Local Approximation Schemes for Topology Control

Local Appr o ximation Schemes f or T opology Contr ol Mirela Damian Dept. of Comp . Sci, Villano va Univ . Villano va, P A 19085 mirela.damia n@villanov a.edu Saur av P andit Sriram P emmar aju Dept. of Comp . Sci, Univ . of Iow a Iow a City , IA 52242-1419 [spandit, sriram]@cs.uiow a.edu ABSTRA CT This pap er presen ts a distributed algori thm on wir eless ad- hoc netw orks that runs in p olylogarithmic n umb er of rounds in the size of the netw ork and constructs a linear size, ligh tw eight, (1 + ε )- sp anner for any giv en ε > 0. A wireless netw ork is mo deled b y a d -dimensional α -quasi unit ball graph ( α - UBG), which is a higher dimensional generalization of t h e standard unit disk graph (UDG) mo del. The d -dimensional α -UBG mo del goes b eyond t he unrealistic “flat w orld” as- sumption of UDGs and also tak es into accoun t transmission errors, fading signal strength , and p hysica l obstructions. The main result in the paper is th is: for an y fi xed ε > 0, 0 < α ≤ 1, and d ≥ 2 there is a d istributed algorithm run- ning in O (log n · log ∗ n ) communication rounds on an n -no de, d -dimensional α -UBG G that compu tes a (1 + ε )-spann er G ′ of G with maxim um degree ∆( G ′ ) = O (1) and total w eight w ( G ′ ) = O ( w ( M S T ( G )). This result is motiv ated by the top ology control problem in wireless ad-ho c netw orks and impro ves on existing top ology con trol algori thms along sev- eral dimensions. The technical contributions of the pap er include a new, sequential, greedy algorithm with relaxed edge ordering and lazy u p dating, and clustering techniques for filtering out un necessary edges. Categories and Sub ject Descriptors: C.2.4 [Computer- Comm unication Netw orks]: Distributed Systems General T e rms: Algorithms, P erformance, Theory . Keywords: Spanners, T op ology control, Wireless ad-h o c netw orks, Unit ball graphs. 1. INTR ODUCTION Let G = ( V , E ) be a graph with edge weigh ts w : E → R + . F or t ≥ 1, a t -spanner of G is a spanning subgraph G ′ of G such that for all pairs of vertice s u, v ∈ V , th e length of a shortest uv -path in G ′ is at most t times the length of a shortest uv -p ath in G . The p roblem of constructing a sparse t -spanner, for small t , of a giv en graph G has b een exten- sive ly studied by researc hers in distributed computing and computational geometry and more recently by researc hers in Permission to make digital or hard copi es of all or part of this work for personal or cl assroom use is granted without fee pro vided that copi es are not m ade or distrib uted for profit or commercial adv antage and that copies bear this notic e and the full cit ation on the first page. T o cop y otherwise, to republi sh, to post on serv ers or to redistrib ute to lists, re quires prior specific permission and/or a fee. Alter ed version of P ODC’06, with a few typ os fixed. July 22-26, 2006, Den ver , Colorado, USA. Copyri ght 2006 A CM 1-59593-384-0/0 6/0007 ... $ 5.00. ad-ho c wireless netw orks. In th is pap er we present a fast d is- tributed algo rithm for constructing a linear size, ligh tw eigh t t -spanner of boun ded d egree for any gi ven t > 1, on wireless netw orks. Below, we describe our result more p recisely . 1.1 Netw ork model W e mo del wireless netw orks using d -dimensional q uasi unit ball graphs. F or any fi x ed α , 0 < α ≤ 1 and in te- ger d ≥ 2, a d -dimensional α -quasi unit b al l gr aph ( α -UBG , in short) is a graph G = ( V , E ) whose vertex set V can be placed in one-one correspond ence with a set of p oints in th e d -dimensional Euclidean space and whose edge set E sat- isfies the constraint: if | uv | ≤ α then { u, v } ∈ E and if | uv | > 1 then { u, v } 6∈ E . Here w e use | uv | to denote the Euclidean distance betw een th e p oin ts corresponding to ver- tices u and v . The α -UBG mo d el do es not prescrib e whether a pair of v ertices whose distance is in th e range ( α, 1] are to b e connected by an edge or not. This is an attempt to take in to account transmission errors, fading signal strength, and physical obstructions. Our algorithm do es not need to know the locations of nod es of the α -UBG in d -dimensional Euclidean space; just the pairwise Euclidean distances. The α - U BG mo del is a higher d imensional generalization of th e somewhat simplistic unit disk graph (UDG) mo del of wireless netw orks that is p opular in literature. Sp ecifi- cally , when α = 1 and d = 2, a d -dimensional α -UBG is just a UDG. UDGs are attractive d ue to their mathemati- cal simplicit y , but hav e been deservedly criticized for b eing unrealistic models of wireless net works [10]. In our view, d - dimensional α -UBGs are a sig nificant step tow ards a more realistic mo del of wireless net works. Tw o-dimensional α - UBGs were prop osed in [1] as a mo del of wireless ad-ho c net- w orks with unstable transmission ranges and the difficult y of doing geometric routing in such netw orks w as shown. Our comm unication mo del is the standard synchronous message passing mo del that do es not account for channel ac- cess and collision issues. In this communicatio n mo del, time is divided in to rounds. In eac h round, each node can send a different message to each of its neigh b ors, receiv e different messages from all neighbors and perform arbitrary (p olyn o- mial) local compu tation. The length of messages exchanged b etw een no des is logarithmic in the num b er of n odes. W e measure the cost of our algorithm in terms of t he n umber of comm unication rounds. Although th is mo del is not widely considered to b e realisti c, it is nevertheless interesting b e- cause it d emonstrates the localit y of compu t ations. 1.2 Our r esult F or any edge w eigh ted graph J , we u se w ( J ) to d enote the sum of the w eights of all the edges in J and M S T ( J ) to denote a minimum weigh t spannin g tree of J . F or an y fixed ε > 0, 0 < α ≤ 1, and d ≥ 2 our algorithm runs in O (log n · log ∗ n ) comm unication round s on an n -n ode, d - dimensional α -UBG and compu tes a (1 + ε ) -spanner G ′ of G whose maximum degree ∆( G ′ ) = O (1) and whose total w eight w ( G ′ ) = O ( w ( M S T ( G )). Since any spann er of G has weigh t b ound ed b elo w b y w ( M S T ( G )), th e w eight of the ou t put pro duced by the algorithm is within a constant times the optimal w eight. A s far as we kn o w, our result significan tly improv es all kn own results of a similar kind along sev eral dimensions. More on th is further b elow . 1.3 T opology control Our result is motiv ated by the top olo gy c ontr ol problem in wireless ad-ho c netw orks. F or an ove rview of top ology contro l, see the survey by Ra jaraman [17]. S ince an ad-hoc netw ork does not come with fixed infrastructure, there is no top ology t o start with and informally speaking, the top ol- ogy control problem is one of selecting neighbors for each nod e so that the resulting top ology has a n umb er of useful prop erties. More precisely , let V be a set of no des that can comm unicate via wireless radios and for eac h v ∈ V , let N ( v ) denote the set of all no des that v can reac h when t ransmit- ting at max im um p ow er. The induced digraph G = ( V , E ) , where E = {{ u, v } | v ∈ N ( u ) } , represents th e netw ork in whic h ev ery no de h as c hosen to transmit at maximum p ow er and has designated every no de it can reac h as its neighbor. The top ology con trol problem is the problem of devising an efficien t and lo cal proto col P for selecting a set of neighbors N P ( v ) ⊆ N ( v ) for each node v ∈ V . The ind uced digraph G P = ( V , E P ), where E P = {{ u, v } | v ∈ N P ( u ) } is typ- ically required the satisfy prop erties suc h as symmetry (if v ∈ N P ( u ) then u ∈ N P ( v )), sparseness ( | E P | = O ( | V | )) or b ounded degree ( | N P ( v ) | ≤ c for all no des v and some constant c ), and the spanner prop erty . Sometimes stronger versi ons of conn ectivit y such as k -vertex connectivity or k - edge connectivity (for k > 1) are desired, b oth for providing fault-tolerance and for improving throughpu t [6, 7]. If the input graph consists of no des in t h e plane, it is quite com- mon to require that th e outpu t graph b e p lanar [13, 14, 15, 18, 19]. This requirement is motiv ated by the existence of simple, memory-less, geometric routing algorithms th at guaran tee messa ge d eliv ery only when t h e underlying graph is planar [9]. Though the topology control p roblem is recen t, there is already an extensive b o dy of literature on the problem to whic h th e ab ov e sample of citatio ns do not d o justice. How - ever, many of the top ology con trol protocols that pro vide w orst case guarantees on the quality of the t op ology , assume that the n et work is mo deled b y a UDG. A recent example [15] p resents a distribut ed algorithm that requires a linear num b er of comm unication rounds in the wo rst case t o com- pute a planar t -spanner of a given UDG with t ≈ 6 . 