Relaxed spanners for directed disk graphs

Symposium on Theoretical Aspects of Computer Science 2010 (Nancy , Fr ance), pp. 609-620 www .st acs-conf .org RELAXED SP ANNERS F OR DIRECTED DISK GR APHS DA VID PELEG 1 AND LIAM RODITTY 2 1 Department of Computer Science and App lied Mathematics, The W eizmann Institute of S cience, Rehov ot 76100 , Israel E-mail addr ess : david.pele g@weizmann. ac.il 2 Department of Computer Science, Bar-Ilan U niversit y , Ramat-Gan 52900, Israel E-mail addr ess : liamr@macs .biu.ac.il Abstra ct. Let ( V , δ ) b e a finite metric space, where V is a set of n p oints and δ is a distance function defi ned for these p oints. Assume that ( V , δ ) has a constant doubling dimension d and assume th at each p oin t p ∈ V has a disk of radius r ( p ) around it. The disk graph that co rresp onds to V and r ( · ) is a di r e cte d graph I ( V , E , r ), whose v ertices are the p oints of V and whose edge set includ es a directed edge from p to q i f δ ( p , q ) ≤ r ( p ). In [8] w e presen ted an a lgorithm for constructing a (1 + ǫ )-spann er of size O ( n/ǫ d log M ), where M is the maximal radius r ( p ). The current pap er presents t wo results. The first show s that the spann er of [8] is essentia lly optimal, i.e., for metrics of constant doubling d imension it is not p ossible to guarantee a spanner whose size is indep endent of M . The second result sho ws that by slightly relaxing the requirements and allo wing a small pertu rbation of the radius assignmen t, considerably b etter spanners can b e constructed. In particular, we s how that if it is allow ed t o use edges of the d isk graph I ( V , E , r 1+ ǫ ), where r 1+ ǫ ( p ) = (1 + ǫ ) · r ( p ) for every p ∈ V , then it is possible to get a (1 + ǫ )-spanner of size O ( n/ǫ d ) for I ( V , E , r ). Our algorithm is simple and can b e implemented efficiently . In tro duction This p ap er concerns efficien t constru ctions of spanners for disk graphs, an imp ortan t family of dir ected graph s. A sp anner is essen tially a skelet on of the graph, namely , a sp arse spanning sub graph that faithfully represents distances. F orm ally , a subgraph H of a graph G is a t -spanner of G if δ H ( u, v ) ≤ t · δ G ( u, v ) for ev ery t w o no des u and v , where δ G ′ ( u, v ) denotes the distance b et w een u and v in G ′ . W e refer to t as the str etch factor of the span n er. Graph sp anners hav e receiv ed considerable attent ion ov er the last t wo d ecades, and w ere used imp licitly or explicitly as key ingredien ts of v arious distribu ted applications. It is kno wn ho w to efficien tly construct a (2 k − 1)-spanner of size O ( n 1+1 /k ) for ev ery weig h ted undirected graph, and this size-stretc h tradeoff is conjectured to b e tig ht . Basw ana and 1998 A CM Subje ct Cl assific ation: F.2 ANA L Y S IS O F ALGORI THMS AND PROBLEM COMPLEX- ITY, F.2.0 General . Key wor ds and phr ases: Spanners, Directed graphs. Thanks: Supp orted in part b y gran ts from t h e Minerv a F oundation and t he Israel Ministry of S cience. c  D . P eleg and L. Roditty CC  Creative Commons Attribution- NoDerivs License 610 D. PELEG AND L. RODITTY Sen [ ? ] pr esen ted a linear time randomized algorithm for computing suc h a spanner. In directed graphs , ho w ev er, the situation is differen t. No such general size-stretc h tradeoff can exist, as indicated b y considerin g the example of a d irected bipartite graph G in which all the edges are directed from one s ide to the other; clearly , the only spanner of G is G itself, as any sp anner f or G must con tain ev ery edge. The m ain difference b etw een u ndirected and directed graphs is that in u ndirected graphs the d istances are symmetric, that is, a path of a certain length fr om u to v can b e u sed also from v to u . In d irected graphs, h o w ev er, the existence of a path f r om u to v do es not imply an ything on th e distance in the opp osite d ir ection f r om v to u . Hence , in order to obtain a spanner for a directed graph one m ust imp ose some restriction either on the graph or on its distances. In order to bypass the problem of asymmetric distances of directed graphs, Co w en and W agner [5] in tro duced the notion of r oundtrip distanc es in wh ic h the distance b et w een u and v is comp osed of the shortest path from u to v plus the shortest path from v to u . It is easy to see that und er th is definition distances are sym m etric also in directed graphs. It is shown by Co wen and W agner [5] and later by Ro d itt y , Thorup and Zwic k [6] that metho ds of path approximati ons from undirected graphs can work us ing more ideas also in directed graphs w hen roun dtrip distances are considered. Bollob´ as, Copp ersmith and Elkin [ ? ] introdu ced the notion of distanc e pr eservers and sho wed that they exist also in directed graphs. In [8] we pr esen ted a spanner construction for