Shallow, Low, and Light Trees, and Tight Lower Bounds for Euclidean Spanners

Shallo w, Lo w, and Ligh t T rees, and Tigh t Lo w er Bounds for Euclidean Spanners Y eﬁm Dinitz ∗ Mic hael Elkin ∗ † Sha y Solomon ∗ † Abstract W e show t hat for every n -p oint metric space M there exists a spa nning tree T with un weigh ted diameter O (log n ) and w eight ω ( T ) = O (log n ) · ω ( M S T ( M )). Mor eov er, there is a designa ted point rt such that for every p oint v , dist T ( rt, v ) ≤ (1 + ǫ ) · dist M ( rt, v ), for an arbitrar ily small constant ǫ > 0. W e extend this r esult, and pr ovide a tradeoﬀ be tw een un weigh ted diameter a nd weigh t, and prov e tha t this tradeoﬀ is tight up to c onstant factors in the en tire range o f par ameters. These results ena ble us to settle a long- standing open question in Computational Geometry . In STOC’95 Arya et al. dev ised a co nstruction of E uclidean Spanners with unw eighted diameter O (log n ) and w eight O (log n ) · ω ( M S T ( M )). T en years later in SOD A’05 Agarwal et al. show ed that this result is tight up to a factor of O (log lo g n ). W e close this gap and s how that the re sult of Arya et a l. is tigh t up to consta n t factors. ∗ Department of Computer S cience, Ben-Gurion Un ivers ity of the N egev, POB 653, Beer-Shev a 84105, Israel. E-mail: { dinitz,e lkinm,shayso } @ cs.bgu.ac.il P artially supp orted by the Lynn and William F rankel Center for Computer Sciences. † This research has been supp orted by the Israeli Academy of Science, grant 483/06. 1 In tro duction 1.1 Bac kground and Main Results Spannin g trees for ﬁn ite metric spaces ha v e b een a sub ject of an ongoing in tensiv e researc h since the b eginning of the nineties [4, 11, 12, 18, 33, 29, 13, 36, 9, 43, 10, 45]. In particular, many researchers studied the notion of shal low-light tr e es , henceforth SL T [13, 36, 9, 10, 45, 5, 43]. Roughly sp eaking, S L T of an n -p oin t metric sp ace M is a spanning tree T of the complete graph corresp ondin g to M w hose total w eigh t is close to the w eigh t w ( M S T ( M )) of the min im um sp an n ing tree M S T ( M ) of M , and whose w eigh ted d iameter is close to that of M . (See S ection 2 for r elev an t deﬁ nitions.) In addition to b eing an app ealing com b inatorial ob ject, SL Ts turned out to b e useful for v arious data gathering and d issemination pr ob lems in the message-passing mo d el of d istributed compu ting [9], in appro ximation algorithms [45], for constructing spanners [10, 5], and for VLSI-circuit design [22, 23, 24]. Near-optimal tradeoﬀs b et w een the w eigh t and diameter of SL T s were established by Khuller et al. [37], and b y Aw erbuch et al. [10]. Ev en though the requirement that the spannin g tree T will ha v e a small w eigh ted-diameter is a natural one, it is no less natural to r equire it to ha v e a small u nweighte d diameter (al so called hop- diameter ). The latter requirement guarant ees th at an y t w o p oints of the metric space will b e connected in T b y a path that consists of only a s mall numb er of e dges or hops . This guaran tee turn s out to b e particularly imp ortan t for routing [34, 1 ], computing almost shortest paths in sequen tial and p arallel setting [20, 21, 27], and in other applicatio ns. An other parameter that p la ys an imp ortan t role in many applications is the maxim um (ve rtex) degree of the constru cted tree [6, 14, 8, 34]. In this pap er w e in tro duce and inv estigate a r elated notion of low-light tr e es , h enceforth LL Ts, that com bine small we igh t with small hop-diameter. W e p resen t n ear-tigh t upp er and lo w er b ounds on the parameters of LL Ts. In addition, our constructions of LL Ts hav e optima l maximum de gr e e . T o sp ecify ou r results, we need some notation. F or a spann ing tree T of a metric M , let Λ = Λ( T ) denote the hop-diameter of T , and Ψ = Ψ( T ) = ω ( T ) ω ( M S T ( M )) denote the ratio b et wee n its weig ht and th e w eigh t of th e minim um spann ing tree of M , henceforth the lig htness of T . In particular, we sho w the follo w in g b ou n ds that are tight up to constant factors in the entir e range of the parameters. 1. F or an y su ﬃcien tly large integ er n and p ositiv e intege r h , and an n -p oin t metric space M , there exists a spannin g tree of M with hop-r adius 1 at most h and ligh tness at most O (Ψ), for Ψ that satisﬁes the follo win g relationship. If h ≥ log n then (Ψ is at most O (log n ) and h = O (Ψ · n 1 / Ψ )). In the complemen tary range h < log n , it holds that Ψ = O ( h · n 1 /h ). Moreo ver, this s p anning tree is a binary one whenever h ≥ log n , and it has the optimal maxim um degree  n 1 /h  whenev er h < log n . In addition, in the ent ire range of p arameters th e r esp ectiv e spanning trees can b e constructed in p olynomial time. 2. F or n and h as ab o ve, and h ≥ log n , ther e exists an n -p oin t metric space M ∗ = M ∗ ( n ) for whic h an y sp anning sub graph with hop-radiu s at most h has lightness at least Ω(Ψ), for some Ψ satisfying h = Ω (Ψ · n 1 / Ψ ). 3. F or n and h as ab o v e, and h < log n , an y spanning s ubgraph w ith hop-radius at most h for M ∗ ( n ) has ligh tness at least Ψ = Ω( h · n 1 /h ). 1 Hop-r adius h ( G, r t ) of a graph G with resp ect to a distinguished vertex rt is the maximum num ber of hops in a simple path connecting the vertex rt with some vertex v in G . O bviously , h ( G, r t ) ≤ Λ( G ) ≤ 2 · h ( G, r t ). F or a ro oted tree ( T , r t ), the hop-radius (called also depth ) of ( T , rt ) is the hop- radiu s of G with resp ect to r t . Hop- radius of G , denoted h ( G ), is deﬁned by h ( G ) = min { h ( G, r t ) | r t ∈ V } . 1 (Note that the equation x · n 1 /x = Θ(log n ) h olds if and only if x = Θ(log n ).) See Figure 1 for an illustration of our results. log n Θ( ) log n Θ( ) h lower bound upper bound A A B C D Ψ lower bound upper bound Figure 1: The dashed line sepa rates t w o sets of pairs (Ψ , h ) . Fo r a pair (Ψ , h ) ab ove the l ine, fo r any n -p oint metric space there exists a spanning t ree with lightness a t most Ψ a n d hop-radius at most h . Fo r a pair (Ψ , h ) b elo w the line, t here exist n -p oint metric spaces fo r which this p rop ert y do es not hold. The tw o areas A and the a rea B are all contained in the forme r set, while the area D is contained in the latter one. The tw o areas A depict our upp er b ound constructions, and t heir extension b y monotonicity is depicted b y t he area B. The a rea D rep resents our low er b ounds. The area C represents the gap b et w een our upp er and low er b ounds. The small maximum degree of our L L Ts ma y b e h elpful for v arious applicatio ns in wh ich the degree of a v ertex v corresp onds to the load on a pro cessor that is lo cated in v . The r equiremen t to ac hiev e small maxim um degree is particularly imp ortan t for applications in C omputational Geometry . (See [6, 14, 8], and the references therein.) 1.2 Lo w er Bounds for Euclidean Spanners While our up p er b ounds apply to all ﬁnite metric sp aces, our lo we r b ound s apply to an extremely basic metric space M ∗ = ϑ n . Sp eciﬁcally , this metric sp ace is the 1-dimens ional Euclidean space with n p oint s v 1 , v 2 , . . . , v n lying on th e x -axis w ith coord in ates 1 , 2 , . . . , n , r esp ectiv ely . The basic nature of ϑ n strengthens our lo we r b ound s, as they are applicable eve n for v ery limited classes of metric sp aces. O ne particularly imp ortan t app lication of our lo wer b ounds is in the area of Euclidean Spanners. F or a set U of n p oints in R 2 , and a parameter α , α ≥ 1, a subset H of the  n 2  segmen ts connecting pairs of p oin ts from U is called an (Euclidean) α -sp anner f or U , if for every p air of p oints u, v ∈ U , the distance b et w een them in H is at most α times the Euclidean d istance b et w een them in the plane. Euclidean spanner is a v ery fundamental geometric construct with n umerous applications in Comp utational Geometry [6, 7, 8] and Net work Design [34, 40]. (See the recent b ook of Narasimhan and Sm id [41] for a detailed accoun t on Euclidean spanners and their applications.) A seminal pap er that w as a culmination of a long line of researc h on Euclidean spanners w as publish ed b y Arya et al. [6] in ST OC’95. On e of the m ain results of this pap er is a construction of (1 + ǫ )-spanners with O ( n ) edges that also ha v e ligh tness and hop-diameter b oth b ounded by O (log n ). As an evidence of the optima lit y of this com bination of parameters, Ary a et al. cited a resu lt by Lenhof et al. [38]. Lenh of et al. show ed that an y construction of E u clidean spanners that emplo ys w ell-separated pair decomp ositions cannot ac h iev e a b etter com b ination of w eigh t and hop-diameter. Ho w ev er, the fundament al question of whether this com bination of parameters can b e impro v ed by other means wa s left op en in Arya et al. [6]. A partial answer to this intriguing problem w as given b y Agarwa l et al. [3] in S OD A’05. S p eciﬁcally , it is sho wn in [3] that an y Euclidean spanner with ligh tness O (log n ) must hav e diameter at least Ω ( log n log log n ), and vice v ersa. Consequ en tly , Agarw al et al. sho w ed that th e upp er b ound of Arya et al. is optimal up to a factor O (log log n ). A simple corollary of our lo w er b ounds is that the resu lt of Arya et al. is tigh t u p to constan ts ev en for one-dimensional s p anners! In other w ords, we sho w th at if the ligh tn ess is 2 O (log n ) then the d iameter is Ω(log n ) and vice v ersa, settling the op en problem of [6, 3]. 1.3 Shallo w-Low- Ligh t-T rees W e sho w that our constructions of LL Ts extend to p ro vide also a go o d appr o x im ation of all weighte d distances fr om an y giv en designated ro ot v ertex r t . The resulting spann ing trees achiev e small w eigh t, hop-diameter, and wei ght ed-diameter simultane ously ! In other w ords, these trees combine the useful prop erties of SL Ts and LL Ts in one c onstruction , and th us w e call them shal low-low-light-tr e es , henceforth SLL Ts. Sp eciﬁcally , w e show that for an y suﬃ cien tly large integ er n , a p ositiv e int eger h , a p ositive real ǫ > 0, an n -p oin t metric space M , and a designated ro ot p oint r t , th ere exists a spanning tree T of M ro oted at r t with h op-radius at most O ( h ) and lightness at most O (Ψ · ( ǫ − 1 )), su c h th at ( h = O (Ψ · n 1 / Ψ ) and Ψ = O (log n )) wheneve r h ≥ log n , and Ψ = O ( h · n 1 /h ) wheneve r h < log n . Moreo ve r, for eve ry p oint v ∈ M , th e wei ght ed d istance b et ween the ro ot r t and v in T is greater by at most a factor of (1 + ǫ ) than the w eigh ted distance b etw een them in M . Th is combinatio n of parameters is optimal up to constant factors. Finally , all our constructions can b e implemented in p olynomial time. W e b eliev e that this construction ma y b e particularly useful in algorithmic ap p lications. I n particular, Aw erbuch et al. [9] p r esen ted the notion of cost-sensitiv e comm unication complexit y to the analysis of distributed algorithms. They used SL Ts to devise eﬃcient algorithms with r esp ect to the cost-sensitiv e comm unication complexit y for a plethora of basic prob lems in the area of Distributed C omp uting, in- cluding n et w ork synchronizatio n, global fun ction computation, and con troller proto cols. Ho wev er, their algorithms m a y p erform quite p o orly w ith resp ect to the standard (not cost-sensitiv e) comm unication complexit y notion. If one could use S LL Ts instead of SL Ts in th e construction of [10], it w ould result in distribu ted algorithms that are eﬃcient w ith resp ect to b oth stand ard and cost-sensitiv e notions of comm unication complexit y . A ma jor d iﬃcult y in implemen ting this scheme is that the construction of Aw erbuch et al. [10] pro vides an S L T w hic h u ses only edges of the original net w ork, while our construction of SLL Ts applies to metric spaces, and thus it ma y emplo y edges that are not pr esen t in the original net wo rk. Moreo ver, w e show (see Section 7) that there are graphs with constant hop-diameter for w hic h an y spann ing tree has either h uge hop-diameter or h uge w eigh t, and th us there is n o hop e that LL Ts or SLL Ts for general netw orks will b e ev er constru cted. Ho w ev er, this approac h seems to b e applicable for d istributed algorithms that run in c omplete net w orks 2 [46, 39], overlay net w orks [28, 2 ], and in other n et work architect ures in whic h either direct or virtual link ma y b e readily established b et wee n eac h pair of pro cessors. T o su mmarize, the problem of un derstanding the inherent tradeoﬀ b et ween diﬀerent parameters of LL Ts is a f undamenta l on e in the in v estigatio n of spanning tr ees for metric spaces and graphs. In addition, this basic and com binatorially app ealing p roblem has imp ortan t applications to Comp utational Ge ometry and Distributed Comp uting. W e b eliev e that furth er in v estigation of LL Ts will exp ose th eir additional applications, and connections to other areas. 