A Unicast-based Approach for Streaming Multicast

Transcription

1 A Unicat-baed Approach for Streaming Multicat Reuven Cohen and Gideon Kaempfer Department of Computer Science Technion, Haifa 32000, Irael Abtract Network layer multicat i know a the mot efficient way to upport multicat eion. However, for ecurity, QoS and other conideration, mot of the real-time application protocol can be better erved by upper layer (tranport or application) multicat. In thi paper we propoe a cheme called M-RTP for multicat RTP eion. The idea behind thi cheme i to et up the multicat RTP eion over a et of unicat RTP eion, etablihed between the variou participant (ource and detination) of the multicat eion. We then addre the iue of finding a et of path with maimum bottleneck for an M-RTP eion. We how that thi problem i NP-Complete, and propoe everal heuritic to olve it. I. INTRODUCTION Over the pat few year the Internet ha een a rie in the number of new application that rely on multicat tranmiion. In thee application, one ender end data to a group of receiver imultaneouly. In a network that upport IP multicat, the ender need to end only a ingle packet, which i then replicated by fork router in the multicat delivery tree. Many protocol for upporting Network layer multicat have been developed, like DVMRP [7], CBT [4], and PIM [8] for Intra-AS multicat and BGMP [3] for Inter-AS multicat. However, multicat can be upported not only in the network layer, but alo in the tranport layer or in the application layer. In uch a cae the replication of data i performed by the upper layer, while uing a unicat routing ervice from the network layer. A traightforward advantage of uch approach i that multicat application can be eecuted in network that do not upport IP layer multicat. However, thi approach i ueful for other reaon alo in cae where multicat i upported in the IP layer. A an eample, the approach preented in [2] can be viewed a upporting multicat in the UDP layer. The main benefit there i reducing the overhead aociated with etting up and maintaining a multicat tree in the network layer. Other reaon for upporting multicat uing unicat in the Network layer i to allow firewall to handle better the traffic created by the multicat application, and to facilitate the allocation of UDP port number (with unicat, each participant i free to elect it port number for each eion, wherea for multicat a global agreement i needed). However, the mot important advantage i Quality of Service (QoS): the proviioning, maintenance and tracking of QoS over a unicat path i known to be a much eaier problem than over a multicat tree. Therefore, by proviioning upper layer multicat over a et of unicat Network layer path it become much eaier to guarantee QoS for a multicat eion. The theoretical problem, the main concept and the algorithm preented in thi paper are general. They can therefore be employed for addreing multicat-related iue in different contet and under different aumption. However, we elected to preent them in the pecific contet of treaming multicat eion. We conider an RTP multicat eion, with a ource S that ditribute real-time traffic to a group of detination node. Providing QoS over a et of unicat path i feaible today uing the mechanim and tool provided by framework like MPLS and DiffServ. However, the proviioning of QoS in a DiffServ network for an IP multicat routing tree i a tough problem which ha not been addreed yet. Moreover, RTP ha a lightweight companion protocol called RTCP (Real Time Control Protocol), whoe main purpoe i to monitor the QoS of the RTP packet. RTCP i baed on the periodic tranmiion of control meage to all participant in the eion, uing the ame ditribution mechanim a the data packet [4]. RTP receiver provide reception quality feedback uing RTCP report. An RTCP report contain information on the highet equence number received, the number of packet lot, a meaure of the interarrival jitter and timetamp needed to compute an etimate of the round-trip delay between the ender and the receiver iuing the report. Thee report can be ued by the ender in order to etimate the QoS perceived by the detination and to adjut it tranmiion accordingly. For intance, when too many packet are lot, the ender can witch to a more aggreive compreion cheme, or require the network to allocate more bandwidth. However, when multicat i performed in the IP layer, analyi the RTCP report become a very difficult tak. A an eample, Figure how a multicat eion, whoe detination group conit of 2 hot: d and, auming that IP multicat i employed. Suppoe that the ender i informed that only 90% of the packet are received by d,and only 95% are received by.evenif know the eact tructure of the tree, it cannot determine how much of the lo hould be attributed to each of the 3 path compriing the tree:, d and. For intance, two poible lo pattern are: (a) no lo on, 0% lo on d and 5% lo on ; (b) 5% lo on, 5%loon d and no lo on. Between thee two etreme pattern, there eit an infinite number of additional different poible pattern. Therefore, there i no efficient method to improve the QoS in thi cae. Thi problem i alo recognized in [], where a mechanim that employ pecial probe from the ender to the receiver i propoed. A trivial way to upport a multicat application over a unicat routing layer i to et up a different unicat eion between the hot and every detination. However, thi approach overload the ource and ubtantially increae the required bandwidth compared to the cae where IP layer multicat i ued. Another approach would be to ue the fork node of the multicat tree (e.g. node in Figure ) a active node that participate in the higher layer protocol. For intance, if node participate in the RTP layer of the eion conidered above, 3 RTP eion are defined: between and, between and d, and between and. Since each RTP eion i aociated with it /0/$ IEEE 440 IEEE INFOCOM 200

