Minimizing the number of critical stages for the on-line steiner tree problem

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1 Mnmzng the number of crtcal stages for the on-lne stener tree problem Ncolas Thbault, Chrstan Laforest IBISC, Unversté d Evry, Tour Evry 2, 523 place des terrasses, EVRY France Keywords: on-lne algorthm, stener tree, optmzaton, rebuldng Abstract Ths paper s devoted to the followng ncremental problem. Intally, a graph and a dstngushed subset of vertces, called ntal group, are gven. Ths group s connected by an ntal tree. The ncremental part of the nput s gven by an on-lne sequence of vertces of the graph, not yet n the current group, revealed on-lne one after one. The goal s to connect each new member to the current tree, whle satsfyng a qualty constrant: the weght of each constructed tree must be at most c tmes the weght of an optmal stener tree (wth c a gven constant). Under ths qualty constrant, our objectve s to mnmze the number of crtcal stages. We call crtcal a stage where the ncluson of a new member mples heavy changes n the current tree. Otherwse, the new member s just added by connectng t wth a (well chosen) path to the current tree. We propose a strategy leadng to at most c ln O() crtcal stages (where s the number of new members and c the constant of the qualty constrant). We also prove that there exsts stuatons where at least 1 Ω() crtcal stages are necessary to any algorthm to mantan the qualty constrant. Our strategy s then worst case optmal n order of magntude for the number of crtcal stages. The Stener tree problem, where the goal s to span a set (called group) of dstngushed vertces (called members) wth a mnmum weght tree, has been extensvely studed. As the problem s NP-complete (see [6]), numerous approxmaton algorthms have been desgned (see [2, 8] for example). In [13], Waxman was the frst to present the on-lne verson of ths problem n whch new members are revealed one by one (see [4, 5] references on on-lne problems). In ths frst paper, he dvdes the problem nto two categores: the model n whch changes n the current tree are not allowed and the model n whch changes are allowed. Imaze and Waxman propose n [9] two dfferent strateges correspondng to the two models above. In the frst one the tree s just ncremented and the degradaton of the weght s evaluated, whereas n the second one they allow changes n the current tree to mantan a certan guaranty on the weght. They prove that they construct wth the frst strategy a tree whose weght s at a logarthmc rato compared to the optmal one (.e. the weght of a Stener tree of the current group), and that they construct wth the second strategy a tree whose weght s at a constant rato compared to the optmal one. They gve for the second strategy an upper bound of O( ) on the average number of elementary changes per stage (where s the number of new members). However, the tree can potentally be changed at each stage; ths means that each addng stage s potentally what we call later a crtcal stage. Then, we can dvde (as Waxman dd n [13]) the other works that have been made snce [9] concernng on-lne stener trees. In [1, 3, 14], the model n whch no changes are allowed s consdered. In [1], the authors gve a lower bound of Ω( log log log ) for the compettve rato ( s the number of added members) for the on-lne stener tree problem n the Eucldean plane. In [3], the authors consder the on-lne generalzed stener tree problem and they propose an algorthm wth a compettve rato of O(log 2 ). In [14], lnear upper bounds and lower bounds are obtaned for the on-lne generalzed stener tree and the on-lne stener tree problem on a drected graph. In [7], the model wth allowed changes s consdered. The am s to mnmze smultaneously the weght of the current tree and the length from a partcular node to all the other ones of the tree. The authors propose a method wth a compettve rato of O(log ) for the weght and constant for the length from the partcular node. Note that n [7, 9], only the number of elementary changes s taken nto account to measure the level of damage due to the allowed changes n the current tree (.e. each stage s potentally a crtcal stage). In our paper we are also concerned by an ncremental group problem where the members of the group are revealed on-lne one by one. We fx a relatve budget on the weght of each successve tree, called qualty constrant, and we propose an algorthm mnmzng the number of crtcal stages necessary to guarantee ths budget constrant at each stage. Our work s the frst whch focus on mnmzng the number of crtcal stages nstead of the number of elementary changes (we already consder ths parameter n [11, 12], but for wth