1.4 Ov erview and Our T ec hniques The most tec hnically c hallenging p art of our pro of is the lo we r b ound for the range of Λ ≥ log n . Th e pro of of this lo w er b ound consists of a n umber of comp onents. First, w e restrict our attent ion to binary trees. Second, w e adapt a linear p rogram for the minimum linear arr angemen t problem from the semin al pap er of Even, Naor, Rao and Schieber [30 ] on spr eadin g metrics to our needs. Third, w e analyze this linear program and sh ow that the problem of p r o viding a low er b ound for its solution reduces to a clean com binatorial pr oblem, and solv e this problem. T h is enables us to establish the d esired lo wer b oun ds for 2 Complete n et work is a net work in which ev ery pair of pro cessors is connected b y a direct link. 3 binary tr e es . Finally , w e extend th ose lo w er b ounds to general trees b y demonstrating that our problem on general trees reduces to the same problem restricted to binary trees. The p ro of of our low er b ounds f or Λ < log n com b ines some ideas from Agarw al et al. [3] with n umerous new ideas. Sp eciﬁcally , Agarw al et al. reduce the problem fr om th e general family of span n ing subgraphs for ϑ n to a certain restricted f amily of stack gr aphs . This reduction of [3 ] p ro vides a v ery elegan t wa y f or ac h ieving somewhat wea k er b ound s, but it is inh eren tly sub optimal. In our p ro of we tac kle the general family of graphs directly . This more dir ect approac h results in a m uc h more tec hn ically in v olv ed pro of, and in m uc h more accurate b ounds. F or upp er b ounds we essen tially reduce the problem of constru cting LL Ts for general metric spaces to the same problem on ϑ n . Somewhat su rprisingly , despite the apparen t simplicit y of the metric sp ace ϑ n , the problem of constructing LL Ts for this space app ears to b e quite complex. 1.5 Related w ork SL Ts were extensive ly s tu died for the last t we nt y years [13, 36, 9, 10, 22, 23 , 24, 37, 5]. Ho wev er, all these constructions of SL T m a y result in trees w ith very large hop-diameter, and the tec hniques u sed in those constructions app ear to b e inapplicable to the problem of constructing LL Ts. Euclidean s p anners are also a su b ject of a r ecen t extensive and inte nsive researc h (see [6, 26, 3 , 7], and the referen ces therein). Ho w ev er, the basic tec hn ique for constructing them r elies h ea v ily on th e metho dology of well -separated p air decomp osition d ue to Callahan and Kosara ju [15 ]. This extremely p o we rfu l metho dology is, ho w ev er, applicable only f or the Eu clidean metric space of constan t d imension, while our constructions apply to general metric spaces. Tight lo w er b ounds on th e hop-diameter of Euclidean spann ers with a giv en num b er of edges were r ecen tly established b y C h an and Gu p ta [16]. Sp eciﬁcally , it is sho wn in [16] that for an y ǫ > 0 there exists an n -p oin t Euclidean metric space M = M ( n, ǫ ) for w hic h any E u clidean (1 + ǫ )-spanner with m edges has hop-diameter Ω( α ( m, n )), where α is the fun ctional in v erse of the Ac k ermann’s fu nction. Moreo v er, the m etric sp ace M is 1-dimensional. (On the other h and, th e space M is still not as restricted as ϑ n .) Ho wev er, this lo we r b ou n d pro vides no indication whatso ev er as to ho w light can b e E u clidean spanners with lo w hop-diameter. In particular, the construction of Arya et al. [6] that p ro vides matc hing up p er b ound s to the lo w er b oun ds of [16] pro du ces spanners that ma y hav e very large w eigh t. In terms of the tec h niques, Chan and Gu pta [16] start with s h o wing their low er b oun ds for m etrics induced b y binary hierarc hically-separated-trees (henceforth, HSTs ), and then translate them in to lo wer b ound s for metrics ind uced by n p oin ts on the real line using known r esu lts. Their pro of of the lo we r b ound for HSTs is an extension of Y ao’s p r o of tec hnique from [47]. As w as discussed ab o ve, our lo w er b ound s are ac h ieved by completely diﬀerent pro of tec h niques that in v olv e analyzing a linear program for the minimum linear arrangement pr oblem. In p articular, our lo wer b ounds are pro v ed d irectly for ϑ n . The study of spannin g trees of the 1-dimensional metric s p ace ϑ n is related to the extremely w ell- studied problem of computing partial-sums. (See the p ap ers of Y ao [47], Chazelle and Rosenberg [17], P˘ atra ¸ scu and Demaine [42], and the references therein.) F or a discussion ab out the relationship b et w een these t w o p roblems we refer to the in tro duction of [3]. The linear program for the minimum linear arran gement problem that w e use for our lo w er b ound s w as studied in [30, 44]. There is an extensiv e literature on the minim um lin ear arran gement p roblem itself [19, 32]. Finally , the extension of our construction of LL Ts to SL L Ts is ac hiev ed b y employing the construction of SL Ts of [10] on top of our construction of LL Ts. 4 1.6 The Structure of the Pa p er In Section 2 w e deﬁn e th e basic notions and presen t the notation that is u sed throughout the pap er. In Section 3 we show that the co ve ring and weig ht functions, deﬁned in Section 2, are monotone non- increasing with the d epth p arameter. This prop erty is emp loy ed in S ections 4 and 5 for proving lo w er b ound s. Section 4 is dev oted to lo wer b ound s. In Section 4.1 w e analyze trees that ha v e dep th h ≥ log n . In Section 4.2 w e turn to the complemen tary range h < log n . In Section 5 we use the lo w er b ounds for LL Ts pro v en in Section 4 to deriv e our lo wer b ound s on the tradeoﬀ b et ween the h op-diameter and w eigh t for Euclidean spann ers. Our upp er boun ds for LL Ts are presented in Section 6. In S ection 7 these upp er b ounds are emplo y ed for constructing SLL T s . Some basic prop erties of the binomial co eﬃcients that w e use in our analysis app ear in App endix A. 2 Preliminaries F or a p ositive in teger n , an n -p oint metric sp ac e M = ( V , dist M ) can b e viewe d as the complete graph G = G ( M ) = ( V ,  V 2  , dist M ) in which for ev ery pair of vertice s u, w ∈ V , th e weigh t of the edge e = ( u, w ) b et we en u and w in G is d eﬁned by ω ( u, w ) = dist M ( u, w ). The distance function dist M is required to b e non-negativ e, equal to zero w h en u = w , and to satisfy the triangle inequalit y ( dist M ( u, w ) ≤ dist M ( u, v ) + dist M ( v , w ), for ev ery triple u, w, v ∈ V of v ertices). A graph G ′ is called a sp anning sub gr aph (resp ectiv ely , sp anning tr e e ; minimum sp anning tr e e ) of M if it is a spanning sub graph (resp., spanning tree; minimum spannin g tr ee) of G ( M ). F or a we igh ted graph G = ( V , E , ω ), and a path P in G , its (weighte d) length is deﬁned as the sum of the we igh ts of edges along P , and its unweighte d length (or hop-length ) is the num b er | P | of edges (or hops ) in P . F or a pair of vertices u, w ∈ V , th e weighte d (resp ective ly , unweighte d) distanc e in G b etwe en u and w , denoted dist G ( u, w ) (resp., d G ( u, w )), is the smallest wei ght ed (resp ., unw eight ed) length of a path conn ecting b etw een u and w in G . The weighte d (resp ectiv ely , u nweighte d or hop- ) diameter of G is the maxim um w eigh ted (resp., unw eigh ted) distance b et wee n a pair of v ertices in V . Whenev er n can b e un dersto o d from the con text, we write ϑ as a shortcut for ϑ n . W e will u se the notion ϑ -tr e e as an abbreviation for a “rooted sp anning tree of ϑ ”. W e say that an edge ( v i , v j ) connecting a p aren t vertex v i with a child verte x v j in a ϑ -tree is a right (resp ectiv ely , left ) e dge if i > j (r esp., i < j ). In this case v j is called a right (resp., left ) child of v i . An edge ( v i , v j ) is said to c over a v ertex v ℓ if i < ℓ < j . F or a ϑ -tree T , the num b er of ed ges e ∈ E ( T ) that co ve r a vertex v of ϑ is called th e c overing of v by T and it is denoted χ ( v ) = χ T ( v ). The c overing of the tr e e T , χ ( T ), is th e maxim um co vering of a ve rtex v in ϑ by T , i.e., χ ( T ) = max { χ T ( v ) | v ∈ V ( ϑ ) } . F or a pair o f p ositiv e in tegers n and h , 1 ≤ h ≤ n − 1, denote b y χ ( h ) (respectiv ely , W ( n, h )) the minim um (v ertex) co v ering (resp., w eigh t) tak en o ver all ϑ n -trees of depth h . As w as sho wn in [3], the notions of co v ering and ligh tness are closely related. Finally , for a pair of non-negativ e in tegers k , n , k ≤ n , w e denote the sets { k , k + 1 , . . . , n } an d { 1 , 2 , . . . , n } b y [ k , n ] and [ n ], resp ective ly . 3 Monotonicit y of W eigh t and Co v ering In this section we restrict our atten tion to ϑ -trees and sho w that b oth the minimum co v ering and the minim um weig ht do not increase as the tree depth gro ws. T his prop erty is very useful for proving low er b ound s. 5 Fix a p ositiv e intege r n . In what follo ws we write χ ( h ) (resp ectiv ely , W ( h )) as a shortcut for χ ( n, h ) (resp., W ( n, h )). Lemma 3.1 The se quenc e ( χ (1) , χ (2) , . . . , χ ( n − 1)) is monotone non-incr e asing. Pro of: Observe that χ ( n − 1) = 0, and for h ≥ 1, χ ( h ) is non-negativ e. Consequ ently , we henceforth restrict our atten tion to the sub sequence ( χ (1) , χ (2) , . . . , χ ( n − 2)). Let T b e a ϑ -tree that has depth h , 1 ≤ h ≤ n − 2, and co ve ring χ = χ ( h ). (In other words, the tree T has the minim um co vering among all trees of depth equal to h .) W e denote its ro ot b y r t . W e constr u ct a tree S ( T ) that h as depth h + 1 and co ve ring at most χ . Consider a v ertex v at distance h fr om r t , and the path P = ( r t = v 0 , v 1 , . . . , v h − 1 , v ) b et wee n th em in T . 1. S ince h ≤ n − 2, ther e exists at least one leaf ℓ in T whic h is not in P . Remov e ℓ along with the edge connecting it to its parent in T . 2. Let ǫ , 0 < ǫ < 1, b e a small real v alue. Assume that v < v h − 1 (resp ectiv ely , v > v h − 1 ). Inser t a new v ertex v ′ , v ′ = v − ǫ (resp., v ′ = v + ǫ ) to b e the left (resp ., righ t) c hild of v . Denote the resulting tree by T ′ . Note that | V ( T ′ ) | = | V ( T ) \ { ℓ } ∪ { v ′ }| = n . C learly , the ﬁ rst step neither c hanges the depth h nor increases the co v ering χ . Since the d istance from r t to the farthest vertex v in T is h , adding v ′ as the left (resp., righ t) child of v in th e s econd step increases th e depth of the tree b y exactly one. Not e that since ǫ < 1, the new edge ( v , v ′ ) do es not co v er an y ve rtex in T ′ . Hence th e co vering of any v ertex v ∈ V ( T ′ ) \ { v ′ } in T ′ is no greater than its co v ering in T . T o conclude that the co vering of T ′ is at most χ , we sh o w that the co vering of the new v ertex v ′ in T ′ is no greate r than χ . In fact, we argu e that an y edge th at co v ers v ′ also co v ers v in T ′ , wh ic h p ro vides the required result. T o see this, note that for an edge e that co v ers v ′ not to co ver v , it must h old that e is inciden t to v . Ho wev er, since v is a leaf in T , the only ed ges whic h are incident to v in T ′ are ( v h − 1 , v ) and the new edge ( v , v ′ ), b oth of whic h do not co v er v ′ , and w e are done. (See Figure 2 for an illustration.) l ε S(T) T T’ 7 3 5 v=6 7 v=5 2 7 v=6 5 3 2 rt=4 2 v’=6 v’=6+ 1 =1 4 3 rt=4 Figure 2: The ϑ 7 -tree T ro oted at r t = 4 is d epi cted on the left. This tree has depth 1 and covering 2. The tree T ′ on the vertices { 2 , 3 , 4 , 5 , 6 , 6 + ǫ, 7 } ro oted at r t = 4 is depicted in the middle. It ha s depth 2 and covering 2. Thi s tree is obtained from T b y removing the vertex ℓ = 1 along with th e edge ( r t, ℓ ) , and adding the vertex v ′ = (6 + ǫ ) a long with t h e edge ( v , v ′ ) . The ϑ 7 -tree S ( T ) ro oted at 3 is depicted on t h e right. It has depth 2 and covering 2. This tree is obtained from T ′ b y relo cating the p oints 2 , 3 , 4 , 5 , 6 , 6 + ǫ, 7 to the p oints 1 , 2 , . . . , 7 , resp ectively . Observe that T ′ do es n ot sp an ϑ . Let v 1 < v 2 < . . . < v n b e the sequence of v ertices of T ′ in an increasing order. T o tran s form T ′ in to a spanning tree S ( T ) of ϑ , for eac h index i , 1 ≤ i ≤ n , relo cate v i to the p oint i . Let T ∗ b e a spanning tree th at has depth h + 1 and min im um co vering χ ( h + 1). By deﬁn ition, χ ( h + 1) is no greate r than the co vering of S ( T ), whic h is at most χ ( h ). Hence χ ( h + 1) ≤ χ ( h ), and we are done. 6 The f ollo win g statemen t is analogous to Lemm a 3.1. Its pro of is very sim ilar (and, in fact, simpler) than that of Lemma 3.1 and is therefore omitted. Lemma 3.2 The se quenc e ( W (1) , W (2) , . . . , W ( n − 1)) is monotone non-incr e asing. W e r emark that the monotonicit y prop er ties d eriv ed in this section apply to an y 1-dimens ional Euclidean space (rather than just to ϑ ). 4 Lo w er Bounds In this section w e devise lo wer b ounds for lightness of ϑ -trees for the ent ire range of parameters. In Section 4.1 w e analyze trees of depth h ≥ log n (h en ceforth high trees), and in Section 4.2 w e study trees with depth in the complemen tary range h < log n (henceforth low trees). 4.1 High T rees In this section w e devise lo we r b oun d s for high ϑ -trees. In Sections 4.1.1 and 4.1.2 w e restrict our atten tion to binary trees, reduce this restricted v ariant of the pr ob lem to a certain question concerning the minim um linear arrangemen t problem, and r esolve th e latter question. In Section 4.1.3 we sh o w that the lo w er b ound for binary trees extends to general high trees, and, in fact, to general spannin g subgraphs . 