directed graphs without symmetric dis- tances. The restriction that w e imp osed on the graph wa s that it must b e a disk graph. More formally , let ( V , δ ) b e a fin ite metric sp ace of constant d oubling dimension d , w here V is a set of n p oin ts and δ is a distance function defin ed for these p oin ts. A metric is said to b e of c onstant doubling dimension if a ball with radius r can b e cov ered by at most a constan t num b er of b alls of radius r / 2. Ev ery p oint p ∈ V is assigned with a radius r ( p ). The disk graph that corresp onds to V and r ( · ) is a dir ected graph I ( V , E , r ), whose v ertices are the p oin ts of V and whose edge set includes a directed edge f rom p to q if q is insid e the d isk of p , th at is, δ ( p, q ) ≤ r ( p ). In [8] we p resen ted an algorithm for constructing a (1 + ǫ )-spanner with size O ( n/ǫ d log M ), where M is the maximal r adius. In the case that w e remo ve the radius restriction the resulted graph is the complete u ndirected graph where the we igh t of ev ery edge is the distance b et w een its end p oint. In su ch a case it is p ossible to create (1 + ǫ )-spanners of size O ( n /ǫ d ), see [4], [2] and [9] f or more details. Moreo ver, when the r adii are all the same and the graph is the unit d isk graph then it is also p ossib le to create (1 + ǫ )-spanner s of size O ( n/ǫ d ), see [3], [8]. As a result of that, a natural question is w hether a sp anner size of O ( n /ǫ d log M ) in the case of directed disk graph is indeed the b est p ossible or maybe it is p ossible to ge t a sp anner of size O ( n/ǫ d ) as in the cases of the complete graph and the unit disk graph. F or the case of the Euclidean metric space, the answer turns out to b e p ositive ; a s im p le mo dification of the Y ao graph constru ction [11] to fit the directed case yields a directed spanner of size O ( n/ǫ d ). Ho wev er, th e question r emains for more general metric sp aces, and in particular for th e imp ortant family of m etric spaces of b ounded doub ling dimension. In this pap er w e pr o vid e an answ er for this question. W e s h o w that our construction from [8] is essen tially optimal b y p ro viding a metric space with a constant doubling dimen- sion and a radius assignm ent whose corr esp onding d isk graph has Ω( n 2 ) edges and n on e of its edges can b e remo v ed. (This do es n ot con tradict our spanner construction from [8] as the maximal r adius in that case is Θ(2 n ) and hence log M = n .) RELAXED SP ANNERS F OR DIRECTED DISK GRAPHS 611 This (essentia lly negativ e) optimalit y r esult m otiv ates our m ain int erest in the current pap er, whic h fo cuses on attempts to slight ly relax the assumptions of the mo del, in order to obtain sparser spanner constructions. Indeed, it turns out that suc h sp arser spanner constructions are feasible under a suitably relaxed mo del. Sp ecifically , w e demonstrate the fact that if a s mall p ertur bation of the radius assignmen t is allo w ed, then a (1 + ǫ )-spanner of size O ( n/ǫ d ) is attainable. More formally , we sho w that if we are allo wed to use edges of the d isk graph I ( V , E , r 1+ ǫ ), w here r 1+ ǫ ( p ) = (1 + ǫ ) · r ( p ) for every p ∈ V , then it is p ossible to get a (1 + ǫ ) -spanner of size O ( n/ǫ d ) for th e original disk graph I ( V , E , r ). This approac h is similar in its nature to the notation of e mulators introd uced b y Dor, Halp erin and Zw ic k [1]. An em ulator of a graph m a y use any edge that do es not exist in the graph in order to app ro ximate its distances. It was used in th e con text of sp anners with an additiv e stretc h . The main application of disk graph sp anners is for top olog y con trol in the wireless ad ho c net w ork mo del. In this mo del th e p o wer requ ired for tr ansmitting from p to q is commonly tak en to b e δ ( p, q ) α , where δ ( p, q ) denotes the distance b etw een p and q and α is a constan t typica lly assumed to b e b et we en 2 an d 4. Most of the ad ho c net w ork literature mak es the assumption that the transmission range of all no des is identica l, and consequently represent s the net w ork b y a unit disk gr aph (UDG), n amely , a graph in whic h tw o no des p, q are adjacen t if their d istance satisfies δ ( p, q ) ≤ 1. A unit disk graph can ha v e as many as O ( n 2 ) edges. There is an extensiv e b o dy of literature on spanners of unit disk graphs. Gao et al. [3], W ang and Y ang-Li [10] and Y ang-Li et al. [7] considered the restricted Delauna y graph, whose worst-c ase stretc h is constan t (larger th an 1 + ǫ ). In [8] we show ed that any (1 + ǫ )- geometric spanner can b e turn ed into a (1 + ǫ )-UDG spann er. Disk graph s are a n atural generalization of un it disk graphs, that provide an in termedi- ate mo del b et wee n the complete graph and the un it disk graph . Our size efficien t s p anner construction for disk