2 a dfferent qualty constrant, offerng guarantees on the maxmum and average dstance between members n the tree nstead of guarantees on the weght of the tree). We dstngush crtcal stages from other stages snce they generate a lot of perturbatons. Indeed, the communcaton routes between members already n the current group have to be changed. All the routng tables of the nodes may be modfed. Ths generates a heavy traffc to update them. Moreover the current communcatons between members ntated before the changes can be nterrupted. For these reasons, the number of crtcal stages must be mnmzed. Note that t s proved n [9] that any on-lne algorthm wthout crtcal stage cannot guaranty a constant qualty constrant. That s why we consder here the model n whch changes are allowed. In Secton 1, we descrbe and motvate the constrants (namely the tree and qualty constrants) that must be satsfed at each stage of addton and we gve the defnton of a crtcal stage. We propose our strategy called OWM (for On-lne Weght Mnmzaton) and prove that t satsfes the constructon constrants n Secton 2. We also prove that our algorthm leads to at most c ln O() crtcal stages (where s the number of new members and c the constant of the qualty constrant). In Secton 3, we prove that there exsts a stuaton n whch at least 1 Ω() crtcal stages are necessary for any on-lne algorthm to satsfy the qualty constrant. These results show that Algorthm OWM s worst case optmal n order of magntude for the number of crtcal stages. 1. Defntons and notatons Let G = (V, E, w) be any connected weghted graph representng a network. V s the set of vertces (modelng the nodes of the network), E the set of edges (modelng the set of physcal lnks) and w a postve weght functon of the edges (modelng the length of the edges). Defnton 1 (Optmal Stener Tree) Let M be a group of members and let T = (V T, E T, w) be a tree spannng M. We denote the weght of T by w(t ) = e E T w(e) and we denote by T opt (M) an optmal stener tree spannng the group M,.e. a tree satsfyng w(t opt (M)) = mn {w(t ) : T spannng M}. Constructon constrants. In our problem, the graph G = (V, E, w) and an ntal group M 0 V are gven (wth M 0 ). For example, n a meetng on network (called net-meetng) ths ntal group M 0 represents the set of partcpants present from the begnnng of the meetng. A structure, denoted as T 0, must be created to connect the members of M 0. However, n the case of an open net-meetng for example, new partcpants can jon the meetng. These new partcpants must be ntegrated to the current group by connectng them to the current connecton structure. We suppose here that these new partcpants are not known n advance and arrve n an on-lne way: a new partcpant, whch s any vertex of the graph, s revealed when t decdes to ntegrate the group, at any moment. The ncremental part of the problem conssts n ntegratng a new member when t s revealed. We call that a stage of addton. If S s a sequence of new members revealed, S = u 1, u 2,..., u, for every k, 1 k, we denote as M k = M k 1 {uk } the k th group. Thus, startng from the ntal connecton structure T 0 for M 0, we must ntegrate, at each addton step k, the new member u k by updatng the current structure T k 1 (spannng M k 1 ) to obtan T k spannng M k. Note that, as the members are revealed one by one, we are n an on-lne model. It means that we do not know the future: nether n whch order the members arrve, nor what s the set of new members that wll be revealed. Hence, each stage can potentally be the last one; ths explans why we are nterested by gvng guarantees at each stage. We are now ready to gve the two constrants that each current structure T k must satsfy. The tree constrant: for every k 0, T k must be a tree wth all leaves n M k (we call that a pruned tree). The qualty constrant: let c 1 be any fxed constant representng the requred level of qualty. Then, for every k, we must have w(t k ) c w(t opt (M k )). As n a net-meetng the current structure T k s used to support the communcatons between members of M k, the tree constrant s set n order to smplfy the mechansms of routng and duplcaton of nformaton n T k. Indeed, there s only one route between any par of members n a tree; moreover as there s no cycle, a smple floodng process can be used to broadcast nformaton from any member. Ths floodng naturally ends at the leaves that are members (because trees are pruned); there s no need of costly process to control t. The 2