2 and in whic h eac h n ode has degree at most 25. These tw o constants can be sligh tly tun ed – t can b e brought dow n to about 3.8 with a sig nificant increase in th e degree b oun d. W e impro ve on th e result in [15 ] along several dimensions. As is gener- ally kno wn among practitioners in ad-hoc wireles s netw orks, the “fl at world” assumption and the identical transmission range assumpt ion of UDGs are unrealistic [10]. By using an α -UBG we significan tly generalize our mo del of wireless netw orks, hop efully moving m uch closer to realit y . F or any ε > 0, our alg orithm retu rns a (1 + ε )- spanner; as far as we know, this is the fi rst distribut ed algorithm that pro duces an arbitr aril y go o d spanner for an α -U BG mod el of wireless n et- w orks. W e also gu arantee th at th e total we ight of the output is within constan t times op t imal – a guaran tee that is n ot provided in [15]. Finally , using algorithmic techniques and distributed data structures that migh t be of indep end ent in- terest, we ensure that our proto col runs in O (log n · log ∗ n ) comm unication rounds. W e are not aw are of any t opology contro l algorithm that runs in p oly-logarithmic number of rounds and provides an ywhere close to the guarantees pro- vided b y our algo rithm. 1.4 Spann ers in computational geometry Starting in the early 1990’s , researchers in computational geometry h ave attempt ed t o find sparse, light w eight span - ners for complete Euclidean graphs. Given a set P of n p oin ts in R d , the tuple ( P, E ), where E is the set of line seg- ments {{ p, p ′ } | p, p ′ ∈ P } , is called the c omplete Euclide an gr aph on P . F or any subset E ′ ⊆ E , ( P, E ′ ) is called a Eu- clide an gr aph on P . The specific problem that researc hers in computational geometry have considered, is this. Given a set P of n p oints in R d and t > 1, compute a Euclidean graph on P t h at is a t -spann er of the complete Euclidean graph on P , whose maximum degree is b ound ed by O (1) and whose w eight is b ounded by the w eigh t of a minim um span- ning tree on P . F or an early example, see [12] in which th e authors sho w that there are “planar graphs almost as goo d as th e complete graphs and almost as cheap as minimum spanning trees.” This w as fol low ed by a series of improv e- ments [2, 3, 4, 5], with the most recen t p ap er [2] presenti ng algorithms for constructing Euclidean subgraphs th at pro- vide the additional prop erty of k -fault tolerance. Most of the papers mentioned ab ov e start with the fol lowi ng simple, greedy algorithm. Algorithm SEQ-GRE EDY ( G = ( V , E ) , t ) 1. O rd er th e edges in E in non-decreasing order of length. 2. E ′ ← φ , G ′ ← ( V , E ′ ) 3. F or eac h edge e = { u, v } ∈ E if there is n o uv -p ath in G ′ of length at most t · | uv | (a) E ′ ← E ′ ∪ { e } (b) G ′ ← ( V , E ′ ) Output G ′ . It is wel l-kn o wn [4] t hat if the input graph G = ( V , E ) is the complete Euclidean graph , then the output graph G ′ = ( V , E ′ ) p rodu ced by SEQ-GREEDY h as t he follo wing u se- ful prop erties: (i) G ′ is a t -spann er of G , (ii) ∆( G ′ ) = O (1), and (iii) w ( G ′ ) = O ( w ( M S T ( G ))). A naiv e implemen tation of SEQ-GREEDY takes O ( n 3 log n ) time b ecause a quadratic num b er of shortest p ath queries need to b e answe red on a dynamic graph with O ( n ) edges. Consequentially , pap ers in this area [4, 5] focus on trying to implemen t SEQ-GRE EDY efficien tly . F or example, Das and Narasimhan [4 ] show h ow to use certain k ind of graph clustering to answ er shortest path q u eries efficiently , thereb y reducing th e running time of SEQ- GREEDY to O ( n log 2 n ). One of the con tribut ions of this pap er is to show ho w a v arian t of the Das-Narasimhan clustering scheme can b e implemen ted and main tained effi- cien tly , in a distributed setting. 1.5 Summary of our contrib utions In obtaining the main result, our pap er mak es th e follo w- ing con tributions. 1. W e first sho w that sparse, ligh tw eight t -spanners for arbitrarily small t > 1, not only exist for d -dimensional α -UBGs, but can b e computed using SEQ-GREEDY . Note that sparse t -spanners for arbitrarily small va lues of t ≥ 1 do not exist for general graphs. F or example, there is a classical graph-theoretic result th at shows that for any t ≥ 1, there exist (infinitely man y) un - w eighted n -v ertex graphs for which every t -spanner needs Ω( n 1+1 / ( t +2) ) edges (see Page 179 in [16]). 2. W e then consider a versio n of SEQ-GREEDY in whic h the requirement that edges b e considered in increasing or- der of length is relaxed. More precisely , the edges are distributed into O ( log n ) bins B 0 , B 1 , B 2 , . . . such t hat edges in B i are all sh orter than edges in B i +1 . It is then shown that any ordering of the edges in which edges in B 0 come first, follo wed by edges in B 1 , fol- lo wed by the edges in B 2 , etc., is go o d enough for th e correctness of SEQ-GREED Y , even for d -dimensional α - UBGs. More imp ortantly , w e sh o w th at the upd ate step in SEQ-GREEDY (Step 3(a)) need not be p erformed after each edge is queried. Instead, a more lazy up- date ma y b e performed, after each bin is completely processed. Being able to p erform a lazy up date is crit- ical for a d istributed implementatio n; roughly speak- ing, we wan t the no des to query all edges in a bin in parallel and not to hav e to rely on answers to queries on other edges in a bin. 3. W e also use a clustering tec hniq u e as a w ay to redu ce the n umb er of edges to b e queried per nod e. Reducing the num b er of query edges p er no de, is critical to b eing able to guarantee t hat the output of our distributed versi on of SEQ-GREEDY do es not ha ve to o many ed ges incident on a node. 4. Ou r next contribution is to show that this relaxed versi on of SEQ-GREEDY can b e implemen ted in a dis- tributed setting in O ( log n ) phases — one phase corre- sp on d ing to eac h bin — su ch that eac h p hase requ ires O (log ∗ n ) communication roun ds. Eac h p hase requires the computation of maximal independ ent sets (MIS) on some deriv ed graphs. W e show that the derived graphs are unit ball graphs of c onstant doub li ng dimen- sion [11] and use the O (log ∗ n )-round MIS algorithm of Kuhn et al [11]. 1.6 Extensions to our main r esult Here we b riefly rep ort on extensions to our main result that w e ha ve obtained. They do n ot app ear in this pap er due to lac k of space. 1. Let G = ( V , E ) b e an edge-weig hted graph. F or any t > 1 and positive integer k , a k -vertex fault-toler ant t -sp anner of G is a spanning subgraph G ′ if for eac h subset S of v ertices of size at most k , G [ V ′ \ S ] is a t - spanner of G [ V \ S ]. A k -edge fault-toleran t t -spanner is defined in a simila r manner. Using ideas from [2] w e can extend our algorithm to produ ce a k -vertex (or a k -edge) fault-tolerant t -spann er in p olylogarithmic num b er of communication roun d s. 2. I n this pap er, w e use Euclidean distances as w eights for the edges of the input graph G . Ho wev er, if the metric c · | uv | γ , for p ositiv e constan t c and γ ≥ 1, is used in place of Euclidean distances | uv | , we can sho w that our algorithm still pro duces a spanner with all three desired properties. Relative Euclidean d istances, such as the fun ction mentioned ab ov e, may b e used to prod uce ener gy sp anners . 