graphs wh ose radii are allo wed to b e sligh tly larger falls exactly into the mo del of n et works in wh ic h the stations can c hange their transmission p ow er. In partic- ular our constriction implies th at if any statio n increases its transmission p o wer b y a sm all fraction then a considerably improv ed top ology can b e b uilt f or the n et work. Our resu lt has b oth practical and th eoretical imp lications. F r om a practical p oin t of view it sho ws that, in certain scenarios, extending the transmission radii even by a small factor can significan tly imp ro v e the ov erall qu alit y of the netw ork top ology . The result is also v ery in triguing from a theoretical standp oin t, as to the b est of our knowledge , our relaxed spanner is th e fi rst example of a spanner constru ction f or directed graphs that enjo ys the same prop erties as the b est constru ctions for u ndirected graph s . (As men tioned ab ov e, it is ea sy to see that for general directed graph s, it is not p ossible to ha v e an algorithm that giv en an y directed graph pro d uces a sp ars e spanner for it.) In th at sense, our result can b e view ed as a significan t step to wards gaining a b etter un derstanding for some of the fundamental differences b et w een d irected and u ndirected graph s. Our result also op ens sev eral n ew researc h directions in the relaxed mo d el of disk graph s . The most ob vious researc h q u estions th at arise are whether it is p ossible to obtain other ob jects that are kno wn to exist in undir ected graphs, suc h as compact routing schemes and distance oracles, for disk graph s as we ll. The rest of this pap er is organized as follo ws. In the next s ection we presen t a metric space of constant doub ling dimension in whic h no edge can b e r emo ved f r om its corre- sp ond in g disk graph. Section 2 fir st describ es a simple v ariant of our constru ction from [8], 612 D. PELEG AND L. RODITTY (a) (b) y 1 x n x 1 x 2 x i y n X y 3 y 2 y 1 Figure 1: (a) First step in constructing the non-sparsifi ab le d isk graph G . (b) Th e non- sparsifiable disk graph G . and then uses it together with new ideas in order to obtain our new r elaxed construction. Finally , in S ection 3 we pr esent some concluding remarks and op en pr oblems. 1. Optimalit y of the spa nner construct ion In this section we build a disk graph G w ith 2 n v ertices and Ω( n 2 ) edges that is non- sparsifiable, namely , wh ose only spann er is G itself. In this graph M = Ω(2 n ) h ence our spanner construction from [8] has a size of Ω( n 2 ) and is essent ially optimal. Giv en a set of p oint s, w e presen t a distance function suc h that for a giv en assignment of radii for th e p oin ts an y spanner of the resulting disk graph must ha ve Ω( n 2 ) edges. W e then pro v e that the und er lyin g metric space has a constan t doublin g dimension. W e p artition the p oints in to tw o t yp es, Y = { y 1 , . . . , y n } and X = { x 1 , . . . , x n } . W e no w d efine the distance fun ction δ ( · , · ) and the radii assignment r ( · ). The main idea is to create a bipartite graph G ( X , Y , E ) in whic h every p oin t of Y is connected b y a directed edge to all the p oints of X . The distance b et we en an y t wo p oints x i and x j is at least 1 + ǫ for some small 0 < ǫ < 1 and the radius assignment of ev ery p oin t x i is exactly 1. Thus, there are no edges b et w een the p oint s of X . W e n o w define the distances b et w een th e p oint s of Y and th e p oints of X . W e start with the p oin t y 1 . Let δ ( y 1 , x i ) = n for ev ery x i ∈ X and let r ( y 1 ) = n . Place the p oin ts of X on the b ound ary of a ball of radius n cen tered at y 1 suc h that the d istance b et wee n an y t wo consecutiv e p oin ts x i and x i +1 is exactly 1 + ǫ . This is depicted in Figure 1(a). T urning to th e p oin t y 2 , let δ ( y 2 , x i ) = 2 n for every x i ∈ X , δ ( y 2 , y 1 ) = 2 n + ǫ , an d r ( y 2 ) = 2 n . Hence there is an edge fr om y 2 to all the p oin ts of X , but no edge connects y 2 and y 1 . W e now turn to define the general case. Consider y i ∈ Y . Let r ( y i ) = 2 i − 1 n and δ ( y i , x j ) = 2 i − 1 n for ev er y x j ∈ X . Let δ ( y i , y i − 1 ) = 2 i − 1 n + ǫ , and in general, f or ev ery 0 < j < i we h a ve δ ( y i , y j ) = i − 1 X k = j δ ( y k +1 , y k ) , (1.1) RELAXED SP ANNERS F OR DIRECTED DISK GRAPHS 613 implying that δ ( y i , y j ) < 2 i n. (1.2) It is easy to v erify that y i has outgoing edges to the p oin ts of X (and to them only) and it do es not ha ve any incoming edges. See Figure 1(b). The r esulting disk graph G has 2 n ve rtices and Ω( n 2 ) edges. Clearly , removi ng an y edge from G will in cr ease the distance b et w een its head and its tail to infinit y , and thus the only s p anner of G is G itself. It is left to sho w that the metric space defin ed ab o ve for G has a constant doubling