3 qualty constrant of level c s set to guaranty that the weght of each tree T k s not too far (up to at most a constant factor of c) from the optmal one. In the rest of the paper we say that an algorthm solves our problem f, for any on-lne sequence M 0,..., M, t returns a sequence of trees T 0,..., T (T spannng M ) satsfyng the tree and qualty constrants. Defnton 2 (Crtcal stage) Let A be any algorthm solvng our problem by returnng a sequence of trees T 0 = (V 0, E 0 ),..., T = (V, E ) satsfyng the tree and qualty constrants. Stage k (1 k ) s a crtcal stage f E k 1 E k. We denote by CS(T 0,..., T ) the total number of crtcal stages after added members. We recall that we dstngush crtcal stages from other stages snce they generate a lot of perturbatons. Indeed, the communcaton routes between members already n the current group M k 1 have to be changed. Potentally all the routng tables of the connectng nodes must be modfed. Ths generates a heavy traffc to update them. Moreover the current communcatons between members of M k 1 ntated before the changes can be nterrupted. The number of crtcal stage must then be mnmzed. On the other hand, a smple (non crtcal) connecton of the new member by just addng a path n the tree (nstead of breakng partally or completely the current tree) generates only local changes. The update of the routng can just be done by broadcastng the dentty of the new member n the new tree T k. Ths does not create any re-routng between the other members. 2. Our Algorthm OWM The man dea of Algorthm OWM (for On-lne Weght Mnmzaton) s to defne partcular stage numbers, called rebuldng stages (determned n functon of the gven level of qualty c) durng whch we (totally) reconstruct the current structure by usng any algorthm solvng the off-lne stener tree problem wth approxmaton rato a. Then, we choose a vertex v of ths a-approxmate stener tree (any vertex), and between two successve rebuldng stages, we add each new member by usng Imaze and Waxman on-lne Algorthm (see [9]), startng from vertex v. Imaze and Waxman Algorthm adds each new member by a shortest path to the current tree. As Imaze and Waxman prove n [9] that ther algorthm buld a tree whose weght s at most O(log j) tmes the weght of an optmal stener tree (where j s the number of new members added n an on-lne way), the resultng spannng structure bult by our Algorthm OWM s bascally the unon of a a-approxmate stener tree spannng a part of the current group and a O(log j)-approxmate stener tree spannng the rest of the current group. By usng ths two propertes, we prove n Secton 3 (see Theorem 1) that f we rebuld completely the structure each 2 c a 1 1 new added members, then the qualty constrant s satsfed (wth c a + 1 the gven constant level of the qualty constrant and a the approxmaton rato of the algorthm used to rebuld the tree off-lne at each rebuldng stage). We gve our Algorthm OWM n Table 1. Important remarks: the rebuldng stages correspond to the crtcal stages of OWM (because the current structure s broken and rebult). The other stages are non crtcal because the algorthm only adds a path to the current structure to connect a new member. Note that OWM s polynomal. Note also that as defned here, OWM does not necessarly buld a tree at each stage. Indeed, when a new path s added to the current structure by usng Imaze and Waxman Algorthm on a subtree of the current structure, a cycle may be created. Ths algorthm can be refned n order to obtan at each stage a tree. Nevertheless, due to space lmtaton, we choose here to avod ths refnement to smplfy the presentaton. Note that even f the bult structure s not a tree (.e. f t s a structure wth heaver weght, by defnton of a tree), we prove that t respects the qualty constrant (see Secton 3). As n the refnement, the tree s ncluded n the structure, t also respects the qualty constrant. Numercal llustraton: f we use the (1 + ln 3 2 )-approxmate algorthm proposed by Robns and Zelkovsky n [10] to rebuld the tree at each rebuldng stage and we set c = 10, then we have 2 c a 1 1 = 173. Ths means that to satsfy the qualty constrant wth a level c = 10, we have to rebuld the current tree each 173 new added members. 3

