A Simple Yao-Yao-Based Spanner of Bounded Degree

A Simple Y ao-Y ao-Bas ed Spanner of Bounded Degree ∗ Mirela Damian † Abstract It is a standing op en ques tion to decide whether the Y ao-Y ao structure for unit disk gra phs (UDGs) is a leng th spanner of not. This question is highly r elev a nt to the topolo g y co ntrol problem for wireles s ad hoc net works. In this pap er we mak e progr ess to wards resolving this question by showing that the Y ao-Y ao structure is a leng th spa nner for UDGs of b ounded asp ect ratio. W e als o prop ose a new lo ca l algo rithm, c a lled Y ao-Sparse-Sink, based on the Y ao-Sink metho d in tro duced b y Li, W an, W ang a nd F rieder, that co mputes a (1 + ε )-spanner of bounded degree for a given UDG and for given ε > 0. The Y ao-Sparse- Sink method enables an efficien t lo cal computation o f spa rse sink trees. Finally , w e s how that all these str uctures for UDGs – Y a o , Y ao-Y ao, Y ao-Sink and Y ao-Spa rse-Sink – hav e arbitr a rily large weigh t. 1 In tro duction Let G = ( V , E ) b e a connected graph with n v ertices emb edded in the Euclidean plane. F or a ny pair of v ertices u, v ∈ V , an uv - p ath is defin ed b y a sequence of edges uu 1 , u 1 u 2 , . . . , u s v . A subgraph H of G is a length sp anner of G if, f or all pairs of v ertices u, v ∈ V , the length of a shortest uv -path in H is no longer than a constan t times the length of a sh ortest uv -path in G ; if the constan t v alue is t , H is called a length t -spanner and t is called length str etch factor . The p ower needed to supp ort a w ireless link uv is | uv | β , where β is a path loss gradien t (a real constan t b et w een 2 and 5) that dep ends on the transm ission en vironment. A subgraph H of a graph G has p ower str etch factor equal to ρ if, for all pairs of v ertices u, v in G , the p ow er of a minim um p ow er uv -path in H is n o higher than ρ times the p o wer of a minimum p o we r uv -path in G . Li et al. [3] sh o w ed that a graph w ith length stretc h factor δ has p o wer stretc h factor δ β , bu t the rev erse is not n ecessarily true: F act 1 [3]. Any su bgraph H ⊆ G with length stretc h factor δ h as p ow er str etch factor δ β . The problem of constructing a sparse sp an n er of a giv en graph h as receiv ed considerable atten tion from researc hers in computational ge ometry and ad-ho c wireless net w orks; w e refer the reader to the recen t b o ok by Narasimhan and Smid [6]. The simplest mo del of a w ireless net wo rk grap h is the Unit Disk Graph (UDG): an edge exists in the graph if and only if the Euclidean d istance b et wee n its endp oin ts is no greater than 1. It is a stand ing op en question to decide w h ether the Y ao-Y ao structure for UDGs in tro d u ced b y Li et al. [3] is a spanner of not. The Y ao-Y ao graph (also kn o wn as Y ao plus r everse Y ao ) is based on the Y ao graph [8], from whic h a n umb er of edges are eliminated through a rev erse Y ao pro cess, to ensure b oun ded degree at ea c h no de. Pr ogress to wa rds resolving this question has b een made b y W ang an d Li [7], wh o sho we d th at the Y ao-Y ao graph h as constan t p ower str etch factor in a civi lize d UDG. F or constan t λ > 0, a λ - civi lize d graph is a graph in w hic h no tw o no des are at distance smaller than λ . Most often wireless devices in a wireless netw ork can not b e to o close, so it is reasonable to mo del a wireless ad h o c net w ork as a civilized UDG. ∗ This w ork has b een sup p orted b y NSF g rant CCF-0728909. † Department of Computer Science, Villano v a Universit y , Vill anov a, P A 19085. E-mail: mirela.dam ian@villanova. edu . 1 In this pap er we sho w that the Y ao-Y ao graph for a civ i lize d UDG has constant length str etch factor as w ell. Although several p ap ers refer to a similar result as app earing in [3], to the b est of ou r kno wledge there is no v ersion of [3] that pub lishes this result. W e also analyze the b ounded degree spanner generated by th e Y ao-Sink tec hn ique int ro duced in [4]. The sink tec h nique replaces eac h directed sta r in the Y ao graph consisting of all li nks directed in to a no de u , by a tree T ( u ) w ith sin k u of b ounded degree. W e prop ose an enhanced tec hnique called Y ao-Sp arse-Sink t hat filters out some of th e edges in the Y ao grap h prior to applying the sin k tec hnique. This