dimension. Giv en a metric space ( V , δ ), its doubling dimension is d efined to b e the minimal v alue d suc h th at ev ery b all B of radius r in the metric space can b e co vered by 2 d balls of radius r / 2. I n the next Theorem we prov e that for the metric s pace describ ed ab ov e, d is constan t. Theorem 1.1. The metric sp ac e ( X ∪ Y , δ ) define d for G has a c onstant doubling dimension. Pr o of. Let B b e a ball with an arb itrary radius r . W e sh o w th at it is p ossible to co v er all the p oint s of X ∪ Y within B using a constant num b er of balls w hose radius is r / 2. The pro of is d ivided in to tw o cases. Case a: There is s ome y j ∈ Y within the ball B . (If there is more th an one such p oint, then let y j b e the p oin t w hose in dex is maximal.) Let B ′ b e a ball of radius R = 2 r cen tered at y j . Clearly B ⊂ B ′ , so B ′ con tains all the p oin ts of B . In what follo w s w e sho w that all the p oin ts of X ∪ Y within B ′ can b e co v ered by a constan t num b er of balls of radius r / 2. Let y i b e the p oint within B ′ whose index is maximal. W e h a ve to consider t wo p ossib le scenarios. The first is that y j = y i . This implies that y j +1 / ∈ B ′ , hence R < δ ( y j +1 , y j ) = 2 j n + ǫ . W e now sh o w that it is p ossib le to co ve r B ′ b y a constan t n umber of balls of radius R / 4. If R < 2 j − 1 n , then only y j is within B ′ and it is cov ered by a ball of radius R / 4 cente red at itself. If 2 j − 1 n ≤ R < 2 j − 1 n + ǫ , then B ′ con tains all the p oints of X and y j . F rom pac king arguments it follo w s that it is p ossible to co ver all the p oints of X b y a constant n u m b er of balls of radius n/ 4, hence also by a constant num b er of b alls of radius R ≥ n . The p oint y j itself is co vered b y a ball cen tered at it. Finally , if 2 j − 1 n + ǫ ≤ R < 2 j n + ǫ , then R / 4 is at least 2 j − 3 n + ǫ/ 4. A ball cen tered at y j − 3 of radius R/ 4 co vers ev ery y k within B ′ , where 1 ≤ k ≤ j − 3, as δ ( y j − 3 , y k ) ≤ 2 j − 3 n . Hence, w e co v er Y ∩ B ′ b y balls of radius R/ 4 whose cen ters are y j , y j − 1 , y j − 2 and y j − 3 . W e co ver X ∩ B ′ as b efore. This completes th e first scenario, wh ere y i = y j . Assume now that y i 6 = y j . This implies that δ ( y i , y j ) ≤ R and that R < δ ( y i +1 , y j ), wh ere the fir st inequalit y f ollo ws from the f act th at y i ∈ B ′ and the second inequalit y follo ws from the f act that y i is th e p oin t with maximal index inside B ′ , h ence, y i +1 / ∈ B ′ . As δ ( y i , y i − 1 ) ≤ δ ( y i , y j ), we get that 2 i − 1 n + ǫ ≤ R . Also, b y (1.2), δ ( y i +1 , y j ) < 2 i +1 n . W e conclude that 2 i − 1 n ≤ R < 2 i +1 n and that R / 4 ≥ 2 i − 3 n . A b all cen tered at y i − 3 of r adius R/ 4 co v ers every y k within B ′ , where k ≤ i − 3, as δ ( y i − 3 , y k ) ≤ 2 i − 3 n . Hence, w e can cov er B ′ ∩ Y b y balls of radiu s R / 4 whose cen ters are y i , y i − 1 , y i − 2 and y i − 3 . W e co ver X ∩ B ′ as b efore. Th is completes the first case. Case b: Th e b all B do es not contai n any p oin t from Y . The p oin ts of X are spread as app ears in Figur e 1 (a), thus b y stand ard pac king argumen ts, an y ball that contai ns only p oints from X is co v ered by a constan t num b er of balls of h alf the radius. 614 D. PELEG AND L. RODITTY 2. Impro v ed spanne r in the relaxed disk graph mo del The (negativ e) optimalit y result from the p revious section motiv ates us to lo ok f or a sligh tly r elaxed defin ition of disk graphs in which it will s till b e p ossible to create a spanner of size O ( n/ǫ d ). Let ( V , δ ) b e a metric space of constant doubling dimension d w ith a radiu s assignment r ( · ) f or its p oin ts and let I = ( V , E , r ) b e its corresp onding disk graph. Assume th at we m ultiply the radiu s assignment of ev ery p oin t by a facto r of 1 + ǫ , for some ǫ > 0, and let I ′ = ( V , E ′ , r 1+ ǫ ) b e the corresp onding disk graph. I t is easy to see that E ⊆ E ′ . In this sectio n w e sho w that it is p ossible to create a (1 + ǫ )-spann er of size O ( n/ǫ d ) if w e are allo wed to use edges of I ′ . As a fi rst step w e p resen t a simple v arian t of our (1 + ǫ )-spanner construction of size O ( n/ǫ d log M ) from [8 ]. This v ariation is needed in order to obtain the efficien t construction in the relaxed mo del whic h is p resen ted r igh t afterw ards . 