4 Let G = (V, E, w) be a graph and M 0 V be the ntal group. At stage 0 : Buld a tree T 0 spannng M 0 wth any a-approxmate polynomal tme off-lne algorthm for the stener tree problem. After the last rebuldng stage k : Let M k+j be the current group and v k be any vertex n M k. Let u k+j be the j th member to add snce the last rebuldng stage k. IF j < 2 c a 1 1. THEN Buld T k+j spannng M k+j = M k+j 1 {u k+j } by usng Imaze and Waxman Algorthm,.e. by addng a shortest path between u k+j and ts closest vertex n the subtree of T k+j 1 spannng M k+j 1 \(M k \{v k }) ELSE, we have j = 2 c a 1 1 (rebuldng stage). Break the current tree and buld a new tree T k+j, spannng M k+j wth any a-approxmate polynomal tme off-lne algorthm for the stener tree problem. k + j s the new last rebuldng stage. Table 1: On-lne Weght Mnmzaton - OWM OWM respects the qualty constrant The followng Theorem shows that OWM respects the qualty constrant, f the requred level of qualty s a constant c a + 1 (where a s the approxmaton rato of the algorthm used to rebuld the tree off-lne at each rebuldng stage). Theorem 1 For any constant c a + 1, for any addng sequence of 0 new members, OWM respects the qualty constrant wth level c. Proof. Let k be the last rebuldng stage. After stage k, there exsts j, 0 j 2 c a 1 1 such that = k + j. Let T = (V, E ) be the subgraph bult by algorthm OWM spannng M. By defnton of OWM, T s the result of the unon between T k spannng M k, bult off-lne at the last rebuldng stage (see defnton of OWM) and the subtree T + of T spannng M + = M \(M k \{v k }), bult on-lne wth the Algorthm of Imaze and Waxman (see defnton of OWM). Thus, we have: w(t ) w(t k ) + w(t + ) a w(t opt(m k )) + w(t + ) (because, by defnton of Algorthm OWM, T k s a a-approxmated tree spannng M k ) a w(t opt (M k )) + log 2 (j + 1) w(t opt (M + )) (as T + s a tree spannng M + bult by Imaze and Waxman Algorthm, Lemma 3 n [9] holds) (a + log 2 (j + 1) ) w(t opt (M )) (a + c a 1 ) w(t opt (M )) c w(t opt (M )) (because M k M, M + M and j 2 c a 1 1 ) OWM leads to at most crtcal stages 2 c a 1 1 Theorem 2 For any constant c a + 1 (representng the requred level of qualty), for any sequence M 0 M of addtons, let T 0,..., T be the sequence of trees constructed by OWM. We have CS(T 0,..., T ) 2 c a 1 1. Proof. By defnton of Algorthm OWM, f there are p rebuldngs (that are crtcal stages), we have: p ( 2 c a 1 1 ) < (p + 1) ( 2 c a 1 1 ) p 2 c a

5 3. Lower bound for the number of crtcal stages of any algorthm In ths secton, we prove that, for any on-lne algorthm, f the tree and qualty constrants are satsfed, then, for any suffcently large, there exsts a partcular addng sequence leadng to 1 Ω() crtcal stages. We frst defne the graph G p,n and the partcular sequence of addtons. Defnton of Graph G p,n. For every p 1, we defne Graph G p,n made of n 1 subgraphs G 0 p,..., G k p,..., G n p. Each subgraph G k p s the graph ntroduced by M. Imase and B. Waxman n Secton 3.1 of [9] wth every edge of weght 2 k. The subgraphs G k p are connected to form G p,n n the followng way. For all k, 0 k n, let u k 1, u k 2 be the two frst members of G k p reveled. For every k, 1 k n, there s an edge of weght 1 n between uk 1 2 and u k 1. Prelmnary results. We use an adaptve adversary, whch chooses new members to add n order to trap any on-lne algorthm. The adaptve adversary add new members n G p,n n the followng order. For all k 1, k 2, 0 k 1 < k 2 n, the adversary add new members n subgraph G k 1 p before addng new members n subgraph G k 2 p. The way members are added n each subgraph G k p (0 k n) s descrbed n [9]. We frst prove that to mantan at each stage a constant level of qualty c, any algorthm needs to rebuld the current tree (.e. leads to a crtcal stage) after the adversary has added chosen members of Subgraph G k p (0 k n). To prove ths result (Lemma 2), we use the followng result, comng from [9], gven here wth our notatons. Lemma 1 [9] For every p 1, let N k be the last group bult by the adaptve adversary (descrbed n [9]) n Subgraph G k p (0 k n) of Graph G p,n descrbed above. Let T opt (N k ) be any optmal stener tree spannng N k and let T k be the tree spannng N k gven by any on-lne