enables an efficien t lo cal computation of sp arse sink t rees, more appr opriate for highly dynamic w ireless netw ork nod es. Our an alysis of the Y ao-Sparse-Sink metho d pro vides additional insigh t into the p rop erties of th e Y ao-Y ao structure. W e also show th at all th ese stru ctures for UDGs – Y ao, Y ao-Y ao, Y ao-Sink an d Y ao-Sparse-Sink – ha v e arbitrarily large weigh t. The rest of the pap er is organized as follo ws. In Section 2 we introdu ce some n otation and definitions and discuss previous r elated wo rk. In Section 3 we show that the Y ao-Y ao graph is a spanner for UDGs of b ound ed asp ect r atio (in particular, for λ -civilized UDGs). In Section 4 we discuss the Y ao-Sink metho d and, based on this, w e p rop ose a n ew technique called Y ao-Sparse-Sink that computes sparse sink trees efficiently . Finally , in S ection 5 w e show that all these str u ctures for UDGs ha v e unboun d ed weigh t. 2 Preliminaries 2.1 Definitions and Notation Throughout the pap er we use the follo wing notation: uv denotes the edge with endp oint s u and v ; − → uv denotes the edge directed fr om sour c e no de u to sink no de v ; | uv | denotes the Euclidean distance b et w een u and v ; p ( u v ) denotes a simple uv -path; and ⊕ denotes the concatenation op erator. F or an y no des u and v , let K u denote an arbitrary cone with ap ex u , and K u ( v ) denote the cone with ap ex u con taining v . F or an y edge set E and an y cone K u , let E ∩ K u denote th e subset of edges in E incident to u that lie in K u . Similarly , for a graph G and a cone K u , G ∩ K u is the subset of edges in G in ciden t to u that lie in K u . Th e asp e ct r atio of an edge set E is th e ratio of the length of a longest edge in E to the length of a shortest edge in E . The asp ect ratio of a graph is defined as the asp ect ratio of its edge set. W e assu me that eac h n o de u has a u nique identi fier ID ( u ). Define the identifier ID ( − → uv ) of a directed edge − → uv to b e the trip let ( | uv | , ID ( u ) , ID ( v )). F or any pair of directed edges − → uv and − − → u ′ v ′ , w e sa y th at ID ( − → uv ) < ID ( − − → u ′ v ′ ) if and only if one of the follo wing conditions holds: (a) | uv | < | u ′ v ′ | (b) | uv | = | u ′ v ′ | and ID ( u ) < ID ( u ′ ) (b) | uv | = | u ′ v ′ | and ID ( u ) = ID ( u ′ ) and ID ( v ) < ID ( v ′ ) F or an und ir ected edge uv , d efine ID ( uv ) = min { ID ( − → uv ) , ID ( − → v u ) } . Note that according to this definition, eac h edge has a unique iden tifier. This enables us to order an y edge set b y increasing ID of edges. 2.2 Previous W ork Y ao [8] d efi ned the Y ao graph Y k ( G ) as follo ws. A t eac h n o de u ∈ V , any k equal-separated ra ys originated at u define k cones; in eac h cone, pic k the edge uv of smallest ID , if suc h an edge exists, and add to the Y ao graph the directed edge − → uv . W e call this the Y ao-Step , describ ed in T able 1. 2 Y ao-Step ( G = ( V , E ) , k ) Set E Y ← φ and Y k ← ( V , E Y ). F o r each no de u Partition the space into k equal-size co nes with a pe x u of angle θ = 2 π /k (assume that eac h cone is half-op en and half-closed). F o r each no de u and each cone K u such that E ∩ K u is nonempty Pick the edge uv ∈ E ∩ K u with low est ID ( uv ). Add the dir ected edge − → uv to E Y . Output Y k = ( V , E Y ) . T able 1: The Y ao step. It has b een shown that the outpu t graph Y k has maxim um no de degree n − 1 and length stretch factor 1 1 − 2 sin π /k . The first pr op ert y (high d egree) is the main dr a wbac k of th e Y ao graph. I n wireless netw orks for example, high degree is u ndesirable b ecause n o des communicati ng with to o man y no des directly ma y exp er ience large o ve rh ead that could otherwise b e distributed among sev eral no des. The Y ao-Y ao graph Y Y k has b een prop osed in [3 ] to o verco me this shortcoming: at eac h no de u in the Y ao graph , discard all directed edges − → v u from eac h cone cente red at u , except for the one with minimum I D . This filtering step is describ ed in T able 2. Reverse Y ao-Step ( Y k = ( V , E Y ) , k ) Set E Y Y ← E Y and Y Y k ← ( V , E Y Y ). Use the same cone partition a s in the Y ao-Step . F o r each no de v and each cone K v Eliminate fr o m E Y Y all edges − → uv with sink v that lie in K v , except for the one with the smallest ID . Output