2 RTP RTP d2 RTCP RTCP d RTP RTP RTCP RTCP Fig.. A imple multicat eion d4 Fig. 2. Eample of M-RTP d3 own RTCP, each of the 3 ender can eaily analyze the RTCP report it receive, in order to maintain the QoS of the tream it generate. Thi approach combine the advantage of IP layer multicat, namely calability and efficiency, with the advantage mentioned above of upper layer multicat. However, thi approach require the the network router to participate in higher layer protocol, in contrat to the fundamental deign principal of the Internet: put mart in the end of the network hot, leaving the core dumb. In thi paper we tudy a third approach, that can be viewed a a combination of the two propoed above. In thi approach, the multicat tree conit of multiple unicat path. However, only member of the multicat eion namely the ource and the detination node can it on the end point of the path. The network router are aumed to perform imple unicat routing only. Thi idea wa propoed in the firt time in [2], in the contet of increaing the calability of multicat tree maintenance. However, in thi paper we concentrate on the etablihment of uchatreewhilefulfilling the bandwidth requirement of a realtime application. We how that the problem of finding uch a tree with ufficient bandwidth i NP-Complete. We then propoe heuritic olution and analyze their performance. The ret of the paper i organized a follow. Section II preent the main concept of the conidered cheme called M- RTP. In Section III the routing problem aociated with M-RTP i tudied. It i hown that finding an M-RTP tree, namely a collection of path that reache each detination node, while guaranteeing the required bandwidth for the application i an NP-Complete problem. In Section IV we preent two heuritic for olving thi problem and in Section V we analyze the performance of thee algorithm. Finally, ection VI conclude the paper. II. THE M-RTP APPROACH FOR A MULTICAST RTP SESSION The main idea behind M-RTP i to replace a ingle multicat RTP eion with multiple unicat RTP eion. In M-RTP each multicat group member i involved in one RTP eion a a receiver, and in 0 or more RTP eion a a proy RTP ender. A an eample, conider a multicat RTP eion where i the ource and M = fd d 3 d 4 g i the detination group. Intead of etablihing a ingle RTP eion, an M-RTP-baed olution i preented in Figure 2. In thi olution node ha a unicat RTP eion with d and another unicat RTP eion with. In thee 2 eion, erve a the RTP ource, while d and erve a two independent RTP receiver. Hot d participate in 2 additional unicat RTP eion, but thi time a a proy RTP ender : with d 3 and d 4. M-RTP retrict the replication of data only to the partie actively participating in the multicat eion. A already indicated, thi retriction bear everal major advantage. The firt advantage i that thi approach require no multicat upport from the underlying routing layer. The econd advantage i that QoS proviioning become a much eaier tak. And finally, the maintenance of QoS i ignificantly treamlined due to the following two reaon. Firtly, the report received by every ender or receiver are related to a path rather than to a tree. Secondly, due to the RTCP bandwidth caling algorithm [4], the number of report received by every ender or receiver i jmj time larger than in the cae of a multicat tree though the total bandwidth conumed by thee report i equal in both cae. A poible diadvantage of M-RTP i the cot of the routing. For intance, in certain cae, M-RTP caue multiple copie of the ame application data to be tranmitted over a ingle link. For intance, Figure 3(a) preent an undirected graph over which a multicat eion from to M = fd g ha to be etablihed. Figure 3(b) preent a olution where a ingle multicat RTP eion i etablihed over a multicat tree. Figure 3(c) preent an M-RTP olution where the multicat eion i carried out uing 2 unicat RTP eion: from to d and from to. Wherea in Figure 3(b) each packet of the multicat eion i ent over the link ; only once, uch a packet i ent twice over the ame link in Figure 3(c). Although wort cae analyi may how thi to be a major problem, imulation ugget that the effect of M-RTP on routing cot i minimal. Unlike RTP mier and tranlator [4], M-RTP doe not impoe proceing of application data at the end point of an RTP con- In the terminology of RTP a defined in [4], a proy-rtp ender can be viewed a a imple RTP tranlator or RTP mier /0/$ IEEE 44 IEEE INFOCOM 200

