we are done with the proof. Now, sppose that in the constrction of constrction of OY k, we choose w for C (γ 1, γ 2 ) with w v. Since k > 24, one shall see that α := max{ϕ(v) γ 1, γ 2 ϕ(v)} max{2π/k + π/4, γ π/4} = 2π/k + π/4 < π/3.

7 J. Li and W. Zhan 62:7 2jπ/k + γ v π/4 w 2jπ/k Figre 3 Illstration of the proof of Lemma 8. The oter cone is C (γ 1, γ 2) where γ 1 = 2jπ/k and γ 2 = 2jπ/k + γ The cone defined by two dashed lines is C (2jπ/k + π/4, 2(j + 1)π/k + π/4). Hence, we can apply Corollary 4 on the sector R = D(, v ) C (γ 1, γ 2 ), which claims w ( v vw τ k = 1 2 sin α ) 1. 2 Since α < π/3, we have vw < v. By the indction hypothesis, we can see that d OY (, v) w + d OY (v, w) w + τ k vw τ k ( v vw ) + τ k vw = τ k v, which completes the proof. We also note that Barba et al. proposed the so-called continos Yao graphs in [1], which play a similar role as OY k. By adapting their method, it might be possible to prove a slightly smaller stretch factor for OY k. However the constant is not or primary goal in this paper, and we leave it to the ftre work. Now we define the trapezoidal Yao graphs based on the crved trapezoid T θ (Definition 5). Definition 9 (Trapezoidal Yao Graph TY k ). Let θ = j = 0,..., k 1, define two crved trapezoids Γ j1 = (T θ ) 2jπ/k and Γ j2 = (T θ ) 2jπ/k. k 8 2π k. For every P and Note that their bottom sides lie on the ray of angle 2jπ/k. For each i = 1, 2, we do the following: We grow a crved trapezoid + λγ ji by gradally increasing λ (initially 0) ntil its bondary hits some point v. If the critical arc hits v, we select v. Otherwise, we select nothing. All the selected edges form the graph TY k (P). Lemma 10. For any integer k > 24, TY k is a τ k -spanner where τ k = ( 1 2 sin ( π k + π ) ) 1. 8 Proof. Note that in Definition 5, we reqire that θ [π/4, π/3). When k > 24, we can see that π k 2π 4 θ = 8 k k + 7 π k 4 < π 3. Hence, the vale θ in Definition 9 satisfies the reqirement of θ in Definition 5. E S A

8 62:8 Almost All Even Yao-Yao Graphs Are Spanners Figre 4 Illstration for Lemma 10: Two crved trapezoids cover a sector. We show that the overlapping Yao graph OY k is actally a sbgraph of TY k, from which the lemma is an immediate conseqence. Given any j and one cone C (2jπ/k, 2jπ/k + γ), we choose two copies of T θ : Γ 1 = (T θ ) 2jπ/k, and Γ 2 = (T θ ) 2jπ/k+γ. See Figre 4. Since 2θ γ π/2, it is clear that Γ 1 and Γ 2 exactly cover the interior of sector D(o, 1) C o (2jπ/k, 2jπ/k + γ). Now, for any edge v OY k selected within this cone, both + v Γ 1 and + v Γ 2 mst have empty interior ( v is the shortest within the cone). Therefore, if ϕ(v) 2jπ/k θ, it shold be selected into TY k with respect to Γ 1. Otherwise it shold be selected with respect to Γ 2. This implies that OY k is a sbgraph of TY k, and by Lemma 8, TY k is a τ k -spanner as well. 4 Yao-Yao Graphs YY 2k are Spanners In this section, we prove or main reslt that YY 2k has a constant stretch factor for large k, by show that YY 2k spans TY 2k. To begin with, we prove an important property of TY 2k, which will be sefl later. The property can be seen as an analoge of the property of Θ 6 shown in [3, Lemma 2]. One of the reasons we can improve YY 6k to more general YY 2k is that we can take advantage of the crved trapezoid rather than the reglar triangle. We state this property by assming, that withot loss of generality, a crved trapezoid T θ is placed at its normal position, i.e., T θ lies in the first qadrant with two vertices o(0, 0) and p(1, 0). From now on, we work with trapezoidal Yao graphs TY 2k. Hence, the associated parameter θ (see Definition 9) is θ = 2k 8 2π 2k = k 4 π k. Lemma 11. Sppose o P and T θ has an empty interior. If there is a point a P sch that 0 < x a < 1, y a 0 and 0 < ϕ(ap) < π/6, then d TY (oa) x a + (2τ 2k + 1) y a where τ k is as defined in Lemma 10, and there is a path in TY 2k from a to o where each edge is shorter than oa. Proof. Note that simply applying Lemma 10 is insfficient to achieve the garantee. Instead, we present an iterative algorithm for finding a path from a to o. The path fond by the iterative algorithm is not necessarily a shortest path from a to o. Nevertheless, we can bond its length as in the lemma.