Y Y k = ( V , E Y Y ) (viewed as an undi rected graph) . T able 2: The r ev erse Y ao step. The outp u t graph Y Y k has maximum no de degree 2 k , a constan t. How ever, the tradeoff is unclear in that the question of wh ether Y Y k is a spanner or not remains op en. 3 YY-Spanner for Civil ized UDGs Note that an y UDG of constan t asp ec t ratio ∆ is a 1 / ∆-civilized UDG, and any λ -civilized UDG has asp ect ratio 1 /λ . Ther efore, fr om here on will refer to λ -civilized UDGs only . The YY-Sp an ner algorithm applied on a UDG G = ( V , E ) comprises the Y ao and rev erse-Y ao steps: 1. Execute Y ao-Step ( G, k ). The result is the Y ao spann er Y k = ( V , E Y ). 2. Execute R everse Y ao-Step ( Y k , k ). T h e result is the Y ao-Y ao graph Y Y k = ( V , E Y Y ). W e no w show that Y Y k is a length spanner for an y civilized UDG. In pro ving this, w e will mak e use of the follo wing lemma: 3 Lemma 1 (Czuma j and Zhao [1]) L et 0 < θ < π 4 and t ≥ 1 cos θ − sin θ . L et u, v , z b e thr e e p oints in the plane with d v uz ≤ θ . Supp ose further that | uz | ≤ | uv | . Then the e dge { u, z } fol lowe d by a t -sp anner p ath fr om z to v is a t -sp anner p ath fr om u to v (se e Figur e 1). u v z θ Figure 1: If θ < π / 4 and p ( z v ) is a t -spanner path, then uz ⊕ p ( z v ) is a t -spanner path. Theorem 2 L et G = ( V , E ) b e a λ -civilize d gr aph, and let Y Y k b e the Y ao-Y ao structur e for G . Then Y Y k is a sp anner with length str etch factor t ≥ λ ( λ +1)(co s 2 π /k − sin 2 π /k ) − 1 , for any inte ger k > 8 satisfying the c ondition (cos 2 π /k − sin 2 π /k ) > 1 λ +1 . Pro of: The pro of is b y ind uction on the rank of edges in the edge set E order ed by increasing ID . The base case corresp onds to the edge uv ∈ E of rank 0 (i.e., w ith smallest ID ( uv )). Ass u me without loss of generalit y that I D ( uv ) = ID ( − → uv ). S ince − → uv has the smallest ID among all ed ges in K u ( v ), − → uv gets added to E Y in th e Y ao-Step . F urthermore, since − → uv has the smallest ID among all edges in E Y Y ∩ K v ( u ) directed into u , − → uv do es n ot get d iscarded in the R everse Y ao-Step . Th us uv is an edge in Y Y k and so the theorem holds for the base case. u θ 1 u θ K ( v ) u K (u) v v 1 u' u θ 1 u θ K ( v ) u K (u) v v 1 v v' 1 1 u' (a) (b) Figure 2: Pro of of Theorem 2: (a) Case 1: p ( u v ) ← p ( u u 1 ) ⊕ u 1 v ; (b) Case 2: p ( u v ) ← p ( u u 1 ) ⊕ u 1 v 1 ⊕ p ( v 1 v ). The in d uctiv e hyp othesis tells that Y Y k con tains t -spanner paths b et w een the endp oin ts of an y edge uv ∈ E of rank no greater than some v alue j ≥ 0. T o pro v e the ind u ctiv e step, consider the edge uv ∈ E of rank j + 1. Assume without loss of generalit y that ID ( uv ) = ID ( − → uv ). W e discu ss t w o cases, d ep endin g on w h ether − → uv b elongs to E Y or not. Let θ = 2 π /k < π / 4. Case 1: − → uv ∈ E Y . If − → uv ∈ E Y Y the pr o of is finished, so assume the opp osite. Note t hat − → uv 6∈ E Y Y happ en s when v eliminates − → uv in the Revers e Y a o-Step in fav or of another edge − → u 1 v , w ith ID ( − → u 1 v ) < ID ( − → uv ) (see Figure 2a). Since u and u 1 b oth lie in a same cone K v , w e h a v e 4 that [ uv u 1 ≤ θ < π / 4 and therefore | u 1 u | < | uv | . It follo ws that ID ( u 1 u ) < ID ( uv ). Conform the inductiv e hypothesis, Y Y k con tains a t -spanner path p ( u u 1 ) from u to u 1 . These toge ther with Lemma 1 sho w that p ( u u 1 ) ⊕ u 1 v is a t -spann er path from u to v in Y Y k . Case 2: − → uv 6∈ E Y . Let uv 1 ∈ E ∩ K u ( v ) b e the edge selected b y u in the Y ao-Step . Th us w e ha v e that ID ( uv 1 ) < ID ( uv ) and therefore | uv 1 | ≤ | uv | . If − → uv 1 ∈ E Y Y , then argument s similar to the ones used for Case 1 show that uv 1 ⊕ p ( v 1 v ) is a t -spanner path from u to v in Y Y k ; the existence of a t -spanner path p ( v 1 v ) in Y Y k is ensured by the inductiv e hyp othesis. Consider now the case where − → uv 1 6∈ E Y Y . Since − → uv 1 ∈ E Y and − → uv 1 6∈ E Y Y , the edge − → uv 1 m ust ha v e b ee n eliminated b y v 1 in the R everse Y ao-Step in fa v or of another edge − − → u 1 v 1 , with ID ( u 1 v 1 ) < ID ( uv 1 ) (r efer to Figure 2b ). Let u ′ 1 b e the p ro jection of u 1 on uv 1 . By the triangle inequalit y , | uu 1 | ≤ | uu ′ 1 | + | u ′ 1 u 1 | = | uv 1 | − | u ′ 1 v 1 | + | u ′ 1 u 1 | ≤ | uv 1 | − | u 