3 y (a) (b) Fig. 3. Inefficient routing for M-RTP. nection. Hence, the only delay an RTP meage encounter due to M-RTP i the hort time required in order to forward the meage from one RTP connection to another, while updating the variou RTP header field. The idea of retricting the multicat tree to contain fork node only from the group of end node wa uggeted in the firt time in [2]. However, the algorithm preented there aimed at finding a collection of unicat route whoe overall cot i minimized. In contrat, the algorithm preented in thi paper, in the contet of M-RTP, aim at finding a collection of unicat route that have ufficient bandwidth to upport an RTP application. Reference [] III. THE PROBLEM OF ROUTING FOR M-RTP A. Introduction Many multicat routing algorithm have been propoed in the literature. Mot of thee eek to optimize the cot of the multicat tree that i normally defined a the um of cot aociated with the communication link ued by the tree. Preferring low cot tree may be a practical model for ome network, but many other model eem jut a practical. For intance, a minimum height tree i relevant for delay enitive application, and a maimum bottleneck tree i relevant for bandwidth enitive application. The actual bandwidth a multicat application can utilize i no more than the bandwidth of the mallet bandwidth link, or bottleneck, in the multicat tree. Thi i mot accurate in the cae of real-time multicat application where all the detination mut receive the ame data from the ource. For thi reaon, although a routing algorithm could contruct a low total cot multicat tree, the low utilization of thi tree, due to a local bottleneck, could render the tree much more epenive than intended. Therefore, maimum bottleneck tree may be of great practical importance. The communication network i often modeled a an undirected graph. Thi implie that only one weight i aigned to an edge, regardle of the direction in which it i ued. Thi implication i in many cae unrealitic and epecially in the cae where the weight aociated to an edge repreent it available bandwidth. For intance, conider a heavily loaded video erver connected to a network by a ingle link. Mot of the traffic going through thi link would be from the erver to the network, while the incoming traffic to the erver, containing mainly RTSP requet and RTCP report, would be only a mall fraction of the total traffic. Thu, even if the phyical bandwidth available in both direction of a link i equal, the available bandwidth on the link connecting the erver would probably be ditributed in (c) d d2 (a) d3 d d2 d3 Fig. 4. An M-RTP eion (a), and the virtual multicat eion tree with which it can be aociated a highly aymmetrical way. Thi i even more likely to occur if the phyical bandwidth i not ymmetrically ditributed, a for ADSL, cable-modem or atellite link. B. Graph theoretical problem related to routing for M-RTP Figure 4 how an eample for an M-RTP eion, and the aociated eion tree. A already eplained, M-RTP ue a et of unicat eion, that can be viewed a a virtual multicat eion tree Each fork node in thi tree mut be either the ource of the multicat RTP eion, or a member of the detination group. The major retriction on the tructure of the underlying eion tree i that every node with an outgoing degree of more than in the tree mut be either the ource or a member in the multicat group M (ee Figure 4(b)). Thi retriction will be referred to a the fork retriction. The formal definition of a new et of optimization problem related to M-RTP i given below: Problem : -Optimal Multicat Path Set Problem (MPSP()). Given a graph G(V E), a weight function w : E!R +, a group of active node A V, a multicat group M A and a multicat ource 2 M, find a et of path T that atifie a given optimality criterion and:. 8p 2 T the endpoint of p are vertice in A. 2. 8v 2 M there eit a path p v = v that i a concatenation of a ubet of path from T. The olution to MPSP() induce a virtual multicat eion tree panning M. Thi tree i a real panning tree in an induced graph G 0 (A E 0 ) where an edge between two vertice in A eit if and only if a path between thee vertice eit in the original graph G. Note, that a olution to MPSP() may actually induce a ubgraph that i not a real tree in G. For intance, the multicat eion tree depicted in Figure 4(b) i not a tree in the original graph hown in Figure 4(c) becaue it contain a directed cycle between d 3 and y. When A V, MPSP() reduce to the problem of finding a real tree rooted at and panning M while atifying. However, in thi paper we conider the verion of thi problem where A M, namely every end-point of the path in the et T i either the ource or belonging to the multicat group M. An intereting generalization i when A contain node in V ; M. Namely, allowing ome of the network router to have the capability of erving a end point of RTP eion. The baic optimization criterion uually defined for multicat tree i the multicat tree cot. The problem relevant to M-RTP i defined a follow: (b) /0/$ IEEE 442 IEEE INFOCOM 200

4 Problem 2: Minimum Sum Multicat Path Set Problem (MPSP(MinSum)). Find a olution T to MPSP()where i defined a minimizing: P p2t P e2p w(e). Thi definition penalize a olution for uing an edge everal time by multiplyingit weight by the number of time it i ued. For eample, conider again Figure 4(a), and aume that all edge in the graph have an equal cot of. The depicted M- RTP eion ha a cot of 8, ince the edge! i ued twice and the edge d 3! y i ued in both direction. If had been in group A, we could have an RTP eion between and, a eion between and d, and a third eion between and. The cot of the reulting eion would have been reduced from 8 to 7. From the ame conideration, the cot could have been reduced by one unit if node y had been in A. For the cae A V, MPSP(MinSum) reduce to the claic Steiner Tree Problem (STP) (ee [0]). Several variation on STP have been defined in the literature. Two uch variation are the Directed STP [3], defined like STP but pecifically for directed graph, and the Dynamic STP [], defined for dynamically changing multicat group. In [3], the SCTF algorithm for the contruction of