2.1. Spanners for general disk graphs Let ( V , δ ) b e a metric space of constan t doubling dimension and assume that an y p oin t p ∈ V is the cen ter of a ball of radius r ( p ), where r ( p ) is tak en from the range [1 , M ]. In this section we describ e a simple v ariant of our construction from [8], whic h computes a (1 + ǫ )-spanner with O ( n/ǫ d log M ) edges for a giv en disk graph. W e then us e this v ariant, together with new ideas, in order to obtain (in the n ext section) our main result, namely , a spanner with only O ( n /ǫ d ) edges. The sp anner construction algorithm receiv es as inpu t a d ir ected graph I ( V , E , r ) and an arbitrarily sm all (constant ) appr o xim ation factor ǫ > 0, and constructs a set of span- ner edges E DIR SP , return ing the spanner subgraph H DIR ( V , E DIR SP ). Th e constru ction of the spanner is based on a hierarc hical p artition of th e p oin ts of V that tak es into accoun t the differen t r adius of eac h p oin t. The constru ction op erates as follo ws . Let α and β b e t w o small constan ts dep ending on ǫ , to b e fixed later on. Assume that the ball r adii are scale d so that the smallest edge in the disk graph is of w eigh t 1. Let i b e an inte ger from the range [0 , ⌊ log 1+ α M ⌋ ] and let M i = M / (1 + α ) i . The edges of I ( V , E , r ) are partitioned int o classes b y length, letting E ( M i +1 , M i ) = { ( x, y ) | M i +1 ≤ δ ( x, y ) ≤ M i } . Let ℓ ( x, y ) b e the lev el of the edge ( x, y ), that is, ℓ ( x, y ) = i suc h that ( x, y ) ∈ E ( M i +1 , M i ). Let p b e a p oin t whose ball is of radius r ( p ) ∈ [ M i +1 , M i ]. It follo ws that leve l i is the first lev el in whic h p can ha ve outgoing edges. W e denote this lev el by ℓ ( p ). F or every i ∈ [0 , ⌊ log 1+ α M ⌋ ], starting from i = 0, the edges of the class E ( M i +1 , M i ) are considered by the algorithm in a non-decreasing order. (Assume that in eac h class the edges are sorted by their w eigh t.) In eac h stage of the construction w e mainta in a set of piv ots P i . Let x ∈ V and let NN ( x, P i ) b e the n earest n eigh b or of x among the p oin ts of P i . F or a pivo t p ∈ P i , define Γ i ( p ) = { x | x ∈ V , NN ( x, P i ) = p, r ( x ) ≥ δ ( x, p ) } , namely , all the p oin ts that ha v e a directed edge to p and p is their nearest neigh b or from P i . W e refer to Γ i ( p ) as the close neighb orho o d of p . The algorithm is giv en in Figure 2. Let ( x, y ) b e an edge considered by the algorithm in the i th iteration. The algorithm fi rst c h ec ks whether x or y or b oth s hould b e add ed to the pivo ts set P i . T he main change with resp ect to [8] is that if y is assigned with a large enough radius it migh t b ecome a pivo t wh en the ed ge ( x, y ) is examined. When considering the edge ( x, y ), the algorithm acts according to the follo wing ru le: If the distance from x to its nearest neigh b or in P i is greater than β M i +1 then x is added to P i . I f the distance f rom y to its nearest neigh b or in P i is great er than β M i +1 and th e r adius of y RELAXED SP ANNERS F OR DIRECTED DISK GRAPHS 615 Algorith m disk-spanner ( I ( V , E , R ) , ǫ ) E DIR SP ← φ P 0 ← φ for i ← 0 to ⌊ log 1+ α M ⌋ for eac h ( x, y ) ∈ E ( M i +1 , M i ) do if δ ( NN ( x, P i ) , x ) > β M i +1 then P i ← P i ∪ { x } if δ ( NN ( y , P i ) , y ) > β M i +1 ∧ r ( y ) ≥ M i +1 then P i ← P i ∪ { y } if r ( y ) ≥ M i +1 if ∄ ( x ′ , y ′ ) ∈ E DIR SP s.t. x ′ ∈ Γ i ( NN ( x, P i )) ∧ y ′ ∈ Γ i ( NN ( y , P i )) then E DIR SP ← E DIR SP ∪ { ( x, y ) } if r ( y ) < M i +1 if ∄ ( x ′ , y ) ∈ E DIR SP s.t. x ′ ∈ Γ i ( NN ( x, P i )) then E DIR SP ← E DIR SP ∪ { ( x, y ) } P i +1 ← P i return H DIR ( V , E DIR SP ) Figure 2: A high leve l imp lementati on of the sp anner constr u ction algorithm for gener al disk graphs is at least M i +1 then y is added to P i . T o d ecide whether the edge ( x, y ) is added to the spanner, the follo wing tw o ca ses are consider ed . The first case is when r ( y ) ≥ M i +1 . In this case, if there is no edge from th e close n eigh b orho o d of x to the close n eigh b orho o d of y then ( x, y ) is added to the spanner. T h e second case is wh en r ( y ) < M i +1 . In this case, if there is no ed ge from the close neigh b orh o od of x to y then ( x, y ) is add ed to the spanner. When i r eac hes ⌊ log 1+ α M ⌋ , the algorithm h andles all the edges that b elong to E ( M ⌊ log 1+ α M ⌋ +1 , M ⌊ log 1+ α M ⌋ ). This includes also edges w hose we igh t is 1, the minimal p ossible w eigh t. The algorithm returns the d irected graph H DIR ( V , E DIR SP ). In wh at follo ws we prov e that for su itably c hosen α and β , H DIR ( V , E DIR SP ) is a (1 + ǫ )- spanner with O ( n/ǫ d log M ) edges of the directed graph I ( V , E , r ). Lemma 2.1 (Stretc h) . L et ǫ > 0 , set α = β < ǫ/ 6 and let H = H DIR ( V , E DIR SP ) b e the gr aph r eturne d by Algorithm disk-spanner ( I ( V , E , r ) , ǫ ) . If ( x, y ) ∈ E then δ H ( x, y ) ≤ (1 + ǫ ) δ ( x, y ) . Pr o of. Recall that th e radii are scaled so that th e sh ortest edge is of w eigh t 1. W e pro v e that ev ery directed edge of an arbitrary no de x ∈ V is appro ximated with 1 + ǫ stretc h. Let i ∈ [0 , ⌊ log 1+ α M ⌋ ]. The pr o of is by induction on i . F or a giv en n o de x , the base of the induction is the maximal v alue of i in whic h x has an edge in E ( M i +1 , M i ). L et j b e th is v alue for x , that is, the set E ( M j +1 , M j ) con tains the sh ortest edge that touches x . Every other no de is at distance at least M j +1 a wa y from x , hence x is a piv ot at this stage and ev ery ed ge th at touc hes x from the set E ( M j +1 , M j ) is add ed to E DIR SP . Let ( x, y ) ∈ E ( M i +1 , M i ) for s ome i < j and let p = NN ( x, P i ). Assume that r ( y ) ≥ M i +1 and let q = NN ( y , P i ). I t follo w s from defin ition that δ ( x, p ) ≤ β M i +1 and δ ( y , q ) ≤ β M i +1 . If the edge ( x, y ) is not in the spann er, then there must b e an edge ( ˆ x, ˆ y ) ∈ E DIR SP , where ˆ x ∈ Γ i ( p ) and ˆ y ∈ Γ i ( q ). Th e cru cial observ ation is that the rad iu s of x and ˆ y is at least M i +1 . By the choic e of β , it follo ws that 2 β M i +1 < M i +1 and ( x, ˆ x ) , ( ˆ y , y ) ∈ E . Thus, 616 D. PELEG AND L. RODITTY there is a (directed) p ath from x to y of the form h x, ˆ x, ˆ y , y i whose length is 4 β M i +1 + M i . Ho wev er, only its middle edge, ( ˆ x, ˆ y ), is in E DIR SP . The length of this edge is b ound ed b y the length of the ed ge ( x, y ) since the algorithm pic ked the m in imal edge that conn ects b et w een the neigh b orho o d s. This implies that the length of ( ˆ x, ˆ y ) is at most M i . By th e ind uctiv e hyp othesis, th e edges ( x, ˆ x ) and ( ˆ y , y ) wh ose w eigh t is at most 2 β M i +1 are approxima ted with 1 + ǫ stretc h. Thus, there is a p ath in the spanner from x to y whose length is at most (1 + ǫ ) δ ( x, ˆ x ) + M i + (1 + ǫ ) δ ( ˆ y , y ) , and th is can b e b ounded b y (1 + ǫ )4 β M i +1 + M i = ((1 + ǫ )4 β + (1 + α )) M i +1 . As th e edge ( x, y ) ∈ E ( M i +1 , M i ) it follo ws th at δ ( x, y ) ≥ M i +1 . It remains to pr o ve that 1 + 4 ǫβ + 4 β + α ≤ 1 + ǫ , whic h follo ws directly from the c hoice of α and β . If r ( y ) < M i +1 then there m ust b e an edge ( ˆ x , y ) ∈ E DIR SP , wh ere ˆ x ∈ Γ i ( p ). F ollo wing similar argumen ts to those used ab ov e it can b e shown that there is a path in the s panner from x to y of length at most (1 + ǫ )2 β M i +1 + M i and hence b ounded by (1 + ǫ ) M i +1 . The size of the s p anner. W e no w p ro v e that the size of the spanner H DIR ( V , E DIR SP ) is O ( n/ǫ d log M ). As a first step, we state the follo wing well-kno wn lemma, cf. [2]. Lemma 2.2. [Packing L emma] If al l p oints in a set U ∈ R d ar e at le ast r ap art fr om e ach other, then ther e ar e at most (2 R/r + 1) d p oints in U within any b al l X of r adius R . The next lemma establishes a b ou n d on the n um b er of incoming spann er edges that a p oint ma y b e assigned on stage i ∈ [0 , ⌊ log 1+ α M ⌋ ] of the algorithm. Lemma 2.3. L et i ∈ [0 , ⌊ log 1+ α M ⌋ ] and let y ∈ V . The total numb er of inc oming e dges of y that wer e adde d to the sp anner on stage i is O ( ǫ − d ) . Pr o of. Let ( x, y ) b e a spanner edge and let NN ( x, P i ) = p . W e asso ciate ( x, y ) to p . F rom the sp anner construction algorithm it follo ws that this is the only incoming edge of y whose source is in Γ i ( p ). Thus, this is the only incoming ed ge of y whic h is asso ciated to p . No w consider all the incoming edges of y on stage i . The source of eac h of these edges is asso ciated to a uniqu e piv ot within d istance of at m ost M i + 2 β M i +1 a wa y from y and any t wo p iv ots are β M i +1 apart fr om eac h other. Usin g Lemma 2.2, we get that the num b er of edges en tering y is ( M i +2 β M i +1 β M i +1 + 1) d = ((1 + α ) /β + 3) d = O ( ǫ − d ). It follo ws from the ab o ve lemma that the total num b er of edges that were added to E DIR SP in the main loop is O ( n/ǫ d log M ). T he total cost of the construction algorithm is O ( m log n ). F or more details on the construction time see [8]. 2.2. Spanner for relaxed disk graphs Let ( V , δ ) b e a metric s pace of constant doub ling d imension d with a radius assignment r ( · ) f or its p oin ts and let I = ( V , E , r ) b e its corresp ond ing disk graph. Assume th at we m ultiply the radiu s assignment of ev ery p oin t by a facto r of 1 + ǫ , for some ǫ > 0, and let I ′ = ( V , E ′ , r 1+ ǫ ) b e the corresp onding disk graph. In this section w e sho w that it is p ossible to create a (1 + ǫ )-spanner of I of size O ( n /ǫ d ) if we are allo wed to use edges of I ′ . Our construction consists of t wo stages: a building stage and a prun ing stage. Th e building stage creates t w o spanners , H and H ′ , using the algorithm of Section 2.1, where H is the spanner of I and H ′ is the spann er of I ′ . In the pr uning stage w e pru ne the union of these t wo spanners. Throughou t the pruning stage we use the r adius assignment of eac h RELAXED SP ANNERS F OR DIRECTED DISK GRAPHS 617 p oint b efore the increase. Let q ∈ V an d let ℓ ( q ) b e the first lev el in whic h q can ha v e outgoing edges, th at is, r ( q ) ∈ [ M ℓ ( q )+1 , M