3. Let G = ( V , E ) b e an edge-w eigh ted graph. The p ower c ost of a vertex u ∈ V is pow er ( u ) = max { w ( u, v ) | v is a neighbor of u } . In other words, the p ow er cost of a vertex u is prop ortional to the cost of u trans- mitting to a farthest n eighb or. The p ower c ost of G is P u ∈ V pow er ( u ) [8]. W e can sho w th at the output of our algorithm is not only ligh tw eigh t with respect to the usual w eight measure (sum of the w eights of all edges) bu t also with resp ect to the p o wer cost measure. 2. SEQ UENTIAL RELAXED GREED Y A L- GORITHM Now w e sho w th at a r elaxe d version of SEQ-GRE EDY pro- duces an output G ′ with all three desired properties, even when the input is n ot a complete Euclidean graph, but is a d -dimensional, α -UBG for fixed d and α . Relaxing the re- quirement in SEQ-GREEDY that the ed ges b e t otally ordered by length and allo wing for the outp ut to b e up dated lazily are critical to obtaining a d istributed algorithm that runs in p olylogarithmic num b er of rounds. Let r > 1 b e a constan t to be fixed later and let W i = r i α/n for eac h i = 0 , 1 , 2 , . . . . Let I 0 = (0 , α/n ] and for eac h i = 1 , 2 , . . . let I i = ( W i − 1 , W i ]. Let m = ⌈ log r n α ⌉ . Then, since no edge has length greater than 1, the length of any edge in E lie s in one of the in terv als I 0 , I 1 , . . . , I m . Let E i = {{ u, v } ∈ E : | uv | ∈ I i } . W e no w eliminate the restriction that edges within a set E i b e pro cessed in increasing order by length. W e run SEQ-GREEDY in m + 1 phases: in phase i , the algorithm pro- cesses edges in E i in arb itrary order and adds a subset of edges in E i to the spanner. F or 0 ≤ i ≤ m , we use G i to denote the sp an n ing subgraph of G consisting of edges E 0 ∪ E 1 ∪ · · · ∪ E i . Thus G i is the p ortion of the inpu t graph that the algori thm has processed in phase i and earlier. W e use G ′ i to den ote the output of the algorithm at the end of phase i . I n other wo rds, G ′ i is the spanning subgraph of G consisting of edges of G that the algorithm has d ecided to retain in phases 0 , 1 , . . . , i . The final output of the algorithm is G ′ = G m . The wa y E 0 is pro cessed is different from the w ay E i , i > 0 is processed. W e now separately describe these tw o parts. 2.1 Pr ocessing Edges in E 0 W e start by stating a prop erty of G 0 that fol lows easily from the fact that all edges in G 0 are small. Lemma 1. Every c onne cte d c omp onent of G 0 induc es a clique i n G . The algorithm PROCESS-SHORT-E DGES for pro cessing edges in E 0 consists of three steps (i) determine the connected com- p onents of G 0 , (ii) use SEQ-GREEDY to compute a t -spann er for each connected comp onent (that is, a clique), and (iii) let G ′ 0 b e the u nion of the t -spanners comp u ted in Step ( 2) and output G ′ 0 . The follo wing theorem states the correctness of the PROCESS-SHOR T-EDGES algorithm. Its pro of follo ws easily from t h e correctness of SEQ-GREEDY . Theorem 2. G ′ 0 satisfies the fol lowing pr op erties. (i) F or every e dge { u, v } ∈ E 0 , G ′ 0 c ontains a uv -p ath of l ength at most t · | uv | , (ii) ∆( G ′ 0 ) = O (1) , and (iii) w ( G ′ 0 ) = O ( w ( M S T ( G ))) . 2.2 Pr ocessing Long Edges W e now d escribe ho w edges in E i are pro cessed, fo r i > 0. The algori thm PROCESS-LONG-ED GES has five steps: (i) computing a cluster cove r for G ′ i − 1 , (ii) selecting query edges in E i , (iii) computing a cluster graph H i − 1 for G ′ i − 1 , (iv) answ ering shortest path queries fo r the q uery edges selected in Step (ii), and (v) removing redundant edges. These steps are described in the next five subsections. F or any graph J , let V ( J ) denote the vertex set for J . F or any pair of vertices u, v ∈ V ( J ) let sp J ( u, v ) denote the length of a shortest uv - path in J . Defi ne a cluster of J with c enter u ∈ V ( J ) and r adius r to be a set of vertices C u ⊆ V ( J ) such th at, for eac h v ∈ C u , sp J ( u, v ) ≤ r . A set of clusters { C u 1 , C u 2 , . . . } of J is a cluster c over of J of r adius r if ev ery cluster in t he set has radius r , ev ery vertex in V ( J ) b elongs to at least one cluster, and for any pair of cluster cen ters u i and u j , sp J ( u i , u j ) > r . 2.2.1 Computing a Cluster Cover for G ′ i − 1 At the b eginning of phase i w e compute a cluster cove r of radius δ W i − 1 , where δ < 1 is a constant that will b e fi xed later. W e start with an arbitrary vertex u ∈ V and run Dijkstra’s shortest path alg orithm with source u on G ′ i − 1 , in order to id entify no des v ∈ V with the property that sp G ′ i − 1 ( u, v ) ≤ δ W i − 1 ; eac h suc h nod e v gets includ ed in the cluster C u . Once C u has b een iden tified, recurse on V \ C u until all nod es b elong to some cluster and w e h a ve a cluster co ver of G ′ i − 1 of radius δ W i − 1 . 2.2.2 Selecting Query Edges in E i As d efined earlier, edges in E i hav e wei ghts in the interv al I i = ( W i − 1 , W i ], while the cluster co ver for G ′ i − 1 has radius δ W i − 1 , with δ < 1. This imp lies that each edge in E i has endp oints in different clusters. Our goal is to select a unique query edge p er pair of clusters. This will guarantee that there are a constant n umber of query edges incident on any nod e (see Lemma 4) and this fact will be critically used by the distributed version of ou r algorithm to gu arantee the degree boun d on the spanner that is constructed. Let θ b e a quantity that satisfies 0 < θ < π 4 and t ≥ 1 / (cos θ − sin θ ). F or any v alue t > 1, no matter ho w small, there alw ays ex ists a θ t h at satisfies these restrictions. De- fine an edge e = { u, v } ∈ E i to b e a c over e d e dge if there is a z ∈ V suc h that (i) { u, z } ∈ G ′ i − 1 , | vz | ≤ α and ∠ v uz ≤ θ or (ii) { v , z } ∈ G ′ i − 1 , | uz | ≤ α and ∠ uv z ≤ θ . Any ed ge in E i that is n ot co vered is a c andidate query edge. The motiv ation for these definitions is the follo wing geometric lemma, due to Czuma j and Zhao [2]. Lemma 3 (Czumaj an d Zhao [2] ). 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 R d with ∠ 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: (a) Edge { u, v } i s c over ed : { u, z } follow ed b y a t -spanner z v -path is a t -spanner uv -path. Now note that for eac h cov ered edge { u, v } ∈ E i , there exists z that satisfies the preconditions of Lemma 3 (by d ef- inition), and using this lemma w e can sho w that G ′ i − 1 al- ready contains a uv -path of length at most t · | uv | . This suggests that cov ered edges need not b e qu eried and there- fore we can start with the complement of the set of co vered edges as candidate query edges. F or each pair of clusters C a and C b , let E i [ C a , C b ] d en ote the subset of candidate qu ery edges in E i with one end- p oin t in C a and the other endp oint in C b . O u r algorithm selects a uniqu e query edge { x, y } from eac h nonempt y sub- set E i [ C a , C b ]. Assuming that x ∈ C a and y ∈ C b , the edge { x, y } is selected so as to minimize t · | xy | − sp G ′ i − 1 ( a, x ) − sp G ′ i − 1 ( b, y ) (1) The quantit y in (1) is caref ully c hosen to guarantee that, if a t -spanner path b etw een the end p oin ts of an edge { x, y } that minimizes (1) exists in G ′ i , then t - spanner paths b etw een the endp oints of al l edges in E i [ C a , C b ] exist in G ′ i (this prop ert y will later b e shown in th e pro of of Theorem 10). This implies that, for each pair of clusters C a and C b , q uerying the edge { x, y } in E i [ C a , C b ] that minimizes (1) rend ers querying any other edge in E i [ C a , C b ] redu ndant. The follo wing lemma sho ws that selecting query edges as described ab ov e filters all bu t a constant number of edges p er cluster. The pro of follo ws from tw o observ ations: (i) if a pair of cluster centers are connected b y an edge in E i , th en the clusters are not to o far from each other in Euclidean space (in p articular, n o farther than (4 δ + r ) W i − 1 ), and (ii) the Eu clidean d istance b etw een an y p air of cluster cen ters is b ounded from b elo w by δ W i − 1 /t , b ecause they would oth- erwise be part of the same cluster. Lemma 4. The numb er of query e dges i n E i that ar e i n- cident on any cluster i s O ( t d ( 4 δ + r δ ) d ) , a c onstant . 2.2.3 Computing a Cluster Graph F or eac h selected query edge { x, y } ∈ E i , we need to know if G ′ i − 1 conta ins an xy - path of length at most t · | xy | . In general, the num b er of h ops in a shortest xy -path in G ′ i − 1 can be qu ite large an d h a ving to trav erse suc h a path w ould mean that the shortest path q u ery corresp onding to edge { x, y } could not b e answ ered q uickly enough. T o get around this problem, w e use an idea fro m [4] in whic h the authors construct an approximatio n to G ′ i − 1 , called a cluster gr aph , and show t h at for an y edge { x, y } ∈ E i , the short- est path qu ery for { x, y } can b e answered appro ximately on H i − 1 in a constant number of steps. The goal of Das and Narasimhan [4] w as to improv e the running time of SEQ-GREEDY on complete Euclidean graphs, but we sho w that the Das-N arasimhan data stru ct u re can b e constructed and main tained in a d istributed fashion for efficiently answ ering shortest path queries for edges belonging to a α -UBG. In the follo wing, w e describe a sequential algori thm that starts with a clu ster co ver of G ′ i − 1 of radius δ W i − 1 , and builds a cluster gr aph H i − 1 of G ′ i − 1 . This algorithm is iden tical to the one in D as and Narasimhan [4] and is included mainly for completeness. The vertex set of H i − 1 is V and t he edge set of H i − 1 conta ins tw o typ es of ed ges: intr a-cluster edges and inter- cluster edges. An edge { a, x } is an intra-cluster edge if a is a cluster center and x is no de in C a . Inter-cluster edges are b etw een cluster centers. A n edge { a, b } is an inter-cluster edge if a and b are cluster centers, and at least one of the follo wing tw o conditions holds: ( i) sp G ′ i − 1 ( a, b ) ≤ W i − 1 , or (ii) th ere is an edge in G ′ i − 1 with one endp oint in C a and the other endpoint in C b . See Figure 2. a b x y b C c C c u a C v Figure 2: Edges interior to dis k s are intr a-cluster edges. Edge { a, b } is an inter-cluster edge b ecause sp G ′ i − 1 ( a, b ) ≤ W i − 1 , and { b, c } is an inter-cluster edge be cause { u, v } is i n G ′ i − 1 . An xy -path in G ′ i − 1 , s ho wn b y the dashed curv e ma y b e appro xi m ated by the path x, a, b, y i n H i − 1 . Regardless of the type of a cluster edge e = { a, b } (in ter- or in tra-), the wei ght of e is the v alue of sp G ′ i − 1 ( a, b ). The follo wing lemma follo ws easily from the defin ition of inter- cluster edges. Lemma 5. F or any inter-cluster e dge { a, b } in H i − 1 , we have that sp G ′ i − 1 ( a, b ) ≤ (2 δ + 1) W i − 1 . The abov e upp er b ound also implies that | ab | ≤ (2 δ + 1) W i − 1 . Using this and argumen ts similar to those used for Lemma 4, w e can sho w that the number of in ter-cluster edges incid ent t o a cluster cen ter is O ((5 + 1 /δ ) d ), so w e hav e t h e follo wing lemma. Lemma 6. The numb er of inter-cluster e dges in H i − 1 in- cident to a cluster c enter is O ((5 + 1 /δ ) d ) , a c onstant . The main reason for constructing the cluster graph H i − 1 is that lengths of paths in H i − 1 are close to lengths of corre- sp on d ing paths in G ′ i − 1 and shortest path q ueries for edges in E i can b e answ ered q uic kly in H i − 1 . The follow ing lemma (whose pro of app ears in D as and Narasimhan [4]) shows that w e can constru ct H i − 1 such that path lengths in H i − 1 ap- proximate path lengths in G ′ i − 1 to any desired extent, de- p ending on t he c hoice of δ . Lemma 7. F or any e dge { x, y } ∈ E i , if ther e is a p ath b etwe en x and y in G ′ i − 1 of length L 1 , then ther e is a p ath b etwe en x and y i n H i − 1 of l ength L 2 such that L 1 ≤ L 2 ≤ 1+6 δ 1 − 2 δ L 1 . 2.2.4 Answering Shortest P ath Queries F or query edges { x, y } ∈ E i , we are in terested in kn o wing whether G ′ i − 1 has an xy -path of length at most t · | xy | . W e ask th is qu estion on the cluster graph H i − 1 . If H i − 1 con- tains an xy -path of length at most t · | xy | , w e do not add { x, y } to G ′ i ; otherwise we do. If H i − 1 conta ins an xy -path of length at most t · | xy | , t h en so do es G ′ i − 1 (by Lemma 7, since L 1 ≤ L 2 ). Therefore, not adding { x, y } to th e span- ner is not a dangerous c hoice. On the other hand, even if H i − 1 does not con tain an xy -path of length at most t · | xy | , G ′ i − 1 migh t contain such a p ath and in this case add in g edge { x, y } is unn ecessary . Adding extra edges is of course not problematic for the t -spanner property . It will tu rn ou t that this is not a problem even for the requiremen t that the spanner sh ould hav e b ounded degree and small weig ht, giv en that p ath s in H i − 1 can app ro ximate paths in G ′ i − 1 to an arbitrary degree. Give n the struct ure of the cluster graph, all b ut at most 2 edges in any simple xy - path are inter-cluster edges. Since the radius of eac h cluster is δ W i − 1 , each inter-cluster ed ge has w eight greater than δ W i − 1 . W e are looking for a path of length at most t · | xy | . Since | x y | ∈ ( W i − 1 , W i ], w e are looking for a path of length at most t · W i = t · r · W i − 1 . Any simple p ath in H i − 1 of length at most t · r · W i − 1 has at most 2 + ⌈ tr /δ ⌉ hops, whic h is a constan t. This yields the foll owing lemma. Lemma 8. F or any e dge { x, y } ∈ E i , if sp H i − 1 ( x, y ) ≤ t · | xy | , then H i − 1 c ontains a shortest xy -p ath with O (1) hops (no mor e than 2 + ⌈ tr /δ ⌉ ). One issue we need t o deal with, especially when attempt- ing to construct and answer q ueries in H i − 1 in a distributed setting, is that edges in H i − 1 need n ot be present in the underlying netw ork G . Specifically , for an intra-cluster edge { u, a } , where C a is a cluster and u ∈ C a , it ma y b e the case that | ua | > α and { u, a } may be absen t from G . S imilarly , an inter-cluster edge { a, b } in H i − 1 ma y be absent in G . How ev er, for an y edge { x, y } in H i − 1 (intra- or in ter-cluster edge), w e hav e t he b ound sp G ′ i − 1 ( x, y ) ≤ (2 δ + 1) W i − 1 . This follo ws from Lemma 5 and the fact th at the radius of each cluster is δ W i − 1 . Thus a shortest xy -path in G ′ i − 1 lies en- tirely in a ball of radius (2 δ + 1) W i − 1 centered at x . Sin ce G ′ i − 1 is a spanning subgraph of G , this implies th at there is a shortest xy -p ath P in G that lies entirely in the d - dimensional ball of radius (2 δ + 1) W i − 1 centered at x . Since any t wo v ertices in P that are tw o hops aw a y from each other are at leas t α apart ( in the d -d imensional Euclidean space), P contains at most ⌈ 2(2 δ + 1) W i − 1 /α ⌉ < ⌈ 2(2 δ + 1) /α ⌉ hops. This argumen t yields the fol lowing theorem. Theorem 9. F or any e dge { x, y } ∈ E i , if sp H i − 1 ( x, y ) ≤ t · | xy | , then G c ontains a shortes t xy -p ath with O (1) hops (no mor e than ⌈ 2(2 δ + 1) /α ⌉ ). This theorem implies that bru te force search initiated from one of th e endpoints, sa y x , will be able to answ er the shortest path query on edge { x, y } in O (1) rounds in a distributed setting. 