directed Steiner Tree i preented. Applying a minor reviion to thi algorithm reult in an algorithm for the directed MPSP(MinSum) which achieve the eact ame approimation ratio that SCTF achieve for STP. Static and dynamic approimation algorithm for the undirected MPSP(MinSum) and it dynamic counterpart are preented in [2]. Thee algorithm achieve approimation ratio that are aymptotically equivalent to the ratio achieved for the non-retricted problem. However, in the contet of real-time RTP eion, the mot important criterion i the availability of bandwidth along the path of every eion. Let be the bandwidth of the data generated by the ource. Auming that the RTP node do not change the coding and the bandwidth, a legal multicat eion tree i one that conit of unicat path whoe available bandwidth i. To find uch a tree, we are eeking for the maimum bottleneck multicat tree. If thi tree ha a bandwidth larger than, then it can be ued for the conidered multicat eion. If the maimum bottleneck multicat eion tree ha a bandwidth maller than, then there i no poible way to etablih the conidered RTP eion in the network without uing multicat within the network. The maimum bottleneck verion of the multicat path et problem i defined a follow: Problem 3: Find a olution T to MPSP() where i defined a maimizing the minimal bandwidth allocated to a path in T. min feje2p p2t g jfpjp2t e2pgj More formally, i defined a maimizing: w(e), where w(e) i the bandwidth available on edge e for the multicat eion. For eample, in Figure 4, aume that all edge in the graph have an equal capacity of on every direction. The routing depicted ha two bottleneck path, d and of capacity 2, that hare a common edge!. Therefore, the bandwidth available for thi tree i =2. If we allowed the network node to upport multicat, namely node could receive a ingle packet from and end one copy to d and another to,thenwe could erve a multicat eion of rather than =2. Solving the non-retricted problem (namely the cae where A V ) for the maimal bottleneck optimization criteria, i.e. finding a tree panning the multicat group with a maimal value for min e2t fw(e)g, i conidered eay. Thi i becaue many polynomial time algorithm olve for thi problem. For intance, the SMMT algorithm in [5] olve thi problem if it i ued for weight. Another option i to run a variation on the Prim w(e) algorithm, a hown in Section IV-C. However, MPSP() inp- Hard for many natural criteria, including the maimal bottleneck criterion. C. On the Hardne of MPSP(MaBottleneck) M-RTP introduce a new perpective to multicat routing problem. It retrict the multicat fork node of the virtual multicat eion tree to the multicat group, thu introducing a baic contraint that mut be atified even before optimization criteria can be approached. Clearly, thi additional contraint leave previouly hard optimization problem, uch a STP, in the NP-Hard domain [0]. The quetion i whether other optimization problem, uch a the maimal bottleneck multicat tree, become any harder. In Section III-B a formal definition of optimization problem ubject to the fork retriction impoed by M-RTP wa introduced. In what follow we how that even without the optimization criterion of MPSP(), if every edge may be ued at mot once, finding a fork retricted routing i NP-Complete. From thi reult follow that MPSP(MaBottleneck) i NP-Har. In order to atify the fork retriction impoed by M-RTP, it i ufficient to find a et of path inducing a virtual tree panning the multicat group with fork node ecluively within thi group. We define the edge dijoint multicat path et problem (MPSP E ) a follow: Problem 4: Edge Dijoint Multicat Path Set Problem (MPSP E ). Given a graph G(V E), a group of active node A V,amulticat group M A and a multicat ource 2 M, findaet T of edge dijoint, but not necearily verte dijoint, path that atifie:. 8p 2 T the endpoint of p are vertice in A. 2. 8v 2 M there eit a path p v = v that i a concatenation of a ubet of path from T. The virtual multicat eion tree hown in Figure 4(a) i not a valid olution for MPSP E becaue the link! i ued twice. However, a olution to MPSP E define a virtual tree panning M but may till induce a ubgraph that i till not a tree. For intance, conider Figure 4(a) again, and uppoe that M conit of only d 3 and d 4. Then, the et of path f! y! d 3 d 3! y! d 4 g i a valid olution for MPSP E that i not a real tree becaue of the directed cycle between d 3 and y. MPSP E i cloely related to MPSP() ecept that the optimization criterion i replaced by the retriction that the path in the olution remain edge dijoint. In what follow, it will be hown that MPSP E i NP-Complete. The conequence of thi reult i that many natural optimization problem temming from MPSP() are NP-Hard too, ince MPSP E can be reduced to them. The hardne of MPSP E will be proven by reducing the 2 Thee reult pertain for directed graph. For undirected graph we have only partial evidence for the hardne of MPSP() /0/$ IEEE 443 IEEE INFOCOM 200