ℓ ( q ) ] (recall that as the lev els get larger the edges get shorter). In the prun ing stage w e only prune incoming edges of q whose level is b elo w ℓ ( q ). In other w ords, we do not touch th e incoming edges of q that are shorter than the radius of q . The prun ing is done as follo w . Let γ = log 1+ α 1 /β + 1. W e keep in the sp anner the incoming edges of q that come fr om the first 4 γ different lev els b elow ℓ ( q ). Let ˆ H b e the resulting sp anner and let ˆ E b e the r emaining set of edges after the pru ning step. In the remaind er of this section we sh o w that the size of ˆ H is O ( n/ǫ d ) and its str etc h with resp ect to the d istances in I ( V , E , r ) is 1 + ǫ . W e start by sho wing that the size of ˆ H is O ( n/ǫ d ). Notice that th e first part of the p ro of b elo w is p ossible only due to the c han ge w e ha v e done in the pr evious section to our spanner construction from [8]. Roughly sp eaking, giv en an edge ( p, q ) ∈ E that is shorter th an r ( q ) we use piv ot selection also on q ’s side (and not on ly on p ’s) to sparisify the graph. T h is allo ws us to deal separately with edges of q of length larger than r ( q ) and those of length smaller than r ( q ). Lemma 2.4. | ˆ E | = O ( n/ǫ d ) . Pr o of. Let ( p, q ) b e a spanner edge that survived the pru ning step. Th er e are t wo p ossible cases to consider. The fi rst case is th at ℓ ( p, q ) > ℓ ( q ). Let i = ℓ ( p, q ) and let x = NN ( p, P i ) an d y = NN ( q , P i ). By pac king considerations sim ilar to Lemma 2.3 it follo ws that th e total num b er of edges at lev el i that connects b et wee n tw o p iv ots as the edge ( p, q ) that are asso ciated with x (and with y ) is O (1 /ǫ d ). The distance b et w een x and y is at most 2 β M i +1 + M i , therefore at level i − 2 γ either x or y are no longer pivots. Let x ∈ P j and x 6∈ P j − 1 , that is, P j is th e fi rst piv ot s et that cont ains x . Th en we c h arge x with every (incoming an d outgoing) edge of this t yp e from lev els [ j, j + 2 γ ] that is in ciden t to x . No w giv en suc h an edge ( p, q ) whose leve l is i , either x or y are not pivots in lev el i − 2 γ , whic h means that either x or y has b een charged f or th is ed ge, since one of them first b ecomes a piv ot b etw een lev els i − 2 γ and i . The second case is that ℓ ( p, q ) ≤ ℓ ( q ). In this case, it must b e that lev el ℓ ( p, q ) is among the 4 γ first differen t lev els b elo w ℓ ( q ) fr om whic h an incoming edge is allo we d to en ter q . Subsequ ently , w e associate th e edge ( p, q ) with q , as th e total num b er of su c h ed ges that q can ha ve is O ( γ /ǫ d ). W e no w tu rn to pro v e that the stretc h of the spanner ˆ H with resp ect to the disk graph I is 1 + ǫ . Lemma 2.5. L et ( p, q ) b e an e dge of the sp anner H that was prune d. We show that ther e is a p ath i n ˆ H whose length is at most (1 + ǫ ) δ ( p, q ) . Pr o of. The pr o of is b y induction on the lengths of the pru n ed edges. F or the ind uction base let ( p, q ) b e the shortest edge that w as pru ned. F or ev ery x ∈ V , let s ( x ) b e the head of an edge wh ose lev el is the γ -th lev el b elo w ℓ ( x ) from whic h x has an incoming edge. Let q 1 , . . . q i , . . . b e a sequence of p oin ts, wh ere q 1 = q and q i = s ( q i − 1 ). As q i +1 = s ( q i ), it follo ws that ℓ ( q i +1 , q i ) ≤ ℓ ( q i ) − γ . Combining this with the fact that ℓ ( q i ) ≤ ℓ ( q i , q i − 1 ) w e get that ℓ ( q i +1 , q i ) ≤ ℓ ( q i , q i − 1 ) − γ . Th erefore, δ ( q i , q i − 1 ) ≤ β δ ( q i +1 , q i ). The analysis d istinguishes b et w een t wo cases. Case a: Th ere is a p oin t q t suc h that δ ( q t , q ) > β δ ( p, q ). This situation is depicted in Figure 3. (If there is more than one p oin t that satisfies this requirement, tak e the one whose index is min imal.) 618 D. PELEG AND L. RODITTY p q β δ ( p, q ) > β / 2 δ ( p, q ) q t − 1 q t Figure 3: The case in whic h q t exists Claim: δ ( q t , q t − 1 ) ≥ β 2 δ ( p, q ). Pr o of. F or the sak e of con tradiction, assume that δ ( q t , q t − 1 ) < β 2 δ ( p, q ). This implies that 2 δ ( q t , q t − 1 ) < β δ ( p, q ) < δ ( q t , q ) ≤ t X i =2 δ ( q i , q i − 1 ) , (2.1) where the last inequalit y follo w s from the triangle inequ alit y as the distance b et w een q and q t is at most P t i =2 δ ( q i − 1 , q i ). F or ev ery 2 ≤ i ≤ t − 1 we h a ve δ ( q i , q i − 1 ) ≤ β δ ( q i +1 , q i ), whic h implies that δ ( q i , q i − 1 ) ≤ β t − i δ ( q t , q t − 1 ). Combined with (2.1), w e get δ ( q t , q t − 1 ) < t − 1 X i =2 δ ( q i , q i − 1 ) ≤ δ ( q t , q t − 1 ) t − 1 X i =2 β t − i . If β < 1 / 2 w e ha v e P t − 1 i =2 β t − i < 1 and this yields a con tradiction. W e n o w f o cus our atten tion on the p oin t q t − 1 . Th e minimalit y of q t implies that δ ( q , q t − 1 ) ≤ β δ ( p, q ). By combining it with the triangle inequalit y we get that δ ( p, q t − 1 ) ≤ δ ( p, q ) + β δ ( p, q ). Therefore, in the graph I ′ there must b e an edge from p to q t − 1 . Let i = ℓ ( p, q t − 1 ). There are t w o p ossible scenarios for the spanner H ′ . The first scenario is when r ′ ( q t − 1 ) < M i +1 . In this case, there is an edge in H ′ from some x ∈ Γ i ( NN ( p, i )) to q t − 1 , whose length is at most δ ( p, q ) + β δ ( p, q ). There are 4 γ d ifferen t leve ls b elo w ℓ ( q t − 1 ) f r om wh ich edges that b elong to th e spanners H and H ′ are not b eing prun ed and survive d to the sp anner ˆ H . W e kno w that the edge ( q t , q t − 1 ) is suc h an ed ge from the γ -th non-empty lev el b elo w ℓ ( q t − 1 ). W e also kno w that δ ( q t , q t − 1 ) > β 2 δ ( p, q ). Therefore, as the length of the edge ( x, q t − 1 ) is at most δ ( p, q ) + β δ ( p, q ) it is within the 4 γ non-empty lev els b elo w ℓ ( q t − 1 ) and it is not p runed. W e can no w build a path f rom p to q b y concatenating three segmen ts as f ollo ws: A path f r om p to x , the edge ( x, q t − 1 ) and a path from q t − 1 to q . T he p oint x is at most 2 β δ ( p, q ) + 2 β 2 δ ( p, q ) RELAXED SP ANNERS F OR DIRECTED DISK GRAPHS 619 a wa y from p and for the righ t choic e of β it is less than δ ( p, q ) / (1 + ǫ ), hence the w eigh t of ev ery edge on the path that app ro ximates the distance b etw een x and p in H ∪ H ′ is less than δ ( p, q ), the shortest pr uned edge, and th e en tire p ath survive d the p unning stage. Similarly , the p oin t q t − 1 is at most β δ ( p, q ) a wa y from q and again for the right c hoice of β ev ery edge on the path that app ro ximates the d istance b et w een q t − 1 and q sur v ived the punn ing stage. Thus, w e get that there is a path wh ose length is at most (1 + ǫ )(3 β δ ( p, q ) + 2 β 2 δ ( p, q )) + δ ( p, q ) + β δ ( p, q ) , whic h is less than (1 + ǫ ) δ ( p, q ) f or β < ǫ/ 11. The second scenario is wh en r ′ ( q t − 1 ) ≥ M i +1 . In this case, there is an ed ge in H ′ from some x ∈ Γ i ( NN ( p, i )) to some y ∈ Γ i ( NN ( q t − 1 , i )) wh ose length is at m ost δ ( p, q ) + β δ ( p, q ), whic h is not b eing p runed. W e can b uild a path from p to q b y concatenating thr ee segments as follo ws: A path from p to x , th e edge ( x, y ) and a path from y to q . As b efore, for the righ t c hoice of β the paths fr om p to x and from y to q are comp osed from ed ges that are shorter from δ ( p, q ), the length of the shortest pruned edge, hence, from the minimalit y δ ( p, q ) every edge on th ese paths survive d th e pun ning stage. W e get that there is a path whose length is at most (1 + ǫ )(4 β δ ( p, q ) + 5 β 2 δ ( p, q )) + δ ( p, q ) + β δ ( p, q ) , whic h is less than (1 + ǫ ) δ ( p, q ) f or β < ǫ/ 19. This completes the pr o of for case a. Case b: Th ere is no p oin t q t suc h that δ ( q t , q ) > β δ ( p, q ). In this case, let q t − 1 b e the last p oin t in the sequence of p oint s q 1 , . . . q i , . . . , wher e q i = s ( q i − 1 ) and q 1 = q . Similarly to b efore, there are t w o p ossible scenarios for the sp anner H ′ . Let i = ℓ ( p, q t − 1 ). The first scenario is wh en r ′ ( q t − 1 ) < M i +1 . In this case, there is an edge in H ′ from some x ∈ Γ i ( NN ( p, i )) to q t − 1 whose length is at most δ ( p, q ) + β δ ( p, q ). This edge could n ot b e prun ed , sin ce if it wa s pr uned then q t − 1 could not ha v e b een the last p oin t in the sequence. Hence w e can construct a path f rom p to q exact ly as we h av e d one in the firs t scenario of case a, d escrib ed ab o ve . The second scenario is when r ′ ( q t − 1 ) ≥ M i +1 . In this case, w e can construct a path from p to q exactly as w e ha ve done in the second scenario of case a, describ ed ab o v e. This completes the pro of of the indu ction base. The pro of of the general in ductiv e step is almost id entical . The only difference is that when a path is constru cted from p to q , its p ortions f rom p to x and from q t − 1 to q in th e fi r st scenario and from p to x and from y to q in the second scenario exist in ˆ H by the induction h yp othesis and not by the minimalit y of δ ( p, q ). W e end this section by s tating its main Theorem. The pr o of of this Theorem stems from Lemma 2.4 and Lemma 2.5. Theorem 2.6. L et ( V , δ ) b e a metric sp ac e of c onstant doubling dimension with a r adius assignment r ( · ) for i ts p oints and let I = ( V , E , r ) b e its c orr esp onding disk gr aph. L et I ′ = ( V , E ′ , r 1+ ǫ ) b e the c orr esp onding disk gr aph in the r elaxe d mo del. It is p ossible to cr e ate a (1 + ǫ ) -sp anner of size O ( n/ǫ d ) for I using e dges of I ′ . 3. Concluding remarks and op en problems This pap er present s a s p anner construction for d isk graphs in a slightly r elaxed mo d el that is as go o d as spann ers for complete graphs and un it disk graphs. 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