1 v 1 | cos θ + | u 1 v 1 | sin θ . (1) Similarly , if v ′ 1 is the pro jection of v 1 on uv , we ha v e | v 1 v | ≤ | v v ′ 1 | + | v ′ 1 v 1 | = | uv | − | uv ′ 1 | + | v ′ 1 v 1 | ≤ | uv | − | uv 1 | cos θ + | uv 1 | sin θ . (2) Since | uu 1 | < | uv 1 | ≤ | uv | and | v 1 v | < | uv | , Y Y k con tains t -spanner paths p ( u u 1 ) and p ( v 1 v ) (b y the in ductiv e h yp othesis). Let P 1 = p ( u u 1 ) ⊕ p ( v 1 v ). W e show that the path P = P 1 ⊕ u 1 v 1 is a t -spanner path from u to v , th us pr oving the inductiv e step. The length of P 1 is | P 1 | ≤ t ( | uu 1 | + | v 1 v | ) . Substituting inequalities (1) and (2 ) yields | P 1 | ≤ t | uv | + t | uv 1 | (1 − cos θ + sin θ ) − t | u 1 v 1 | (cos θ − sin θ ) . (3) Th us the length of P = P 1 ⊕ u 1 v 1 is | P | ≤ t | uv | + t | uv 1 | (1 − cos θ + sin θ ) − | u 1 v 1 | ( t cos θ − t sin θ − 1) . (4) Since the input graph G is λ -civilize d, we h a v e that | u 1 v 1 | ≥ λ . This along with the inequalit y (4) and the fact that | uv 1 | ≤ 1 imp lies | P | ≤ t | uv | + ( t (1 − cos θ + sin θ ) − λ ( t cos θ − t sin θ − 1)) . Note that the s econd term on the right side of the inequalit y ab o ve is non-p ositi ve for an y t ≥ λ ( λ +1)(co s θ − sin θ ) − 1 and for an y θ satisfying the cond ition cos θ − sin θ > 1 λ +1 . This completes the pro of. Theorem 2 imp lies that, for fixed small λ > 0 and for any ε > 0, one can c ho ose θ such that cos θ − sin θ = λ + ε +1 ( λ +1)( ε +1) > 1 λ +1 , to pro d u ce a t -spann er Y Y k with t ≥ λ ( λ +1)(co s θ − sin θ ) − 1 = 1 + ε . So w e hav e the follo w ing r esu lt: Corollary 3 The Y Y k structur e pr o duc e d by the YY-Sp a nner al gorithm for a given civilize d UDG is a sp anner with max imum de gr e e 2 k , length str etch factor (1 + ε ) , and p ower str etch factor (1 + ε ) β , for any r e al ε > 0 and inte ger k > 8 satisfying the c ondition cos 2 π /k − sin 2 π /k = λ + ε +1 ( λ +1)( ε +1) . 5 4 Efficien t Lo c al Y a o-Sp arse -Sink Algori thm for UDGs W e ha v e established in S ection 3 that the Y ao-Y ao graph is a length span n er for civilized UDGs. The question of whether the Y ao-Y ao graph is a length spanner for arb itrary UDGs remains op en. In ord er to guaran tee b ot h b ound ed degree and the length spanner prop erty , Li et al. [4] suggest a sparse top ology , called Y ao-Sink . Let G = ( V , E ) b e a UDG. The Y ao-Sink algorithm app lied on G consists of tw o steps: (1) Execute Y ao-Step ( G, k ) to pro duce the Y ao s p anner Y k = ( V , E k ), and (2) Execute Sink-Step ( Y k , k ) to reduce the degree of Y k . T h e Sink-Step is describ ed in detail in T able 3 . In [5] the authors show that th e outp ut Y S k generated by the Y ao-Sink metho d has maxim um degree k ( k + 2) and length stretc h fact or  1 1 − 2 sin π /k  2 . In fact, the authors sho w a more general result that app lies to mutual inclu si on g r aphs , whic h allo w for non-uniform transmission ranges at no d es. Sink-Step ( Y k = ( V , E Y ) , k ) Use the same cone partition a s in the Y ao-Step . 1. Set E Y S ← ∅ and Y S k ← ( V , E Y S ). 2. F o r each node v and each cone K v { Build the tree T ( v ) corresp onding to K v . } 2.1 Let I b e the set o f vertices u such that − → uv ∈ E Y ∩ K v . Set I ( v ) ← I . Initialize the ordered vertex sequence J ← ( v ). 2.2 Initialize T ( v ) ← ∅ . Repea t until I is empt y 2.2.1 Remov e the fir s t vertex u from the s equence J . 2.2.2 F or each cone K u Let w ∈ I ( u ) ∩ K u be the no de that minimizes ID ( − → wu ) (if an y). Add − → wu to T ( v ) and move w fro m I to J . Set I ( w ) ← I ( u ) ∩ K u . 2.3 Add all edges of T ( v ) to E Y S . Output Y S k = ( V , E Y S ) (viewed as an undi rected graph) . T able 3: Th e Sin k step. The follo wing t w o lemmas (Lemmas 4 and 5) iden tify tw o imp ortant p rop erties of the outp ut spanner Y S k generated by the Y ao-Sink m etho d. S p ecifically , they show the existence of a particular path in Y S k corresp ondin g to eac h Y ao edge remo v ed in the Sink-Step . Lemma 4 F or e ach e dge − → uv ∈ E Y , ther e is a uv - p ath Π = w 0 w 1 , w 1 w 2 , . . . , w h − 1 w h in K v ( u ) , with w 0 = v and w h = u , such that − − − − → w i w i − 1 ∈ E Y S ∩ K w i − 1 ( u ) and ID ( − − − − → w i w i − 1 ) < ID ( − − − → uw i − 1 ) , for e ach i = 1 , 2 , . . . , h . Pro of: Let I = I ( v ) b e the v ertex set