algorthm. We have: ( w(t k ) log2 ( N k 1) ) w(t opt (N k )) 2 Lemma 2 Let c > 1 be any constant. Let p = 4c and let k, 1 k n. Let N (resp. N ) be the group after all members n subgraphs G 0 p,..., G k p (resp. G 0 p,..., G k 1 p ) to be add have been added by our adaptve adversary. Let T opt (N) be any optmal stener tree spannng N and let T (resp. T ) be the tree spannng N (resp. N ) gven by any on-lne algorthm. If we have W (T ) c W (T opt (N)), then CS(T,..., T ) 1. Proof. We prove Lemma 2 by contradcton. Suppose that there exsts an on-lne algorthm such that there exsts k, 1 k n satsfyng W (T ) c W (T opt (N)) wth CS(T,..., T ) = 0 (.e. wthout crtcal stage). We frst upper bound W (T opt (N)) and lower bound W (T ). Let N 0,..., N k respectvely be the groups of members added by the adaptve adversary n subgraphs G 0 p,..., G k p. By structure of graph G p,n : w(t opt (N)) = w(t opt (N 0 )) + 1 n + w(t opt(n 1 )) + 1 n + + w(t opt(n k 1 )) + 1 n + w(t opt(n k )) = k 1 n + w(t opt(n k )) + w(t opt(n k )) + + w(t opt(n k )) 2 k 2 k w(t opt(n k )) (because for every k, 0 k n, all edges of subgraph G k p has weght 2 k ) = k n + w(t opt(n k )) k l=0 1 2 k = k n + ( k ) w(topt (N k )) 2w(T opt (N k )) (because as for every k, 0 k n, all edges of subgraph G k p has weght 2 k, w(t opt(n k )) 2 k 1 k n ) Let T k be the subtree of T spannng N k. As we also have w(t ) w(t k ), we obtan: W (T ) W (T opt(n)) w(t k ) 2w(T opt ( log 2 ( N k 1) )w(t opt(n k )) (N k )) 2w(T opt (by Lemma 1) (N k )) = log2 ( N k 1) = log 2(2 p + 1 1) = p c > c (because by [9], the adversary adds 2 p + 1 members n each subgraph G k p and because p = 4c ) Ths result contradcts W (T ) c W (T opt (N)), thus, Lemma 2 s proved by contradcton. 5

6 Man result of the secton. The followng Theorem shows that f we want a constant level of qualty, any on-lne algorthm leads to 1 Ω() crtcal stages (where s the number of added members). Theorem 3 Let c 1 be any constant. For any on-lne algorthm, for every 2 4c + 1, there exsts a graph G, there exsts M 0 M, such that f the algorthm returns a sequence of trees T 0,..., T (such that for every l, wth 0 l, T l spans M l ) respectng the qualty constrant wth level c, we have CS(T 0,..., T ) 2 4c Ω(). Proof. Let c be any constant c 1. We set p = 4c. Let 2 p + 1 = 2 4c + 1. Let G be the graph G p,n and M 0 M be the sequence of addtons of the adversary defned above. Thus, there exsts k, 1 k n such that k (2 p + 1) (k + 1) (2 4c + 1). Thus, we have: k 1 (1) For every k, 0 k k, let T k be the subtree of T spannng all the members added by the adaptve adversary n Subgraph G k p. CS(T 0,..., T 1 ) 1 CS(T 1,..., T 2 ) 1 Then, by Lemma 2, we have:. CS(T k 1,..., T k ) 1 CS(T 0,..., T k ) k CS(T 0,..., T ) k (because k 2 p + 1) CS(T 0,..., T ) 1 Ω() (by (1) and because c s constant) References [1] N. Alon and Y. Azar. On-lne stener trees n the eucldean plane. In Symposum on Computatonal Geometry, pages ACM Press, [2] G. Ausello, P. Crescenz, G. Gambos, V. Kann, A. Marchett-Spaccamela, and M. Protas. Complexty and approxmaton. Sprnger, [3] B. Awerbuch, Y. Azar, and Y. Bartal. On-lne generalzed stener problem. In SODA: ACM-SIAM Symposum on Dscrete Algorthms, pages 68 74, [4] A. Borodn and R. El-Yanv. Onlne computaton and compettve analyss. Camb. Unv. press, [5] A. Fat and G. J. Woegnger. Onlne algorthmes: The state of the art. LNCS no. 1442, Sprnger, [6] M. Garey and D. Johnson. Computers and ntractablty. In Freeman and compagny, [7] A. Goel and K. Munagala. Extendng greedy multcast routng to delay senstve applcatons. Algorthmca, 33(3): , [8] D. Hochbaum. Approxmaton algorthms for NP-hard problems. PWS publshng compagny, [9] M. Imase and B.M. Waxman. Dynamc stener tree problem. SIAM J. Dscr. Math., 4(3): , [10] G. Robns and A. Zelkovsky. Improved stener tree approxmaton n graphs. In SODA, pages Socety for Industral and Appled Mathematcs, [11] N. Thbault and C. Laforest. An optmal rebuldng strategy for a decremental tree problem. In SIROCCO, LNCS 4056, pages Sprnger, [12] N. Thbault and C. Laforest. An optmal rebuldng strategy for an ncremental tree problem. In Journal of Interconnexon Networks, accepted, [13] B. Waxman. Routng of multpont connectons. IEEE Journal on Selected Areas n Communcatons, 6(9): , [14] J. Westbrook and D. Yan. Lnear bounds for on-lne stener problems. Informaton Processng Letters, 55(2):59 63,