4.1.1 The Minimum Linear Arrangemen t Problem In this section we describ e a relationship b et we en the pr ob lem of constru cting LL Ts and th e minimum line ar arr angement (henceforth, MINLA ) problem [30, 44]. The MINLA problem is deﬁned as follo ws. Giv en an un d irected graph G = ( V , E ), we would like to ﬁnd a p ermutatio n (called also a line ar arr angement ) of the no des σ : V → { 1 , . . . , n = | V |} that minimizes the cost of the linear arrangemen t σ , LA ( G, σ ) = X ( i,j ) ∈ E | σ ( i ) − σ ( j ) | . The m inim um linear arrangement of the graph G , denoted M I N LA ( G ), is d eﬁned as the minim um cost of a linear arrangemen t, i.e., M I N L A ( G ) = m in { LA ( G, σ ) | σ ∈ S n } , where S n is the set of all p ermutati ons of [ n ]. Let G = ( V , E ) b e an n -vertex graph. F or a p ermuta tion σ ∈ S n , let G σ = ([ n ] , E σ ) d enote the graph with v ertex set [ n ] and edge set E σ = { ( σ ( u ) , σ ( w )) | ( u, w ) ∈ E } , equipp ed with the we igh t function w ( i, j ) = | i − j | for every i 6 = j , i, j ∈ [ n ]. ( G σ is an isomorp hic cop y of G .) Observe that LA ( G, σ ) = ω ( G σ ) = P e ∈ E σ ω ( e ). Also, let G ∗ = ([ n ] , E ∗ ) denote the graph G σ ∗ for the optimal p ermutati on σ ∗ = σ ( G ), th at is, for σ ∗ suc h that ω ( G σ ∗ ) = min { ω ( G σ ) | σ ∈ S n } . It follo ws that M I N L A ( G ) is equal to ω ( G σ ∗ ). Moreo v er, for a family F of n -v ertex graphs, the minimum weigh t ω ( F ) = { ω ( G σ ∗ ) | G ∈ F } is precisely equal to the m inim um v alue M I N LA ( F ) = { M I N LA ( G ) | G ∈ F } of the MINLA prob lem on one of the graphs of the f amily F . Next, w e s tu dy the family B n ( h ) of binary ϑ -trees of depth no greater than h , and sho w a lo w er b ound on the v alue B in ( n, h ) of the min im um w eigh t of a ϑ -tree from B n ( h ). Observe that B in ( n, h ) = M I N LA ( B n ( h )) . (1) 7 Hence, it is suﬃ cien t to p ro vide a lo wer b ound for the v alue M I N LA ( B n ( h )) of the MINLA for graphs of this family . In a seminal w ork on spreading metrics, Ev en et al. [30] studied the follo wing linear program relax- ation LP 1 for th e MINLA p roblem. The v ariables of this linear program { ℓ ( e ) | e ∈ E } can b e view ed as edge lengths. F or a pair of v ertices u and v , dist ℓ ( u, v ) stands for the distance b et w een u and v in the graph G equipp ed with length function ℓ ( · ) on its edges. LP 1 : min X e ∈ E ℓ ( e ) s.t. ∀ U ⊆ V , ∀ v ∈ U : X u ∈ U dist ℓ ( v , u ) ≥ 1 4 · ( | U | 2 − 1) ∀ e ∈ E : ℓ ( e ) ≥ 0 . It is well-kno wn that the optimal solution of this linear pr ogram is a lo w er b ound on M I N LA ( G ) [30, 44]. As w as already men tioned, w e are only interested in the MINLA problem for binary trees. Next, w e present a v arian t LP 2 of th e linear program LP 1 wh ic h inv olv es only a small subset of constrain ts that are used in LP 1. Consequent ly , the optimal solution of LP 2 is a low er b ound on the optimal solution of LP 1. Con s ider a r o oted tree T = ( T , r t ). F or a v ertex v in T , let U v b e the v ertex set of the subtree of T r o oted at v . While in LP 1 there is a constrain t for eac h pair ( U, v ), U ⊆ V , v ∈ U , there are only the constrain ts that corresp ond to p airs ( U v , v ) present in LP 2. LP 2 : min X e ∈ E ( T ) ℓ ( e ) s.t. ∀ v ∈ V : X u ∈ U v dist ℓ ( u, v ) ≥ 1 4 · ( | U v | 2 − 1) ∀ e ∈ E : ℓ ( e ) ≥ 0 . W e will henceforth use the sh ortcut dist ( u, v ) for dist ℓ ( u, v ). F or a vertex v in T , let T otalD ist ( v ) = P u ∈ U v dist ℓ ( v , u ), I n eq ( v ) b e the inequalit y T otalD ist ( v ) ≥ 1 4 ( | U v | 2 − 1), and E q ( v ) b e th e equation T otal D ist ( v ) = 1 4 ( | U v | 2 − 1). Next, we restrict our atten tion to binary trees. The next lemma shows that if all in equalities I neq ( v ) are replaced by equations E q ( v ), the v alue of the linear program LP 2 do es not c hange. Lemma 4.1 F or a binary ϑ -tr e e T , i n any optimal solution to LP 2 al l ine q u alities { I n eq ( v ) | v ∈ V } hold as e q ualities. Pro of: First, observ e that for a leaf z in T , T otal D ist ( z ) = X u ∈ U z dist ( u, z ) = 0 , implying that I neq ( z ) holds as equalit y . Let n denote the n umber of vertice s of T . Or der the ( n − 1) edges e 1 , e 2 , . . . , e n − 1 arbitrarily , and consider the subset C of all v alue assignmen ts ψ to the v ariables ℓ ( e 1 ) , ℓ ( e 2 ) , . . . , ℓ ( e n − 1 ), that constitute an o ptimal 8 solution to the linear program LP 2, and suc h that there exists a ve rtex v ∈ V ( T ) for whic h I neq ( v ) holds as a strict inequ alit y un d er ψ . Su pp ose for cont radiction that C 6 = φ . F or an assignment ψ ∈ C , let the level of ψ , denoted L ( ψ ), b e the minimum leve l of a v ertex v in T f or whic h I neq ( v ) holds as a strict inequalit y . Consider a optimal solution ψ ∗ ∈ C of minimum level, that is, L ( ψ ∗ ) = min { L ( ψ ) | ψ ∈ C } . Let v b e an inn er v ertex of lev el L ( ψ ∗ ) for which I neq ( v ) holds as a strict inequalit y under the assignmen t ψ ∗ . By deﬁnition, T otal D ist ( v ) > 1 4 ( | U v | 2 − 1) . It is con ve nient to imagine that th e ve rtices of T v are colo red in t w o colors as follo ws. The r o ot vertex v of T v is colored white. All lea ves are colored blac k. An inner v ertex u ∈ U v \ { v } is colored wh ite, if the follo wing three conditions hold. • I ts paren t π ( u ) in T v is colored white. • I neq ( u ) holds as a strict inequalit y . • All ed ges e connecting u to its c h ildren satisfy ℓ ( e ) = 0 under ψ ∗ . Otherwise, u is colored blac k. Remark: Ob serv e that for a w h ite v ertex u ∈ U v \ { v } , all v ertices of the path P v,π ( u ) connecting v with π ( u ) are colored white. Claim 4.2 At le ast one vertex of depth 1 in T v is c olor e d white. Pro of: Sup p ose for contradict ion that all wh ite vertice s in T v ha v e d epth at least 2. Let w b e a wh ite v ertex of minimum depth d , d ≥ 2. S ince w is colored white, all v ertices in the path P v,w connecting v with w in T v are colored w hite as well , implying that for eac h vertex x along that path, I neq ( x ) holds as a strict inequalit y , and all edges that connect x to its children ha v e w eigh t zero. Denote the left (resp ectiv ely , righ t) child of w by w L (resp., w R ). Since the w eigh t of the edges that connect w to its c h ildren ha v e weigh t zero, it holds that T otal D ist ( w ) = T otal D ist ( w L ) + T otal D ist ( w R ) . Since I neq ( w ) h olds as a s tr ict inequalit y , T otal D ist ( w ) > 1 4 · ( | U w | 2 − 1) = 1 4 · (( | U w L | + | U w R | + 1) 2 − 1) > 1 4 · ( | U w L | 2 − 1) + 1 4 · ( | U w R | 2 − 1) . Th us, at least one among the t w o in equalities I neq ( w L ) and I neq ( v R ) holds as a strict one. W e assu me without loss of generalit y that I n eq ( w L ) holds as a strict inequalit y . T o complete the pro of we need the follo wing claim. Claim 4.3 Al l e dges that c onne ct w L to its childr en have v alue zer o under ψ ∗ . 9 Pro of: Su pp ose for contradictio n that there is a c hild y of w L suc h that the length of the edge e = ( w L , y ) under the assignment ψ ∗ is s ome δ > 0. Consider the p ath P v,w = ( v = v 0 , v 1 , . . . , v j = w ), j ≥ 0, connecting the v ertices v and w . The an alysis splits in to tw o cases dep ending on wh ether v is the ro ot r t of T or not. First, supp ose that v = r t . Observe that for ev ery index i , i ∈ [0 , j ], f i ( δ ) = f i ( ℓ ( e )) = T ot alD is t ( v i ) − 1 4 · ( | U v i | 2 − 1) is a con tin uous fun ction of the v ariable ℓ ( e ). Since for every i ∈ [0 , j ], f i ( δ ) > 0, w e can sligh tly decrease the v alue of ℓ ( e ) and set it to some δ ′ , 0 < δ ′ < δ , so that all f i ( δ ′ ) are still non-negativ e. Ho w ev er, this c h an ge in the v alue of ℓ ( e ) results in a new feasible assignment ψ ′ of v alues to the v ariables { ℓ ( e ) | e ∈ E ( T ) } . Moreo ver, obviously P e ∈ E ( T ) ℓ ( e ) of the ob jectiv e function of LP 2 is smaller under ψ ′ than under ψ ∗ . This is a contradictio n to the assumption that ψ ∗ is a optimal solution for LP 2. The case that v 6 = r t is hand led similarly . In this case the diﬀerence ǫ = δ − δ ′ is added to the v alue of ℓ ( e ′ ), for the edge e ′ = ( v , π ( v )) connecting v to its paren t in T . It is easy to verify that the resu lting assignmen t ˜ ψ is f easible, and th at the v alue of the ob jectiv e fu nction P e ∈ E ( T ) ℓ ( e ) is the same un der ψ ∗ and ˜ ψ . Also, sin ce f or ev ery i ∈ [0 , j ], f i ( ℓ ( e )) = f i ( δ ′ ) > 0 under ˜ ψ , it follo ws that the inequalities I neq ( v i ) hold as strict inequalities for all i ∈ [0 , j ], and thus b oth assignments ˜ ψ and ψ ∗ b elong to the set C . Ho w ev er, I neq ( π ( v )) holds as a strict inequalit y und er ˜ ψ as we ll, and thus L ( ˜ ψ ) < L ( ψ ∗ ). This is a contradictio n to the assump tion that ψ ∗ has the minimum lev el in C . Hence under ψ ∗ , all edges that connect w L to its c hildren ha ve v alue zero. Recall that I neq ( w L ) holds as a strict inequalit y , and w = π ( w L ) is a white v ertex. C onsequen tly , w L should b e colored white as w ell. Ho w ev er, its depth is smaller than the minimum depth of a white ve rtex in T v , con tradiction. This completes the pro of of Claim 4.2. Consider a white verte x x in T v of depth 1. The edges connecting x to its c hildren are assigned v alue zero, imp lying that T otal D ist ( x ) = 0. Ho wev er, since x is colored wh ite, the inequalit y I neq ( x ) holds as a strict inequalit y , i.e., T otal D ist ( x ) > 1 4 · ( | U x | 2 − 1) > 0 . This is a con tradiction to the assump tion th at C is not empt y , proving Lemma 4.1. Consider a subtree T v ro oted at an inner v ertex v . Without loss of generalit y , v has a left c hild v L , and p ossibly a righ t child v R , eac h b eing the r o ot of the corresp onding subtrees T v L and T v R , resp ectiv ely . Let U L = U v L , U R = U v R , n L = | U L | , n R = | U R | , e L = ( v , v L ), e R = ( v , v R ), x L = ℓ ( e L ), an d x R = ℓ ( e R ). If v has only a left c hild, then we wr ite v R = T v R = U v R = N U LL , and n R = x R = 0. Also, without loss of generalit y assume that n L ≥ n R . The next lemma provides a lo w er b ound on the sum x L + x R of v alues assigned by a minimal optimal solution for LP 2 to the edges e L and e R . Lemma 4.4 F or an optima l solution for LP 2 , x L + x R > 1 2 · (min { n L , n R } + 1) = 1 2 · ( n R + 1) . Pro of: It is easy to v erify that X u ∈ U v dist ( v , u ) = x L · n L + X u ∈ U L dist ( v L , u ) + x R · n R + X u ∈ U R dist ( v R , u ) . (2) 10 By Lemma 4.1, b oth inequalities I neq ( v ) and I neq ( v L ) hold as equalities, i.e, X u ∈ U v dist ( v , u ) = 1 4 · ( | U v | 2 − 1) = 1 4 · (( n L + n R + 1) 2 − 1) . (3) X u ∈ U L dist ( v L , u ) = 1 4 · ( n L 2 − 1) . (4) The analysis splits in to tw o cases. Case 1: v has two childr en. By Lemma 4.1, the inequalit y I n eq ( v R ) holds as equalit y as we ll, i.e., X u ∈ U R dist ( v R , u ) = 1 4 · ( n R 2 − 1) . (5) Plugging the equations (3), (4), and (5) in equation (2) implies x L · n L + x R · n R = 1 2 · ( n L · n R + ( n L + n R ) + 1) . (6) Since n L ≥ n R , and n L > 0, it follo ws that x L + x R · n R n L > 1 2 · ( n R + 1) , and so, x L + x R ≥ x L + x R · n R n L > 1 2 · ( n R + 1) = 1 2 · (min { n L , n R } + 1) . Case 2: v has only a left child. Then n R = x R = 0, and X u ∈ U v dist ( v , u ) = x L · n L + X u ∈ U L dist ( v L , u ) . (7) Plugging equations (3) and (4) in equation (7), we obtain x L · n L = 1 4 · (2 · n L + 1) . Hence x L + x R = x L > 1 2 = 1 2 · (min { n L , n R } + 1) . 4.1.2 The C ost F unction In this section we deﬁn e and analyze a cost function on b inary ϑ -trees. W e will sh o w that in ord er to pro vide a lo wer b ound for M I N LA ( B n ( h )), it is suﬃcien t to pr o vide a lo wer b oun d for th e minimum v alue of this cost function on a tree from B n ( h ). Consider a binary ϑ -tree ( T , rt ) in which for ev ery inner v ertex v that has tw o children, one of those c hildren is designated as the left child v .l ef t and the other as the right one v .r ig ht . If v h as ju st one c hild th en this child is designated as the left one. Also, for an inner vertex u , let | u | denote the num b er 11 of v ertices in the subtree of T r o oted at u . Let I = I ( T ) denote the set of inner vertice s of T . By Lemma 4.4, for an y optimal assignment ψ for the v alues { ℓ ( e ) | e ∈ E ( T ) } of the lin ear p rogram LP 2, X e ∈ E ( T ) ℓ ( e ) = X v ∈ I ( T ) ( ℓ ( v , v .lef t ) + ℓ ( v , v .r ig ht )) ≥ 1 2 · X v ∈ I ( T ) (min {| v.l ef t | , | v .r ig ht |} + 1) . (8) W e call the r igh t-hand side expression the c ost of the tr ee T , and denote it C ost ( T ). L et M I N C OS T ( B n ( h )) denote min { C ost ( T ) | T ∈ B n ( h ) } . It f ollo ws th at M I N LA ( B n ( h )) ≥ M I N C OS T ( B n ( h )), and in the sequel w e pr ovide a lo wer b oun d for M I N C O S T ( B n ( h )). Note that by (1), this lo wer b oun d will apply to B in ( n, h ) as wel l. The s ubtree r o oted at the left (resp ectiv ely , right) child of T is called the left subtr e e (resp., right subtr e e ) of T . W e w ill use the notation T .lef t and r t.l ef t (respective ly , T .r ig ht and r t.r ig ht ) in terc hange- ably to denote the left (resp., right ) sub tree of T . Also, let | T | denote the size of the tree T , that is, the n umb er of ve rtices in T . Consider the follo w in g cost function on bin ary trees, C ost ′ ( T ) = C ost ′ ( T .l ef t ) + C ost ′ ( T .r ight ) + min {| T .lef t | , | T .r ight |} . It is easy to v erify that C ost ( T ) can b e equiv alen tly expressed as C ost ′ ( T ) = X v ∈ I ( T ) min {| v .l ef t | , | v .r ig ht |} . Since for an y binary tree T , 2 · C ost ( T ) ≥ C ost ′ ( T ), w e will henceforth fo cus on pro ving a low er b oun d for C ost ′ ( T ), and use the notion “cost” to refer to the function C ost ′ . Fix a pair of p ositiv e int egers n and h , n − 1 ≥ h . A ro oted binary tree on n vertice s that has dep th at most h will b e calle d an (n,h)-tr e e . Let R ( n, h ) denote the minim um cost tak en o ver all ( n, h )-trees. It follo ws that B in ( n, h ) ≥ 1 2 · R ( n, h ) . (9) This section is dev oted to pro ving the follo wing th eorem that establishes lo w er b oun ds on R ( n, h ), for all h ≥ log n . Theorem 4.5 1. If log n ≤ h ≤ 2 ⌊ log n ⌋ , then R ( n, h ) ≥ 2 3 · n · ⌊ 1 8 log n ⌋ . 