2.2.5 Removing Redundant Edges Let t 1 b e such th at 1 < t 1 < t . R ecall that shortest path queries for edges in E i are answered on H i − 1 , and so up d ates to G ′ i in phase i do not infl uence sub sequent shortest p ath queries in phase i . Thus it is p ossible that in phase i tw o edges { u, v } and { u ′ , v ′ } get add ed t o G i , yet b oth of the follo wing h old: (i) sp H i − 1 ( u, u ′ ) + | u ′ v ′ | + sp H i − 1 ( v ′ , v ) ≤ t 1 · | uv | (ii) sp H i − 1 ( u ′ , u ) + | uv | + sp H i − 1 ( v , v ′ ) ≤ t 1 · | u ′ v ′ | Note that, since sp G ′ i − 1 ( x, y ) ≤ sp H i − 1 ( x, y ) h olds for any pair of no des x and y , and since t 1 < t , conditions (i) and (ii) ab ov e imply that G ′ i conta ins t -spanner paths from u to v and from u ′ to v ′ . W e call tw o edges { u, v } and { u ′ , v ′ } satisfying conditions (i) and (ii) ab ov e mutual ly r e dundant : one of them could p otentially be eliminated from G i , with- out compromising t he t -spanner p roperty of G i . In fact, such mutually redundant pairs of edges need to b e elimi- nated from G ′ i b ecause our pro of that G ′ has small w eight (Theorem 13 ) dep ends on the absence of suc h pairs of edges. T o do this, we b u ild a graph J that h as a no de for each edge in a mutually redundant pair and an edge b etw een ev- ery p air of no des that corresp ond to a m utu ally redundant pair of edges in G ′ i . W e construct an MIS I of J and elim- inate from G ′ i all edges associated with no des in J that do not app ear in I . 2.3 The Three Desired Pr operties Let G ′ = G ′ m b e the spanner at the end of phase m . W e now prov e that G ′ satisfies the three prop erties that the output of SEQ-GREEDY was guaran teed to hav e. The pro ofs of th ese theorems form the tec hn ical core of the pap er and are presented next in this section. Theorem 10. F or any 0 < δ ≤ t − t 1 4 , the output G ′ is a t -sp anner. Pr oof. W e first p ro ve that th e theorem holds for all query edges in E , then w e extend the argument t o non- query edges as wel l. Let { x, y } be an arbitrary query edge and let i ≥ 1 b e such that { x, y } ∈ E i . Then either (i) { x , y } is added to the spanner in phase i , or (ii) sp H i − 1 ( x, y ) ≤ t · | x y | . If the former is true and { x, y } is not a redu n dant edge, th en the theorem holds. If { x, y } is a redu ndant edge but does not get remov ed from G i , then again the theorem holds. If { x, y } is a redundant edge that gets remove d from G i , th en at least one mutually redundant counterpart edge must re- main in G i (since remov ed edges form an indep endent set), ensuring a t -spanner x y -path in G i . If (ii) is tru e, then from Lemma 7, sp G ′ i − 1 ( x, y ) ≤ sp H i − 1 ( x, y ) (first part of the in- equality) and therefore sp G ′ i − 1 ( x, y ) ≤ t · | xy | . F or non-query edges, the proof is by indu ction on t h e length of edges in G . The base case corresponds to edges in E 0 , for whic h SEQ-GREEDY en su res that the theorem holds. Assume th at the theorem is true for any edge in E of length n o greater than some v alue q , and consider a smallest non-query edge { x, y } in G of length greater th an q . W e prov e that sp G ′ ( x, y ) ≤ t · | xy | . Let i b e such that { x, y } ∈ E i . W e now consider tw o cases, dep ending on whether { x, y } is a c andidate query edge in ph ase i or not. If { x, y } is not a candidate query edge, then it is a cov ered edge. That is, there exists an edge { x, z } in G ′ i − 1 such that | y z | ≤ α and ∠ y xz ≤ θ , or an edge { y , z } in G ′ i − 1 such that | xz | ≤ α an d ∠ xy z ≤ θ . The tw o cases are symmetric and so without loss of generality , assume that the former is true. Here θ satisfies t h e h yp othesis of the Czuma j-Zhao lemma (Lemma 3), that is, 0 < θ < π 4 and t ≥ 1 cos θ − si n θ . Since | y z | ≤ α and G is an α -UBG, this implies that { y , z } is an edge is E . F urt hermore, since 0 < θ < π 4 , w e have | y z | < | xy | . R efer to Figure 3 a. If { y, z } is a qu ery edge, then by the argumen t ab ov e w e have that G ′ conta ins a t -spanner y z -path p . Otherwise, if { y, z } is not a query edge, since its length is less than the length of { x, y } , by the in d uctive hypothesis w e get that there is a t -spanner yz - path p . In either case, Lemma 3 tells us that { x, z } follo w ed by p is a t - spanner path from x to y , completing this case. x y z < θ a u b v x y (a) (b) Figure 3: (a) { x, y } is a cov ered e dge (b) { u, v } is a query edge: if G i con tains a t -spanner uv -path, then G i con tains a t -spanner xy -path. W e now consider the case when { x, y } is a candidate q uery edge in phase i , b ut not a query edge. Let a and b b e such that x ∈ C a and y ∈ C b , and let { u, v } b e th e q uery edge selected in phase i , with u ∈ C a and v ∈ C b . Refer to Figure 3b. Du e to the criteria for selecting { u, v } , w e hav e t · | uv | − sp G ′ i − 1 ( a, u ) − sp G ′ i − 1 ( b, v ) ≤ t · | xy | − sp G ′ i − 1 ( a, x ) − sp G ′ i − 1 ( b, y ) . (2) Recall that G ′ i is the partial sp an n er at th e end of ph ase i . W e show th at sp G ′ i ( x, y ) ≤ t · | x y | . W e discuss tw o cases, dep ending on whether { u, v } w as added to G ′ i or not. Assume fi rst that { u, v } w as n ot added to G ′ i . This means that sp H i − 1 ( u, v ) ≤ t · | uv | . N ote ho wev er that sp H i − 1 ( u, v ) = sp G ′ i − 1 ( u, a ) + sp H i − 1 ( a, b ) + sp G ′ i − 1 ( b, v ) ≤ t · | uv | . (3) W e no w ev aluate sp G ′ i − 1 ( x, y ) ≤ sp G ′ i − 1 ( x, a ) + sp G ′ i − 1 ( a, b ) + sp G ′ i − 1 ( b, y ) ≤ sp G ′ i − 1 ( x, a ) + sp H i − 1 ( a, b ) + sp G ′ i − 1 ( b, y ) ≤ t · | x y | . This latter inequality inv olves simple substitutions that u se inequalities (2) an d ( 3), and completes this case. Now assume th at { u, v } was added to G ′ i . Since u ∈ C a and C a has radius δ W i − 1 , we hav e that sp G ′ i − 1 ( a, u ) ≤ δ W i − 1 . Similarly , sp G ′ i − 1 ( b, v ) ≤ δ W i − 1 . These together with (2) yield t · | uv | − 2 δ W i − 1 ≤ t · | xy | − sp G ′ i − 1 ( a, x ) − sp G ′ i − 1 ( b, y ) . (4) If the edge { u, v } turns out to b e redundant and eliminated from G i , the existence of a m utu ally redundant counterpart edge in G ′ i ensures th at sp G ′ i ( u, v ) ≤ t 1 · | uv | . This enables us to construct in G ′ i a path from a to b of weigh t sp G ′ i ( a, b ) ≤ sp G ′ i ( a, u ) + t 1 · | uv | + sp G ′ i ( v , b ) ≤ 2 δW i − 1 + t 1 · | uv | , (5) since sp G ′ i ( a, u ) ≤ sp G ′ i − 1 ( a, u ) ≤ δ W i − 1 , and same for sp G ′ i ( v , b ). W e can n o w construct a path in G ′ i from x to y of w eight sp G ′ i ( x, y ) ≤ sp G ′ i ( a, x ) + sp G ′ i ( b, y ) + sp G ′ i ( a, b ) ≤ t · | xy | + 2 δ W i − 1 − t · | uv | + sp G ′ i ( a, b ) ≤ t · | xy | + 4 δ W i − 1 − ( t − t 1 ) · | uv | < t · | xy | + 4 δ W i − 1 − ( t − t 1 ) W i − 1 In deriving th is chain of inequalities, we hav e used (4), (5) and the fac t that | uv | > W i − 1 . N ote t h at for any δ ≤ t − t 1 4 , the quantit y 4 δ W i − 1 − ( t − 1) · W i − 1 above is negative, yielding sp G i ( x, y ) < t · | xy | . This completes the pro of. 2 α 2 α χ u v z θ u 1 v 1 u 2 v 2 u 3 v 3 Figure 4: (a) Region χ con tains tw o neighbors v and z of u . (b) Defini tion of the t -leapfrog prop erty with S = {{ u 1 , v 1 } , { u 2 , v 2 } , { u 3 , v 3 }} . Theorem 11. G ′ has O (1) de gr e e. Pr oof. Let θ b e a quan tity satisfying the conditions of Lemma 3. Fix a vertex u and consider t he d -dimensional unit radius ball cen tered at u . F or some T that dep ends only on θ and d , th is b all can b e partitioned into T cones, eac h with ap ex u , such t hat for any x , y in a cone, ∠ xuy ≤ θ . Y ao [20] shows ho w to construct such a partition with T = O ( d 3 / 2 · sin − d ( θ / 2) · log( d sin − 1 ( θ / 2))) cones. Place an infin ite axis-parallel grid of d -dimensional cub es, each of dimension α √ d × α √ d × · · · × α √ d , on the plane. S ee Fig- ure 4(c) for a 2-dimensional v ersion of th is picture. There are O (1 /α d ) cells that intersect the u nit ball centered at u , and therefore there are O (1 /α d ) cells that intersect eac h cone in the cone partition of this u nit ball. Th us the cones and th e square cells t ogether partition the unit ball cen- tered at u into O ( T /α d ) regions. W e sho w that in G ′ , u has O ( t d (4 δ + r ) d δ d ) neigh b ors in eac h region, whic h is a constant. Let v 1 , v 2 , . . . , v k b e neighbors of u in G ′ that lie in a region χ . Without loss of general ity , assume that | uv 1 | ≥ | uv j | , for j = 2 , . . . , k , and let i b e suc h that { u, v 1 } ∈ E i . S ince | uv j | ≤ | uv 1 | , w e hav e that for all j = 2 , . . . , k , { u, v j } ∈ E ℓ , with ℓ ≤ i . W e no w pro ve that { u, v j } is in fact in E i for all j . T o derive a con