defined in Step 2.1 of S ink-Step for node v and cone K v ( u ). If u ∈ I ( v ) ∩ K v ( u ) minimizes ID ( − → uv ), then the p ath sough t is Π = v u and the p ro of is finished. Otherwise, let Π = w 0 w 1 , w 1 w 2 , . . . , w p − 1 w p b e a longest path in K v ( u ) that sat isfies the conditions of the lemma: − − − − → w i w i − 1 ∈ E Y S ∩ K w i − 1 ( u ) and I D ( − − − − → w i w i − 1 ) < ID ( − − − → uw i − 1 ), for eac h i = 1 , 2 , . . . , p . W e pro ve b y cont radiction that w p = u . Assume to the con trary th at w p and u are distinct. Since − − − − − → w p w p − 1 ∈ E Y S , it must b e that w p ∈ I ( w p − 1 ). F ur th ermore, since w p and u lie in a s ame cone K w p − 1 ( u ), the set I ( w p ) defin ed in Step 2.2.2 of the Sink-Step is I ( w p ) = I ( w p − 1 ) ∩ K w p − 1 ( u ) and includes b oth w p and u . 6 K (u) v w 1 w 2 w p θ θ v = w 0 w u = w h K (u) v w 1 w 2 w 3 θ v = w 0 w l u = w h w' 1 (a) (b) Figure 3: (a) Lemma 4: path Π = w 0 w 1 , w 1 w 2 , . . . , w h − 1 w h (b) Lemma 5: [ w 1 uv ≤ θ . Consider no w the instance when w p gets pro cessed (i.e , it gets remo v ed from J in Step 2.2.1 of Sink-Step ). See Figure 3a . First observe that I ( w p ) ∩ K w p ( u ) is nonempty , since it con tains at least the no de u . This imp lies that there exists w ∈ I ( w p ) ∩ K w p ( u ) that minimizes ID ( − − → ww p ). It follo ws that − − → ww p ∈ E Y S ∩ K w p ( u ) and either w = u or ID ( − − → ww p ) < ID ( − − → uw p ). Either case con tradicts our assumption that Π is a longest path that satisfies the conditions of the lemma. In the conte xt of Lemma 4, w e next pro ve the existence of a long enough s u bpath of Π from v to one of the v ertices w ℓ ∈ Π that closely appro ximates th e dir ect link v w ℓ . Lemma 5 L et Π = w 0 w 1 , w 1 w 2 , . . . , w h − 1 w h , with w 0 = v and w h = u , b e the p ath identifie d in L emma 4 c orr esp onding to a given e dge − → uv ∈ E Y . Then ther e exists ℓ ≤ h su c h that | v w ℓ | ≥ | uv | / (2 cos θ ) and ℓ − 1 X i =0 | w i w i +1 | ≤ | v w ℓ | cos 2 θ . Pro of: Let ℓ ≤ h b e the sm allest ind ex in the sequence 1 , 2 , . . . , h suc h that | vw ℓ | ≥ | uv | / (2 cos θ ). Since | v w h | = | uv | > | uv | / (2 cos θ ), such an index alwa ys exists. Let w ′ i b e the pro jection of w i on uv , for eac h i . W e fir st pro v e that the follo w ing inv arian t holds: (a) [ w i v u ≤ θ , for ea c h i = 0 , 1 , . . . , ℓ . (b) [ w i uv ≤ θ , for eac h i = 1 , . . . , ℓ − 1. (c) | w i w i − 1 | ≤ | w ′ i w ′ i − 1 | / cos 2 θ , for eac h i = 1 , . . . , ℓ . Prop erty (a) follo ws immediately from the fact that w i and u b elo ng to a same cone K v ( u ), for eac h i . The pro of for prop erties (b ) and (c) is b y induction on the index i . The base case corresp ond s to i = 1 (i.e , Π = w 0 w 1 ). See Figur e 3b. W e pro v e th at [ w 1 uv ≤ θ (claim (b) f or the case when ℓ ≥ 2). First observ e that | w 1 v | < | uv | / (2 cos θ ), otherwise it would contradict our c h oice of ℓ . Th us w e ha v e that tan [ w 1 uv = | w 1 w ′ 1 | | uv | − | w ′ 1 v | = | w 1 v | sin [ w 1 v u | uv | − | w 1 v | cos [ w 1 v u ≤ | w 1 v | sin θ | uv | − | w 1 v | cos θ < ta n θ . If follo ws that [ w 1 uv < θ , s o claim (b) holds. F or claim (c), note that | v w 1 | ≤ | v w ′ 1 | / cos θ < | v w ′ 1 | / cos 2 θ . Assume that the claim holds for an y index less than i , for some i > 1. T o p ro v e the inductiv e step, consid er a path Π = w 0 w 1 , . . . , w i − 1 w i , with i ≤ ℓ . W e distinguish three cases: 7 w i -1 w i u v = w 0 u w i v = w 0 w -1 i w i u v = w 0 w i -1 w' i (a) (b) (c) w' i w' i -1 Figure 4: Lemma 5 pro of: (a) w i ∈ △ uw i − 1 v (b) w i − 1 ∈ △ uw i v (c) Neither (a) not (b) holds. (i) w i ∈ △ uw i − 1 v (see Figure 4a). T hen [ w i v u < \ w i − 1 v u ≤ θ (this latter inequalit y is tru e b y the inductiv e h yp othesis). Also note that the angle formed by w i − 1 w i and uv is no greater than \ uw i − 1 w i + \ w i − 1 uv ≤ 2 θ and therefore | w i w i − 1 | ≤ | w ′ i w ′ i − 1 | / cos 2 θ . (ii) w i − 1 ∈ △ uw i v (see Figure 4b). W e sho w that [ w i uv ≤ θ (claim (b) for the ca se when i ≤ ℓ − 1). First observe that the condition | w i v | < | uv | / (2 cos θ ) m u st hold for eac h i ≤ ℓ − 1; otherwise, w e could find a lo wer ind ex i < ℓ satisfying the condition | v w i | ≥ | uv | / ( 2 cos θ ), con tradicting our c hoice of