2. If 2 ⌊ log n ⌋ < h ≤ n − 1 , let f ( h ) b e the minimum inte ger such that  h +1 f ( h )  > 2 3 · n . Then R ( n , h ) > 2 3 · n · ( f ( h ) − 2) . Remark 1: Note that for h > 2 ⌊ lo g n ⌋ ,  h +1 log n  > 2 3 · n , and th us f ( h ) is wel l-deﬁned in this range. Remark 2: By (9), the lo we r b ound s of Theorem 4.5 apply (up to a factor of 2) to B in ( n, h ) as well. Let n and h b e n on-negativ e int egers. Giv en a binary tree T , we restructur e it without changing its c ost and depth, so that for eac h vertex v in T , th e size of its righ t sub tree v .r ig ht w ould not exceed the size of its left su btree v .l ef t . Sp eciﬁcally , if in the original tree T it h olds that | v .l ef t | ≥ | v .r ig ht | , then no adjus tment o ccurs in v . Ho we v er, if | v .l ef t | < | v .r ig ht | , then the restructuring pro cess exc h an ges b et we en the left and righ t su btrees of v . W e refer to this restru cturing pro cedure as the right-adjustment of T , and denote the resulting binary tr ee by ˜ T . (See Figure 3 for an illustration.) Since T and its right-adjuste d tree ˜ T h a v e the same cost, we h enceforth restrict our atten tion to right -adjusted trees. By deﬁnition, in a r igh t-adjusted tree ˜ T , for any v ∈ V ( ˜ T ), it h olds that | v.r ig ht | ≤ | v .lef t | , and consequently , C ost ( ˜ T ) = X v ∈ V ( ˜ T ) | v .r ig ht | . 12 1 011 101 001 01 10 1001 0011 0001 0 00 000 0000 Figure 3: A cost-optimal binary tree for n = 20 and h = 4 . A set of n binary words w ith at most h bits eac h will b e called an (n,h)-vo c abulary . Next, w e deﬁne an injection S from the set of ( n, h )-tree s to the set of ( n, h )-v o cabularies. F or a vertex v in a b inary tree T , denote by P v = ( r t = v 0 , v 1 , . . . , v k = v ) the path from r t to v in T , and deﬁne B v = b 1 b 2 . . . b k to b e its corresp onding binary w ord, where for i ∈ { 1 , 2 , . . . , k } , b i = 0 if v i is the left c hild of v i − 1 , and b i = 1 otherwise. Giv en an ( n , h )-t ree T , let S ( T ) b e the ( n, h )-v o cabulary that consist of the | T | binary w ords th at corresp ond to the set of all ro ot-to-v ertex paths in T , namely , S ( T ) = { B v | v ∈ V ( T ) } . (See Figure 3 for an illustration.) F or a bin ary word W , denote its Hamming weight (the num b er of 1’s in it) b y H ( W ). F or a set S of bin ary w ords, deﬁne its total Hamming weigh t, henceforth Hamming c ost , H C ost ( S ), as the su m of Hamming w eigh ts of all w ords in S , n amely , H C ost ( S ) = X B ∈ S H ( B ). Finally , denote the minim um Hamming cost of a set of n distinct binary words with at most h bits eac h by H ( n, h ). Observe that the function H ( n, h ) is monotone non-increasing with h . In the next lemma w e show that it is suﬃcient to pr o v e the desired lo w er b ound for H ( n, h ). Lemma 4.6 F or al l p ositive inte gers n and h , n − 1 ≥ h , H ( n, h ) ≤ R ( n, h ) . Pro of: It is easy to v erify by a doub le counting that in a righ t-adjusted tree ˜ T , X v ∈ V ( ˜ T ) | v .r ig ht | = X v ∈ V ( ˜ T ) |{ u ∈ V ( P v ) : v ∈ u.r ig ht }| . Consequent ly , C ost ( ˜ T ) = X v ∈ V ( ˜ T ) | v .r ig ht | = X v ∈ V ( ˜ T ) |{ u ∈ V ( P v ) : v ∈ u.r ig ht }| = X v ∈ V ( ˜ T ) H ( B v ) = H C ost ( S ( ˜ T )) . 13 Let T ∗ b e a righ t-adjusted ( n, h )-tree realizing R ( n, h ), that is, C ost ( T ∗ ) = R ( n, h ). It follo ws that R ( n, h ) = C ost ( T ∗ ) = H C ost ( S ( T ∗ )) ≥ H ( n, h ) . In what follo ws we establish lo w er b ounds on H ( n, h ). Consider a set S ∗ = S ∗ ( n, h ) realizing H ( n, h ), that is, a set that satisﬁes H C ost ( S ∗ ) = H ( n, h ). F or a n on-negativ e in teger i , i ≤ h , let S ( h, i ) b e the set of all distinct binary words with at most h bits eac h, so that eac h word of whic h con tains pr ecisely i 1’s. T o con tain the minimum total n um b er of 1’s, the set S ∗ needs to contai n all bin ary wo rds with no 1’s, all b inary wo rds that con tain just a single 1, etc. In other w ords, there exists an integ er r = r ( h ) for whic h r [ i =0 S ( h, i ) ⊂ S ∗ ⊆ r +1 [ i =0 S ( h, i ) . (See Figure 3 for an illustration.) By F act A.1, |S ( h, i ) | = h X k = i  k i  =  h + 1 i + 1  . Note that for a pair of distinct in dices i and j , 0 ≤ i, j ≤ h , the sets S ( h, i ) and S ( h, j ) are disjoin t. S ince |S ∗ | = n , it holds that r X i =0  h + 1 i + 1  < n ≤ r +1 X i =0  h + 1 i + 1  . (10) Recall th at for ev ery non-negativ e int eger i , i ≤ h , eac h w ord in S ( h, i ) con tains pr ecisely i 1’s, and thus H C ost ( S ( h, i )) = i ·  h + 1 i + 1  . Let N = n − r X i =0  h + 1 i + 1  > 0 (11) b e the n umber of wo rds with Hamming we ight r + 1 in S ∗ . Hence H ( n, h ) = H C ost ( S ∗ ) = r X i =0 H C ost ( S ( h, i )) + N · ( r + 1) = r X i =0 i ·  h + 1 i + 1  + N · ( r + 1) . (12) The next claim establishes a helpful relationship b et wee n th e parameters h , r , and n . Claim 4.7 F or h ≥ ⌊ 2 log n ⌋ , r ≤  h +1 4  − 1 . Pro of: Supp ose for con tradiction that r ≥ ⌊ h +1 4 ⌋ . Then r + 1 ≥  ⌊ 2 log n ⌋ + 1 4  + 1 . 14 Hence n ≤  ⌊ 2 log n ⌋ + 1 j ⌊ 2 l og n ⌋ +1 4 k + 1  ≤  h + 1 r + 1  < r X i =0  h + 1 i + 1  . Ho w ev er, b y (10) the righ t-hand side is smaller than n , cont radiction. The next lemma establishes lo w er b ound s on H ( n , h ) f or h ≥ ⌊ 2 log n ⌋ . Since H ( n, h ) is monotone non-increasing with h , it follo ws that the lo w er b oun d for h = 2 ⌊ log n ⌋ ap p lies for all smaller v alues of h . Lemma 4.8 1. H ( n, 2 ⌊ lo g n ⌋ ) ≥ 2 3 · n · ⌊ 1 8 log n ⌋ . 2. F or any 2 ⌊ log n ⌋ < h ≤ n − 1 , H ( n, h ) > 2 3 · n · ( f ( h ) − 2) . Pro of: By Lemma A.3, r X i =0  h + 1 i + 1  < 3 2 ·  h + 1 r + 1  . Since r X i =0  h + 1 i + 1  = n − N , it follo w s that  h + 1 r + 1  > 2 3 · ( n − N ) , and so,  h + 1 r + 1  + N > 2 3 · n. Hence, b y equation (12), H ( n, h ) = r X i =0 i ·  h + 1 i + 1  + N · ( r + 1) > r ·  h + 1 r + 1  + N · ( r + 1) > 2 3 · n · r. (13) The analysis splits in to tw o cases. Case 1: h = ⌊ 2 log n ⌋ . W e will pr o ve the ﬁrst assertion of Lemma 4.8, that is, H ( n, ⌊ 2 log n ⌋ ) ≥ 2 3 · n · ⌊ 1 8 log n ⌋ . W e assum e that n ≥ 256, as otherwise the r igh t-hand side v anishes and the statemen t h olds trivially . The next claim sho ws that in this case r is quite large. Claim 4.9 r ≥  1 8 log n  . Pro of: Supp ose for contradictio n that r ≤  1 8 log n  − 1. Ob s erv e that for n ≥ 3,  1 8 log n  +1 ≤ j ⌊ 2 log n ⌋ +1 4 k , and th us Lemma A.3 is applicable. Hence, ⌊ 1 8 log n ⌋ +1 X i =0  ⌊ 2 log n ⌋ + 1 i  < 3 2 ·  ⌊ 2 log n ⌋ + 1 ⌊ 1 8 log n ⌋ + 1  . (14) 15 By (10), n ≤ r +1 X i =0  h + 1 i + 1  = r +1 X i =0  ⌊ 2 log n ⌋ + 1 i + 1  ≤ ⌊ 1 8 log n ⌋ X i =0  ⌊ 2 log n ⌋ + 1 i + 1  = ⌊ 1 8 log n ⌋ +1 X i =0  ⌊ 2 log n ⌋ + 1 i  − 1 . Ho w ev er, (14 ) implies that the righ t-hand side is strictly smaller than 3 2 ·  ⌊ 2 log n ⌋ +1 ⌊ 1 8 log n ⌋ +1  − 1 ≤ n, con tra- diction. Consequent ly , by (13), w e ha v e H ( n, ⌊ 2 log n ⌋ ) > 2 3 · n · r ≥ 2 3 · n · ⌊ 1 8 log n ⌋ . This completes the pro of of the ﬁrst assertion of Lemma 4.8. T o prov e the second assertion we analyze the case h > ⌊ 2 log n ⌋ . Case 2: h > ⌊ 2 log n ⌋ . W e start th e an alysis of this case by s h o wing that for h in this range, the upp er b ound on the v alue of r established in Claim 4.7 can b e impro v ed. Claim 4.10 r ≤  h +1 4  − 2 . Pro of: It is easy to verify that n ≤  ⌊ 2 log n ⌋ +2 j ⌊ 2 log n ⌋ +2 4 k  . Sup p ose for contradict ion that r ≥  h +1 4  − 1. Then r + 1 ≥ j ⌊ 2 log n ⌋ +2 4 k . Hence n ≤  ⌊ 2 log n ⌋ + 2 j ⌊ 2 log n ⌋ +2 4 k  ≤  h + 1 r + 1  < r X i =0  h + 1 i + 1  . Ho w ev er, b y (10), the righ t-hand side is smaller than n , con tradiction. T o complete the pro of, we pr ovide a lo we r b ound on r . Recall that f ( h ) is deﬁned to b e the minimum int eger th at satisﬁes  h +1 f ( h )  > 2 3 · n . Claim 4.11  h +1 r +2  > 2 3 · n . By Claim 4.10, r ≤ ⌊ h +1 4 ⌋ − 2. Hence, Lemma A.3 im p lies that r +1 X i =0  h + 1 i + 1  < 3 2 ·  h + 1 r + 2  . Consequent ly , by (10), n ≤ r +1 X i =0  h + 1 i + 1  < 3 2 ·  h + 1 r + 2  . The assertion of the claim follo ws. Claim 4.11 implies that r ≥ f ( h ) − 2. By equation (13), it follo ws th at H ( n, h ) > 2 3 · n · r ≥ 2 3 · n · ( f ( h ) − 2) , 16 implying the second assertion of Lemma 4.8. Lemmas 4.6 and 4.8 imp ly Theorem 4.5. W e are n o w ready to deriv e the d esired lo wer b ound for binary trees. Theorem 4.12 F or suﬃci e ntly lar ge inte ge rs n and h , h ≥ log n , the minimum weight B in ( n, h ) of a binary ϑ -tr e e that has depth at most h is at le ast Ω(Ψ · n ) , for some Ψ satisfying h = Ω(Ψ · n 1 / Ψ ) . Pro of: First, observ e that B in ( n, h ) ≥ 1 2 · R ( n, h ). The analysis sp lits in to three cases, dep end ing on th e v alue of h . Case 1: log n ≤ h ≤ 2 ⌊ log n ⌋ . By the ﬁrst assertion of Theorem 4.5, B in ( n, h ) ≥ 1 2 · R ( n, h ) = Ω(log n · n ) . Observe that for h ∈ [log n, 2 ⌊ log n ⌋ ], and k = log n , it holds that h = Ω  k · n 1 /k  , and w e are done. Case 2: 2 ⌊ log n ⌋ ≤ h ≤ n 1 / 4 . By the s econd assertion of Theorem 4.5, B in ( n, h ) ≥ 1 2 · R ( n, h ) ≥ 1 3 · n · ( f ( h ) − 2) , where f ( h ) is the m in im um int eger that satisﬁes  h +1 f ( h )  > 2 3 · n . Obs erv e that for a suﬃcient ly large n , and h ∈ [2 ⌊ log n ⌋ , n 1 / 4 ], we ha ve that f ( h ) ≥ 4, and so, f ( h ) − 2 ≥ 1 2 · f ( h ). It follo w s that B in ( n, h ) = Ω( f ( h ) · n ) . Also, it holds that  e · ( h + 1) f ( h )  f ( h ) >  h + 1 f ( h )  > 2 3 · n. Consequent ly , h > 1 e · f ( h ) ·  2 3 · n  1 f ( h ) ! − 1 = Ω  f ( h ) · n 1 f ( h )  . Case 3: h > n 1 / 4 . In this range an y constan t k ≥ 4 satisﬁes h = Ω( k · n 1 /k ). Note th at the w eigh t of the minim um spann ing tree of ϑ is n − 1, which implies that for a constan t k , B in ( n, h ) ≥ n − 1 = Ω ( k · n ) . 4.1.3 Lo w er Bounds for General High T rees In this s ection w e sho w that our lo wer b ound for binary trees implies an analogous lo wer b ound f or g eneral high tr ees. W e do this in tw o stages. First, w e sh ow that a lo w er b oun d for trees in which ev ery v ertex has at most four c hildr en, henceforth 4 -ary tr e es , will su ﬃce. Second, w e s h o w that the lo w er b ound for binary trees implies the desired lo we r b ound for 4-ary trees. F or an inn er no de v in a tr ee T , we denote its children b y c 1 ( v ) , c 2 ( v ) , . . . , c ch ( v ) ( v ), wh er e ch ( v ) denotes the n umber of its c hildren in T . Supp ose without loss of generalit y that th e c h ild ren are ordered so that the sizes of their corresp ond ing sub trees form a monotone non-increasing sequence, i.e., | T c 1 ( v ) | ≥ | T c 2 ( v ) | ≥ . . . ≥ | T c ch ( v ) ( v ) | . F or a vertex v in T , we d eﬁ ne its star sub gr aph S v as th e sub graph of T connecting v to its c hildren in T , namely , S v = ( V v , E v ), where V v = { v , c 1 ( v ) , . . . , c ch ( v ) ( v ) } and E v = { ( v , c i ( v )) | i = 1 , 2 , . . . , ch ( v ) } . F or con v enience, we denote T i = T c i ( rt ) , for i = 1 , 2 , . . . , ch ( v ). 17 The Pro cedure F ul l accepts as input a star sub graph S v of a tree T an d transforms it in to a full binary tree B v ro oted at v that has d epth ⌊ log ( ch ( v ) + 1) ⌋ , su c h th at c 1 ( v ) , c 2 ( v ) , . . . , c ch ( v ) are arranged in th e resulting tree by increasing order of lev el. (See Figure 4 for an illustr ation.) Sp eciﬁcally , c 1 ( v ) and c 2 ( v ) b ecome the left and r ight c hildren of v in B v , resp ectiv ely . More generally , for eac h ind ex i = 1 , 2 , . . . , ⌊ ch ( v ) − 1 2 ⌋ , c 2 i +1 ( v ) and c 2 i +2 ( v ) become the left and right c hildr en of the v ertex c i ( v ), resp ectiv ely . B S (v) v c (v) c (v) c (v) c (v) c c (v) v (v) c v v 1 4 2 c c (v) (v) c (v) 1 2 3 4 3 5 5 Figure 4: The tree S v on the left is the star subgraph ro oted at v , and having the ﬁve vertices c 1 ( v ) , c 2 ( v ) , . . . , c 5 ( v ) as its l eaves. The full binary tree B v on the right is obtained a s a result of the invo cation of the p ro cedure F ul l on S v . The P ro cedure 4 E xtension accepts as inp u t a tree T and transforms it in to a 4-ary tree sp an n ing the original set of vertices. Basically , the p ro cedure in v ok es the Pro cedu re F ull on every star of the original tree T . More sp eciﬁcally , if the tree T contai ns only one no de, then the Pro cedur e 4 E xtenstion lea v es the tree in tact. Oth erwise, it is inv ok ed recursiv ely on eac h of th e sub trees T 1 , T 2 , . . . , T ch ( r t ) of T . A t this p oint the Pro cedure F ull is in v ok ed with the parameter S r t and transform s the star subgraph S r t of the ro ot in to a full binary tree B r t as describ ed ab o v e. (See Figure 5 for an illustration.) T T’ c (rt) a= (rt) c d= e= f= g= h= i= j= k= l= m= n= p= o= c (j) c (j) rt rt b= b d e j k h c m n f g i o l p a c (c) c (c) c (b) c (b) c (a) c (b) c (a) c (a) c (a) c (rt) c= c (a) c (a) 1 2 3 1 2 3 4 6 1 2 3 1 2 2 1 5 Figure 5: A spanning tree T of { r t, a, b, . . . , p } ro oted at the vertex r t i s depicted on the left. The 4-ary spanning tree T ′ of { r t, a, b, . . . , p } ro oted a t r t is depicted on the right. The t ree T ′ is obtained a s a result of the invo cation of the p ro cedure 4 E xtension on T . 