tradiction, assume that there is a j > 1 su ch that { u, v j } ∈ E ℓ , with ℓ < i . This means that just before edge { u, v 1 } is pro cessed, G ′ conta ins edge { u, v j } . Also note that since v 1 and v j lie in t h e same region, | v 1 v j | ≤ α . But, this means that { u, v 1 } is a cov ered edge in p h ase i and will not b e q ueried. This con tradicts the presence of edge { u, v 1 } in G ′ . W e hav e sh o wn that { u, v j } ∈ E i for all j . R ecall that our algo rithm picks a unique query edge p er pair of clusters. This along with Lemma 4 prov es that k is constan t. In the next th eorem, w e sho w that the spanner produced by the algorithm has small weigh t. The pro of relies on the line segmen ts in th e spann er satisfying a p rop erty known as the le apfr o g pr op erty [2, 5]. F or any t ≥ t 2 > 1, a set of line segmen ts, d en oted F , has the ( t 2 , t ) -le apfr o g pr op erty if for every subset S = {{ u 1 , v 1 } , { u 2 , v 2 } , . . . , { u s , v s }} of F t 2 · | u 1 v 1 | < s X i =2 | u i v i | + t · “ s − 1 X i =1 | v i u i +1 | + | v s u 1 | ” . (6) Informally , this definition says that if there exists an edge b etw een u 1 and v 1 , then any path not including { u 1 , v 1 } must hav e length greater than t 2 | u 1 v 1 | (see Figure 4(c) for an illustration of t his definition). The follow ing imp lica- tion of the ( t 2 , t )-leapfrog property wa s sho wn by Das and Narasimhan [4]. Lemma 12. L et t ≥ t 2 > 1 . I f the line se gments F in d -dimensional sp ac e satisfy the ( t 2 , t ) -le apfr o g pr op erty, then wt ( F ) = O ( w t ( M S T )) , wher e M S T is a minimum sp anning tr e e c onne cting the endp oints of line se gments in F . The c onstant in the asymptotic notation dep ends on t , t 2 and d . Theorem 13. L et 0 < δ < min { ( t − 1) / (6 + 2 t ) , ( t − t 1 ) / 4 } . L et t δ denote t 1 · (1 − 2 δ ) / (1 + 6 δ ) . L et 1 < r < ( t δ + 1) / 2 . W hen the r elaxe d gr e e dy algorithm is run with these values of δ and r , the output G ′ satisfies w ( G ′ ) = O ( w t ( M S T ( G ))) . Pr oof. Let β > 1 be a constan t pick ed as follo ws. When tα < 1, pic k β satisfying 1 < β < min { 2 , 1 / (1 − t α ) } . Oth - erwise, pick β satisfying 1 < β < 2. Partiti on the edges of G ′ into subsets F 0 , F 1 , . . . such that F 0 = {{ u, v } ∈ G ′ | | uv | ≤ α } and for eac h j > 0, F j = {{ u, v } ∈ G ′ | αβ j − 1 < | uv | ≤ αβ j } . Let ℓ = ⌈ log β 1 α ⌉ . Then every edge in G ′ is in some subset F j , 0 ≤ j ≤ ℓ . W e will now show that each F j satisfies the ( t 2 , t )-leapfrog property , for an y t 2 satisfying: 1 ≤ t 2 < min { t δ + 1 r − 1 , 2 r , t r , 2 β , tα + 1 β } . (7) It is easy to c heck th at our choi ce for δ , r , and β guaran tee that each qu anti ty in side the min operator is strictly greater than 1. Sho wing the ( t 2 , t )-leapfrog prop erty for F j w ould imply that w ( F j ) = O ( w ( M S T ( G ))), and since the edges of G ′ are partitioned into a constant num b er of sub sets F j , w ( G ′ ) = O ( w ( M S T ( G ))). Consider an arbitrary sub set S = {{ u 1 , v 1 } , { u 2 , v 2 } , . . . , { u s , v s }} ⊆ F 0 . T o pro ve inequ alit y (6) for S , it suffices to consider the case when { u 1 , v 1 } is a longest edge in S . W e consider F 0 separately from F j , j > 0. The F 0 case. If for any 1 ≤ k < s , | v k u k +1 | > | u 1 v 1 | or | v s u 1 | > | u 1 v 1 | , then th e leapfro g prop erty holds. S o w e assume that for all 1 ≤ k < s , | v k u k +1 | ≤ | u 1 v 1 | and | v s u 1 | ≤ | u 1 v 1 | . Let i b e the phase in which { u 1 , v 1 } gets processed, i.e. , { u 1 , v 1 } ∈ E i . Since | u 1 v 1 | ≤ α , it is the case that for all 1 ≤ k < s , | v k u k +1 | ≤ α and | v s u 1 | ≤ α . Hen ce, {{ v s , u 1 }} ∪ {{ v k , u k +1 } | 1 ≤ k < s } is a subset of edges of G and each edge in this set gets pro cessed in phase i or earlier. Assume first that at least one edge in the set {{ v s , u 1 }} ∪ {{ v k , u k +1 } | 1 ≤ k < s } gets processed in phase i . Then the righ t hand side of inequality (6) is at least tW i − 1 , since edges in E i hav e w eigh ts in the in terv al I i = ( W i − 1 , rW i − 1 ]. Also since t 2 | u 1 v 1 | ≤ t 2 r W i − 1 , and since the inequ alit y t 2 r W i − 1 < tW i − 1 is guaran teed by the v alues of r and t 2 in (7), the leapfrog prop erty h olds for this case. Assume no w that all edges in {{ v s , u 1 }} ∪ {{ v k , u k +1 } | 1 ≤ k < s } ha ve b een processed in phase i − 1 or earl ier, meaning that t -span n er paths betw een th eir endp oin ts exist in G ′ i − 1 at the time { u 1 , v 1 } gets processed. F or 1 ≤ k < s , let P k b e a shortest v k u k +1 -path in G ′ i − 1 , and let P s b e a shortest v s u 1 -path in G ′ i − 1 . Let P b e the follo wing u 1 v 1 - path in G ′ i : P = P 1 ⊕ { u 2 , v 2 } ⊕ P 2 ⊕ { u 3 , v 3 } ⊕ · · · ⊕ P s . Here, w e use ⊕ to denote concatenation. W e distinguish three cases, depend ing on the size of the subset S ∩ E i . (i) | S ∩ E i | > 2. Then, w ( P ) ≥ 2 W i − 1 . W e also ha ve that | u 1 v 1 | ≤ r W i − 1 , since { u 1 , v 1 } ∈ E i . It follo ws that w ( P ) > t 2 | u 1 v 1 | for an y t 2 < 2 r . F urthermore, w ( P ) is n o greater than the right h an d side of th e ( t 2 , t )- leapfrog inequalit y (6), so lemma holds for th is case as w ell. (ii) | S ∩ E i | = 2. In addition to { u 1 , v 1 } , assume that { u k , v k } ∈ E i for some k , 1 < k ≤ s . I t the ( t 2 , t )- leapfrog inequality (6) holds, we are done and so let us assume the opp osite of that: t 2 · | u 1 v 1 | ≥ s X i =2 | u i v i | + t · “ s − 1 X i =1 | v i u i +1 | + | v s u 1 | ” . (8) Since all edges { u j , v j } , 1 ≤ j ≤ s , except for { u 1 , v 1 } and { u k , v k } are in G ′ i − 1 , and since G ′ i − 1 conta ins t - spanner v j u j +1 -paths for all j , 1 ≤ j < s , and a t - spannner v s u 1 -path, the ab ov e inequality yields t 2 · | u 1 v 1 | ≥ sp G ′ i − 1 ( v 1 , u k ) + | u k v k | + sp G ′ i − 1 ( v k , u 1 ) . Multiplying b oth sides b y (1 + 6 δ ) / (1 − 2 δ ) and using t 2 < t δ (whic h is implied b y our choice of t 2 ) and Lemma 7, w e get t 1 · | u 1 v 1 | ≥ sp H i − 1 ( v 1 , u k ) + | u k v k | + sp H i − 1 ( v k , u 1 ) . (9) Let ∆ = P s − 1 i =1 | v i u i +1 | + | v s u 1 | . W e now observ e that t δ · | u k v k | < k − 1 X i =1 | u i v i | + s X i = k +1 | u i v i | + t · ∆ (10) implies t he ( t 2 , t )-leapfrog prop erty . T o see this u se the fact that b oth { u 1 , v 1 } and { u k , v k } belong to E i and therefore | u 1 v 1 | < r · | u k v k | , whic h substituted in ( 10) yields: t δ · | u k v k | − ( r − 1) · | u k v k | < s X i =2 | u i v i | + t · ∆ . W e get the low er b oun d t 2 · | u 1 v 1 | on the left hand side of the abov e inequality b y using | u k v k | > | u 1 v 1 | /r again and our c hoice of t 2 < ( t δ + 1) /r − 1. T his yields t h e ( t 2 , t )-leapfrog prop erty . So we assume that inequality (10) do es not h old, that is, t δ · | u k v k | ≥ k − 1 X i =1 | u i v i | + s X i = k +1 | u i v i | + t · ∆ . Since all edges { u j , v j } , 1 ≤ j ≤ s , except for { u 1 , v 1 } and { u k , v k } are in G ′ i − 1 , and since G ′ i − 1 conta ins t - spanner v j u j +1 -paths for all j , 1 ≤ j < s , and a t - spannner v s u 1 -path, the abov e inequalit y yields t δ · | u k v k | ≥ sp G ′ i − 1 ( v 1 , u k ) + | u 1 v 1 | + sp G ′ i − 1 ( v k , u 1 ) . Multiplying both sides b y (1 + 6 δ ) / (1 − 2 δ ) and using Lemma 7, we get t 1 · | u k v k | ≥ sp H i − 1 ( v 1 , u k ) + | u 1 v 1 | + sp H i − 1 ( v k , u 1 ) . (11) Inequalities (9 ) and (11) imply that edges { u 1 , v 1 } and { u 2 , v 2 } are mutually redundant and therefore cann ot b oth exist in the spanner — a contradiction. (iii) | S ∩ E i | = 1. This means that P exists in G ′ i − 1 at the time { u 1 , v 1 } is pro cessed. F urthermore, w ( P ) > t · | u 1 v 1 | > t 2 · | u 1 v 1 | , oth erwise { u 1 , v 1 } w ould not h ave b een ad d ed to th e spanner, a contradiction. The F j case, j > 0 . In this case, | u k v k | > | u 1 v 1 | /β for all k = 2 , 3 , . . . , s . If | S | ≥ 3, th en the righ t hand side of the ( t 2 , t )-leapfrog inequality (6) is at least 2 · | u 1 v 1 | /β and therefore the ( t 2 , t )-leapfrog inequalit y goes through for any 1 < t 2 < 2 /β . Otherwise, if | S | = 2, then w e need to sho w that t 2 · | u 1 v 1 | < | u 2 v 2 | + t · ( | u 1 v 2 | + | u 2 v 1 | ). If eac h of | u 1 v 2 | and | u 2 v 1 | is at most α , then using the same ar- gument as in the F 0 -case with | S ∩ E i | = 2, w e can sho w that { u 1 , v 1 } and { u 2 , v 2 } are mutually redundant and will not b oth exist in the spanner. Otherwise, if one of | u 1 v 2 | or | u 2 v 1 | is greater than α , then the right hand side of the ( t 2 , t )-leapfrog inequalit y (6) is greater than | u 1 v 1 | /β + tα . T o en sure th at the inequalit y goes through, w e req uire th at t 2 · | u 1 v 1 | ≤ | u 1 v 1 | β + tα . Since | u 1 v 1 | ≤ 1, the ab o ve inequal- it y is satisfied for any 1 < t 2 ≤ tα + 1 β , which holds true cf. (7). 