ℓ . As b efore, we ha v e that tan [ w i uv = | w i w ′ i | | uv | − | w ′ i v | = | w i v | sin [ w i v u | uv | − | w i v | cos [ w i v u ≤ | w i v | sin θ | uv | − | w i v | cos θ < ta n θ . If follo ws that [ w i uv < θ . Also n ote that in this ca se the angle formed by w i − 1 w i and uv is no greater than \ w i w i − 1 u < θ and therefore | w i w i − 1 | ≤ | w ′ i w ′ i − 1 | / cos θ < | w ′ i w ′ i − 1 | / cos 2 θ . (iii) Ne ither (i) not (ii) holds (see Figure 4c). Argument s identica l to the ones u s ed in case (ii) ab o ve sho w that [ w i uv ≤ θ . Also n ote that th e angle formed b y w i − 1 w i and uv is n o greater than max { \ w i − 1 uv , [ w i v u } ≤ θ and therefore | w i w i − 1 | ≤ | w ′ i w ′ i − 1 | / cos θ < | w ′ i w ′ i − 1 | / cos 2 θ . W e h a v e shown that | w i w i − 1 | ≤ | w ′ i w ′ i − 1 | / cos 2 θ for eac h i = 1 , 2 , . . . , ℓ . Sum m ing up ov er i yields ℓ − 1 X i =0 | w i w i +1 | ≤ | w ′ ℓ v | cos 2 θ < | w ℓ v | cos 2 θ . This completes the pro of. W e will sho w that Lemmas 4 and 5 en ab le us to disca rd some Y ao edges from Y k prior to execut- ing the Sink-Step , without compromising the span n er p rop erty . This leads to the construction of efficien t sparse sink trees in the Sink-Step . T able 4 describ es our metho d called Y ao-Sp arse-Sink that incorp orates this intermediate edge fi ltering step. In the filtering step, eac h no de u partitions the set of Y ao edges incident to u in to a n umber of subsets, su c h that all edges in a same subset F i ha v e sim ilar s izes. The asp ect ratio of eac h subset F i is con trolled by the in p ut parameter r > 1. F rom eac h sub set F i , only the Y ao edge w ith smallest ID is carried on to the Sink-Step ; all other Y ao edges f r om F i are discarded. It can b e verified that the result Y E k of the filtering step is a sp anner for G of maxim um degree O (log ∆ ), w here ∆ is the asp ect ratio of G . Because of space co nstraints we skip this pro of and turn instead t o sho wing that the output Y E S k generated b y the Y ao-Sp arse-Sink m etho d is a spanner of constan t maximum d egree. In tuitiv ely , if Y E S k con tains short p aths b et w een the end p oints of an edge p ro cessed in the Sink-Step , then Y E S k con tains short paths b et wee n the endp oin ts of all nearb y edges of similar sizes. 8 Algorithm Y ao-Sp arse-Sink ( G = ( V , E ) , k , r ) 1. Execute Y ao-Step ( G, k ). The result is the Y ao spa nner Y k = ( V , E Y ). 2. F o r each node v ∈ V and each cone K v Let F ⊆ E Y ∩ K v be the subset of Y ao e dges from K v directed into v Let − → uv ∈ F b e the edge of minimum ID . Let ∆ be the asp ect ra tio of F . Partition F in to disjoint subsets F 1 , F 2 , . . . , F s , with s = ⌈ log r ∆ ⌉ , such that F i = { ab ∈ F | | uv | r i − 1 ≤ | ab | < | uv | r i } . F o r each i = 1 , 2 , . . . , s Add to E Y E (initially ∅ ) the edge from F i of smallest ID . Result is Y E k = ( V , E Y E ) of degree O (log ∆) . 3. Execute Sink-Step ( Y E k , k ). The result is Y E S k = ( V , E Y E S ). Output Y E S k (view ed as an undirected graph) . T able 4: Th e Y ao-Sparse-Sink algorithm. Theorem 6 L et G = ( V , E ) b e a UDG and let r > 1 , k ≥ 8 , θ = 2 π /k and λ = 1 2 r cos θ b e c onstants such that (cos θ − sin θ ) > λ λ +1 . When run with these values of r and k , the output of the Y ao-Sp arse-Sink algorithm i s a t -sp anner of de gr e e k ( k + 2) , for any t ≥ λ/ cos(2 θ ) ( λ +1)(co s θ − sin θ ) − 1 . Pro of: The d egree of Y E S k is no greater than the degree of the Y ao-Sink spann er , whic h is k ( k + 2) [5]. W e n ow prov e that Y E S k is a t -sp an n er. The pr o of is b y ind uction on the rank of edges in the set E ord ered b y increasing ID . The base case corresp onds to the edge uv ∈ E of minimum ID . Argum en ts similar to the ones u sed for the base case in Th eorem 2 sho w th at uv ∈ Y Y S k . The inductiv e hyp othesis tells that Y E S k con tains t -spanner paths b et w een th e end p oints of an y edge uv ∈ E w hose rank is no greate r than some v alue j ≥ 0. T o prov e the inductiv e step, consider th e edge uv ∈ E of rank j + 1. Assume without loss of generalit y that I D ( uv ) = ID ( − → uv ). W e d iscuss t w o cases, dep ending on w hether − → uv b elongs to E Y or not. Case 1: − → uv ∈ E Y . Ass ume first that − → uv ∈ E Y E . By Lemma 5, Y E S k con tains an edge − − → w 1 v ∈ K v ( u ). Th is imp lies that [ uv w 1 ≤ θ < π / 4. F urtherm ore, since ID ( w 1 v ) < ID ( uv ) (and therefore | w 1 v | ≤ | uv | ), w e h a v e that | uw 1 | < | uv | (see Figure 5a). Thus w e can use the inductiv e hypothesis to sho w th at Y E S k con tains a t -spanner path p ( u w 1 ). By Lemma 1, p ( u w 1 ) ⊕ w 1 v is a t -spanner path in Y E S k from u to v . Assume no w that − → uv 6∈ E Y E . Let F be the edge set corresp ond ing to cone K v ( u ) and let i b e suc h that − → uv ∈ F i . Since uv 6∈ E Y E there is an edge − → u 1 v ∈ F i of smaller ID that gets add ed to E Y E . Note ho we ve r that u 1 and u b elong to one same cone K v ( u ) (see Fig ur e 5b). By the same argumen ts as ab ov e, there is − − → w 1 v ∈ K v ( u ) corresp ond in g to the edge − → u 1 v ∈ E Y E whic h en ab les us to iden tify the t -spann er path p ( u w 1 ) ⊕ w 1 v from u to v in Y E S k . Case 2: − → uv 6∈ E Y . Let uv 1 ∈ E Y b e the edge selected b y u in the Y ao-Step . Thus w e ha v e that ID ( uv 1 ) < ID ( uv ) and therefore | uv 1 | ≤ | uv | . If − → uv 1 ∈ E Y E S , then argument s similar to the ones used for Case 1 sho w that uv 1 ⊕ p ( v 1 v ) is a t -sp anner path from u to v in Y E S k . Consider no w the case when − → uv 1 6∈ E Y E S . Let F b e the edge set corresp ondin g to cone K v 1 ( u ) and let i b e suc h that − → uv 1 ∈ F i . Let − − → u 1 v 1 ∈ F i b e th e edge that gets add ed to E Y E in Step 2 9 u K (u) v v = w 0 w 1 θ u K (u) v v = w 0 w 1 θ u 1 u K (u) v v = w 0 u 1 1 v w 1 w 2 w l w 3 (b) (a) (c) 1 Figure 5: Pro of of Th eorem 6: (a) Case ~ uv ∈ E Y E (b) Case ~ uv ∈ E Y \ E Y E (c) Case ~ uv 6∈ E Y . of the Y a o-Sp arse-Sink algorithm. Since b oth uv 1 and u 1 v 1 b elong to a same set F i , and s ince ID ( − − → u 1 v 1 ) ≤ ID ( − → uv 1 ) (equalit y h app ens when u 1 = u ), w e ha v e that | u 1 v 1 | ≥ | uv 1 | /r . (5) Lemma 5 indicates th at, corresp onding to the edge − − → u 1 v 1 ∈ E Y E , there exists a path P 0 ∈ Y E S k ∩ K v 1 ( u 1 ) extending from w 0 = v 1 to some v ertex w ℓ , suc h that | P 0 | ≤ | w ℓ v 1 | / cos 2 θ . (6) | w ℓ v 1 | ≥ | u 1 v 1 | / (2 cos θ ) (7) No w n ote that, since | v 1 w ℓ | ≤ | v 1 u 1 | and since \ w ℓ v 1 u ≤ θ ≤ π / 4, we ha v e that | uw ℓ | < | uv 1 | ≤ | uv | . Similarly , | v 1 v | ≤ | uv | . Th us w e can u s e th e inductiv e h yp othesis to claim the existence of t -spanner paths p ( u w ℓ ) and p ( v 1 v ). Let P 1 = p ( u w ℓ ) ⊕ p ( v 1 v ). W e show that P = P 0 ⊕ P 1 is a t -spanner path from u to v . Calculations identic al to the ones used to derive the inequalit y (3) yield | P 1 | ≤ t | uv | + t | uv 1 | (1 − cos θ + sin θ ) − t | w ℓ v 1 | (cos θ − sin θ ) . This along with (6) shows that the lengt h of P = P 0 ⊕ P 1 is | P | ≤ t | uv | + t | uv 1 | (1 − cos θ + sin θ ) − | w ℓ v 1 | ( t cos θ − t sin θ − 1 / cos 2 θ ) . Substituting (5) and (6) yields | P | ≤ t | uv | + ( t | uv 1 | (1 − cos θ + sin θ ) − | uv 1 | 2 r cos θ ( t cos θ − t sin θ − 1 / cos 2 θ )) . (8) Note th at the second term in the right h an d side of the inequalit y (8) is non-p ositive for an y t ≥ λ/cos 2 θ ( λ +1)(co s θ − sin θ ) − 1 and for an y θ satisfying the condition cos θ − sin θ > 1 λ +1 . Argumen ts similar to the ones u sed for C orollary 3 s ho w that, for appr op r iate v alues of r and k corresp ondin g to a fixed ε > 0, Y Y S k is a (1 + ε )-spanner. Efficien t Lo cal Implemen tation. F or a lo cal implementat ion of the Y ao-Step , the authors prop ose in [3] to ha ve eac h sink no de u build T ( u ) and then b roadcast T ( u ) to all no des in T ( u ). It can b e easily v erified th at, for eac h n o de u and eac h cone K u , the neighbors of u that lie in K u (including u ) form a clique. This suggests a more efficien t alternate lo cal implementa tion of the 10 Y ao-Step : eac h no de collects the co ord inate in formation from its immediate neigh b ors, then sim- ulates the execution of the Y ao-Step locally , on the colle cted neigh b orho od. T his implemen tation a v oids broadcasting messages of size O ( n ) (enco ding the sink trees) by eac h n o de, th us sa ving some battery p o wer. This idea can b e extend ed to the Y ao-Sp arse-Sink algorithm as w ell: eac h no d e collect s its neigh b orh o o d information in one round of comm unication, then simulat es the exec ution the Y ao-Sp arse -Sink algorithm on the collected neigh b orhoo d . 