18 It is easy to v erify that for an y tree T , the tree T ′ obtained as a resu lt of the in vocation of the Pro cedure 4 E xtension on the tree T h as the same vertex s et. Moreo v er, in the follo wing lemma w e show that no v ertex in T ′ has more than four c hildren. Lemma 4.13 T ′ is a 4-ary tr e e suc h that its r o ot r t has at most two childr en. Pro of: The v ertices c 1 ( r t ) and c 2 ( r t ) are the only c hildren of r t in T ′ . F or a vertex v ∈ V , v 6 = r t , let u denote the parent of v in the original tree T . Let i b e the in d ex such that v = c i ( u ). If i ∈ { 1 , 2 } , then u remains the parent of v in T ′ . Otherwise, c ⌊ ( i − 1) / 2 ⌋ ( u ) b ecomes the p aren t of v in T ′ . Moreo ve r, v will ha v e at most four c h ildren in T ′ , sp eciﬁcall y , c 2 i +1 ( u ), c 2 i +2 ( u ), c 1 ( v ) and c 2 ( v ). If v is a leaf in T then it will hav e at most t w o children in T ′ , c 2 i +1 ( u ) and c 2 i +2 ( u ). If i > j ch ( u ) − 1 2 k , then v ma y ha v e only t wo c hildren in T ′ , c 1 ( v ) and c 2 ( v ). In this case if v is a leaf in T th en it is a leaf in T ′ as w ell. Remark: Observe that this pr o cedure nev er increases v ertex degrees, and thus for a binary (resp ec- tiv ely , ternary) tree T , th e r esulting tree T ′ is binary (resp., ternary) as we ll. In the next lemma w e show that the heigh t of T ′ is not m uc h greater than that of T . Lemma 4.14 h ( T ′ ) ≤ h ( T ) + log | T | . Pro of: The pro of is b y induction on h = h ( T ). Basse: h = 0 . The claim holds v acuously in this case, since T ′ = T . Induction Step: W e assume the correctness of the claim for all trees of depth at most h − 1, and p ro v e it for trees of depth h . Consider a tree T of d epth h . By th e induction h yp othesis, for all indices i, 1 ≤ i ≤ ch ( r t ), h ( T ′ i ) ≤ h ( T i ) + log | T i | . (15) Note that the ro ot c i ( r t ) of T ′ i has depth ⌊ log( i + 1) ⌋ in the tree T ′ . Hence the depth h ( T ′ ) of T ′ is giv en b y h ( T ′ ) = max { h ( T ′ i ) + ⌊ log ( i + 1) ⌋ : i ∈ { 1 , 2 , . . . , ch ( r t ) }} . (16) Let t b e an index in { 1 , 2 , . . . , ch ( r t ) } realizing the maximum, i.e., h ( T ′ ) = h ( T ′ t ) + ⌊ log( t + 1) ⌋ . By equations (15) and (16), h ( T ′ ) ≤ h ( T t ) + log | T t | + ⌊ log ( t + 1) ⌋ . Note also that h ( T t ) ≤ h − 1, and ⌊ log ( t + 1) ⌋ ≤ ⌊ log t ⌋ + 1 . Hence, it holds that h ( T ′ ) ≤ h + log | T t | + ⌊ log t ⌋ . Recall that the c hildren of r t are ordered su ch that | T 1 | ≥ | T 2 | ≥ . . . ≥ | T ch ( r t ) | . Hence, t · | T t | ≤ t X i =1 | T i | ≤ | T | . Consequent ly , log | T t | + ⌊ log t ⌋ ≤ log | T | , and we are done. In the follo wing lemma w e argue that the weig ht of the resulting tree T ′ is not m uc h greater than the w eigh t of the original tr ee T . Lemma 4.15 ω ( T ′ ) ≤ 3 · ω ( T ) . 19 Pro of: F or eac h vertex v in T , its s tar subgraph S v = ( V v , E v ) is r ep laced by a full b inary tree F v = F ul l ( S v ) = ( V v , E ′ v ). The w eigh t of S v is giv en by ω ( S v ) = ch ( v ) X i =1 ω ( v , c i ( v )) . Next, w e sho w that the weig ht of F v is at most three times greate r th an the w eigh t of S v . T h is will imply the statemen t of the lemma. By the triangle inequalit y , for eac h edge e = ( x, y ) in E ′ v , w e ha v e ω ( e ) ≤ ω ( v , x ) + ω ( v , y ) . Therefore, ω ( F v ) = X e ∈ E ′ v ω ( e ) ≤ X e =( x,y ) ∈ E ′ v ( ω ( v , x ) + ω ( v, y )) . Denote the degree of a ve rtex z in F v = ( V , E ′ v ) by deg ( z , F v ). It is easy to v erify b y double counting that X e =( x,y ) ∈ E ′ v ( ω ( v , x ) + ω ( v, y )) = X u ∈ V v deg ( u, F v ) · ω ( v , u ) . Since F v is a binary tree, for eac h v ertex z in F v , deg ( z , F v ) ≤ 3. Hence, ω ( F v ) ≤ 3 · X u ∈ V v ω ( v , u ) = 3 · ch ( v ) X i =1 ω ( v , c i ( v )) + 3 · ω ( v , v ) = 3 · ω ( S v ) . The next corollary summarizes the prop erties of the Pro cedur e 4 E xtension . Corollary 4.16 F or a ϑ -tr e e T = ( V , E ) , the Pr o c e dur e 4 E xtension ( T ) c onstructs a 4 -ary ϑ -tr e e T ′ = ( V , E ′ ) such that h ( T ) ≤ h ( T ′ ) ≤ h ( T ) + log | T | and ω ( T ′ ) ≤ 3 · ω ( T ) . Next, w e demons tr ate that a 4-ary tree can b e transformed to a binary tree, while incr easing the depth and we igh t only b y constan t factors. W e did n ot mak e an y sp ecial eﬀort to minimize these constant factors. The Pro cedu re P ath accepts as input a star sub grap h S v of a tree T and trans f orms it into a path P v = ( c 0 ( v ) = v , c 1 ( v ) , . . . , c ch ( v ) ( v )). (See Figure 6 for an illustration.) The Pro cedure B inE xtension accepts as input a 4-ary tree T ′ = ( V , E ′ ) ro oted at a giv en v ertex r t ∈ V and transforms it in to a bin ary tree spanning the original set of vertic es. If the tree T ′ con tains only one no de, then the P ro cedure B inE xtenstion lea ve s th e tree intact. Otherwise, for ev ery inn er v ertex v ∈ V , the pro cedu re replaces the s tar S v = ( v , c 1 ( v ) , . . . , c ch ( v )) b y the path P v = P ath ( S v ). (See Figure 7 for an illustration). In the next lemma w e argue that T ′′ := B inE xtension ( T ′ ) is a binary tree. Lemma 4.17 T ′′ is a binary tr e e such that its r o ot r t has at most one child. Remark: Th e notation c i ( v ) refers to the i th c h ild of v in the tree T ′ that is p r o vided to the Pro cedure B inE xt ension as input . Pro of: The v ertex c 1 ( r t ) is the only c hild of r t in T ′ . F or a v er tex v ∈ V , v 6 = r t , let u denote the paren t of v in the tree T ′ . L et i b e the index su c h that v = c i ( u ). If i = 1 then u is the p aren t of v in T ′′ . 20 P S v 1 (v) c (v) c c 2 (v) v c (v) c (v) c (v) 1 2 3 v v 3 Figure 6: Th e tree S v on the left is the star subgraph ro oted at v and having the three vertices c 1 ( v ) , c 2 ( v ) , c 3 ( v ) a s it s leaves. The path P v on the right is obtained as a result of the in v o cation of the p ro cedure P ath on S v . T’ T’’ c (rt) a= (rt) c b= h= i= l= k= c (h) c (h) rt rt b d e f g h i k l c j a 1 2 c (rt) c= g= f= e= d= c (a) c (a) c (a) c (a) c (b) c (b) j= c (c) 3 1 2 3 4 1 1 1 2 2 Figure 7: A 4-ary spanning tree T ′ of { r t, a, b, . . . , l } ro oted at the vertex r t is depicted on the left. The bina ry spanning tree T ′′ of { rt, a, b, . . . , l } ro oted at r t is depicted on the right. The tree T ′′ is obta ined as a result of the invo cation of the p ro cedure B inE xtension on T ′ . 21 Otherwise, c i − 1 ( u ) is the paren t of v in T ′′ . Moreo ver, if i < ch ( u ) then v ma y ha v e t w o c hildr en in T ′′ , sp eciﬁcally , c i +1 ( u ) and c 1 ( v ). If v is a leaf in T ′ then it will ha ve only one c hild, c i +1 ( u ). If i = ch ( u ) then v ma y hav e at most one c hild in T ′′ , sp eciﬁcally , c 1 ( v ). In this case if v is a leaf in T ′ then it is a leaf in T ′′ as w ell. The next t wo statemen ts imply that b oth the depth and th e w eigh t of T ′ and T ′′ are the same, u p to a constan t factor. Lemma 4.18 h ( T ′ ) ≤ h ( T ′′ ) ≤ 4 · h ( T ′ ) . Pro of: F or eac h v ertex v in T ′ , its star s u bgraph S v is transformed into a path P ath ( S v ) of depth ch ( v ). Since T ′ is a 4-ary tree, ch ( v ) ≤ 4. Th us the ratio b et ween the depth of P ath ( S v ) and the d epth of S v is at most 4. It follo w s that for every ve rtex v ∈ V , the distance b etw een the ro ot and v in T ′′ is at most four times larger than the distance b et we en them in T ′ . Lemma 4.19 ω ( T ′′ ) ≤ 2 · ω ( T ′ ) . The pro of of this lemma is v ery similar to that of Lemma 4.15, and it is omitted. W e summ arize the p rop erties of the r eduction from 4-ary to binary trees in the follo w in g corollary . Corollary 4.20 F or a 4-ary ϑ -tr e e T ′ = ( V , E ′ ) , the Pr o c e dur e B inE xtension ( T ′ ) c onstructs a binary ϑ -tr e e T ′′ such that h ( T ′ ) ≤ h ( T ′′ ) ≤ 4 · h ( T ′ ) and ω ( T ′′ ) ≤ 2 · ω ( T ′ ) . Corollary 4.16 and Corollary 4.20 imply the follo win g statemen t. Lemma 4.21 F or a high ϑ - tr e e T = ( V , E ) , the invo c ation B inE xtension (4 E xtension ( T )) r eturns a binary ϑ -tr e e T ′′ such that h ( T ) ≤ h ( T ′′ ) ≤ 8 · h ( T ) and ω ( T ′′ ) ≤ 6 · ω ( T ) . W e are no w ready to prov e the d esired lo wer b ound for high ϑ -trees. Theorem 4.22 F or suﬃciently lar ge inte gers n and h , h ≥ log n , the minimum weight W ( n, h ) of a gener al ϑ -tr e e that has depth at most h is at le ast Ω( k · n ) , for some k satisfying h = Ω( k · n 1 /k ) . Pro of: Giv en an arbitrary ϑ -tree T of depth h ( T ) at least log n , Lemma 4.21 implies the existence of a binary tree ˜ T s uc h that h ( ˜ T ) ≤ 8 · h ( T ) and ω ( ˜ T ) ≤ 6 · ω ( T ). By Theorem 4.12, the w eigh t ω ( ˜ T ) of ˜ T is at least Ω( k · n ), f or some k s atisfying h ( ˜ T ) = Ω( k · n 1 /k ). Since ω ( T ) = Ω( ω ( ˜ T )) and h ( T ) = Ω( h ( ˜ T )), it follo ws th at the weig ht ω ( T ) of T is at least Ω( k · n ) as w ell, and k satisﬁes the required relation h ( T ) = Ω( h ( ˜ T )) = Ω( k · n 1 /k ). Clearly , the ligh tness of a graph G is at least as large as that of an y BFS tree of G , implyin g the follo w in g result. Corollary 4.23 F or suﬃciently lar ge inte ge rs n and h , h ≥ log n , any sp anning su b gr aph with hop-r adius at most h has lightness at le ast Ω(Ψ) , for some Ψ satisfying h = Ω(Ψ · n 1 / Ψ ) . 4.2 Lo w er Bounds for Low T rees In this s ection we devise low er b ounds for the wei ght of low ϑ -trees, that is, trees of dep th h at most logarithmic in n . Our str ategy is to sho w lo wer b ounds for the co vering (Sectio n 4.2. 1 ), and then to translate them into lo w er b ounds for the we igh t (Section 4.2.2). 22 4.2.1 Lo w er Bounds for Cov ering In this section w e prov e lo w er b ounds for the co vering of trees that hav e depth h ≤ 1 5 · log n . The follo w in g lemma shows that for an y ϑ -tree T , the cov ering of T cannot b e m uc h smaller than th e maxim um degree of a ve rtex in T . Lemma 4.24 F or a tr e e T and a vertex v i n T , the c overing χ ( T ) of T is at le ast ( deg ( v ) − 2) / 2 . Pro of: Consider an edge ( v , u ) in T . Since ev ery edge adjacen t to v is either left or righ t with r esp ect to v , by the pigeonhole pr in ciple either at least l deg ( v ) 2 m edges are left with resp ect to v or at least l deg ( v ) 2 m edges are righ t with resp ect to v . S upp ose without loss of generalit y that at least l deg ( v ) 2 m edges are righ t with resp ect to v , and den ote b y R the set of edges which are adj acen t to v and right with resp ect to v . Note that all these edges except p ossibly the edge ( v , v + 1) co v er the ve rtex ( v + 1), and thus the co vering of ( v + 1) is at least |R| − 1 ≥  deg ( v ) 2  − 1 ≥ deg ( v ) − 2 2 . Though the ultimate goal of this section is to establish a lo w er b ound for the range h ≤ 1 5 · log n , our argu m en t provides a relationship b et w een co v ering and d ep th h in a wider r ange h ≤ ln n . The next lemma tak es care of the simple case of h close to ln n . Th e far more complex r an ge of small v alues of h is analyzed in Lemma 4.26. Lemma 4.25 F or a tr e e that has depth h and c overing χ , su c h that ln n 2 < h ≤ ln n , χ > 1 e 2 · h · n 1 /h − h. Pro of: W e claim that for eac h ln n 2 < h ≤ ln n , th e righ t hand -sid e is of a negativ e v alue. T o see this, note that in the range [1 , ln n ], th e function g ( h ) = 1 e 2 · h · n 1 /h − h is monotone decreasing with h . Thus, g ( h ) is smaller than g ( ln n 2 ) = 0 in the range ( ln n 2 , ln n ], as required. Lemma 4.26 F or a tr e e that has depth h and c overing χ , su c h that h ≤ ln n 2 , χ > 1 10 · h · n 1 /h − h. Pro of: The pro of is b y induction on h , for all v alues of n ≥ e 2 h . Base: The statemen t h olds v acuously f or h = 0. F or h = 1, the degree of the ro ot is n − 1. Hence Lemma 4.24 implies that the co vering is at least n − 3 2 ≥ 1 10 · n − 1, for an y n ≥ e 2 . Induction Step: W e assume the claim for all trees of depth at m ost h , and pro v e it for trees of depth h + 1. Let T b e a tree on n vertic es th at has depth h + 1 and co vering χ , suc h that h + 1 ≤ ln n 2 . I f deg ( r t ) ≥ √ n , then Lemma 4.24 imp lies that the co vering is at least √ n − 2 2 . It is easy to verify that for h ≥ 1 and n ≥ e 2( h +1) , √ n − 2 2 ≥ 1 10 · ( h + 1) · n 1 / ( h +1) − ( h + 1) . Henceforth, w e assume that deg ( r t ) < √ n . 23 F or a c h ild u of r t , denote by T u the su b tree of T ro oted at u . Su c h a subtr ee T u will b e called light if | T u | < e h . Denote the set of all light su btrees of T by L , and deﬁn e T ′ as the tree obtained from T by omitting all ligh t su btrees from it, namely , T ′ = T \ [ T u ∈L T u . Obs erv e that the co v ering χ ′ of T ′ is at most χ . Next, w e obtain a lo w er b ound on χ ′ . Observe that      [ T u ∈L T u      = X T u ∈L | T u | < deg ( r t ) · e h < √ n · √ n e = n e , implying that n ′ = | T ′ | = | T | −      [ T u ∈L T u      > n − n /e. (17) Denote the subtrees of T ′ b y T 1 , . . . , T k , where k ≤ deg ( r t ), and deﬁne f or eac h 1 ≤ i ≤ k , n i = | T i | , χ i = χ ( T i ), and h i = depth ( T i ). Fix an in d ex i , i ∈ { 1 , . . . , k } . Obser ve that n i ≥ e h , or equiv alentl y , h ≤ ln n i . Since h i ≤ h , we hav e that h i ≤ ln n i . By th e induction h yp othesis and Lemma 4.25, χ i > 1 10 · h i · n 1 /h i i − h i . As the function f ( x ) = 1 10 · x · n 1 /x i − x is monotone decreasing in the range [1 , ln n i ], an d s ince h i ≤ h ≤ ln n i , it follo w s that χ i > 1 10 · h i · n 1 /h i i − h i ≥ 1 10 · h · n 1 /h i − h. Therefore, for eac h index i , 1 ≤ i ≤ k , n i < χ i + h 1 10 · h ! h . (18) F or eac h i , 1 ≤ i ≤ k , c ho ose some ve rtex v ∗ i in T i with m axim um co vering, i.e., χ T i ( v ∗ i ) = χ i . W e assu m e without loss of generalit y that v ∗ 1 < v ∗ 2 < . . . < v ∗ k . (Note that this ord er is not necessarily the ord er of the children u 1 , u 2 , . . . , u ch ( r t ) of the ro ot of T . In other words, it may b e the case th at v ∗ 1 < v ∗ 2 but u 1 > u 2 .) Let p b e the ind ex for whic h v ∗ p < r t and v ∗ p +1 > r t . Claim 4.27 χ ′ ≥ max { χ 1 , χ 2 + 1 , . . . , χ p + ( p − 1) } . Pro of: Consider the path in T ′ b et we en r t and v ∗ i , for eac h 1 ≤ i ≤ p − 1. F or eac h index j , i + 1 ≤ j ≤ p , the vertex v ∗ j m ust b e co v ered by at least one edge in that p ath. It follo ws that χ T ′ ( v ∗ j ) ≥ χ j + j − 1, for eac h 1 ≤ j ≤ p , as required. A symmetric argumen t yields the follo wing in equalit y . Claim 4.28 χ ′ ≥ max { χ k , χ k − 1 + 1 , . . . , χ p +1 + ( k − p − 1) } . Supp ose without loss of generalit y that p X i =1 n i ≥ k X i = p +1 n i . (The argumen t is symmetric if this is n ot the case.) Sin ce n ′ = | T ′ | = k X i =1 n i , it follo ws that n ′ 2 ≤ p X i =1 n i . By (18), P p i =1 n i < P p i =1  χ i + h 1 10 · h  h . 