3. DISTRIBUTED RELAXED G REED Y AL- GORITHM W e now d escrib e a distribut ed versi on of the relaxed greedy algorithm from Section 2. Like the sequential relaxed greedy algorithm, this algorithm also runs in O (log n ) phases — with edges in E i b eing processed in phase i . W e will sho w that edges in E 0 can be pro cessed in O (1) rounds. Re- call that eac h subsequent phase consists of th e follo wing five steps: (i) computing a cluster cov er of G ′ i − 1 , (ii) se- lecting q uery edges in E i , (iii) computing a cluster graph H i − 1 of G ′ i − 1 , (iv) answ ering sh ortest path queries for se- lected query edges, and (v) deleting some redun dant edges. W e wil l sho w that Steps (ii), (iii), and (iv) can b e completed in O (1) rounds and Steps (i) and (v ) tak e O (log ∗ n ) rounds. Step (i) and Step (v) will eac h inv olve comput ing an MIS in a certain d eriv ed graph and in b oth cases, we will show that the derived graph is a UBG that resides in a metric sp ace of constan t doubling dimension. Put ting this all together, w e will show that th e algorithm runs in O (log n · log ∗ n ) comm unication roun ds. 3.1 Dis trib uted Processing of Short Edges Lemma 1 implies that vertices in th e same component of G 0 = G [ E 0 ] ind uce a clique and therefore can communicate in one hop with eac h oth er. In t he distributed version of the algorithm, each ve rtex u obtains the t op ology of its close d neighborho od along with pairwi se distances b etw een neigh- b ors in one hop. Using this information, u d etermines the connected component C of G 0 that it b elongs to. Then u simply runs SEQ-GREEDY on C and comput es a t - spanner of C . Finally , u identifies the edges of the t -spanner incident on itself and informs all its neighbors of this. Theorem 14. The e dges in E 0 c an b e pr o c esse d in O (1) r ounds of c ommunic ation. 3.2 Dis trib uted Processing of Long Edges In this section, w e show how long edges, that is, edges in E i , i > 0, can b e pro cessed in a d istributed setting. The first step of this process is the computation of a cluster co ver for the spanner G ′ i − 1 up dated at the end of t he previous phase. 3.2.1 Distributed Cl us ter Cover for G ′ i − 1 Recall that in this step our goal is to compu te a clus- ter co ver { C u 1 , C u 2 , . . . } of G ′ i − 1 of radius δ W i − 1 . T o do this, each n ode u first identifies all no des v in G satis- fying sp G ′ i − 1 ( u, v ) ≤ δ W i − 1 . Using arguments similar to those in Section 2.2.4, we can sho w that an y no de v satisfy- ing sp G ′ i − 1 ( u, v ) ≤ δ W i − 1 must be at most 2 δ W i − 1 /α hops from u . S o eac h n ode u constructs the sub graph of G ′ i − 1 induced by nod es that are at most 2 δ W i − 1 /α hops aw ay from it in G . No d e u then runs a ( sequential ) single source shortest path algorithm with source u on the lo cal view of G ′ i − 1 it has obtained and identifies all nod es v satisfying sp G ′ i ( u, v ) ≤ δ W i − 1 . At the end of the ab o ve p ro cess, every node u in the net- w ork is a cluster center. W e no w force some nodes to cease b eing cluster cen ters, so that all pairs of cluster centers are far enough from each other. Let J be the graph with vertex set V and whose edges { x, y } are such that x ∈ C y (and b y symmetry , y ∈ C x ). Lemma 15. J is a UBG that r esides in a metric sp ac e of c onstant doubling dimension. Pr oof. F or an y edge { x, y } in J , w e ha ve that x ∈ C y and therefore sp G ′ i − 1 ( x, y ) ≤ δ W i − 1 . A ssign t o ev ery p air of no des { x, y } in V a weigh t w ( x , y ) = sp G ′ i − 1 ( x, y ) . The w eights w form a metric simply because shortest path d is- tances in an y graph form a metric. Th us J is a graph whose nod es reside in a metric space an d whose ed ges conn ect pairs of nod es separated by distance of at most δ W i − 1 (in the met- ric space). By scaling the quantit y δ W i − 1 up to one, we see that J is a UBG in the u nderlying metric space defin ed by the weig hts w . Recall from [11] that the doubling dim en- sion of a metric space is t h e smallest ρ such that eve ry ball can b e co vered by at most 2 ρ balls of h alf the radius. T o see that the metric sp ace induced b y the weigh ts w has constant doubling d imension, start with a ball of B radius R centered at an arbitrary vertex u . Every vertex v in ball B satisfies sp J ( u, v ) ≤ R . Now co ver the ve rtices in B using balls of radius R / 2 as follo ws: repeatedly pick an uncov ered vertex v in B and gro w a ball of radius R/ 2 centered at v , until all vertices ha ve been cover ed. W e n o w sh o w that t he num b er of balls of radius R / 2 is constant. Let a and b b e tw o arbitrary centers of different balls of radius R/ 2. Then sp J ( u, v ) > R / 2, otherwise a and b would b elong to the same b all of radius R/ 2. W e distinguish three situations: • { a, b } is not an edge in G . This imp lies that | ab | > α . • { a, b } is an edge in G th at has n ot b een processed prior to phase i . This implies that | ab | > W i − 1 ≥ 1 /δ (after scaling δ W i − 1 up to 1). • { a, b } is an edge in G that has been pro cessed prior t o phase i . This implies that G ′ i − 1 conta ins a t -spann er path from a to b and th erefore | ab | ≥ sp G ′ i − 1 ( a, b ) /t ≥ R/ 2 t . W e have established that | ab | ≥ min { α, 1 δ , R 2 t } , so no tw o ball centers can b e too close to eac h other. I t follo ws that the num b er of balls of radius R / 2 that fit inside B is constant, proving the lemma true. Let I b e an MIS of J constructed using the MIS algorithm in [11]. This algorithm runs in O (log ∗ n ) communicatio n rounds on a U BG that resides in a metric space of constant doubling dimension. Then each no de in V \ I has one or more neighbors in I . Eac h nod e u ∈ I is declared a cluster center, and eac h node v ∈ V \ I attaches itself to the n eigh b or in I with th e highest iden tifier. This gives us the desired cluster co ver of radius δ W i − 1 . Theorem 16. A cluster c over of G ′ i − 1 of r adius δ W i − 1 c an b e c ompute d i n O ( log ∗ n ) r ounds of c ommunic ation. 3.2.2 Distributed Query Edge Selec tion Only nod es t hat are cluster heads need to participate in the pro cess of selecting query edges. Eac h cluster head a seeks to gather information on all edges in E i b etw een th e cluster C a and any other cluster C b . U sing the argumen t in Section 2.2.4, we know th at every node in C a is at most 2 δ W i − 1 /α h ops a w ay from a in G . Therefore, if t h ere is an edge { u, v } ∈ E i , u ∈ C a and v ∈ C b , then v is at most 1 + 2 δ W i − 1 /α hops aw a y from a . So a gets information from nod es that are at most 1 + 2 δ W i − 1 /α hops a w ay from it and it identifies all edges in E i [ C a , C b ]. Recall that th is is the set of edges in E i whic h connect a no de in C a and a no de in C b . No de a then discards all co vered edges from E i [ C a , C b ], lea ving only candidate query edges in E i b etw een C a and C b . Finally , from among the candidate qu ery edges, no de a selects an edge { u, v } that minimizes t · | uv | − sp G ′ i − 1 ( a, u ) − sp G ′ i − 1 ( b, v ). Theorem 17. Query e dges fr om E i c an b e sele cte d in O (1) r ounds of c ommunic ation. 3.2.3 Distributed Construction of the Cl uster Graph As in the query edge selection step, only th e cluster heads need to p erform actions to compute the cluster graph. Any member u of a cluster C a lies at most 2 δ W i − 1 /α hops a wa y from a in G . Thus a can identify intra-cluster edges inci- dent on it b y gathering informatio n from at most 2 δ W i − 1 /α hops a wa y . If C b is a cluster with sp G ′ i − 1 ( a, b ) ≤ W i − 1 , then nod e a can identify the inter-cluster edge { a, b } by gather- ing informatio n from at most 2 W i − 1 /α h ops a w ay . I f C b is a cluster such that there is an edge { u, v } in G ′ i − 1 with u ∈ C a and v ∈ C b , then nod e a can iden tify the inter- cluster edge { a, b } by gathering informatio n from at most 2(2 δ + 1) W i − 1 /α hops aw a y . Note that the information that a gathers contains a local v iew of G ′ i − 1 along with all p air- wise distances. Using this information, no de a is able to run a single source shortest path alg orithm with source a and determine the wei ghts of all in ter-cluster and intra-cluster edges incident on a . Theorem 18. Computing the cluster gr aph H i − 1 of G ′ i − 1 takes O (1) c ommunic ation r ounds. 3.2.4 Answering Shortes t P ath Queries Eac h node u knows all the query edges incident on it. As pro ved in Section 2.2. 4 , n ode u only needs to gather information from no des that are at most a constan t number of hops aw a y , to b e able to determine lo cally , for all inciden t query edges { u, v } ∈ E i , whether sp H i − 1 ( u, v ) ≤ t · | uv | . Thus, after constant number of comm unication rounds, u knows th e subset of incident query ed ges { u, v } for which sp H i − 1 ( u, v ) > t · | uv | and u identifies t h ese as the incident edges to b e added to G ′ i . Theorem 19. A nswering shortest p ath queries takes O (1) c ommunic ation r ounds. 3.2.5 Distributed Removal of Red undant Edges Tw o edges { u, v } and { u ′ , v ′ } in G ′ i are mutually redu n- dant if (i) sp H i − 1 ( u, u ′ ) + | u ′ v ′ | + sp H i − 1 ( v ′ , v ) ≤ t 1 · | uv | and (ii) sp H i − 1 ( u ′ , u )+ | uv | + sp H i − 1 ( v , v ′ ) ≤ t 1 ·| u ′ v ′ | . Eac h no de u takes charge of all edges { u, v } added to G i in phase i and for whic h th e identi fier of u is higher than the identifier of v . F or each suc h edge { u, v } that u is in charge of, u deter- mines all edges { u ′ , v ′ } such that { u, v } and { u ′ , v ′ } form a mutually redu ndant pair. N ote that the no des u and v ′ are a constant number of hops a w ay from each other in G , and similarly for nod es v and u ′ . No de u then contributes to the construction of the graph J by adding to V ( J ) a vertex for each redundant edge u is in c harge of, and to E ( J ) an edge connecting n odes in V ( J ) that correspond to m utually redundant edges in G i . W e now show the follow ing prop erty of J : Lemma 20. J is a UBG that r esides in a metric sp ac e of c onstant doubling dimension. Pr oof. Let a and b be v ertices in J corresp onding to edges { u a , v a } and { u b , v b } in G ′ i . Assign to the vertex pair ( a, b ) a weigh t equal to d J ( a, b ) = min( sp H i − 1 ( u a , u b ) + sp H i − 1 ( v a , v b ) , sp H i − 1 ( u a , v b ) + sp H i − 1 ( v a , u b )) . First we show that th e wei ghts defin ed by d J form a met- ric. Clearly d J ( a, a ) = 0 and d J ( a, b ) = d J ( b, a ). T o prov e the triangle inequality , consider three vertices a , b, c ∈ J . Assume w.l.o. g. that d J ( a, b ) = s p H i − 1 ( u a , u b ) + sp H i − 1 ( v a , v b ) d J ( b, c ) = sp H i − 1 ( u b , u c ) + sp H i − 1 ( v b , v c ) W e identify tw o possible scenarios: (1) d J ( a, c ) = sp H i − 1 ( u a , u c ) + sp H i − 1 ( v a , v c ) (see Fig- ure 5b). Since sp is itself a metric, it follow s immedi- ately that d J ( a, c ) ≤ d J ( a, b ) + d J ( b, c ). u a v a u b v b u c v c u a v a u b v b u c v c c a b (a) (b) (c) Figure 5: d J is a metric: (a) No des a, b, c ∈ J cor- resp ond to edges { u a , u b } , { u b , u c } , { u a , u c } ∈ G ′ i . (b) d J ( a, c ) = sp H i − 1 ( u a , u c ) + sp H i − 1 ( v a , v c ) (c) d J ( a, c ) = sp H i − 1 ( u a , v c ) + sp H i − 1 ( v a , u c ) . (2) d J ( a, c ) = sp H i − 1 ( u a , v c ) + sp H i − 1 ( v a , u c ) (see Fig- ure 5c). Then it m ust b e that d J ( a, c ) ≤ sp H i − 1 ( u a , u c ) + sp H i − 1 ( v a , v c ) ≤ d J ( a, b ) + d J ( b, c ), cf. scenario (1). W e ha ve sh own that d J defines a metric. W e no w show that J is a quasi-UBG residing in the metric space defined by d J . F or each edge { a, b } in J , t he follo wing redund an cy conditions hold: (a) sp H i − 1 ( u a , u b ) + sp H i − 1 ( v b , v a ) ≤ t 1 · | u a v a | − | u b v b | (b) sp H i − 1 ( u b , u a ) + sp H i − 1 ( v a , v b ) ≤ t 1 · | u b v b | − | u a v a | Recall that { u a , v a } and { u b , v b } are b oth in E i , for some i ≥ 0. This implies that their lengths differ by a factor of r at the most: W i − 1 < | u a v a | ≤ r · W i − 1 and W i − 1 < | u b v b | ≤ r · W i − 1 . Thus th e righ t hand side of inequalities (a) and (b) ab o ve is a quan tity t h at lies in the interv al (( t 1 − r ) W i − 1 , ( t 1 r − 1) W i − 1 ). By scaling ( t 1 r − 1) W i − 1 up to one w e can say that J is an t 1 − r t 1 r − 1 - qUBG in the underlying metric space d efined b y d J . It remains to sho w that the metric space defined by d J has constant doubling dimension. Throughout t h e rest of the pro of we use B J ( B H ) to d enote a ball in th e metric space defined b y d J ( sp H i − 1 ). R R/4 u x v x u a v a x a (a) R R/2 R (b) u b v b b R/2 Figure 6: (a) The metric space defined b y (a) sp H i − 1 and (b) d J , has doubling di mension. Consider a ball B J ( x, R ) of radius R cen tered at an arbi- trary vertex x ∈ J corresponding t o edge { u x , v x } ∈ H i − 1 . Let a, b ∈ J be suc h that d J ( a, b ) > R/ 2 (see Figure 6). A s- sume w.l.o.g that d J ( a, b ) = sp H i − 1 ( u a , u b ) + sp H i − 1 ( v a , v b ). Then at least one of the follo wing must b e true: (i) sp H i − 1 ( u a , u b ) > R/ 4. (ii) sp H i − 1 ( v a , v b ) > R/ 4. W e use these observ ations, along with the fact that sp H i − 1 defines a metric space of constant doubling dimension, to sho w that d J defines a metric space of constant doub ling dimension. T o cove r all vertices in B J ( x, R ), do the follo wing re- p eatedly: (i) pick an uncov ered vertex a ∈ J (ii) gro w a ball B J ( a, R/ 2), and (iii) gro w tw o balls B H ( u a , R/ 4) and B H ( v a , R/ 4) in H i − 1 , where a = { u a , v a } . Arguments simila r to the ones used in Lemma 15 sho w that, for any ball centers u a , u b ∈ B H ( u x , R ), we ha ve that | u a u b | ≥ min { α, 1 δ , R 2 t } . Since no t wo ball centers can b e to o close to eac h other, it follo ws that B H ( u x , R ) gets cov ered by a constant num b er of balls of radius R/ 4, and similarly for B H ( v x , R ). Conform observ ation (ii) above , correspond- ing to eac h unco vered a ∈ J , there is an unco vered vertex u a or v a in H i − 1 . These together show that the n umber of balls cov ering B J ( x, R ) is constant, thus completing the proof. Let I b e an MIS of J constructed u sing the MIS algorithm in [11] that takes O (log ∗ n ) communicatio n roun ds on a UBG that resides in a metric space of constan t doub ling d imen- sion. Eac h n ode u then remov es from G i all incident edges in V ( J ) \ I . Theorem 21. R emoving r e dundant e dges tak es O (log ∗ n ) c ommunic ation r ounds. 4. FUTURE WORK The results presented in this pap er apply to α -UDGs em- b edded in constant-dimension Euclidean spaces, and do not directly generalize to doubling metric spaces. F or lo w d i- mensional doubling metric spaces, we b elieve it possible to construct an O (log n log ∗ n ) distributed algori thm that p ro- duces a (1 + ε )- sp anner with constan t maximum d egree. How ev er, new techniques m ay b e needed for light wei ght spanners; the techniques presented in this p aper u se a key prop erty ( the leapfrog prop erty) that do es not seem to gen- eralize to metrics of doubling dimension. 5. REFERENCES [1] L. Barri ´ ere, P . 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