5 T otal W eigh t of Y k , Y Y k , Y S k and Y E S k Define the total weight w t ( G ) of a graph G as the sum of the lengths of its constituent edges. W e first sho w that the total w eigh t of the Y ao graph Y k constructed by Y ao-Step is arb itrarily high compared to the w eigh t of the Minimum Spanning T ree (MST) for V . Although this result is fairly straigh tforw ard, to the b est of our kno wledge it has not app eared in the literature. Theorem 7 L et G b e a UDG and let Y k = Y ao-Step ( G, k ). Then w t ( Y k ) = Ω( n ) · w t ( M S T ) . Pro of: Consider a set of n = 2 s nod es equally distributed along the top and b ottom sides of a unit square, as in Figure 6. Let u 1 , u 2 , . . . u s denote the top no des and v 1 , v 2 , . . . v s the b otto m no des. Observe that for eac h n o de u i , the angular d istance b et ween an y of its left/righ t neigh b ors and v i 1 u 2 u s u 1 v 2 v s v θ 1 u 2 u s u 1 v 2 v s v (a) (b) Figure 6: Y k has unboun ded weigh t: (a) wt ( Y k ) = n/ 2 + 2; (b) wt ( M S T ) = 3. is π/ 2. This means that the only edge inciden t to u i that lies in the cone of angle 2 π /k ≤ π/ 3 (for k ≥ 6) cen tered at u i and con taining v i is u i v i (see Figure 6a). Consequently , u i adds − − → u i v i to E Y in the Y ao-Step . S imilarly , v i adds − − → v i u i to E Y . Thus the total weigh t of Y k is no less than s X i =1 | u i v i | = n/ 2 . Ho w ev er, the w eigh t of the sp anning tree illustrated in Figure 6b is 3. T his completes the pro of. W e ha v e sho wn th at, for an y no de u in the top ology from Figure 6a, at most one edge from Y k lies in an y cone K u cen tered at u . T his implies that: (a) No edges fr om Y k get discarded in the Reve rse Y ao-Step . (a) No edges fr om Y k get discarded in the filtering step (Step 2) of Y ao-Sp arse-Sink . 11 (b) No edges fr om Y k get altered in the Sink-Step . This implies that Y Y k , Y S k and Y E S k are all iden tical to Y k and therefore ha v e unb ounded weigh t as w ell. It is w orth noting that any civilized UDG G = ( V , E ) h as w eigh t within a constan t factor of w t ( M S T ( V )) and therefore the structures Y k , Y Y k , Y S k and Y E S k for civilized UDGs ha v e b ound ed weigh t as well. This follo ws immediately from a result obtained by Das et al. [2]: F act 2 ( T heorem 1.2 in [2 ]). If a set of line segmen ts E satisfies the isolation prop erty , then wt ( E ) = O (1) · w t ( M S T ). A set of line segmen ts E is said to satisfy the isolation pr op erty if eac h segmen t uv ∈ E can b e asso ciated with a cylinder B of heigh t and width equal to c | uv | , for s ome constan t c > 0, suc h that the axis of B is a subsegment of uv and B do es not in tersect any line s egment other than uv . In the case of λ -civilized UDGs, this p rop erty is satisfied by c = λ . 6 Conclusions W e ha ve shown that the Y ao-Y ao graph is a spanner for UDGs of b ounded asp ec t ratio. W e ha v e also prop ose d an extension of the Y ao-Sink metho d, called Y ao-Sparse-Sink, that enables an efficien t local computation of sparse sink trees. The Y ao-Sparse-Sink method is preferable to the Y ao-Sink metho d fo r topology con trol in highly dynamic wireless en vironm en ts. Our analysis of the Y ao-Sparse-Sink method pro vides additional insight in to the prop erties of the Y ao-Y ao stru cture. Ho w ev er, the main question of whether the Y ao-Y ao graph for arbitrary UDGs is a length spanner or not remains op en. Ac kno wledgemen t. W e thank Mic hiel Smid for helpful discussions on these p r oblems. References [1] A. Cz uma j and H. Zhao . F ault-tole r ant geometric spanners. Discr ete & Computational Ge ometry , 32(2):207 –230 , 20 04. [2] Ga utam Das, Gir i Naras imha n, and Jeffrey Sa low e. A new w ay to weigh ma lno urished Euclidea n g raphs. In SODA ’95: Pr o c e e dings of the sixth annual ACM-SIAM symp osium on Discr ete algorithms , pages 215–2 22, Philadelphia, P A, USA, 19 9 5. So ciety for Industrial a nd Applied Mathematics. [3] X. Li, P . W an, Y. W ang, and O. F r ieder. Sparse p ow er efficient top olo gy for wireless netw orks . In HICSS’02: Pr o c. of the 35th Annual Hawaii In ternational Confer enc e on System S cienc es , volume 9, page 29 6.2, 2002 . [4] X. Y. Li, P . J. W an, and Y. W ang. 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