24 Notice that for eac h 1 ≤ i ≤ p , χ i + i − 1 ≤ χ ′ . Consequen tly , p X i =1 χ i + h 1 10 · h ! h ≤ p X i =1 χ ′ + h − ( i − 1) 1 10 · h ! h = 10 h ·  1 h  h · p X i =1  χ ′ + h − ( i − 1)  h . Since P p i =1 ( χ ′ + h − ( i − 1)) h < ( χ ′ + h +1) h +1 h +1 , w e conclude that n ′ 2 < 10 h ·  1 h  h · ( χ ′ + h + 1) h +1 h + 1 = 1 10 ·  h + 1 h  h · χ ′ + h + 1 1 10 · ( h + 1) ! h +1 ≤ e 10 · χ ′ + h + 1 1 10 · ( h + 1) ! h +1 . (19) By (17), n − n/e < n ′ , and th us (19) implies that n · (1 − 1 /e ) 2 < e 10 · χ ′ + h + 1 1 10 · ( h + 1) ! h +1 , and consequently , n <  χ ′ + h +1 1 10 · ( h +1)  h +1 . It follo ws that χ ≥ χ ′ > 1 10 · ( h + 1) · n 1 h +1 − ( h + 1) , as required. The next corollary follo w s easily from the ab ov e analysis. Corollary 4.29 F or a tr e e that has depth h and c overing χ , such that h ≤ 1 5 · log n , it holds that χ > 1 20 · h · n 1 /h . Pro of: It is easy to v erify that for an y h ≤ 1 5 · log n , it holds that h < 1 2 ·  1 10 · h · n 1 /h  . Hence, th e statemen t follo ws fr om Lemma 4.26 . 4.2.2 Lo w er Bounds for W eigh t In this section we emplo y the lo wer b oun d for th e co vering (Corollary 4.29) to sh o w lo w er b ounds for the w eigh t of ϑ -trees of depth h < log n . F or the range h ≤ 1 10 · log n , w e emp lo y a tec hn ique due to Agarw al et al. [3 ] to translate the lo we r b ound on the co v ering established in the previous section into the desired lo w er b ound f or high trees. (Our low er b ound on the co vering is, ho wev er, signiﬁcan tly stronger than that of [3].) Somewh at su rprisingly , our lo w er b ound for the complemen tary range 1 10 · log n < h < log n relies on our lo wer b oun d for the range h ≥ log n (Theorem 4.22). The follo w ing claim establishes a relation b et ween the weigh t of a tree, and the sum of co verings of its v ertices. Lemma 4.30 F or any ϑ -tr e e T of depth h , it holds that X e ∈ E ( T ) ω ( e ) = X v ∈ V ( T ) χ ( v ) + n − 1 . 25 Pro of: F or an edge e ∈ E ( T ), let ϕ ( e ) denote the num b er of vertic es co ve red b y e . Clearly , ω ( e ) = ϕ ( e )+ 1. It is easy to v erify by doub le counti ng that X v ∈ V ( T ) χ ( v ) = X e ∈ E ( T ) ϕ ( e ) . Consequent ly , X e ∈ E ( T ) ω ( e ) = X e ∈ E ( T ) ( ϕ ( e ) + 1) = X v ∈ V ( T ) χ ( v ) + n − 1 . Denote the minimum cov ering of a ϑ -tree of dep th at most h by ˜ χ ( n, h ). Since the sequ ence ( χ ( n, h )) n − 1 h =1 is monotone non-increasing (b y Lemma 3.1), it h olds that ˜ χ ( n, h ) = χ ( n, h ) . (20) The p ro of of the f ollo wing lemma is closely r elated to the pro of of Lemma 2.1 from [3], and is pro vided for completeness. Lemma 4.31 F or a su ﬃciently lar ge inte ger n , and a p ositive inte ger h , h ≤ 1 10 · log n , it holds that W ( n, h ) = Ω( n · h · n 1 /h ) . Pro of: Let T b e a ϑ -tree of m in im um w eigh t W ( n, h ), th at has depth at most h , h ≤ 1 10 · log n , an d co vering χ . Denote the ro ot v ertex of T by r t . Let g ( n, h ) = 1 20 · h · n 1 /h . By Corollary 4.29 and (20), χ > ˜ χ ( n, h ) ≥ g ( n, h ) . (21) A verte x v is called he avy if χ ( v ) ≥ 1 6 · g ( n, h ), and it is calle d light otherwise. Let H b e the set of hea vy v ertices in T . Next, we p ro v e that |H| ≥ n/ 2. Since b y Lemma 4.30, X e ∈ E ( T ) ω ( e ) ≥ X v ∈ V ( T ) χ ( v ) ≥ |H | · 1 6 · g ( n, h ) , this w ould complete the pr o of of the lemma. Supp ose to the con trary that |H | < n/ 2. W e need the follo win g d eﬁnition. A set B = { i, i + 1 , . . . , j } of consecutiv e v ertices is said to b e a blo ck if B ⊆ H . If ( i − 1) 6 = B and ( j + 1) 6 = B then B is called a maximal blo ck . Decomp ose H int o a set of (disjoint) maximal blo cks B = { B 1 , B 2 , . . . , B k } . By con tracting the in duced subgraph of eac h B i in to a single no de w i , for 1 ≤ i ≤ k , we obtain a new graph G ′ = ( V ′ , E ′ ). Ob serv e that G ′ is not n ecessarily a tree and m a y con tain m ultiple edges connecting the same pair of vertice s. F or eac h pair of v ertices u and v in V ′ , omit all edges except for the ligh test one that connect u and v in T ′ . If r t ∈ w i , for some in d ex i ∈ [1 , k ], then designate w i as the n ew ro ot v ertex r t . Let T ′ = ( V ′ , E ′′ ) b e the BFS tree of G ′ ro oted at r t . Clearly , n ′ = | V ′ | ≥ | V \ H | > n / 2, and h ( T ′ ) ≤ h ( T ) = h . Let v 1 , v 2 , . . . , v n ′ b e the vertice s of T ′ in an increasing ord er. T r ansform T ′ in to a spanning tree ˜ T of ϑ n ′ b y mapp ing eac h v i to the p oin t i on the x -axis, for i = 1 , 2 , . . . , n ′ . Claim 4.32 χ ( ˜ T ) ≤ 1 3 · g ( n, h ) + 1 . Pro of: First, observe that all light v ertices in T remain light in ˜ T . F or a con tracted hea vy v ertex w i , w e argue that χ ( w i ) ≤ χ ( w − i ) + χ ( w + i ) + 1, wh ere w − i (resp ectiv ely , w + i ) is the v er tex to the immediate left (resp., right ) of w i . T o see this, note that for eac h edge e 6 = ( w + i , w − i ) that co v ers w i , e co v ers either w − i 26 or w + i . Since b oth v ertices w + i and w − i are ligh t, it follo ws that χ ( w i ) ≤ 2 ·  1 6 · g ( n, h )  + 1, and w e are done. Observe that for n ≥ 4, w e ha v e that 1 10 · log n ≤ 1 5 · log ( n/ 2) . Since h ≤ 1 10 · log n , and n ′ > n / 2, it follo w s that h ≤ 1 5 · log n ′ . Hence b y Corollary 4.29 and (21), χ ( ˜ T ) ≥ ˜ χ ( n ′ , h ) ≥ g ( n ′ , h ) = 1 20 · h · n ′ 1 h > 1 20 · h · ( n/ 2) 1 h =  1 2  1 h · g ( n, h ) . Ho w ev er, for suﬃcien tly large n ,  1 2  1 h · g ( n, h ) > 1 3 · g ( n, h ) + 1 , con tradicting Claim 4.32. Lemma 4.33 F or a suﬃciently lar ge inte ger n , and a p ositive i nte ger h , 1 10 · log n ≤ h < log n , it holds that W ( n, h ) = Ω ( n · h · n 1 /h ) . Pro of: By Theorem 4.22, the minim um w eigh t W ( n, log n ) of a ϑ n -tree that h as depth h , h ≤ log n , is at least Ω( k · n ), for some k satisfying log n = Ω( k · n 1 /k ). It f ollo ws that W ( n, log n ) = Ω(log n · n ). Since b y Lemma 3.2, the sequence ( W ( n, h )) | n − 1 h =1 is monotone n on increasing, it holds that for h ∈ [ 1 10 · log n, log n ], W ( n, h ) = Ω(log n · n ) = Ω( n · h · n 1 /h ) . Lemmas 4.30 and 4.33 imply the follo w ing lo we r b ound for h < log n . Corollary 4.34 F or a suﬃcie ntly lar ge inte ger n , and a p ositive inte ger h , h < log n , i t holds that W ( n, h ) = Ω( n · h · n 1 /h ) . Clearly , the ligh tness of a graph G is at least as large as that of an y BFS tree of G , implyin g the follo w in g result. Theorem 4.35 F or a suﬃciently lar ge inte ger n and a p ositive inte ger h , h < log n , any sp anning sub gr aph with hop-r adius at most h has lightness at le ast Ψ = Ω( h · n 1 /h ) . 5 Euclidean Spanners The follo win g theorem is a direct corollary of Theorem 4.22 and Corollary 4.34. It implies that no construction that pr o vides Euclidean sp an n er with hop-diameter O (log n ) and lightness o (log n ), or vice v ersa, is p ossib le. This settles the op en p roblem of [6 , 3]. Theorem 5.1 F or a suﬃciently lar ge inte ger n , any sp anning sub gr aph of ϑ with hop-diameter at most O (log n ) has lightness at le ast Ω(log n ) , and vic e versa. 27 Pro of: First, we s ho w that any ϑ -tree that has depth at most O (log n ) has lightness at least Ω(log n ). If the depth h is at least log n , Theorem 4.22 implies that the lightness is at least Ω ( k ), for some k satisfying h = Ω( k · n 1 /k ). Ob serv e that f or h = O (log n ), an y k that satisﬁes h = Ω ( k · n 1 /k ) is at least Ω(log n ), as required. By the monotonicit y (Lemma 3.2), the low er b ound of Ω(log n ) applies for all smaller v alues of h . Consider a sp anning subgraph G of ϑ that has hop-d iameter Λ = O (log n ). Consider a BFS tree T ro oted at some v ertex r t . Obviously , h ( T , r t ) ≤ Λ = O (log n ), and thus ω ( T ) = Ω(log n ) · ω ( M S T ( ϑ )) = Ω( n log n ). Since ω ( G ) ≥ ω ( T ), it follo ws that the lightness of G is Ω(log n ) as wel l. Next, we argue that an y ϑ -tree that has lightness at most O (log n ) h as depth at least Ω(log n ). Indeed, by Corollary 4.34, an y ϑ -tree of depth h = o (log n ) has lightness at least Ω( h · n 1 /h ) = ω (log n ). Moreo ver, if a spann ing su bgraph G of ϑ h as ligh tness Ψ ( G ) = O (log n ), then its BFS tree T satisﬁes Ψ( T ) = O (log n ) as we ll. Th erefore, h ( T , rt ) = Ω(log n ), and th us Λ( G ) ≥ h ( T , rt ) = Ω (log n ). 6 Upp er B ounds In this section we devise an upp er b oun d for the tradeoﬀ b et w een v arious parameters of binary LL T s. This upp er b ound is tigh t up to constan t factors in the enti re range of parameters. Consider a general n -p oint metric sp ace M . Let T ∗ b e an MST for M , and D b e an in-order trav ersal of T ∗ , starting at an arbitrary v ertex v . F or ev ery v ertex x , remo v e from D all o ccur rences of x except for the ﬁrst one. It is w ell-kno wn ([25], c h. 36) that this w a y w e obtain a Hamiltonian p ath L = L ( T ) of M of tota l w eigh t ω ( L ) = X e ∈ L ω ( e ) ≤ 2 · ω ( M S T ( M )) . Let ( v 1 , v 2 , . . . , v n ) = L b e the order in whic h the p oints of M app ear in L . Consid er an edge e ′ = ( v i , v j ) connecting tw o arbitrary p oin ts in M , and an edge e = ( v q , v q +1 ) ∈ E ( L ), q ∈ [ n − 1] . The edge e ′ is said to c over e with r esp e c t to L if i ≤ q < q + 1 ≤ j . When L is clear f rom the con text, we write that e ′ c overs e . F or a spannin g tree T of M , the num b er of edges e ′ ∈ E ( T ) that co v er an edge e of E ( L ) is called the lo ad of e by T and it is denoted ξ ( e ) = ξ T ( e ). The lo ad of the tr e e T , ξ ( T ), is the maxim um load of an edge e ∈ E ( L ) by T , i.e., ξ ( T ) = max { ξ T ( e ) | e ∈ E ( L ) } . Observe that ω ( T ) ≤ X e ∈ L ( T ) ξ T ( e ) · ω ( e ) ≤ ξ ( T ) · w ( L ) , and so, ξ ( T ) ≥ ω ( T ) ω ( L ) ≥ 1 2 · Ψ( T ) . (22) (See Figure 8 for an illustration.) In the sequ el w e provide an upp er b ound for the load ξ ( T ) of a tree T , whic h yields the same up p er b ound for the ligh tness Ψ( T ) of T , up to a factor of 2. Since the we igh t of a tree T do es n ot aﬀect its load, w e ma y assume without loss of generalit y th at v i is located in the p oin t i on the x -axis. Lemma 6.1 Su pp ose ther e exists a ϑ -tr e e T with lo ad ξ ( T ) and depth h . Then ther e exists a sp anning tr e e T ′ for M with the same lo ad (with r esp e ct to L ( T ) ) and depth. 28 L 3 L 5 L 11 5 6 4 2 3 2 4 5 3 3 T 4 T 3 * 2 5 1 b) a) a c d b 4 5 1 2 5 4 1 2 3 4 5 2 1 r 3 3 Figure 8: a) The construction of the Hamiltonian path L . b) The load of T on the edge (3 , 4) is 3. Pro of: The edge s et E ′ of the tree T ′ is deﬁned b y E ′ = { ( v i , v j ) | ( i, j ) ∈ T } . It is easy to v erify that its depth and load are equal to those of T . Consequent ly , the problem of pro viding upp er b ou n ds for general metric spaces r educes to the problem of pro viding upp er b oun ds for ϑ . 6.1 Upp er Bounds for High T rees In this section w e devise a construction of ϑ n -trees w ith dep th h ≥ log n . T his construction exhibits tigh t up to constant factors tradeoﬀ b et ween load and depth , when the tree depth is at least logarithmic in the num b er of v ertices. In addition, th e constructed trees are binary , and so their maxim um degree is optimal . W e start with d eﬁning a certain comp osition of binary trees. L et n ′ and n ′′ b e t w o p ositiv e intege rs, n = n ′ + n ′′ . Let ϑ ′ , ϑ ′′ , and ϑ b e the n ′ -, n ′′ - and n -p oint metric s paces ϑ n ′ , ϑ n ′′ , and ϑ n , resp ectiv ely . Also, let { v ′ 1 , v ′ 2 , . . . , v ′ n ′ } , { v ′′ 1 , v ′′ 2 , . . . , v ′′ n ′′ } , and { ˜ v 1 , ˜ v 2 , . . . , ˜ v n } denote the set of p oin ts of ϑ ′ , ϑ ′′ , and ϑ , resp ectiv ely . Cons id er spannin g tr ees T ′ and T ′′ for ϑ ′ and ϑ ′′ , resp ectiv ely . Let v ′ = v ′ i b e a vertex of T ′ . Consider a tree ˜ T th at sp ans the v ertex s et { ˜ v 1 , ˜ v 2 , . . . , ˜ v n ′ + n ′′ } formed out of th e tr ees T ′ and T ′′ in the follo wing w a y . Th e ro ot r t ′′ of T ′′ is added as a righ t c hild of v ′ in ˜ T . The v ertices v ′ 1 , v ′ 2 , . . . , v ′ i = v ′ of T ′ retain their indices, and are tr anslated int o v ertices ˜ v 1 , ˜ v 2 , . . . , ˜ v i in ˜ T . Th e v ertices v ′′ 1 , v ′′ 2 , . . . , v ′′ n ′′ of T ′′ get the in dex i of v ′ i , added to their indices, and are translated in to v ertices ˜ v i +1 , ˜ v i +2 , . . . , ˜ v i + n ′′ in ˜ T . Finally , the v ertices v ′ i +1 , v ′ i +2 , . . . , v ′ n ′ of T ′ get the n umber n ′′ of v ertices of T ′′ added to their indices, and are translated into ve rtices ˜ v i +1+ n ′′ , ˜ v i +2+ n ′′ , . . . , ˜ v n ′ + n ′′ . W e say th at the tree ˜ T is comp osed by adding T ′′ as a right subtr e e to the vertex v in T ′ . Adding a left subtree T ′′ to a v ertex v in T ′′ is deﬁned analogously . Consider a family of binary ϑ -trees T ( ξ , h ) with n = N ( ξ , h ) v ertices, load ξ an d depth h , h ≥ ξ − 1, ξ ≥ 1. These trees are all ro oted at the p oin t 1. F or ξ = 1, and h = 0, th e tree T (1 , 0) is a singleton v ertex, and so N (1 , 0) = 1. F or con v enience, we deﬁne the load of T (1 , 0) to b e 1. F or ξ = 1 and h ≥ 1, the tree T (1 , h ) is the path P h +1 = ( v 1 , v 2 , . . . , v h +1 ). The d epth of T (1 , h ) is equal to h , and its load is 29 1. Hence N (1 , h ) = h + 1. F or ξ ≥ 2 an d h ≥ ξ − 1, th e tree T ( ξ , h ) is constr u cted as follo ws. Let T ′ = P h +1 b e the path ( v 1 , v 2 , . . . , v h +1 ), an d for eac h index i , i ∈ [ h ], deﬁn e for tec h n ical con v enience v ′ i = v i . F or eac h in dex i , i ∈ [ h ], let T ′′ i b e the tree T ( ξ ′′ i , h ′′ i ), w ith ξ ′′ i = min { ξ − 1 , h − i + 1 } , h ′′ i = h − i . Obs erv e that for eac h i ∈ [ h ], h ′′ i ≥ ξ ′′ i − 1, and th us the tree T ′′ i is w ell-deﬁned. F or eve ry i ∈ [ h ], we add the tree T ′′ i as a righ t subtree of v ′ i in T ′ . (See Figure 9 for an illustration.) T(1,0) T(2,2) T(2,1) r Figure 9: The tree T (3 , 3) . Lemma 6.2 The depth of the r esulting tr e e T ( ξ , h ) with r esp e ct to the vertex v ′ 1 is h , and its lo ad is ξ . Pro of: The pro of is b y induction on ξ , for all v alues of h ≥ ξ − 1. Base ( ξ = 1) : The tree T (1 , h ) has load 1 and depth h , as required. Induction Step: W e assume the correctness of the statemen t for all s maller v alues of ξ , and pro v e it for ξ . First, the v ertex v ′ h +1 is connected to the r o ot v ′ 1 via the path P h +1 , and th us the d epth h ( T ( ξ , h )) of T ( ξ , h ) is at least the length of this path, that is, h . C onsider a v ertex u ∈ V ( T ′′ i ), for some i ∈ [ h ]. Th e path P u connecting v ′ 1 and u in T ( ξ , h ) starts with a subp ath ( v ′ 1 , v ′ 2 , . . . , v ′ i , r t ( T ′′ i )), and contin ues with the uniqu e path P ′′ u connecting b et ween the ro ot of T ′′ i and the vertex u in the tree T ′′ i . The length of the latte r path is no greate r than the depth of T ′′ i , whic h, b y the induction hypothesis, is equal to h − i . Hence | P u | = i + | P ′′ u | ≤ h . Hence h ( T ( ξ , h )) = h . T o analyze th e load ξ ( T ( ξ , h )) of the tree T ( ξ , h ), consider some edge e = ( v q , v q +1 ) of the path P n , where n is the n umber of ve rtices in T ( ξ , h ). The path T ′ con tributes at most on e un it to the load of e . In addition, th er e exists at most one in dex i , i ∈ [ h ], suc h that the tree T ′′ i co vers the edge e . By induction, the load ξ ( T ′′ i ) is equal to min { ξ − 1 , h − i + 1 } . Consequently , the load of e is ξ T ( ξ ,h ) ( e ) ≤ max { min { ξ − 1 , h − i + 1 } + 1 | i ∈ [ h ] } = min { ξ − 1 , h } + 1 = ξ . Finally , w e analyze the num b er of vertice s N ( ξ , h ) in th e tree T ( ξ , h ). By construction, N ( ξ , h ) = h + 1 + h X i =1 N (min { ξ − 1 , h − i + 1 } , h − i ) . 