5 Directed Hamiltonian Circuit (DHC) problem [0] to it. DHC i defined a follow: Definition : Directed Hamiltonian Circuit (DHC). Given a directed graph G(V E), doe G contain a directed Hamiltonian circuit (i.e. a imple circuit that pae through all vertice in V )? The following theorem prove the hardne of MPSP E. Theorem : MPSP E on directed graph i NP-Complete Proof: We how a reduction from DHC: Given a directed graph G(V E), contruct the following directed graph G 0 (V 0 E 0 ) (ee Figure 5). Firt, chooe an arbitrary ource verte 2 V. V V V 0 4 = v in v mid v out jv 2 V nfg [ in mid mid2 out E 0 4 = (v in v mid ) (v mid v out )jv 2 V nfg [ ( in mid ) ( mid mid2 ) ( mid2 out ) [ (v out w in )j(v w) 2 E : Fig. 5. Contruction eample for the reduction from DHC to MPSP E. Define a multicat group M 0 = 4 v mid jv 2 V [ mid mid2 and a multicat ource 0 4 = mid2 in G 0. It i now hown that a olution to DHC on G eit if and only if a olution to MPSP E on G 0 eit auming A 0 M 0. A directed Hamiltonian circuit C in G can be tranformed into a directed Hamiltonian path P 0 in G 0 a follow. For every edge (u v) 2 C add the edge (u out v in ) to P 0. For every verte v 0 2 V 0 n 0 add the edge (v in v mid ) (v mid v out ) to P 0. Finally, add the edge ( in mid ) ( mid2 out ) to P 0.SinceP 0 touche all vertice in V 0 n mid mid2 eactly twice and touche mid mid2 eactly once, it mut be a directed Hamiltonian path in G 0 tarting at 0.Now,P 0 can be cut into a et T 0 of ubpath connecting vertice in M 0, thu creating a valid olution to MPSP E on G 0. To prove the other direction, let T 0 be a olution to MPSP E on G 0. T 0 i by definition a et of edge dijoint path, and by contruction, for every verte v 0 2 V 0, it in-degree or it out-degree (or both) i eactly. Therefore, T 0 mut be a et of path that form a ingle imple path P 0 when united. Thi path mut tart at mid2, travere every verte in M 0 and finally reach mid. Thu, it induce a directed Hamiltonian circuit in G. Clearly, MPSP E 2 NP, and thu MPSP E i NP-complete. MPSP E hold the eence of the fork retriction impoed by M-RTP. Therefore, it can be epected that any optimization problem baed on MPSP E i likely to be NP-Hard. Specifically, MPSP(MaBottleneck) i uch a cae. Theorem 2: MPSP(MaBottleneck) i NP-Hard. Proof: The following i a reduction from MPSP E to MPSP(MaBottleneck). Given an input tuple (G A M ) for MPSP E, we create an input tuple (G A M w 0 ) for MPSP(MaBottleneck) where w 0 (e). If the bottleneck of an optimal olution T for MPSP(MaBottleneck) i, it can be deduced that every edge in T i ued eactly once. Thu, T i a olution to MPSP E too. On the other hand, if the bottleneck of T i le than, no olution to MPSP(MaBottleneck) eit that ue every edge at mot once. Hence, no olution to MPSP E eit. IV. MAXIMUM BOTTLENECK ROUTING ALGORITHMS FOR M-RTP In thi ection we propoe two algorithm for approimating MPSP(MaBottleneck). In what follow the approimation ratio of an algorithm i defined a follow: Definition 2: Approimation ratio of algorithm A. For every intance p of a problem P,letB(p) be the value of the olution found by A and B (p) the value of the optimal olution. The approimation ratio of algorithm A i either: min p2p B(p) B (p) for maimization problem, or: B(p) ma p2p B (p) for minimization problem. The firt algorithm i referred to a the Widet Path Heuritic (WPH). It achieve a olution that i no wore than time O(jMj) the bottleneck ize of the optimal olution, where jmj i the ize of the multicat group. Although thi approimation ratio can be achieved by other algorithm, imulation how that WPH perform nearly optimally (ee Section V). The econd algorithm, referred to a the Double Tree Heuritic (DTH), achieve an approimation ratio that i dependent on the graph aymmetry rather than the ize of the multicat group. DTH aume that every directed edge u! v ha an antiymmetric edge v! u. In ymmetrically loaded network, the ratio guaranteed by DTH can be better than the ratio guaranteed by WPH, even if the multicat group i of mall ize. For undirected graph, the approimation ratio DTH achieve i eactly. 2 A. The Widet Path Heuritic (WPH) WPH build a virtual tree panning M. The algorithm tart with the ource verte. In each iteration of the algorithm, /0/$ IEEE 444 IEEE INFOCOM 200

6 amaimumreidual bottleneck path i conidered from every coveredverteinm to every verte in M that ha not yet been covered. The reidual weight of an edge e ued n time by the algorithm i defined a w(e),wherew(e) i the available bandwidth of the edge. The path with the maimal bottleneck i n+ added to the olution. Thu, after each iteration, at leat one more verte from M i covered. A hown later, thi algorithm achieve an approimation ratio of in the wort cae. A O(jMj) formal definition of the algorithm i given below: Algorithm : Widet Path Heuritic (WPH).. Initialization: T X fg. 2. While MnX = do (a) Find the maimal reidual bottleneck path p from X to a verte m in MnX. (b) X X [ m. (c) Update all reidual weight of edge in p (i.e the new reidual weight of an edge e ued n time o far i w(e) ). n+ (d) T T [ p. 3. T i the deired olution. The implementation of finding the maimal reidual bottleneck path in tep (2a) i imilar to the contruction of a maimal bottleneck tree. A variation of the Prim algorithm for finding a minimum panning tree [2] can be ued a follow. Two et of vertice are ued: a et of known vertice and a et of unknown vertice. In each iteration, the unknown verte connected by the edge of greatet capacity to a known verte i added to the known et. The algorithm terminate when all unknown vertice become known. If the initial et of known vertice i a ingle verte, the algorithm contruct a maimal bottleneckpanningtree rootedat (ee Section IV-C). We ue thi algorithm lightly differently for the implementation of tep (2a). The initial et of known vertice i the et of vertice X already covered by the algorithm. The algorithm top a oon a an unknown verte of M become known. Every time a verte i added to the known et, the edge through which it i dicovered i remembered. At the end of the algorithm, a maimal bottleneck path from a verte in the initial known et can be recontructed by backtracking thee tored edge from any initially unknown verte. In thi manner, a minimal bottleneck path i found at the end of the Prim algorithm variant. For a formal proof, ee Section IV-C. The algorithm can be implemented in O(jEj + jv j log jv j) time uing a Fibonacci heap [9]. Hence, the total time compleity of WPH i no more than O(jMj(jEj + jv j log jv j)). Lemma : WPH achieve an approimation ratio of. O(jMj) Proof: Denote by B(X) the bottleneck of a olution X for MPSP(MaBottleneck), and by B(p) thereidual bottleneck of a path p. Let T be an optimal olution, and T a olution found by WPH. Let p = y be the lat path added to T with B(p) =B(T ). According to tep (2a) of the algorithm, when p i added to T it ha a larger bottleneck than any other path p 0 connecting a verte from the multicat group already covered by WPH with a verte from the multicat group yet to be covered, namely: 8p 0 2f 0 y 0 j 0 2 X y 0 2 MnXg B(p) B(p 0 ): () d K d K- d K-2 d 2 3 K- K Fig.. A wort cae network for WPH Since 2 X at all tage of the algorithm, () implie that: 8p 0 = y where y 2 MnX B(p) B(p 0 ): (2) Let p be a imple path embedded over the path connecting to y in the optimal olution T (e.g. in Figure 4(a), the path from to d 4 i! y! d 3! y! d 4 wherea a imple path embedded on thi path i! y! d 4 ). By (2), at the time p i elected, B(p) B(p ) hold. Hence, we have: B(T )=B(p) B(p ): (3) Let e be an edge in p with the minimal weight. Every edge can be ued no more than jmj; time during the algorithm, ince all maimal bottleneck path (found in tep (2a)) are imple path, and no more than jmj; iteration of the while loop are performed. Hence: B(p ) w(e ) jmj; B(T ) jmj; : (4) Thu, by (3) and (4), B(T ) B(T ). O(jMj) The ratio of i a trict bound a demontrated in O(jMj) the following eample. Conider the directed graph in Figure. miing antiymmetric edge may be defined a having a weight of le than jmj. For implicity of preentation, thee edge are ignored. Clearly, an optimal olution for MPSP(MaBottleneck) on the above defined network, mut have a bottleneck of at the mot, and uch a olution indeed eit: T = 4 f!! d K g[fd i! d i; j2 i Kg. However, WPH may cover verte d i in the i th iteration and the olution it find i: T = f!! d i j i Kg. Thi olution ha a bottleneck of ince the edge ( ) i ued K jmj; time. Thu, the approimation ratio achieved in thi eample i O(jMj). B. The Double Tree Heuritic (DTH) DTH i aimed at achieving an approimation ratio that i independent of the ize of the multicat group. The aymmetry of agraphgi defined a the maimal ratio between the weight of /0/$ IEEE 445 IEEE INFOCOM 200

7 a pair of antiymmetric edge. Formally, the following notation propoed in [3] i ued: Definition 3: Maimum edge aymmetry. Let G = (V E) be a directed graph with a weight function w : E!R +.Define 4 mafw(u v) w(v u)g m(g) = ma (u v)2e minfw(u v) w(v u)g a the maimum edge aymmetry. Unlike WPH, DTH aume that every directed edge u! v ha an antiymmetric edge v! u. The algorithm ha three phae. In the firt phae, the weight of the bottleneck of a maimal bottleneck tree panning M i computed. Thi can be done by contructing a maimal bottleneck tree, uing the variation on Prim MST algorithm preented in Section IV-C. In the eample depicted in Figure 7, the bottleneck ize i. In the econd phae, from all the tree whoe bottleneck i equal to the maimal bottleneck, a tree with a maimal revere bottleneck i found. The revere bottleneck of an outgoing directed tree i defined a the minimum of the antiymmetric weight. Note that thee tree need not atify the fork retriction. The econd phae can again be implemented uing a variation on the Prim algorithm. In the eample, thi tree (c) differ from the firt tree contructed in (b) ince the revere bottleneck achieved i 3 (becaue of d! ) wherea the revere bottleneck of the firt tree contructed i 2 (becaue of! d ). In the third phae, the tree found in the econd phae i tranformed into a et of path with endpoint in the multicat group. The path are contructed by touring the econd tree in a depthfirt manner. Thi tour i broken into ubpath uch that every time a verte belonging to the multicat group i paed, the previou ubpath i ended and a new one i initiated. Finally, unneeded path that lead to multicat detination already reached by previou path from our olution can be removed. In the eample, the full tour include three path:!! d, d!! and!!. The lat path return to and may therefore be omitted (d). The bottleneck of the olution found by DTH in the eample i 3 (becaue of the edge d! ) wherea the optimal olution (e) achieve a bottleneck of. Note that any olution produced by DTH ue every edge once at the mot. The algorithm can be implemented to run in O(jEj + jv j log jv j) time. Lemma 2: DTH achieve an approimation ratio of. O( m (G)) Proof: Denote by B(X) and B r (X) the bottleneck and the revere bottleneck of X repectively. Denote by T an optimal olution to MPSP(MaBottleneck), by T the olution found by DTH, and by T 2 the tree found in the econd phae of DTH. DTH guarantee that: B(T ) minfb(t 2 ) B r (T 2 )g Note that the bottleneck of an optimal non-retricted panning tree i clearly greater or equal to the bottleneck of an optimal olution to MPSP(MaBottleneck). Thu, and ince T 2 i an optimal non-retricted panning tree, if B(T 2 ) B r (T 2 ) then B(T ) B(T 2 ) B(T ), namely B(T )=B(T ). If B(T 2 ) >B r (T 2 ),let~e r be a revere edge in T 2 atifying w(~e r ) = B r (T 2 ). Let ~e be the antiymmetric partner of ~e r. Since w(~e) B(T 2 ) > B (T 2 ) = w(~e ), bydefinition 3: m(t 2 ) w(~e) w(~e r ). Therefore: B r (T 2 )=w(~e r ) Thu, B r (T 2 ) B(T2) m(t 2) hold, and w(~e) m(t 2 ) B(T 2) m(t 2 ) B(T ) B r (T 2 ) B(T 2) m(t 2 ) B(T ) m(t 2 ) B(T ) m(g) Therefore, regardle of the relation between B(T 2 ) and B r (T 2 ), it alway hold that: B(T ) B(T ) m(g) Although the wort cae performance of DTH can be guaranteed by performing the third phae on any maimal bottleneck tree rooted at and panning M, and not necearily the one that ha the maimal revere bottleneck, the econd phae i performed in order to improve the average performance. Thi i clearly demontrated in imulation (ee Section V). C. An Algorithm for Finding a Maimal Bottleneck Tree In Section IV-A a variant of the Prim algorithm [2] wa introduced. For a given graph G(V E), thi algorithm build a maimal bottleneck panning tree T rooted at a given verte. A formalization of thi algorithm and proof of it correctne i given hereafter. Algorithm 2: Maimum Bottleneck Tree (MBT) (Prim variant).. K U V 2. predeceor[] :=NULL bottleneck[] := 8v 2 Unfg bottleneck[v] :=0 Q fg 3. while Q = (a) Let v beaverteinq with the maimal bottleneck[v] (b) K K [fvg U Unfvg (c) for every edge v! e u uch that u 2 U: if bottleneck[u] < min fbottleneck[v] w(e)g then i. bottleneck[u] :=minfbottleneck[v] w(e)g ii. predeceor[u] :=v iii. Q Q [fug (d) Q Qnfvg Thi algorithmcontruct a predeceor array that define a tree rooted at. Clearly, if the graph i connected, thi tree pan the entire graph G(V E). Otherwie, only the vertice found in K at the end of the algorithm are reached by thi tree. In the following lemma it will be hown that thi i a maimal bottleneck tree. Lemma 3: MBT contruct a maimal bottleneck tree panning K. Proof: It uffice to how that for every verte v, ifapath from to v containing only edge in predeceor eit, it i a maimal bottleneck path connecting to v with a bottleneck /0/$ IEEE 44 IEEE INFOCOM 200

8 d d d (a) Input graph (b) Phae I: Ma bottleneck panning tree (c) Phae II: Ma bottleneck panning tree with ma revere bottleneck d d (d) Phae III: Tree i tranformed into path. (e) Optimal Fig. 7. The DTH algorithm equal to bottleneck[v]. We prove by induction on the order by which vertice are added to the et of known vertice K. For thi i trivial. Aume the correctne of the induction aumption for all vertice in K at the beginning of tep (3). We will prove the aumption for the verte v added to K in tep (3b). Let w 4 = predeceor[v]. By tep (3c), w i already in K. Thu, by the aumption, bottleneck[w] i the ize of a maimal bottleneck path from to w contained in predeceor. Alo, by tep (3(c)i) and (3(c)ii), thi path can be etended to v by the edge predeceor[v] thu creating a path with a bottleneck of bottleneck[v]. Aume toward a contradiction, that a maimal bottleneck path p from to v ha a bottleneck b that i greater than bottleneck[v]. p mut pa from the et of known vertice K to the et of unknown vertice U by mean of an edge e! y, ince 2 K and v 2 U. Sinceb w(e) and 2 K, bytep (3(c)i): bottleneck[y] minfbottleneck[] w(e)g b > bottleneck[v]. Thu, by tep (3a), y hould have been added to K before v -a contradiction. A variant of MBT can be ued to contruct a maimal bottleneck tree panning only a ubet X of the vertice in the graph. To achieve thi, the algorithm mut be topped immediately after the lat verte in X ha become known. A formal proof of the correctne of thi algorithm i imilar to the proof for SMMT in [5]. Conider a minor change to MBT that allow more than one verte to be initially put in the known et of vertice K: Inthe decription of the algorithm, change every appearance of into M, wherem i a et of vertice. The reulting algorithm, find a maimal bottleneckpath fromm to any verte reachable from M. The proof for thi algorithm can be contructed by changing every appearance of into M in the above lemma. V. SIMULATION RESULTS In the previou Section IV two approimation algorithm for MPSP(MaBottleneck) were preented and analyzed for wort cae performance. In thi ection we preent the reult thee algorithm achieve on actual graph that hopefully repreent realworld communication network. The main concluion of thee imulation i that the propoed algorithm, and in particular WPH, behave in practice almot optimally. Two verion of DTH are conidered. DTH i the verion preented in the previou ection, wherea DTH i DTH without the econd phae (i.e. the path in the olution are created by running the third phae of the algorithm on an arbitrary maimal bottleneck tree). DTH i preented only in order to ae the benefit of the econd phae of DTH. The algorithm were teted on 500 randomly generated network, baed on the model ued in [2]. The weight (capacitie) of the edge were initialized at random value uniformly ditributed between 2 an2. For every edge e with initial weight w(e) ued i time in the above path, the edge weight wa updatedtobe w(e). On every graph, the algorithm were run on i+ every multicat group ize between 3 and 99 (thi ize include the ource node). The multicat group were reelected at random for every graph and every group ize. Since MPSP(MaBottleneck) i NP-Hard (ee Section III-C), the optimal olution could not be computed. Intead, the bottleneck of a maimal bottleneck tree rooted at the multicat ource and panning the detination wa computed. Thi bottleneck i an upper bound on the value of the optimal olution ince i diregard the fork retriction impoed by M-RTP. A hown in the following, it frequently equaled the optimal olution. In Figure 8, the performance of the evaluated algorithm i preented a a function of the multicat group ize. The mot prominent property of the graph i that the optimal olution and the performance of the algorithm rapidly decline with the multicat group ize. Thi behavior i epected ince the greater the multicat group ize, the more edge need to be ued on the /0/$ IEEE 447 IEEE INFOCOM 200

9 %RWWOHQHFN VL]H RI RSWLPDO XSSHU ERXQG 0XOWLFDVW JURXS VL]H 237 :3+ Fig. 8. Algorithm performance veru multicat group ize 0XOWLFDVW JURXS VL]H '7+ '7+ :3+ '7+ '7+ Fig. 9. Algorithm relative performance veru multicat group ize average, and thu the probability of uing a low capacity edge increae. In practice, WPH perform almot optimally, regardle of the multicat group ize, depite the theoretical approimation ratio of. DTH perform relatively wore than WPH averaging O(jMj) 78% of the optimal olution. DTH i conitently wore than DTH, averaging 45% of the optimal olution, thu jutifying the econd phae of DTH. In Figure 9, the relative performance of the evaluated algorithm i preented a a function of the multicat group ize. Thi graph highlight the reult dicued previouly. Our main concluion i that WPH i in practice almot optimal for any multicat group ize. Although WPH i the only algorithm propoed that may caue the ame data to be tranmitted more than once over the ame link, in practice thi rarely occur. For very large group ize, DTH may be preferable, ince it too achieve almot optimal olution within le time. Throughout our tudy we have alo meaured the cot penalty induced by the retriction that data i replicated only by the member of the multicat eion. We have compared the cot of every tree to the cot incurred by the MBT algorithm, propoed in Section IV-C, which find the maimal bottleneck tree without thi retriction. It turn out that the average cot of the tree found by both WPH and DTH i only 5-5% higher than the average cot of the optimal tree found by MBT. Thee reult are in accordance with thoe preented in [2]. VI. CONCLUSIONS We have propoed a cheme called M-RTP for multicat RTP eion. The main idea behind thi cheme i to et up the multicat eion over a et of unicat path between the participant of the multicat eion. The main advantage of thi approach i the eae of QoS proviioning and maintenance. In particular, the information carried by RTCP report can be more accurately analyzed in order to increae the allocated bandwidth over a path whoe QoS fall hort of epectation. We have tudied the problem of etting up a multicat tree under the contraint impoed by the new cheme. Thi gave rie to a new family of problem called MPSP (Minimum Path Set Problem). We have concentrated on the MPSP(MaBottleneck) variation of thi problem, which eek for a et of path that guarantee maimum bottleneck. Thi problem wa hown to be NP-Complete an heuritic were propoed: the Widet Path Heuritic (WPH) and the Double Tree heuritic (DTH). For each algorithm we analyzed the approimation ratio analytically, and the average performance by mean of imulation. The imulation how that WPH i almot optimal for any group ize. REFERENCES [] A. Adam, T. Bu, J. Horowitz, D. Towley, R. Cacere, N. Duffield, F. Lo- Preti, S. Mon, and V. Paon. The ue of end-to-end multicat meaurement for characterizing internal network behaviour. IEEE Communication Magazine, 38(5), [2] E. Aharoni and R. Cohen. Retricted dynamic Steiner tree for calable multicat in datagram network. IEEE/ACM Tranaction on Networking, (3), June 998. [3] A. Almeroth. The evolution of multicat: from the mbone to internet2 deployment. IEEE Communication Magazine, 4(), [4] T. Ballardie, P. Franci, and J. Crowcroft. Core baed tree (CBT). In SIGCOMM Sympoium on Communication Architecture and Protocol, page 85 95, September 993. [5] Charle Chiang, Majid Sarrafzadeh, and C. K. Wong. Global routing baed on Steiner min-ma tree. IEEE Tranaction on Computer Aided Deign, 9(2):38 325, December 990. []. Chu, S. Rao, and H. Zhang. A cae for end ytem multicat. In ACM Sigmetric, Santa Clara, CA, June [7] S. Deering, C. Partridge, and D. Waitzman. Ditance vector multicat routing protocol. RFC-075, November 988. [8] Stephen Deering, Deborah L. Etrin, Dino Farinacci, Van Jacobon, Liu Ching-Gung, and Liming Wei. The PIM architecture for wide-area multicat routing. IEEE/ACM Tranaction on Networking, 4(2):53 2,April 99. [9] Michael L. Fredman and Robert Endre Tarjan. Fibonacci heap and their ue in improved network optimization algorithm. Journal of the ACM, 34(3):59 5, July 987. [0] Michael R. Garey and David S. Johnon. Computer and Intractability - A Guide to the Theory of NP-Completene. W. H. Freeman and company, 979. [] Makoto Imae and Bernard M. Waman. Dynamic Steiner tree problem. SIAM Journal on Dicrete Mathematic, 4(3):39 384, Augut 99. [2] R.C. Prim. Shortet connection network and ome generalization. Bell Sytem Technical Journal, 3():389 40, November 957. [3] S. Ramanathan. An algorithm for multicat tree generation in network with aymmetric link. In Proceeding of the Conference on Computer Communication (IEEE Infocom), page , San Franico, California, March 99. [4] H. Schulzrinne, S. Caner, R. Frederick, and V. Jacobon. RTP: a tranport protocol for real-time application. RFC-889, January /0/$ IEEE 448 IEEE INFOCOM 200