9 J. Li and W. Zhan 62:9 ψ θ ψ π 2 T θ ψ o v (ii) v (i) (iii) (T θ ) ψ θ R 2 π/6 p Figre 5 Illstration of three cases in the proof of Lemma 11. Let be the crrent point in P; Initially is set to be a. While o and ϕ(o) > π 6 do Let ψ = min{jπ/k jπ/k > ϕ(o), j = 0, 1,..., 2k 1}; Grow a crved trapezoid + λ(t θ ) ψ ntil its bondary hits some point v. If v C (ψ π 2, ψ θ): (i) TY 2k mst contain the edge v. We add v to or path; Otherwise: (ii) Add the path from to v in OY 2k constrcted in Lemma 8; Set the crrent point to be v and proceed to the next iteration. If now = o, the path is already fond; (iii) Otherwise take the path from to o in OY 2k constrcted in Lemma 8. See Figre 5 for an illstration of the three cases (i),(ii) and (iii). Let l() be the length of the path from to o generated by the algorithm above. To analyze its behavior, we define a potential fnction Φ() = x + (2τ 2k + 1) y l(). We claim that the potential fnction never increases as the algorithm proceeds and is eventally 0. The potential only changes in the three ( labeled ] steps (i),(ii) and (iii). When 5 execting step (i) and (ii), it is clear that ϕ(o) 6 π, π, ϕ(p) [0, 16 ) π. Therefore ψ ( ] 5 also falls in the range 6 π, π. Moreover, v mst be contained in R 2 de to Lemma 6, that is, y v 0. In step (i), v is simply the nearest neighbor in the cone C (ψ θ, ψ). Since ϕ(p) < π/6, we know v < vp. Since ψ θ > π/2 and ψ π, we can see that x v x and y v < y. Hence the change in potential is Φ = Φ(v) Φ() = v (x x v ) (2τ 2k + 1)( y y v ) v (x x v ) ( y y v ) 0. E S A

10 62:10 Almost All Even Yao-Yao Graphs Are Spanners In step (ii), since ϕ(v) [ψ π, ψ θ), we know ϕ(v) π/2 is at most π/4, so 2 y v < y and x v x < y y v. That frther implies v < 2( y y v ), and ths d TY (v) τ 2k v < 2τ 2k ( y y v ). Therefore the change in potential is Φ = Φ(v) Φ() = d TY (v) (x x v ) (2τ 2k + 1)( y y v ) = (d TY (v) 2τ 2k ( y y v )) + (x v x ( y y v )) 0. In step (iii), sppose the node arrived before is, with ϕ( o) > 5 π. Since is 6 ( π chosen, it mst be the case that o. It is immediate that ϕ(o) 3, 5 ] 6 π, and ths o 2 y, x y. Therefore, we have that Φ = Φ(o) Φ() = (d TY (o) 2τ 2k y ) + ( x y ) 0. Now that Φ(o) = 0, and the above argments show that the potential cannot increase dring the path from to o, we can conclde that Φ(a) 0. That is, l(a) x a + (2τ 2k + 1) y a. Also notice that in steps (i) and (ii), it is ensred that v < o and ov < o since v is contained in the crved trapezoid. On the other hand, the paths from OY 2k in steps (ii) and (iii), which are constrcted according to Lemma 8, contains only edges shorter than the direct distance. Ths the edges we selected are all shorter than oa. Lemma 12. On a Yao-Yao graph YY 2k (P) with k 42, if v TY 2k, then d YY (v) τ k v, where τ k = 2 + O(k 1 ) is a constant depending only on k. Proof. Let τ k 1 be a constant to be fixed later. Or proof is by indction on the length v. The base case is simple: the shortest edge is in both TY 2k and YY 2k. Now, consider an edge v TY 2k. Withot loss of generality, we assme that v = 1. We frther assme that T θ, which generates v in TY 2k, in its normalized position, i.e., is o(0, 0) and another vertex is p(1, 0). See Figre 6. Noticing that θ is a mltiple of π/k, we can see that v mst be the nearest neighbor of in its corresponding cone of YY 2k. If v YY 2k, then d YY (v) = v and we are done with the proof. Otherwise, there mst be another edge wv YY 2k where and w are in the same v-apex cone, and vw v (this is de to the constrction of Yao-Yao graph). It follows that w < v since k 42. Now, we prove by indction that d YY (v) τ k v. Now noticing that T θ has an empty interior, there are only two cases for w we need to consider. See Figre 6 for an example. 1. y w > 0: See the left part of Figre 6. By Lemma 3, we can see that within all possible positions of w, vw /( v τ 2k w ) can only be maximized when vw = v or when w is on the left bondary of T θ. Since when k 42, it holds 2τ 2k sin π 2k < 1, therefore vw v τ 2k w 1 1 2τ 2k sin π. 2k Hence, if τ k 1/(1 2τ 2k sin π 2k ), we have that vw τ k ( v τ 2k w ). Consider the shortest path from to w in TY 2k. Notice that d TY (w) τ 2k w < v. Applying the indction hypothesis to every edge of the shortest path, we can see d YY (w) τ k d TY(w). Combining the above ineqalities, we can finally obtain that d YY (v) d YY (w) + vw τ k(d TY (w) + v τ 2k w ) τ k v. 2. y w 0: See the middle and right parts of Figre 6. Consider the two cases of angle ϕ(v):

11 J. Li and W. Zhan 62:11 w T θ v T θ T θ p v p w p w v Figre 6 Proof for Lemma 12. Left: w is in the pper half-plane; Middle: w is in the lower half-plane and ϕ(v) π/k; Right: w is in the lower half-plane and ϕ(v) > π/k. If 0 ϕ(v) π/k, then ϕ(wp) ϕ(wv) π/k; If π/k < ϕ(v) θ, then pv (k 1)π 2k wv. Ths p is otside the circmcircle of vw, and ϕ(wp) = pw vw π/k. Therefore in both cases, we have 0 < x w < x p and 0 ϕ(wp) π/k. We notice the following two ineqalities: y w (x p x w ) tan ϕ(wp) ( v x w ) tan π k, ( vw (x v x w ) sec ϕ(wv) ( v x w ) sec θ + π ). k Consider the path from w to as we constrcted in Lemma 11 (w and correspond to a and o in the lemma, respectively). Since all edges in the constrcted path are shorter than w (ths shorter than v as well), we can apply the indction hypothesis to get d YY (w) τ k d TY(w). Applying Lemma 11, we have d YY (v) d YY (w) + vw τ kd TY (w) + vw τ k(x w + (2τ 2k + 1) y w ) + vw τ kx w + (τ k(2τ 2k + 1) tan π ( k + sec θ + π )) ( v x w ). k Sppose we choose τ k sch that τ k (2τ 2k + 1) tan π k + sec(θ + π k ) τ k, (which is eqivalent to τ ((1 (2τ 2k + 1) tan π k ) cos(θ + π k )) 1 ). Note that the above holds also becase (2τ 2k + 1) tan π k < 1 when k 42. Finally, we can conclde that d YY (v) τ k v. We can conclde from the above two cases that τ k can be chosen to be sec π 4 + O(k 1 ) = 2 + O(k 1 ). Combining Lemma 10 and 12, with the observation that τ k = (1 2 sin(π/8)) 1 + O(k 1 ), we can get or main reslt that almost all YY 2k are spanners. Theorem 2 (restated). For any integer k 42, YY 2k is a t k -spanner, where t k = τ k τ 2k = 2(1 2 sin(π/8)) 1 + O(k 1 ) = O(k 1 ). 5 Conclsion and Ftre Work In this paper we proved that Yao-Yao graphs YY 2k are geometric spanners for k large enogh, making a positive progress to the long-standing Conjectre 1. For Yao-Yao graphs with odd E S A

12 62:12 Almost All Even Yao-Yao Graphs Are Spanners parameters, the resoltion of the conjectre is still elsive and seems to reqire additional new ideas. We did not try very hard to optimize the constant stretch ratio for different k vales. Hence, the constant claimed in Theorem 2 may well be frther improved. One potential approach is to se techniqes developed in [1]. Obtaining tighter bonds (and lower bonds as well!) is left as an interesting ftre work. We propose some other potential ftre directions, which are interesting in their own rights and might lead to sparse spanners with bonded degrees and small stretch factors. Consider the variant of Yao-Yao graphs where the cones are not divided niformly. One cold expect that sch variants are spanners if the apex angles of the cones are all small. It seems that or techniqes in the paper can be adapted to this sitation if the separated cones are centrosymmetric: every separation ray of polar angle ϕ has a corresponding one at polar angle π + ϕ. Bt the more general case is still open. Instead of insisting on one polar division for every node, we can let each point choose its polar axis individally by niformly and independently randomizing a direction. We conjectre that sch graphs are also spanners (in expectation or with high probability) for k large enogh. References 1 Lis Barba, Prosenjit Bose, Jean-Lo De Carfel, Mirela Damian, Rolf Fagerberg, André van Renssen, Peroz Taslakian, and Sander Verdonschot. Continos Yao graphs. In Proceedings of the 26th Canadian Conference on Comptational Geometry, CCCG 2014, Halifax, Nova Scotia, Canada, 2014, Lis Barba, Prosenjit Bose, Mirela Damian, Rolf Fagerberg, Wah Loon Keng, Joseph O Rorke, André van Renssen, Peroz Taslakian, Sander Verdonschot, and Ge Xia. New and improved spanning ratios for Yao graphs. JoCG, 6(2):19 53, Matthew Baer and Mirela Damian. An infinite class of sparse-yao spanners. In Proceedings of the Twenty-Forth Annal ACM-SIAM Symposim on Discrete Algorithms, pages SIAM, Prosenjit Bose, Paz Carmi, Sebastien Collette, and Michiel Smid. On the stretch factor of convex delanay graphs. Jornal of comptational geometry, 1(1):41 56, Prosenjit Bose, Mirela Damian, Karim Doïeb, Joseph O rorke, Ben Seamone, Michiel Smid, and Stefanie Whrer. π/2-angle Yao graphs are spanners. International Jornal of Comptational Geometry & Applications, 22(01):61 82, Prosenjit Bose, Joachim Gdmndsson, and Michiel H. M. Smid. Constrcting plane spanners of bonded degree and low weight. Algorithmica, 42(3-4): , Pal Chew. There is a planar graph almost as good as the complete graph. In Proceedings of the second annal symposim on Comptational geometry, pages ACM, Artr Czmaj and Hairong Zhao. Falt-tolerant geometric spanners. Discrete & Comptational Geometry, 32(2): , Mirela Damian. A simple Yao-Yao-based spanner of bonded degree. arxiv preprint arxiv: , Mirela Damian, Nawar Molla, and Val Pinci. Spanner properties of π/2-angle Yao graphs. In Proc. of the 25th Eropean Workshop on Comptational Geometry, pages Citeseer, Mirela Damian and Kristin Radonis. Yao graphs span theta graphs. In Combinatorial Optimization and Applications, pages Springer, 2010.

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