30 Lemma 6.3 F or h ≥ ξ − 1 , N ( ξ , h ) ≥  h ξ  . Pro of: The pro of is b y induction on ξ . Base: F or ξ = 1, N (1 , h ) = h + 1 ≥  h 1  , as required. Step: F or ξ ≥ 2, an d i ≤ h − ξ + 1, min { ξ − 1 , h − i + 1 } = ξ − 1. Hence N ( ξ , h ) ≥ h + 1 + h X i =1 N (min { ξ − 1 , h − i + 1 } , h − i ) ≥ h − ξ +1 X i =1 N (min { ξ − 1 , h − i + 1 } , h − i ) = h − ξ +1 X i =1 N ( ξ − 1 , h − i ) . Observe that for eac h index i , i ∈ [ h − ξ + 1], h − i ≥ ξ − 1. Hence, by the ind uction h yp othesis and F act A.1, the latter sum is at least h − ξ +1 X i =1  h − i ξ − 1  =  h ξ  . The follo w ing theorem su mmarizes the pr op erties of the trees T ( ξ , h ). Theorem 6.4 F or a suﬃciently lar ge n , and h , h ≥ 2 log n , ther e exists a binary ϑ n -tr e e that has depth at most h and lo ad at most ξ , wher e ξ satisﬁes ( h = O ( n 1 /ξ · ξ ) and ξ = O (log n ) ). Pro of: First, n ote that for h in this range,  h log n  > n . By Lemma 6.3, for an y p air of p ositiv e int egers h and ξ such that ξ − 1 ≤ h , it holds that N ( ξ , h ) ≥  h ξ  . F urther, observe that N ( ξ , h ) is monotone increasing with b oth ξ and h , in the en tire range 0 ≤ ξ − 1 ≤ h . I t follo ws that for h ≥ 2 log n , there exists a p ositiv e inte ger ξ , ξ ≤ h , suc h that N ( ξ , h − 1) ≤ n < N ( ξ , h ). Consider the binary tree obtained from the tr ee T ( ξ , h ) by remo ving N ( ξ , h ) − n lea v es from it, one after another. Clearly , the resulting tree has depth at most h , load at most ξ , and its n umber of ve rtices is equal to n . Observe that n ≥ N ( ξ , h − 1) ≥  h − 1 ξ  ≥  h − 1 ξ  ξ , implying that h ≤ n 1 /ξ · ξ + 1 = O ( n 1 /ξ · ξ ) . The next corollary emplo ys Theorem 6.4 to dedu ce an analogous resu lt f or general metric spaces. Corollary 6.5 F or suﬃciently lar g e n and h , h ≥ 2 log n , ther e exists a binary sp anning tr e e of M that has depth at most h and lightness at most 2 · Ψ , wher e Ψ satisﬁes ( h = O ( n 1 / Ψ · Ψ) and Ψ = O (log n ) ). Mor e over, this binary tr e e c an b e c onstructe d i n time O ( n 2 ) . Remark: If M is an Euclidean 2-dimensional metric space, the run n ing time can b e fur ther impro v ed to O ( n · log n ). This is b ecause the ru nning time of th is construction is dominated by the runnin g time of the su broutine for constructing MST, and an MST of an Euclidean 2-dimensional metric space can b e constructed in O ( n · log n ) time [31]. By th e same considerations, for Euclidean 3-dimens ional spaces our algorithm can b e implemente d in a r andomized time of O ( n · log 4 / 3 n ), and more generally , for dimension 31 d = 3 , 4 , . . . it can b e implemented in deterministic time O ( n 2 − 2 ⌈ d/ 2 ⌉ +1 + ǫ ), for an arbitrarily sm all ǫ > 0 (cf. [31], page 5). E v en b etter time b ounds can b e p ro vided if one u ses a (1 + ǫ )-approximat ion MST instead of the exact MST for Euclidean metric spaces (cf. [31], page 6). Pro of: By Th eorem 6.4 a nd Lemma 6.1, for a suﬃcien tly large n , and h , h ≥ 2 log n , there exists a b in ary spanning tree T of M that has d epth at most h an d load at most Ψ, where Ψ satisﬁes h = O ( n 1 / Ψ · Ψ). By (22), th e ligh tness Ψ( T ) is at m ost 2Ψ. Note that sin ce th e complete graph G ( M ) indu ced by the metric M con tains at most O ( n 2 ) edges, its MST can b e computed within O ( n 2 ) time (cf. [25], c hapter 23). The in-order tra v ersal, and the constr u ction of the tree T ( ξ , h ) can b e p erformed in O ( n ) time in the straight- forward wa y . Hence the o ve rall run ning time of the algorithm for compu ting an LL T with the sp eciﬁed prop erties is O ( n 2 ). Finally , w e pr esent a simp le constr u ction for the range log n ≤ h < 2 log n . C on s ider a fu ll 3 balanced binary sp anning tr ee T n of { 1 , 2 , . . . , n } . The ro ot of T n is ⌈ n/ 2 ⌉ . Its left (resp ectiv ely , righ t) su btree is the full balanced b inary tree constructed recursive ly f r om the v ertex set { 1 , . . . , ⌈ n/ 2 ⌉ − 1 } (resp., {⌈ n/ 2 ⌉ + 1 , . . . , n } ). (See Figure 10 for an illus tration.) 1 2 3 4 5 6 7 8 9 10 11 12 Figure 10: The full bala nced bi na ry tree fo r n = 12 a nd h = 3 . Lemma 6.6 Both the depth and the lo ad of T n ar e no gr e ater than log n . Pro of: The pro of is b y induction on n . The base is trivial. Induction Step: W e assume the correctness of the claim for all smaller v alues of n , and pro ve it for n . By the induction hyp othesis, the d epth of b oth the left and the right subtr ees of the r o ot ⌈ n/ 2 ⌉ is at most log( n/ 2), implyin g that the d epth h ( T n ) of T n is at most log( n/ 2) + 1 = log n , as required. T o analyze th e load ξ ( T n ) of the tr ee T n , consider some edge e = ( v q , v q +1 ) of the p ath L n . T he t w o edges conn ecting r t to its c hildren con tribute at m ost one unit to the load of e . In add ition, at most one among the su btrees of the r o ot ⌈ n/ 2 ⌉ co v ers the ed ge e . By induction, b oth of these subtrees has load no greater than log( n/ 2). Consequently , the load of e is at most log( n/ 2) + 1 = log n . Theorem 6.7 F or a suﬃcie ntly lar ge n , and h , log n ≤ h < 2 log n , ther e exists a binary ϑ n -tr e e that has depth at most h and lo ad at most ξ , wher e ξ satisﬁes h = O ( n 1 /ξ · ξ ) . (In this c ase ξ = O (log n ) .) Pro of: By Lemma 6.6, for a s uﬃcien tly large n , and h , log n ≤ h < 2 log n , the b inary tree T n has depth at most h , and load at most ξ = log n , where h = O ( n 1 /ξ · ξ ). The next corollary emplo ys Theorem 6.7 to dedu ce an analogous resu lt f or general metrics. Corollary 6.8 F or suﬃciently lar ge n and h , log n ≤ h < 2 log n , ther e exists a binary sp anning tr e e of M that has depth at most h and lightness at most 2 · Ψ , wher e Ψ satisﬁes h = O ( n 1 / Ψ · Ψ ) . 3 Strictly speaking, this tree is full when n = 2 k − 1, for in teger k ≥ 1. How ever, for simplicit y of presentation, we call it “full” for other v alues of n as well. 32 Pro of: By Theorem 6.7 and L emm a 6.1, for a suﬃciently large n , and h , log n ≤ h < 2 log n , there exists a bin ary spann in g tree of M that has depth at most h and load at most Ψ, where Ψ satisﬁes h = O ( n 1 / Ψ · Ψ). By (22), the ligh tness is at most t wice the load, and we are done. Remark: By the same considerations as those of Corollary 6.5 , this construction can b e implemen ted within time O ( n 2 ) in general metric s paces, and in time O ( n · p olylog( n )) in Eu clidean lo w-dimensional ones. 6.2 Upp er Bounds for Lo w T rees In this section we devise a construction of ϑ n -trees with depth in the range of h < log n . Lik e the construction in Section 6.1, this construction provides a tigh t up to c onstant factors u pp er b ound for the tradeoﬀ b etw een w eigh t and depth. In addition, the maxim um degree of the constructed trees is optimal as w ell. (It is ⌈ n 1 /h ⌉ ). W e devise a family of d -regular trees ˜ T ( d, h ) with ˜ N ( d, h ) = h X i =0 d i v ertices, load ξ ≤ d · h , depth h , and d ≥ 2. Since d ≥ 2, it holds that ˜ N ( d, h ) = d h +1 − 1 d − 1 . T he family ˜ T ( d, h ) is constructed recursiv ely as follo w s . F or h = 0, the tree T 0 = ˜ T ( d, 0) is a sin gleton vertex, and so ˜ N ( d, 0) = 1. F or h ≥ 1, the tree ˜ T ( d, h ) is constructed as follo ws. F or eac h i ∈ [ d ], let T i b e a cop y of ˜ T ( d, h − 1), and let T 0 = ˜ T ( d, 0). F or ev ery i ∈ [ ⌈ d/ 2 ⌉ ] , we add the tree T i as a left sub tr ee of th e ro ot v ertex r t of T 0 , and for eac h i ∈ [ ⌈ d/ 2 ⌉ + 1 , d ], we add the tree T i as a right subtr ee of r t . Lemma 6.9 The depth of the r esulting tr e e ˜ T ( d, h ) with r esp e ct to the vertex r t is h , its lo ad ξ is at most d · h , and its size is e q u al to h X i =0 d i . Pro of: The pro of is b y induction on h . The base h = 0 is trivial. Induction Step: W e assume the correctness of the claim for all sm aller v alues of h , and pr o v e it for h . By construction, the ro ot r t has d children, eac h b eing the ro ot of a tree ˜ T ( d, h − 1). By th e in duction h yp othesis, ˜ T ( d, h − 1) has dep th h and size h − 1 X i =0 d i . Obviously , the d epth of ˜ T ( d, h ) is equal to h . T o estimate the size ˜ N ( d, h ) of ˜ T ( d, h ), note that ˜ N ( d, h ) = 1 + d · ˜ N ( d, h − 1) = 1 + d · h − 1 X i =0 d i ! = h X i =0 d i . T o analyze the load ξ ( ˜ T ( d, h )) of th e tree ˜ T ( d, h ), consider some edge e = ( v q , v q +1 ) of the path L ˜ n , wh ere ˜ n is the num b er of ve rtices in ˜ T ( d, h ). Th e d edges connecting r t to its children contribute altoge ther at most d units to the load of e . In addition, ther e exists at most one in d ex i , i ∈ [ h ], suc h that the tree T i co vers the edge e . By th e induction hypothesis, the load ξ ( T i ) of T i is no greater than d ( h − 1). Consequen tly , the load of e is at most d ( h − 1) + d = d · h , and we are done. Theorem 6.10 F or any p ositive inte gers n and h , h < log n , ther e exists an  n 1 /h  -ary ϑ n -tr e e that has depth at most h , and lo ad at most  n 1 /h  · h . The case h = 0 is trivial. F or h > 0, observe that ˜ N ( d, h ) is monotone increasing w ith b oth d and h . Giv en a pair of p ositiv e in tegers n and h , 1 ≤ h < log n , let d b e the p ositiv e in teger that satisﬁes ˜ N ( d − 1 , h ) ≤ n < ˜ N ( d, h ). 33 Consider the d -ary tree obtained from ˜ T ( d, h ) b y remo ving ˜ N ( d, h ) − n lea ve s from it, on e after another. Clearly , the depth, the load, and the maxim um degree of the resulting tree are n o greater than those of the original tree ˜ T ( d, h ). Moreo ver, the size of the resulting tree is precisely n . Ob serv e that n ≥ ˜ N ( d − 1 , h ) = h X i =0 ( d − 1) i > ( d − 1) h , implying that d < n 1 /h + 1. It follo ws that d ≤  n 1 /h  . Th us by Lemma 6.9, the resu lting tree is a spanning  n 1 /h  -ary tree of size n that has depth at most h , and load at most  n 1 /h  · h , and we are d one. The next corollary is an extension of Theorem 6.10 to general metric spaces. Corollary 6.11 F or a suﬃciently lar ge n , and h , h < log n , ther e exists a sp anning  n 1 /h  -ary tr e e of M that has depth at most h , and lightness at most O ( n 1 /h · h ) . Pro of: By T heorem 6.10 and L emma 6.1, for a suﬃcien tly large n , and h , h < log n , there exists a spanning  n 1 /h  -ary tree of M that has d epth at most h and load at most  n 1 /h  · h . By (22), the ligh tness is at most t wice the load, and we are done. Remark: By the same considerations as in Section 6.1, this construction can b e implemente d within time O ( n 2 ) in general metric spaces, and in time O ( n · p olylog( n )) in Euclidean lo w-dimensional ones. Corollaries 6.5, 6.11 and 6.8 imply the follo w in g theorem. Theorem 6.12 F or any suﬃciently lar ge inte ge r n and p ositive inte ger h , and n - p oint metric sp ac e M , ther e exists a sp anning tr e e of M of depth at most h and lightness at most O (Ψ) , that satisﬁes the fol lowing r elationship. If h ≥ log n then ( h = O (Ψ · n 1 / Ψ ) and Ψ = O (log n ) ). In the c omplementa ry r ange h < log n , Ψ = O ( h · n 1 /h ) . Mor e over, this sp anning tr e e is a binary one for h ≥ log n , and it has the optima l maximum de gr e e  n 1 /h  , for h < log n . 7 Shallo w-Lo w-Ligh t-T rees In this section we extend our construction of lo w-ligh t-trees and construct sh allo w-lo w-ligh t trees. Our argumen t in this section is closely related to that of Awerbuc h et al. [10 ]. Consider a spanning tree T = ( V , E ) of an n -p oin t metric space M , ro oted at a ro ot v ertex r t , h a ving depth h ( T ) and w eigh t ω ( T ). W e construct a spann ing tree S ( T ) of M that h as the same depth and w eigh t, up to constant factors, and that also approxi mates the distances b et ween the r o ot v ertex r t and all other v ertices in V . (Th us, for a lo w-ligh t tree T , S ( T ) is a shallo w -lo w -light tree.) Let L = ( v 1 , v 2 , . . . , v n ) b e the sequence of v ertices of T , ordered acco rdin g to an in-order trav ersal of T , starting at r t . Fix a p arameter θ to b e a p ositiv e r eal num b er. Th e v alue of θ determines the v alues of other parameters of the constructed tree. W e start with iden tifying a set of “break-p oin ts” B = { B 1 , B 2 , . . . , B k } , B ⊆ V . The break-p oin t B 1 is the v ertex v 1 . T h e break-p oin t B i is the ﬁrst v ertex in L after B i − 1 suc h that dist T ( B i − 1 , B i ) > θ · dist M ( r t, B i ) . Let S b e the set of edges conn ecting the b reak-p oin ts with r t in M , namely , S = { ( r t, B i ) | B i ∈ B } . Let ˜ G = ( V , E ∪ S ) b e the graph obtained from T by adding to it all edges in S . Finally , w e deﬁn e S ( T ) to b e the sh ortest-path-tree (henceforth, SPT) tree of ˜ G r o oted at r t . 34 The f ollo wing claim imp lies that the sum of distances in T , tak en o v er all pairs of consecutiv e b reak- p oint s, is not to o large. Claim 7.1 P B i ∈B dist T ( B i − 1 , B i ) ≤ 2 · ω ( T ) . Pro of: First, obs erv e that any in-order trav ersal of T visits eac h edge exactly twice . Hence, sin ce th e v ertices in L are ord er ed according to an in -order tra v ersal of T , w e ha v e that P n i =2 dist T ( v i − 1 , v i ) ≤ 2 · ω ( T ) . Clearly , for any pair of v ertices v i and v j , 1 ≤ i ≤ j ≤ n , it holds that dist T ( v j , v k ) ≤ P k i = j +1 dist T ( v i − 1 , v i ). It follo ws that P k i =2 dist T ( B i − 1 , B i ) ≤ P n i =2 dist T ( v i − 1 , v i ), and w e are done. The follo wing thr ee lemmas im p ly that the depth , the weigh t, and the w eigh ted diameter of the constructed tree S ( T ) are not muc h greater than those of the original tr ee T . In fact, if w e set θ to b e a small constan t, the three p arameters do not increase by more than a small constan t factor. Moreo ve r, not only the weigh ted diameter do es not gro w to o m uch, but, in fact, all d istances b et w een the ro ot v ertex r t and other v ertices in the graph are roughly the same in T and in S ( T ). Lemma 7.2 h ( S ( T )) ≤ 1 + 2 · ( h ( T ) − 1) . Pro of: A path from the ro ot v ertex r t to a v ertex v in S ( T ) ma y use an edge of S on ly as its ﬁ rst ed ge. If it do es so, its hop-length is at most 1 + 2( h ( T ) − 1). Otherwise its hop-length is at most h ( T ). Lemma 7.3 ω ( S ( T )) ≤ (1 + 2 /θ ) · ω ( T ) . Pro of: Observe that ω ( S ( T )) ≤ ω ( ˜ G ) = ω ( T ) + ω ( S ) = ω ( T ) + X B i ∈B dist M ( r t, B i ) . (23) By the c h oice of the br eak-p oin ts, for eac h B i ∈ B , dist M ( r t, B i ) < 1 θ · dist T ( B i − 1 , B i ) . By Claim 7.1, X B i ∈B dist T ( B i − 1 , B i ) ≤ 2 · ω ( T ) . Therefore X B i ∈B dist M ( r t, B i ) < 1 θ · X B i ∈B dist T ( B i − 1 , B i ) ≤ 2 θ · w ( T ) , and th us (23) implies that ω ( S ( T )) ≤ (1 + 2 /θ ) · ω ( T ). Lemma 7.4 F or a vertex v ∈ V , it holds that dist S ( T ) ( r t, v ) ≤ (1 + 2 θ ) · dist M ( r t, v ) . Pro of: Since S ( T ) is an SP T with resp ect to r t in ˜ G , it suﬃces to b ound dist ˜ G ( r t, v ). Let i b e the index suc h that v is lo cated b etw een B i and B i +1 in L , v 6 = B i +1 . Since B i is a break-p oint, we ha v e dist ˜ G ( r t, B i ) = dist M ( r t, B i ). Then dist ˜ G ( r t, v ) ≤ dist ˜ G ( r t, B i ) + dist ˜ G ( B i , v ) = di st M ( r t, B i ) + dist ˜ G ( B i , v ) ≤ di st M ( r t, B i ) + dist T ( B i , v ) 35 Since v w as not selected as a br eak-p oin t, n ecessarily dist T ( B i , v ) ≤ θ · dist M ( r t, v ) . (24) Hence dist ˜ G ( r t, v ) ≤ dist M ( r t, B i ) + θ · dist M ( r t, v ) . (25) Observe that dist M ( r t, B i ) ≤ dist M ( r t, v ) + dist M ( B i , v ) ≤ di st M ( r t, v ) + dist T ( B i , v ) . Ho w ev er, b y (24), the righ t-hand side is at most dist M ( r t, v ) + θ · dist M ( r t, v ) = (1 + θ ) · dist M ( r t, v ) , whic h implies dist M ( r t, B i ) ≤ (1 + θ ) · dist M ( r t, v ) . (26) Plugging (26) in (25), w e obtain dist ˜ G ( r t, v ) ≤ (1 + 2 θ ) · dist M ( r t, v ). Set ǫ = θ 2 . Lemmas 7.2, 7.3 and 7.4 imp ly the follo win g theorem. Theorem 7.5 Given a r o ote d sp anning LL T ( T , rt ) of a metric sp ac e M that has depth at most h and weight at most W , the tr e e S ( T ) is a sp anning SLL T of M that has depth at most 2 h − 1 and weight O ( W · ( ǫ − 1 )) . In addition, for every p oint v , dist S ( T ) ( r t, v ) ≤ (1 + ǫ ) · dist M ( r t, v ) . Theorems 6.12 and 7.5 imply the follo w ing corollary , wh ic h is the main result of this section. Corollary 7.6 F or a suﬃciently lar ge inte g er n , a p ositive inte ger h , a p ositive r e al ǫ > 0 , an n -p oint metric sp ac e M , and a designate d r o ot p oint r t , ther e exists a sp anning tr e e T of M r o ote d at r t with hop-r adius at most O ( h ) and lightness at most O (Ψ · ( ǫ − 1 )) , such that ( h = O (Ψ · n 1 / Ψ ) and Ψ = O (log n ) ) whenever h ≥ log n , and Ψ = O ( h · n 1 /h ) whenever h < log n . Mor e over, for every p oint v ∈ M , the weighte d distanc e b etwe e n the r o ot r t and v in T is gr e ater by at most a factor of (1 + ǫ ) than the weig hte d distanc e b etwe en them in M . This construction can also b e imp lemen ted within O ( n 2 ) time in general metric spaces, and w ithin O ( n · p olylo g( n )) time in Eu clidean lo w -dimensional ones. The only ingredient of this construction that is not p resen t in the constructions of Sections 6 and 6.2 is the construction of the shortest-path-tree for ˜ G . W e observe, ho w ev er, that ˜ G h as only a linear n umber of edges, and th us, this step requ ires only O ( n · log n ) additional time. Finally , w e sho w that there are graphs for whic h an y sp an n ing tree has either huge hop-diameter or h uge wei ght. Sp eciﬁcally , consider th e graph G = ( V , E , ω ) form ed as un ion of the ( n − 1)-v ertex path P n − 1 = ( v 1 , v 2 , . . . , v n − 1 ) with the star S = { ( z , v i ) | i ∈ [ n − 1 ] } . All edges of the path ha v e unit weig ht, and all edges of S hav e weig ht W , for s ome large in teger W , n ≪ W . (See Figure 11 for an illustration.) Note that G is a metric gr aph , that is, for ev ery ed ge ( u, w ) ∈ E , ω ( u, w ) = dist G ( u, w ). I t is easy to v erify th at any spanning tree T of G that has weigh t at most q · W , for some in teger parameter q , q ≥ 1, has hop-diameter Ω( n/q ). Symmetrically , for an integ er parameter D , D ≥ 1, an y spann ing tree T of G that h as hop-diameter at m ost D con tains at least Ω( n/D ) edges of w eigh t W . Since the weigh t of the MST of G is only sligh tly greater than W , it follo ws that the lightness of T is at least Ω( n /D ). Consequent ly , for any general construction of LL Ts for g r aphs , Ψ · Λ = Ω( n ). As w e ha v e shown, for metric sp ac e s the situation is drastically b etter, and, in particular, one can hav e b oth Ψ and Λ no greater than O (log n ). 36 1 n−1 v 2 v n−2 v 1 z 1 1 1 W W W W G v Figure 11: The graph G = ( V , E , ω ) . Ac kno wledgemen ts The s econd-n amed a uthor thanks Mic hael Segal and Hanan Shpungin for approac hin g h im with a problem in th e area of wireless net w orks that is related to the prob lem of constructing L L Ts. Corresp ondence with them triggered this researc h. 37 References [1] I. Abra ha m, D. Malkhi, Compact routing on euclidian metrics, PODC’04 , 141-1 49. [2] I. Abraham, B. Aw erbuch, Y. Azar, Y. Bartal, D. Ma lkhi, E. Pavlo v, A Generic Scheme for Building Overlay Net works in Adv ersar ial Scenar ios, IPD PS’03 , 40. [3] P . K. Agarwal, Y. W ang, P . Yin. Low er bound for sparse Euclidean spanners, SODA’05 , 6 70-67 1. [4] N. Alon, R. M. Karp, D. P eleg, D. B. W est. A Graph-Theor etic Game and Its Application to the k-Server Problem, SIAM J. Comput. 24(1) (1995), 78-100 . [5] C. J. Alp ert, T. C. Hu, J. H. Huang, A. B. Kahng, D. Karg e r . Pr im-Dijkstra tradeoﬀs for improved per formance-dr iven routing tree design, IEEE T r ans. on CAD of Inte gr ate d Cir cuits and S ystems 14(7) (199 5), 890-8 96. [6] S. Arya, G. Das, D. M. Mo un t, J. S. Sa lowe, M. H. M. Smid. Euclidea n spanners: short, thin, and lanky , STOC 1995 , 489-49 8. [7] S. Arya, D. M. Mount, M. Smid, Randomized and deterministic algorithms for geometric spa nners of sma ll diameter, FOCS’9 4 , 7 03-71 2. [8] S. Ary a and M. Smid, E ﬃcie nt construction of a bo unded degr ee spanner with low weigh t, Alg orithmic a , 17 , (1997), pp. 33-5 4. [9] B . Aw erbuch, A. Ba ratz and D. Peleg, Co st-sensitive a nalysis of commu nication proto cols, PODC’90 , pp. 177-1 87. [10] B. Aw erbuch, A. B aratz, D. Peleg. E ﬃcien t Broadca st a nd Light-W eight Spa nner s, Manuscript, 19 91. [11] Y. Bartal, Pro babilistic Approximations of Metric Spaces and Its Algor ithmic Applications , F OCS’96 , 184-193. [12] Y. Bartal, On Approximating Ar bitrary Metrices by T ree Metrics, STOC’98 , 1 61-16 8 . [13] K. B ha rath-Kumar , J. M. Jaﬀe, Routing to multiple destinations in co mputer netw orks , IEEE T r ans. on Commun. , COM-31 , (198 3), pp. 3 43-35 1. [14] P . Bos e , J. Gudm undsson, M. H. M. Smid, Co nstructing Plane Spanners o f Bounded Degree and Low W eight, Algo rithmic a 42(3-4) ,(2005), pp. 249 -264. [15] P . B. Callaha n, S. R. Kosa r a ju, A Decomp osition o f Multi-Dimensional Poin t-Sets with Applications to k - Nearest-Neighbo r s and n -Bo dy Potential Fields, STOC’92 , 546 -556. [16] T-H. H. Chan, A. Gupta, Small hop-dia meter sparse spanners for doubling metrics , SODA’0 6 , 70- 78. [17] B. Chazelle, B. Rosenberg, The complexity of computing partial sums oﬀ-line, Int. J. Comput. Ge om. Appl. 1 , (1991), 33-45 . [18] M. Charik ar, C. Chekuri, A. Goel, S. Guha, S. A. Plotkin, Approximating a Finite Me tric by a Small Num b er of T ree Metrics. FOCS’98 , 379-3 88. [19] M. Charik a r, M. T. Ha jiaghayi, H. J. Ka rloﬀ, S. Rao , l 2 2 spreading metrics for vertex order ing pr oblems, SODA’06 , 101 8 -1027 . [20] E. Cohen, F ast a lgrotihms for constructing t -spa nners a nd paths with stretch t , F OCS’93 , 648 -658. [21] E. Cohen, Polylog -time and near-linear work approximation s cheme for undirected shortest paths, STOC’94 , 16-26 . [22] J. Cong, A. B. Ka hng, G. Robins, M. Sarra fzadeh, C. K. W ong , P erfor mance-Driven Global Routing for Cell Based ICs, ICCD’91 , 170- 173. [23] J. Cong, A. B. Kahng, G. Robins, M. Sarra fzadeh, C. K. W ong, Prov ably go o d p er fo rmance-driven glo bal routing, IEEE T r ans. on CAD of Inte gr ate d Cir cuits and Systems 11(6) , (1992 ), 73 9-752 . [24] J. Cong, A. B. Kahng, G. Robins, M. Sarrafzadeh, C. K. W ong, P rov ably go o d alg orithms for p erfor mance- driven glo bal r outing, Pr o c. IEEE In tl. Symp. on Cir cuits and Systems , (19 92), pp. 2240-2 243. [25] T. H. Co rman, C. E . Leiser son, R. L. Rivest, C. Stein, Intro duction to Algo rithms, 2 nd edition, McGr aw-Hil l Bo ok Comp any , Boston, MA, 2001. [26] G. Das, G. Naras imhan, A F ast Algo r ithm for Constructing Sparse Euclidean Spanners, SOCG’94 , 1 32-13 9 [27] D. Dor, S. Halp erin, U. Zwick, All P airs Almost Shor test P aths, F OCS’96, 452-4 61. [28] Shlomi Dolev, Ev angelos K ranakis, Danny Kr izanc, David Peleg: Bubbles : adaptive r outing scheme for high- sp eed dynamic netw orks STOC’95 , 528-5 37. [29] M. Elkin, Y. Emek, D. Spielman, S. T eng, Low er Stretch Spanning T rees, STOC’05 , pp. 4 94-503 [30] G. Even, J. Naor, S. Rao, B. Schieber, Divide-and-C o nquer Approximation Algor ithms via Sprea ding Metrics, F OCS’95 , 62-7 1. [31] D. Eppstein, Spanning trees and s pa nners, tec hnical rep ort 9 6-16, Dept. of Information and Computer-Science, Univ ersity o f California, Irvine. [32] U. F eige, J. R. Lee, An impr ov ed approximation ratio for the minimum linear ar rangement problem, Inf. Pr o c ess. L ett. , 101(1) ,(2007), pp. 2 6 -29. [33] J. F a kc haro enphol, S. Ra o, K . T a lw ar, A tight b ound on approximating arbitrar y metrics by tr e e metrics, STOC 2003 , 448-45 5. [34] Y. Hassin, D. Peleg, Spar se commun ication netw orks and eﬃcient r outing in the plane, PODC’00 , 41 -50. [35] R. L. Graham, D. E. K n uth, O. Patashnik, Concrete Mathematics, Re ading, Massachusetts, A ddison-Wesley (1994), xiii+657 pp. [36] J. M. Ja ﬀe, Distr ibuted multi-destination routing: the constraints of lo c a l information, SIAM journal on Computing , 14 , (1985), pp. 87 5-888 [37] S. Khuller, B. Raghav achari, N. E. Y o ung, Approximating the Minim um E quiv alent Diagra ph, S ODA’94 , 177-1 86. [38] H. P . Lenhof, J. S. Salowe, D. E. W reg e, New metho ds to mix shortest-pa th and minim um spanning tr e es, manuscript, 19 94. [39] Z. Lotker, B. P att-Shamir , E. Pavlo v, D. Peleg, Minim um-W eight Spanning T re e Construction in O(log log n) Communication Rounds. SIAM J. Comput. 3 5(1): 1 20-13 1 (2005) [40] Y. Mansour, D. Peleg, An Approximation Algorithm for Min-Cost Netw ork Design, D IMA CS Series in Discr. Math and TCS 5 3 , (2000), pp. 97-10 6 . [41] G. Narasimhan, M. Smid, Geometric Spanner Netw ork s, P ublis hed by Cambridge University Pr ess , 2 007. [42] M. P˘ atra¸ scu, E . D. Demaine, Tight b ounds for the partial-sums problem, SO D A’04, 20- 29. [43] D. Peleg, Distributed Computing: A Lo cality-Sensitive Approach, S IAM , Philadelphia, P A, 2 000. [44] S. Rao, A. W. Richa, New Approximation T echniques for Some Ordering Pro blems, SODA’98 , 211-2 18. [45] R. Ravi, R. Sundaram, M. V. Marathe, D. J. Ro s enkrantz, S. S. Ravi. Spanning T r ees Short or Small, SODA’94 , 546-5 55. [46] G. Singh, Leader Election in Co mplete Net works, PODC’92 , 1 7 9-190 . [47] A. C. Y ao, Space-time tradeoﬀ for answering ra nge q ueries, STOC’82 , 128 - 136. App endix A Prop erties of the Binomial Co eﬃcien ts In this section w e present a num b er of u s eful prop erties of the binomial coeﬃcient s. The follo w ing t wo statemen ts are w ell-kno wn [35]. F act A.1 (P ascal’s 2nd iden tity) F or any non-ne gative inte gers h and i , such that i ≤ h , h X k = i  k i  =  h + 1 i + 1  . F act A.2 (P ascal’s 7th iden tity) F or any non-ne gative inte ge rs n and k , such that k + 1 ≤ n ,  n k + 1  = n − k k + 1 ·  n k  . The next lemma sho ws th at the sequence of bin omial co eﬃcien ts B = {  n i  | i = 1 , 2 , . . . , n } gro ws exp onent ially with i as long as i ≤ ⌊ n 4 ⌋ . Lemma A.3 F or any non-ne gative inte gers n and k , such that k ≤ ⌊ n 4 ⌋ , k X i =0  n i  < 3 2 ·  n k  . Pro of: By F act A.2, for an y 0 ≤ i ≤ k − 1, it holds that  n i + 1  = n − i i + 1 ·  n i  ≥ n − ⌊ n 4 ⌋ + 1 ⌊ n 4 ⌋ ·  n i  > 3 ·  n i  . Hence, k X i =0  n i  ≤  n k  · k X i =0  1 3  i < 3 2 ·  n k  . i

Shallow, Low, and Light Trees, and Tight Lower Bounds for Euclidean Spanners

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment