THE deployment of router-based IP multicast has been

Transcription

1 88 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO. 9, SEPTEMBER 2007 ACOM: Ay-Source Capacity-Costraied Overlay Multicast i No-DHT P2P Networks Shipig Che, Baile Shi, Shigag Che, ad Ye Xia Abstract Applicatio-level multicast is a promisig alterative to IP multicast due to its idepedece from the IP routig ifrastrucre ad its flexibility i costructig the delivery trees. The existig overlay multicast systems either support a sigle data source or have high maiteace overhead whe multiple sources are allowed. They are iefficiet for applicatios that require ay-source multicast with varied host capacities ad dyamic membership. This paper proposes ACOM, a ay-source capacity-costraied overlay multicast system, cosistig of three distributed multicast algorithms o top of a o-dht overlay etwork with simple strucres (radom overlay with a o-dht rig) that are easy to maage as odes joi ad depart. The odes have differet capacities, ad they ca support differet umbers of direct childre durig a multicast sessio. No explicit multicast trees are maitaied o top of the overlay. The distributed executio of the algorithms arally defies a implicit, roughly balaced, capacity-costraied multicast tree for each source ode. We prove that the system ca deliver a multicast message from ay source to all odes i expected Oðlog c Þ hops, which is asymptotically optimal, where c is the average ode capacity ad is the umber of members i a multicast group. Idex Terms Ay-source overlay multicast, peer-to-peer etworks, distributed multicast algorithms. Ç INTRODUCTION THE deploymet of router-based IP multicast has bee slow due to the requiremet for global deploymet of multicast-capable routers, the lack of ISP support, ad the state scalability problem caused by a large umber of multicast groups. Applicatio-level multicast becomes a promisig alterative that ca be quickly deployed without the depedecy o the router ifrastrucre [], [2], [3], [4]. Recet papers [5], [6], [7], [8], [9], [0] sdied overlay multicast from differet aspects. However, they are isufficiet i supportig applicatios that require aysource multicast with varied host capacities ad dyamic membership. May works focus o desigig a optimized overlay multicast tree for a sigle data source [], [2], [3], [4]. They are suitable for video or software distributio but ot for distributed games, telecoferecig, viral classrooms (discussio sessios), or multimedia chat groups, where may or all odes ca potetially be the data sources at ay time. A multicast tree that is optimal for oe source may be bad for other sources. O the other had, oe tree per member will be too costly. To add to the problems, member hosts may vary widely i their capacities i terms of upload badwidth, memory, ad CPU power. Some are able to support a large umber of direct childre, but others support few. Furthermore, members may joi ad leave at ay time, which makes the maiteace of the overlay etwork ad the multicast trees. S. Che is with the Network Ceter, Uiversity of Shaghai for Sciece ad Techology, Shaghai, , Chia. chesp@usst.edu.c.. B. Shi is with the Departmet of Computer ad Iformatio Techology, FuDa Uiversity, Shaghai, , Chia. bshi@fuda.edu.c.. S. Che ad Y. Xia are with the Departmet of Computer ad Iformatio Sciece ad Egieerig, Uiversity of Florida, Gaiesville, FL {sgche, yx}@cise.ufl.edu. Mauscript received 2 Nov. 2005; revised 7 Jue 2006; accepted 0 July 2006; published olie 9 Ja Recommeded for acceptace by Y. Pa. For iformatio o obtaiig reprits of this article, please sed to: tpds@computer.org, ad referece IEEECS Log Number TPDS Digital Object Idetifier o. 0.09/TPDS a critical issue. Fially, the system should be fully distributed ad scalable to a large Iteret group. The goal of this paper is to sdy low-maiteace overlay systems that support distributed applicatios requirig ay-source capacity-costraied multicast with dyamic membership. Noe of the existig systems to be surveyed i the ext sectio meets all these requiremets. I order to hadle ay-source multicast i dyamic groups, ovel proposals were made to implemet multicast o top of strucred P2P etworks [5], [6], [7]. However, the overhead of maitaiig DHT-based multicast overlays is a major desig cocer [8]. Moreover, these systems assume each ode has the same umber of childre. Host heterogeeity is ot addressed. Eve though overlay multicast ca be implemeted o top of overlay uicast, they have very differet requiremets. I overlay uicast, low-capacity odes affect oly traffic passig through them ad, therefore, they create bottleecks of limited impact. I overlay multicast, all traffic will pass all odes i the group ad the multicast throughput is decided by the ode with the smallest throughput, particularly i the case of reliable delivery. The strategy of assigig a equal umber of childre to each itermediate ode is far from optimal. If the umber of childre is set too big, the low-capacity odes will be overloaded, which slows dow the etire sessio. If the umber of childre is set too small, the highcapacity odes will be uderutilized. CAM-Chord [9], a multicast extesio of Chord, allows odes to have a differet umber of eighbors. It achieves roughly balaced multicast trees, which helps reduce both delay ad delay jitter, but its maiteace overhead is large for highly dyamic multicast groups. This paper proposes ACOM, a ay-source capacitycostraied overlay multicast service, cosistig of three distributed multicast algorithms o top of a o-dht overlay etwork with simple strucres (radom overlay with a o- DHT rig) that are easy to maage as odes joi ad depart. At rutime, the distributed executio of a proposed algorithm arally defies a implicit, roughly balaced, capacity-costraied multicast tree for deliverig a message /07/$25.00 ß 2007 IEEE Published by the IEEE Computer Society

2 CHEN ET AL.: ACOM: ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST IN NON-DHT P2P NETWORKS 89 from a arbitrary source ode. No explicit multicast trees are built ad maitaied. I ACOM, the odes may have differet capacities ad the umbers of direct childre that they ca support i a multicast sessio may vary. Each ode keeps a umber of radom eighbors equal to its capacity. We prove that the system ca deliver a multicast message from ay source to all odes i a expected Oðlog c Þ hops, which is asymptotically optimal, where c is the average ode capacity ad is the umber of members i a multicast group. I compariso, CAM-Chord requires Oðlog c Þ times more eighbors per ode. Our observatio is that the DHT-based overlay strucres desiged for file lookup are ot optimal for multicast; other, simpler strucres should be explored. We perform extesive simulatios to evaluate the performace of the proposed system, ad we provide detailed discussios o a variety of implemetatio issues, icludig how to maitai the overlay, how to measure the system parameters, ad how to hadle low-capacity odes, proximity, as well as dyamic capacities. The rest of the paper is orgaized as follows: Sectio 2 discusses the related work. Sectio 3 defies the problem ad the etwork model. Sectio 4 motivates our approach, gives a basic versio of the distributed multicast algorithm, ad provides a thorough aalysis. Sectio 5 discusses the implemetatio issues. Sectio 6 describes two improved multicast algorithms. Sectio 7 presets the simulatio results. Sectio 8 draws the coclusio. 2 RELATED WORK Shi et al. proved that costructig a miimum-diameter degree-limited spaig tree is NP-hard [20]. Note that the terms degree ad capacity are iterchageable i the cotext of this paper. Cetralized heuristic algorithms were proposed to balace multicast traffic amog multicast service odes (MSNs) ad to maitai low ed-to-ed latecy [20], [2]. The algorithms do ot address the dyamic membership problem, i.e., MSN joi/deparre. A umber of overlay multicast systems have bee proposed to optimize a sigle-source multicast tree. Bullet [] is desiged to improve the throughput of data dissemiatio from oe source to a group of receivers. A overlay tree rooted at the source is first established. Disjoit data objects are delivered from the source via the tree to differet receivers. The receivers the commuicate amog themselves to retrieve the missig objects. The direct commuicatio betwee receivers adds more badwidth to the tree dissemiatio. Overlay Multicast Network Ifrastrucre (OMNI) [2] dyamically adapts its degree-costraied multicast tree to miimize the latecies to the etire cliet set. Riabov ad Zhe Liu proposed a cetralized costatfactor approximatio algorithm for the problem of costructig a sigle-source degree-costraied miimumdelay multicast tree [3]. Yamaguchi et al. described a distributed algorithm that maitais a degree-costraied delay-sesitive multicast tree for a dyamic group [4]. The above algorithms are desiged for a sigle source. They are ot suitable for multisource scearios (e.g., distributed games ad multimedia chat groups), which are the target applicatios of this paper. Buildig oe tree for each possible source is too costly. Usig a sigle tree for all sources is also problematic. First, a miimum-delay tree for oe source may ot be a miimum-delay tree for other sources. Secod, the sigle-tree approach cocetrates the traffic o the liks of the tree. Third, a sigle tree may be partitioed beyod repair for a dyamic group. Bayeux [5] ad Borg [6] were proposed to support applicatio-level multicast based o Tapestry [22] ad Pastry [23], respectively, ad CAN-based Multicast [7] was implemeted o top of CAN [24]. El-Asary et al. sdied efficiet broadcast i a Chord etwork, ad their approach ca be adapted for the purpose of multicast [25]. Castro et al. compares the performace of tree-based ad floodig-based multicast i CAN-style versus Pastry-style overlay etworks [8]. Oe of the coclusios is that the overhead of maitaiig DHT-based multicast overlays is a major desig cocer. With each ode havig the same umber of childre, the above systems do ot cosider the heterogeeity i host capacities. 3 MODEL Cosider a multicast group G of odes. Each ode x 2 G has a capacity c x, specifyig the umber of direct child odes to which x is willig to forward the received multicast messages. The value of c x should be made roughly proportioal to the upload badwidth of ode x. The upload badwidth is likely to be the bottleeck because the commo cable-modem or DSL liks are highly asymmetric ad biased agaist upload. Moreover, a ode x may have to forward c x copies for each received message. I a heterogeeous eviromet, the capacities of differet odes may vary i a wide rage. The capacity of the same ode may also chage over time (Sectio 5.6). We wat to costruct a multicast service, which meets the capacity costraits of all odes, allows frequet membership chages, ad delivers multicast messages from ay source to the group members via a dyamic, balaced multicast tree. To achieve this goal, a overlay etwork is established for each multicast group, which trasforms the multicast problem to a broadcast oe withi the scope of the overlay. A sigle delivery tree for all data sources is ot a good idea because it cocetrates traffic o a small umber of overlay liks. O the other had, oe tree per source is too costly to maitai for highly dyamic groups. A ideal approach is to use implicit per-source multicast trees that are ot explicitly built ad thus icur o maiteace overhead. These implicit trees chage arally at zero cost as the overlay topology chages. Before presetig how this ca be achieved, we shall describe the o-dht (Distributed Hash Table) overlay etwork that we use. The overlay etwork cosists of two compoets. Oe is a urestricted rig that coects all odes. The other is a radom graph amog the odes. The urestricted rig is fudametally differet from the DHT-based rig foud i may strucred P2P etworks. I the DHT-based rig, each ode has a specific locatio, which makes the maiteace of the rig difficult whe a ew ode jois the etwork. I the urestricted rig, a ode ca be aywhere i the rig. The rig maiteace is trivial. Each ode keeps the ext ode o the rig as oe of its eighbors, called the successor. As log as a ew ode x kows aother ode z that is already i the rig, it iforms z to set x as the successor, ad the it sets its ow successor to be z s previous successor. I additio to the successor, a ode x has c x or more other eighbors radomly selected from the set of odes. Details about topology maiteace ad proximity will be discussed later i Sectio 5. We assume that x is willig to support both the successor ad the. These copies follow differet Iteret paths to their respective destiatios, which makes the upload lik of x the oly shared bottleeck.

3 90 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO. 9, SEPTEMBER 2007 c x radom eighbors as its childre i a multicast sessio. A TCP coectio is established betwee two eighbors for data ad cotrol commuicatio. Whe there is o data commuicatio, TCP trasmits keep-alive segmets. The deparre of a eighbor is detected whe the TCP coectio times out due to the lack of keep-alive segmets. To prevet the rig from beig broke after a abrupt ode deparre, each ode should also kow a umber of odes after the successor dow the rig. It also kows the previous ode o the rig, called the predecessor. However, these odes will ot be used as eighbors for data commuicatio. I the existig literare, the strucred P2P etworks all use DHT, ad the ustrucred P2P etworks do ot use DHT. I this sese, our overlay etwork belogs to the ustrucred category. O the other had, it does have (o- DHT) strucres, a rig, ad a radom etwork. We may reclassify P2P etworks i three categories, DHT-based strucred, o-dht strucred, ad ustrucred, with the secod category icludig all P2P etworks that do ot use DHT but have topological strucres, ad the third category icludig all P2P etworks that assume arbitrary topologies. 4 ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST 4. Motivatio Radom Walk, Limited Floodig, ad Probabilistic Floodig. A overlay etwork for multicast is fudametally differet from a overlay etwork for file sharig such as Gutella or Kazaa. I file-sharig applicatios, the existig approaches of radom walk, limited floodig, ad probabilistic floodig work well whe the searched object has may copies stored i dispersed locatios of the etwork. Searchig oly a small portio of the odes will r up the object with a good probability. Missig a existig object with a small probability is ot cosidered to be a serious problem. For multicast, the requiremet is differet. A overlay etwork is costructed amog the odes of a multicast group. Multicast is implemeted as broadcast. Every ode of the group is supposed to receive a copy of every message, which either radom walk or limited floodig ca guaratee. By forwardig a message (received for the first time) to eighbors with a certai probability, the approach of probabilistic floodig also caot guaratee reachig every ode, eve whe the overlay topology itself is coected. Efficiet broadcastig i a o-dht overlay has bee a challegig problem. Hop Complexity ad Commuicatio Complexity. We itroduce two performace metrics for overlay multicastig with odes havig differet capacities. Let c be the average ode capacity. Other performace cosideratios will be discussed later. The hop complexity is defied as the expected umber of overlay hops it takes for a multicast message to reach ay ode. The commuicatio complexity is the expected umber of copies of a multicast message that are trasmitted i order to esure that the message is delivered to all odes. Wheever a ode forwards a message to a eighbor, that couts as oe copy. A smaller hop complexity meas a more balaced multicast tree ad a smaller average delivery latecy. A smaller commuicatio complexity meas less etwork badwidth cosumed. I the followig, we use two multicast schemes to demostrate a trade-off betwee these two complexities. Oe complexity ca be optimized at the expese of the other. TABLE Compariso of Complexities Full Floodig. I this scheme, each ode forwards a message to its radom eighbors whe the message is received for the first time. It ca be show that ) the hop complexity is Oðlog c Þ, which is optimal because floodig follows the shortest paths, ad 2) the commuicatio complexity is c, with each ode x forwardig c x copies. Rig Traversal. O the other ed of the spectrum, we ca achieve the best commuicatio complexity of by forwardig a message alog the rig that serially coects all odes. However, the hop complexity is =2, which is the worst amog all schemes. ACOM. We demostrate that there are other alteratives betwee these two extremes. I particular, we desig a multicast scheme, called ACOM, that realizes a favorable trade-off betwee the hop complexity ad the commuicatio complexity. It achieves asymptotically optimal hop complexity of Oðlog c Þ at suboptimal commuicatio complexity of þ Oð log c Þ. A quick referece to the complexity compariso is give Table. 4.2 Idea Give the overlay topology defied i Sectio 3, we wat to desig a multicast routie such that. it ca deliver a message from ay source to every ode, 2. its hop complexity is Oðlog c Þ, which is asymptotically optimal, 3. its commuicatio complexity is close to, which is optimal, ad 4. it does so without buildig ad maitaiig explicit multicast trees o top of the overlay. The idea is to perform two-phase multicast. Phase oe is partial radom distributio, ad phase two is segmeted rig traversal. Combied, they defie implicit per-source trees that are embedded i the strucre of the existig overlay without additioal cost. These implicit trees are roughly balaced ad capacity-costraied. They are differet for each source ad, cosequetly, spread the workload to as may overlay liks as possible. Because we do ot establish ay explicit multicast tree, we avoid the problem (ad the overhead) of tree maiteace o top of the uderlyig overlay etwork. Phase oe. It forwards the message via radom eighbors to odes withi K hops from the source, where K is a system parameter that will be determied later. This phase delivers the message to a subset of odes that are radomly located across the etwork. They are called phase-oe odes. A illustratio is give by the first two plots of Fig.. Suppose the capacity of the source ode s is 3 ad the capacities of odes i, j, ad l are 2, 2, ad 3, respectively. I

4 CHEN ET AL.: ACOM: ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST IN NON-DHT P2P NETWORKS 9 Fig.. Two-phase multicast. phase oe, the message is delivered to 0 odes i a tree strucre, where the umber of childre of a ode is costraied by its capacity. Phase two. Each phase-oe ode will deliver the message to its eighborhood. If we had used the traditioal cocept of a eighborhood cosistig of odes withi a certai umber of hops away, phase two would become distributed limited floodig performed by all phase-oe odes. I order to cover the etire etwork, the phase-oe odes would have to flood the messages deep eough i their eighborhoods, causig overlaps where some odes would receive multiple copies of the message. To avoid this problem, we use the urestricted rig istead. The phaseoe odes partitio the rig ito segmets. As illustrated i the third plot of Fig., each phase-oe ode is resposible for oe adjacet segmet. It forwards the message to its successor, which further forwards the message to the successor s successor,..., util the message reaches a ode that has already received the message. The two phases are completely decetralized with o global coordiatio. Together, they deliver a message i a tree strucre, as illustrated i the fourth plot of Fig.. The tree follows the fabric of the uderlyig overlay etwork ad does ot eed maiteace. Phase oe restricts floodig i K hops. The problem with full floodig is at its fial stage whe most odes forward the message to eighbors that have already received the message. The segmeted rig traversal i phase two avoids such collisios while still esurig that every ode receives a copy of the message. If the value of K is small eough, the partial radom distributio i phase oe is likely to deliver the message to distictive radom odes, ad the probability of a ode receivig more tha oe copy will be small, which helps to reduce the commuicatio complexity. If the value of K is large eough, there will be eough phase-oe odes to partitio the rig ito small segmets, which reduces the hop complexity i phase two. These two coflictig requiremets o K lead to a iterestig questio. Ca we choose a right value for K such that the hop complexity is asymptotically optimal, Oðlog c Þ, ad the commuicatio complexity is close to? Nodes are allowed to have differet umbers of eighbors, which makes the questio more iterestig ad more realistic, but also much harder. Below, we first give a basic versio of the algorithm. Its simple strucre serves the purposes of brigig out the idea ad alleviatig aalysis from optimizatio details. We the aalyze the properties of the algorithm, address the implemetatio issues, ad, fially, discuss two improved versios of the algorithm. 4.3 Basic Algorithm (ACOM-) A multicast message is deoted as Mðk; idþ, where M is the message, k is the TTL field whose iitial value is K, ad id is a globally uique idetifier, cosistig of the source ID (e.g., IP address) ad a sequece umber. To deliver a multicast message, the source ode s begis by forwardig MðK;idÞ to c s radom eighbors. Whe a ode x receives Mðk; idþ, if it has ot received M with id before ad k>0, it decreases k by oe ad forwards the message to c x radom eighbors. If k ¼ 0, it oly forwards the message to its successor. The pseudocode of the algorithm is give i Fig. 2. The key questio is how to determie the value of K. For this purpose, we itroduce some otatios below that will also be used i the aalysis. I phase oe, whe a ode receives a multicast message for the first time, it forwards the message to its radom eighbors. Collectively, the message is delivered i a tree strucre, called the phase-oe tree, formed by radom eighbors. The depth of the tree is K. The source ode s is at level 0. Let q i be the umber of odes at level i. q 0 ¼, q ¼ c s, ad q i, i 2½2::KŠ, is a radom umber because the radom eighbors of odes at level ði Þ may overlap at level i. The umber of odes at levels zero through i i the phase-oe tree is Q i ¼ P i q j. The total umber of odes i the phase-oe tree is Q K. All of the leaves of the phase-oe tree are at level K. The umber of iteral odes is Q K. The umber of childre of a iteral ode is bouded by its capacity. Let c i;, c i;2,..., ad c i;qi be the capacities of the odes at level i. The summatio of the capacities of all iteral odes, plus oe for the source, i.e., þ P K P qi i¼0 c i;j, gives a upper boud for the tree size Q K. To avoid excessive duplicate receipts of the same message, our desig philosophy is for phase oe to reach oly a small fractio of odes i the Fig. 2. Basic algorithm (ACOM-).

5 92 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO. 9, SEPTEMBER 2007 multicast group. From those odes, the message is delivered i a circular fashio to reach all other odes i phase two. Therefore, we wat to choose K coservatively such that the tree size Q K is far smaller tha. Defie a system parameter as follows: X K i¼0 c i ¼ : characterizes the iteded percetage of odes i N that are covered by the phase-oe tree. We emphasize that is a system parameter that we pick. After we pick the value of, the value of K ca be calculated from (). Hece, the problem of determiig a appropriate value for K becomes the problem of determiig a appropriate value for. I the followig, we show that, if we choose ¼ ð log c Þ, the hop complexity will be Oðlog c Þ ad the commuicatio complexity will be þ Oð log c Þ. How to measure ad c will be addressed i Sectio Aalysis The mai aalytical results are give below. The ode capacities are ot fixed but, istead, are treated as radom variables, which makes the aalysis cosiderably tougher. The proofs ca be foud i the Appedix. Theorem (Hop Complexity). The expected umber of overlay hops for a multicast message to reach ay ode is bouded by log c ðþþ 2 : c c c Corollary (Hop Complexity). If ¼ ð log c Þ ad c 2, the the expected umber of overlay hops for a multicast message to reach ay ode is Oðlog c Þ. Theorem 2 (Commuicatio Complexity). The expected umber of copies of a multicast message that are trasmitted is bouded by ð þ Þ. Corollary 2 (Commuicatio Complexity). If ¼ ð log c Þ, the the expected umber of copies of a multicast message that are trasmitted is bouded by þ Oð log c Þ. 5 IMPLEMENTATION ISSUES 5. Maitaiig the Rig Ulike Chord [26], i our overlay etwork, there is o restrictio o where a ode should be placed o the rig. Whe a ew ode x jois the multicast group, it must kow a ode z that is already i the group. 2 x jois as z s successor, ad z s previous successor becomes x s successor. I compariso, Chord or most other DHT etworks must perform a lookup i order to determie the specific locatio where x ca joi i the etwork. 2. Whe a ode jois a P2P etwork, it must kow a existig ode i the overlay. The ode caot joi if it has zero clue about where the overlay etwork is. This requiremet is true for Gutella, Kazaa, or ay other P2P etwork. I the IP multicast, a ew member may query a directory server for the root of the multicast tree. I ACOM, a ew member may query the server for a existig ode i the overlay. The ower (creator) of a multicast group is a permaet member that is resposible for updatig the server with a umber of existig odes. ðþ As we discussed i Sectio 3, each ode kows a umber of odes after the successor dow the rig to prevet the rig from beig broke after the successor s abrupt deparre. This method was used by Chord to keep its successor rig from beig broke, ad it was iherited by other systems [27], [9] that used Chord s etwork maiteace protocols. I the rare case that all of those odes depart abruptly together, a broadcast is performed to idetify the odes that do ot have predecessors or successors so that they ca be coected to form a rig. 5.2 Establishig Radom Neighbors After a ew ode x jois the rig, it ca ask a multicast source s to help fid its c x radom eighbors. Whe s multicasts a message, it attaches c x tokes, cotaiig x s IP address. Each toke, piggybacked i the message, idepedetly chooses a radom path i the phase-oe tree, ad the traverses a rig segmet i phase two, durig which it will pick a radom ode from the segmet. More specifically, the toke tetatively picks the first ode of the segmet, sets L ¼, ad begis the segmet traversal. At each hop, L is icreased by oe ad the toke has a probability of =L of replacig the previously picked ode with the curret ode. At the ed of the rig segmet, the ode that is picked last is reported to x as a radom eighbor. Because a multicast message is received by every ode i the group, every ode will have a chace to be x s eighbor. As a low-probability special case i phase oe, whe a message carryig a toke arrives at a ode that has already received the message, the ode discards the message ad reports itself to x. Without the kowledge of a multicast source, whe x jois as z s successor, it ca ask z s help to fid radom eighbors. z seds out c x tokes. Each toke idepedetly performs a radom walk of Oðlog c Þ hops ad the reports the reached ode to x. I total, the tokes travel Oðc x log c Þ hops to fid c x radom eighbors. Whe a ode loses a radom eighbor (that departs), it adds a ew eighbor by radom walk. The eighbor liks are directed, which meas, eve if y is a radom eighbor of x, x may ot be a radom eighbor of y.i a radom etwork, each ode is expected to serve as a eighbor for roughly the same umber of other odes. Based o this observatio, we desig a simple optimizatio method to improve the radomess i eighbor selectio. Let I x be the set of odes for which x is a radom eighbor. x has the kowledge of I x. Periodically x exchages ji x j with its eighbors. If ji x j is greater tha the smallest ji y j value that x receives from eighbors y, x will iform a ode i I x to use y (with the smallest ji y j value) istead of x as a radom eighbor. This will gradually equalize ji x j ad ji y j. I additio, x selects a arbitrary eighbor z ad swaps a ode i I x with a ode i I z. All of the above commuicatio ca be piggybacked i the keep-alive messages betwee eighbors. 5.3 Estimatig ad c Because a multicast message will reach every ode, the measuremet of ca be carried out durig the process of multicast. Periodically, a multicast source piggybacks a flag i a multicast message. As the message with the flag is distributed, each ode temporarily remembers who the paret is. The ode cout iformatio is propagated backward from the ed odes of the Phase-two rig segmets back to the source ode, where the value of is

6 CHEN ET AL.: ACOM: ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST IN NON-DHT P2P NETWORKS 93 Fig. 3. Improved algorithm (ACOM-2). determied. The source ode the distributes to all other odes, piggybacked i the ext multicast message. To guard agaist some odes leavig the etwork durig the process, whe a ode x fids the coectio to a child ode is broke or it has t received the ode cout from the child after a timeout period, it will artificially assig that child the average ode cout reported by other childre mius oe (due to the deparre of that child). x will the report the total ode cout from all childre plus oe to its paret. To further improve the accuracy, the source may perform a umber of measuremets of i a period of time. It may also receive the measuremets from other sources. It will take the average measured value of durig that period as the ew estimatio of. The value of c is measured alog with. Together with the ode cout, the iformatio of the average ode capacity will also be propagated backward from the odes to the source. 5.4 Group Members with Very Small Upload Badwidth A ode x with very small upload badwidth should oly be a leaf i the implicit multicast trees ulessit is the data source. I order to make sure that we do ot select x as a itermediate ode i ay phase-oe tree, x should ot be a regular member of the overlay etwork. Istead, it jois as a exteral member. x asksaode y kowto be i the overlay to perform aradom walk of Oðlog c Þ hops to idetify a radom ode z. x the attempts to joi z as a exteral member. If z caot support x, z forwards x to successorðzþ. Ifz admits x as a exteral member, z will forward the received multicast messages to x ad x will multicast its messages via z.ifz leaves the group, x must rejoi via aother ode i the overlay. 5.5 Proximity The purpose of phase oe is to spread a message across the etire etwork such that each regio is likely to receive a copy. Therefore, radom eighbors are desirable. I phase two, the message travels alog rig segmets. There is o restrictio o where a ode should be located o the rig. Hece, proximity measures ca come ito play. We wat to reduce the latecy of the overlay liks o the rig. Oe approach uses viral coordiates [28]. Whe a ew ode x jois the overlay, it pigs a set of ladmark machies. From the delays to the ladmarks, a set of viral coordiates are calculated. Whe c x radom walks are performed to idetify x s radom eighbors (Sectio 5.2), x s viral coordiates are carried. As a radom-walk cotrol message travels aroud, it fids the ode o its path that is closest to x based o the distace calculated from the viral coordiates. This ode is reported to x at the ed of the radom walk. Amog all reports, x picks the best oe to joi as its successor. 5.6 Dyamic Capacities The capacity of a ode does ot have to be fixed. It may chage as the ode s upload badwidth chages. Accordig to the algorithms i Figs. 2, 3, ad 4, a ode x forwards a multicast

7 94 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO. 9, SEPTEMBER 2007 aroud, the phase-oe tree should have levels 0 through bkcþ, but the ðbkcþþth level should have bee oly partially populated. Not all odes at level ðbkcþþ will be icluded i the tree. Istead, each ode at this level oly has a probability p of beig icluded. p is calculated below. Fig. 4. Ticket algorithm (ACOM-3). message to c x selected radom eighbors based o the curret value of c x, which may be differet for fure messages. If c x becomes greater tha the curret umber m of radom eighbors, the x will oly forward the message to the m eighbors. If c x is cosistetly greater tha m for a period of time, x will perform radom walks to add more eighbors. 5.7 Termiatig Duplicate Copies Cosider a ode x forwardig a message to a eighbor y. Oce y receives the message header, it kows whether the message has already bee received based o the message idetifier id. If this is a duplicate copy ad the message is very log, y immediately seds a cotrol message to x ad asks x to stop forwardig, which sigificatly reduces the overhead. 6 IMPROVED ALGORITHMS (ACOM-2 AND ACOM-3) By the desig of the algorithm ACOM-, a source ode first chooses a value for i the order of log c. For a estimated phase-oe tree size i the order of, we calculate K from () as follows: K ¼ log c ððc ÞþÞ: Because K may ot be a iteger, the source has to sed either MðbKc;idÞ or MðdKe;idÞ to iitiate the executio of the distributed algorithm ACOM- i Fig. 2. The roudig itroduces iaccuracy. If MðbKc;idÞ is used, the estimated size of the phase-oe tree is o loger. Istead, it is P bkc i¼0 ci, which ca be as small as c. O the other had, if MðdKe;idÞ is used, the estimated size of the phase-oe tree is P dke i¼0 ci, which ca be as large as c. To keep the tree size ð2þ X bkc i¼0 c i þ c bkcþ p ¼ ; p ¼ cbkcþ c : c bkcþ The source s seds out MðbKc;p;idÞ with a additioal field p. Whe a ode x receives a message Mð0;p;idÞ, it forwards the message to each of its c x radom eighbors with a probability p. This algorithm, called ACOM-2, is give i Fig. 3. Alteratively we ca use a differet cocept, called tickets, to cotrol the phase-oe tree size. Istead of usig a k field to carry the depth of the tree, we use a t field to carry the size of the tree. The message format is Mðt; idþ, where t is the umber of tickets, each represetig the permissio of icludig oe ode i the subtree rooted at the receiver of the message. The source ode seds out Mð ;idþ. Whe a ode x receives Mðt; idþ, it cosumes oe ticket by subtractig oe from t. It the evely distributes the remaiig tickets amog its c x childre at the ext level of the tree. The algorithm is called ACOM-3 ad is give i Fig. 4. Whereas gives a upper boud o the expected size of the phase-oe tree i ACOM- ad ACOM-2 accordig to Lemma 5, it sets the upper boud o the acal tree size i ACOM-3. 7 SIMULATION 7. Sep A overlay etwork is costructed amog 0,000 odes. We assume the odes come from all over the Iteret, rather tha beig clustered i a few places. I [29], Mukherjee foud that the ed-to-ed packet delay o the Iteret ca be modeled by a shifted Gamma distributio, which is a log-tail distributio. The shape parameter varies from approximately.0 durig low loads to 6.0 durig high loads o the backboe. I this paper, we set the shape parameter to be 4.0 ad the average packet delay to be 50 ms. 3 Because every ode ca be a potetial data source, we do ot maitai oe explicit multicast tree per ode. Istead, the multicast is performed (as broadcast) directly o top of the overlay etwork, cosistig of a radom graph ad a optioal augmeted rig, as described i Sectio 3. No cetral elemet is available to coordiate the overlay costructio or the multicast procedure. Nodes have differet capacities. Whe the average ode capacity is set to be c, the ode capacities are take from ½2; 2ðc ÞŠ with uiform probabilities. If ot specified otherwise, c ¼ 6 by default. We will vary the value of c i a wide rage i some simulatios. We implemet seve multicast algorithms ad their variats. The algorithms are floodig, probabilistic floodig, limited-degree (LD) floodig, ACOM- with bkc, ACOM-2, 3. Chagig the average packet delay i the simulatio will proportioally scale the curves i the latecy compariso, but will ot chage the shape of the latecy curves.

8 CHEN ET AL.: ACOM: ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST IN NON-DHT P2P NETWORKS 95 Fig. 5. Compariso of differet algorithms of traffic volume for deliverig oe multicast message. ACOM-3, ad CAM-Chord [9]. Floodig is oe of the most popular broadcast algorithms o the IP etworks; reverse path forwardig is oe example. I our case, each ode forwards a copy of a message to all radom eighbors whe it receives the message for the first time. I order to cotrol the overhead, probabilistic floodig forwards a copy of the message to a radom eighbor with a certai probability, which meas ot all eighbors of a ode will receive the message forwarded by the ode. O the other had, LD floodig oly forwards a copy of the message to a fixed umber of radom eighbors. ACOM- with bkc is the algorithm i Fig. 2 with K calculated from (2). Because ACOM-2 ad ACOM-3 are better algorithms (Sectio 6), we oly preset their results i most cases. Proximity o the rig is implemeted by default. The above six algorithms are all desiged based o a radom etwork. CAM-Chord is desiged based o a more complex DHT etwork. It has Oðlog c Þ times more eighbors per ode, which meas multiple times higher overhead for maitaiig the etwork, but it is more efficiet i deliverig a multicast message. ACOM ad CAM-Chord represet a trade-off betwee cheaper etwork maiteace ad more efficiet multicastig (Sectio 7.6). The performace metrics that we sdy iclude. the traffic volume for deliverig oe message, which is measured by the umber of copies that are forwarded by all odes, 2. the average legth of the delivery paths, which is measured by the average umber of overlay hops it takes a message to reach a ode, 3. the average delivery latecy, which is measured by the average time it takes a message to reach a ode, 4. hop distributio, which is the distributio of the delivery-path legths for all odes, 5. delay distributio, which is the distributio of the latecy values for all odes, ad 6. the maiteace overhead, which is the average umber of cotrol messages set for each ode joi/deparre. 7.2 Performace Evaluatio The first set of simulatios compares the average performace of various algorithms i deliverig messages. Fig. 5 shows the traffic volumes caused by these algorithms with respect to the average ode capacity. I probabilistic Fig. 6. Number of odes reached by a multicast message. Not all algorithms deliver the message to all odes. floodig, the probabilistic for a ode to forward a message to a eighbor is 30 percet. I LD floodig, each ode forwards a message to two eighbors. I ACOM-2 ad ACOM-3, ¼ 0:25. Accordig to the figure, floodig is the clear loser because of too much traffic. The traffic volume of probabilistic floodig is small whe the average ode capacity is small. The traffic volume of LD floodig is modest ad isesitive to the ode capacity. The problem is that either of them esures that a message will reach all odes, as demostrated i Fig. 6. Oe way to solve this problem is to icorporate the segmeted rig traversal proposed i this paper. A rig is augmeted to the overlay etwork. Whe a ode receives a message for the first time, it ot oly forwards the message to radom eighbors but also starts traversig the adjacet rig segmet util reachig a ode that has also received the message. But, the segmeted rig traversal comes with a cost; it forwards ð¼ 0;000Þ copies of the message, which brigs up the traffic volume as show i Fig. 7. I compariso, the traffic volumes caused by ACOM-2 ad ACOM-3 are both much smaller. Fig. 8 compares the average legths of the delivery paths. It takes the least umber of hops for floodig to reach the odes. ACOM-2 performs slightly better tha ACOM-3. Both ACOM-2 ad ACOM-3 perform better tha LD floodig, but worse tha floodig. Probabilistic floodig performs well Fig. 7. Compariso o traffic volume for deliverig oe message. All algorithms perform segmeted rig traversal.

9 96 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO. 9, SEPTEMBER 2007 Fig. 8. Compariso of the algorithms o average legth of delivery paths. Fig. 0. Scalability compariso o traffic volume for deliverig oe message with respect to etwork size. Fig. 9. Compariso of the algorithms o average latecy of deliverig a message. whe the average ode capacity is large, but that comes with a large overhead, show i Fig. 7. Due to the proximity measure that is implemeted o the rig (Sectio 5.5), ACOM-2 ad ACOM-3 perform closer to floodig i terms of latecy tha i terms of hops. As show i Fig. 9, proximity has less impact o LD floodig because it traverses o shorter rig segmets tha ACOM-2 ad ACOM-3. The average latecy values of ACOM-2 ad ACOM-3 are slightly worse tha that of floodig but much better tha that of LD floodig. We also perform scalability evaluatio with the umber of odes i the etwork ragig from 2,000 to 20,000. Fig. 0 shows that the traffic volume icreases liearly with the etwork size for all four algorithms uder compariso. Fig. shows that, except for probabilistic floodig, the average latecy values of the algorithms are ot very sesitive to the etwork size. 7.3 Performace Trade-off i ACOM The secod set of simulatios sdies the performace of ACOM with respect to the value of. Figs. 2, 3, ad 4 demostrate a trade-off i ACOM betwee commuicatio complexity (traffic volume) ad hop complexity (umber of hops ad latecy). By icreasig, ACOM ca reduce the latecy at the cost of higher traffic volume, ad vice versa. I Fig. 2, a larger value meas a larger phase-oe tree, which icreases the traffic volume that costructs the Fig.. Compariso of the algorithms o average legth of delivery paths with respect to etwork size. Fig. 2. Compariso of ACOM algorithms o traffic volume with respect to. tree. Note that the traffic volume of phase two is always ð¼ 0; 000Þ forwarded copies. O the other had, a larger phase-oe tree meas smaller rig segmets, which reduces the umber of hops ad, thus, latecy i phase two, also causig overall hop/latecy decrease (Figs. 3 ad 4). The performace curve of ACOM- with bkc is a staircase with respect to. That is because chagig does ot cotiuously chage the value of bkc, accordig to (2).

10 CHEN ET AL.: ACOM: ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST IN NON-DHT P2P NETWORKS 97 Fig. 3. Compariso of ACOM algorithms o average legth of delivery paths with respect to. Fig. 5. Hop distributio. Fig. 4. Compariso of ACOM algorithms o average latecy with ad without implemetig proximity. 7.4 Delay ad Proximity The third set of simulatios sdies the impact of proximity. Our rig is a urestricted rig, where a ode ca be located aywhere. This gives us the freedom of movig earby odes adjacet o the rig (Sectio 5.5), which reduces the latecy of performig the segmeted rig traversal i phase two. Fig. 4 shows that the impact of proximity is greater whe is smaller. That is because a smaller value for meas a smaller phase-oe tree ad, cosequetly, larger rig segmets i phase two, which gives proximity more impact. 7.5 Hop ad Delay Distributios The fourth set of simulatios sdies the hop distributio ad the delay distributio, which are show i Fig. 3 ad Fig. 4, respectively. Because the average delay of a overlay lik is 50 ms, 20 hops i Fig. 5 correspods to,000 ms i Fig. 6. The value of is chose to be I Fig. 5, over 93.2 percet of all odes receive the message withi twice the average umber of hops, which is 9.43 for ACOM-2 ad 8.4 for ACOM-3. The delay distributio is more cocetrated. I Fig. 6, over 96.4 percet of all odes receive the message withi twice the average latecy, which is ms for ACOM-2 ad ms for ACOM-3. Fig. 6. Delay distributio. 7.6 Network Maiteace Overhead ad ACOM versus CAM-Chord The algorithms evaluated so far are all based o radom overlay etworks. Because these algorithms use the same uderlyig overlay topology i the simulatios, the etwork maiteace overhead is the same for them. The goal of ACOM is to support ay-source capacitycostraied multicast with dyamic membership. We have elaborated i the itroductio ad the related work that most existig systems do ot meet these requiremets. CAM- Chord is a exceptio, but it is DHT-based ad requires Oðc log c Þ eighbors per ode, which is Oðlog c Þ times more tha ACOM. The fifth set of simulatios compares CAM- Chord ad CAM i terms of multicast efficiecy ad etwork maiteace overhead. Thaks to its more complex topological strucre, CAM- Chord forwards oly oe copy of a multicast message to each ode. Therefore, it is more efficiet i terms of multicastig (Fig. 7), but less efficiet i terms of etwork maiteace (Fig. 8). Assume a TCP coectio is established betwee two eighbors. The maiteace overhead is the summatio of ) the umber of cotrol messages used to idetify the eighbors, 2) the umber of cotrol messages for establishig ew TCP coectios betwee eighbors, ad 3) the umber of cotrol messages for tearig dow the TCP coectios betwee odes that cease to be eighbors. The maiteace overhead of ACOM icreases slowly with the etwork size, ad it is far less tha that of CAM-Chord.

11 98 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO. 9, SEPTEMBER 2007 Fig. 7. CAM-Chord forwards oe copy of the multicast message for each ode, which is optimal. I total, ACOM forwards about 24 percet more copies tha ecessary due to its radom strucre. CAM-Chord achieves the optimality at the cost of Oðlog c Þ times more eighbors per ode ad the more complex DHT-based etwork strucre, which is harder to repair oce it is fracred. CAM-Chord ad ACOM represet a trade-off betwee topology maiteace overhead ad multicast overhead. They provide differet optios for practitioers to choose based o the system requiremets that may be i favor of more efficiet multicastig or low overhead maiteace. The average latecy values of CAM-Chord ad ACOM are comparable ad the simulatio results are omitted. 8 CONCLUSION This paper proposes a ay-source overlay multicast service o top of a radom overlay etwork with a augmeted rig. It is able to deliver a multicast message from ay source with a hop complexity of Oðlog c Þ ad a commuicatio complexity of þ log c. Remarkably, this is achieved without maitaiig ay explicit multicast trees. We provide a rigorous aalysis ad perform extesive simulatios to evaluate the performace of the system. We preset three distributed multicast algorithms ad address a variety of issues, icludig overlay maiteace, parameter measuremet, low-capacity odes, proximity, ad dyamic capacities. APPENDIX PROOFS We first give the upper ad lower bouds of Q K ad the derive the hop complexity ad the commuicatio complexity of the algorithm. The defiitios for Q K ad other symbols ca be foud i Sectio 4.3 ad Sectio 3. Lemma. EðQ i Þ P i cj. Proof. The umber of odes at level i is bouded by the summatio of the capacities of all odes at level ði Þ. Namely, q i Xq i c i;j : j¼ The expected value of q i, uder the coditio that there are q i odes at level ði Þ, is calculated below. c i;j, j 2½::q i Š, is a radom variable whose expected value is c. Fig. 8. CAM-Chord has may more eighbors per ode tha ACOM. Its overhead, i terms of umber of cotrol messages per ode joi/ deparre, is much higher tha that of ACOM, which is a strucrallysimpler o-dht etwork. Therefore, CAM-Chord ad ACOM represet a trade-off betwee topology maiteace overhead ad multicast overhead. Eq ð i j q i Þ E Xq i c i;j ¼ Xq i Ec i;j ¼ qi c: j¼ j¼ Take the expected value o both sides with respect to q i ad, because EðEðq i jq i ÞÞ ¼ Eðq i Þ, we have Eðq i ÞEðq i Þc: By recursively applyig the above iequality, we have It follows that Eðq i Þc i Eðq 0 Þ¼c i : EðQ i Þ¼E Xi q j Xi Lemma 2. For ay set S of coditios, it must be true that EQ ð i jsþ E Q i j Q i Xi c j ;S : Proof. EQ ð i j SÞ is the expected value of Q i amog all possible multicast istaces that satisfy S. E Q i Qi Xi c j ;S is the expected value of Q i amog all possible multicast istaces that ot oly satisfy S but also Q i P i cj. The differece betwee them is that EðQ i Q i P i cj ;SÞ excludes all Q i istaces that are greater tha P i cj. Sice it excludes large values, we certaily have that EQ ð i j SÞ E Q i Qi Xi c j ;S : c j : ut

12 CHEN ET AL.: ACOM: ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST IN NON-DHT P2P NETWORKS 99 Lemma 3. For ay set S of coditios that place o artificial restrictio o a ode s capacity, it must be true that E Q i j Q i Xi c j ;S > c E Q i Qi Xi C Ki c j ;S : Proof. Let T i be the subtree that cosists of levels 0 through i of a phase-oe tree. The size of T i is Q i. Durig the formatio of T i, each ode x at levels 0 through ði Þ makes c x attempts to iclude its radom eighbors i the tree at the ext level. Because there are less tha Q i odes i the tree, each attempt has a probability of at least ð Q i Þ of fidig that a radom eighbor is outside the tree ad thus ca be brought i, which icreases the tree size by oe. P There are, i total, i P ql l¼0 j¼ c l;j attempts at levels 0 through ði Þ. Therefore, Q i > Xi X q l c l;j l¼0 j¼ Q i : ð4þ Oe may be puzzled by the above iequality amog radom variables. The correct iterpretatio is that the iequality will hold for ay give multicast istace. Give ay set of values, q l, l 2½::i Š, which appears i fq ;...;q i jq i P i cj ;Sg, we compute the coditioal expected value of Q i. E Q i j q l ;l2½::i Š; Q i Xi c j ;S >E Xi X q l c l;j l¼0 j¼ E ¼ Xi X ql c l;j l¼0 j¼ P i cj Q i ql ;l2½::i Š; Q i Xi c j ;S P i cj Q i Xi X i X ql l¼0 j¼ Q i Xi by ð4þ ql ;l2½::i Š; c j ;S by coditio Q i Xi E c l;j ql ;l2½::i Š; c j ;S c j Basedotheaboveiequality,taketheexpectedvalueofQ i over q l, l 2½::i Š, uder the coditio of Q i P i cj ad S. E Q i j Q i Xi c j ;S > cð C KiÞEðQ i j Q i Xi Lemma 4. EðQ i Þ > ð c Ki ðcþ Þci. Proof. By Lemma 2, EðQ i ÞE Let S 0 ¼;. By Lemma 3, Q i j Q i Xi c j ;SÞ: c j : EðQ i Þ >c E Q i Qi Xi C Ki c j ;S 0 : Let S ¼ S 0 þfq i P i cj g. By Lemmas 2 ad 3, EðQ i Þ >c C Ki EQ i S c E Q i Qi Xi c j ;S C Ki >c 2 C Ki E C Kiþ Q i2 Qi Xi c j ;S : Repeat this process by recursively applyig Lemmas 2 ad 3. We have EðQ i Þ >c i Yi Eðq 0 Þ j¼ ¼ c i Yi j¼ c Kj c Kj : It ca be show by iductio that Q i j¼ð Þ > c Kj c Ki ðcþ. Therefore, EðQ i Þ > c Ki c i : ðc Þ ¼ c P i cj P i cj X i ¼ c >c c Ki Q i X q l l¼0 j¼ Q i by lemma assumptio; S places o restrictio o c l;j by ðþ: Lemma 5. ð c cþð c Þ < EðQ KÞ. Proof. By Lemma, EðQ K Þ P K i¼0 ci ¼. By Lemma 4, EðQ K Þ > c c K : ð5þ c From (), c K ¼ ðcþþ c have > ðcþ c. Applyig this to (5), we

13 200 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 8, NO. 9, SEPTEMBER 2007 EðQ K Þ > c : c c X is the summatio of Q K idepedet radom variables, where Q k ¼ P K i¼0 q i. Q K is itself a radom variable. Give a value of Q K, the coditioal expected value of X is Theorem (Hop Complexity). The expected umber of hops for a multicast message to reach ay ode is bouded by log c ðþþ 2ð c cþð c Þ : Proof. By (), P K i¼0 ci ¼. Hece, c K <. K<log c ðþ, which meas that phase oe termiates i o more tha log c ðþ hops. By Lemma 5, after phase oe, the rig is partitioed ito more tha ð c cþð cþ segmets. Phase two delivers the message i parallel alog these segmets. The expected legth of a segmet is bouded by c ¼ c c c : c I phase two, the average umber of hops for the message to reach a ode is half of the segmet legth. Combiig phase oe ad phase two, the expected umber of hops for a multicast message to reach ay ode is bouded by log c ðþþ 2 : c c c Corollary (Hop Complexity). If ¼ log c ad c 2, the the expected umber of hops for a multicast message to reach ay ode is Oðlog c Þ. Proof. By Theorem, the upper boud o the expected umber of hops is log c ðþþ 2 c c c ¼ ðlog c ðlog c log c ÞÞ þ 2 c ¼ ðlog c Þþ log c log c c c c log c ¼ ðlog c Þþðlog c Þ¼ðlog c Þ: Theorem 2 (Commuicatio Complexity). The expected umber of copies of a multicast message that are trasmitted is bouded by ð þ Þ. Proof. The umber X of copies that are trasmitted i phase oe is equal to the summatio of the capacities of all odes at levels zero through K. X ¼ XK i¼0 X q i j¼ c i;j : EX QK ¼ X K i¼0 X qi j¼ Ec i;j ¼ X K i¼0 X qi j¼ c ¼ cq K : Take the expected value o both sides with respect to Q K. By (3) ad (), we have EðXÞ ¼cEðQ K Þ c XK c i < : Phase two seds a fixed umber of copies i the segmeted rig traversal. Therefore, the expected umber of copies that are trasmitted for a multicast message is ð þ Þ. Corollary 2 (Commuicatio Complexity). If ¼ ð log c Þ, the the expected umber of copies of a multicast message that are trasmitted is bouded by þ Oð log c Þ. Proof. Directly from Theorem 2. ACKNOWLEDGMENTS This work is i part supported by the Natioal Naral Sciece Foudatio of Chia uder grat ad by Shaghai Leadig Academic Disciplie Project (Project Number T0502). REFERENCES [] G. Baavar, M. Chadra, B. Nagarajaro, R. Strom, ad C. Srma, A Efficiet Multicast Protocol for Cotet-Based Publish- Subscribe System, Proc. It l Cof. Distributed Computig Systems (ICDCS 98), May 998. [2] Y.H. Chu, S. Rao, S. Sesha, ad H. Zhag, A Case for Ed System Multicast, IEEE J. Selected Areas i Comm., vol. 20, o. 8, Oct [3] J. Jaotti, D. Gifford, K. Johso, M. Kaashoek, ad J. O Toole, Overcast: Reliable Multicastig with a Overlay Network, Proc. Symp. Operatig Systems Desig ad Implemetatio (OSDI 00), Oct [4] C.K.S. Baerjee ad B. Bhattacharjee, Scalable Applicatio Layer Multicast, Proc. ACM SIGCOMM 02, Aug [5] B. Zhag, S. Jami, ad L. Zhag, Host Multicast: A Framework for Deliverig Multicast to Ed Users, Proc. INFOCOM 02, Jue [6] G.-I. Kwo ad J.W. Byers, ROMA: Reliable Overlay Multicast with Loosely Coupled TCP Coectios, Proc. INFOCOM 04, Mar [7] P.M. Zhi Li, Impact of Topology o Overlay Routig Service, Proc. INFOCOM 04, Mar [8] Y. Shavitt ad T. Takel, O the Curvare of the Iteret ad Its Usage for Overlay Costructio ad Distace Estimatio, Proc. INFOCOM 04, Mar [9] F. Baccelli, A. Chaitreau, Z. Liu, A. Riabov, ad S. Sahu, Scalability of Reliable Group Commuicatio Usig Overlays, Proc. INFOCOM 04, Mar [0] V. Pappas, B. Zhag, A. Terzis, ad L. Zhag, Fault-Tolerat Data Delivery for Multicast Overlay Networks, Proc. It l Cof. Distributed Computig Systems (ICDCS 04), Mar [] J.A. Deja Kosti, A. Rodriguez, ad A. Vahdat, Bullet: High Badwidth Data Dissemiatio Usig a Overlay Mesh, Proc. Symp. Operatig Systems Priciples (SOSP 03), Oct i¼0

14 CHEN ET AL.: ACOM: ANY-SOURCE CAPACITY-CONSTRAINED OVERLAY MULTICAST IN NON-DHT P2P NETWORKS 20 [2] S. Baerjee, C. Kommareddy, B.B.K. Kar, ad S. Khuller, Costructio of a Efficiet Overlay Multicast Ifrastrucre for Real-Time Applicatios, Proc. INFOCOM 03, Mar [3] A. Riabov ad L.Z. Zhe Liu, Overlay Multicast Trees of Miimal Delay, Proc. It l Cof. Distributed Computig Systems (ICDCS 04), Mar [4] H. Yamaguchi, A. Hiromori, T. Higashio, ad K. Taiguchi, A Autoomous ad Decetralized Protocol for Delay Sesitive Overlay Multicast Tree, Proc. It l Cof. Distributed Computig Systems (ICDCS 04), Mar [5] S. Zhuag, B. Zhao, A. Joseph, R. Katz, ad J. Kubiatowicz, Bayeux: A Architecre for Scalable ad Fault-Tolerat Wide- Area Data Dissemiatio, Proc. th It l Workshop Network ad Operatig System Support for Digital Audio ad Video (NOSSDAV 0), Jue 200. [6] R. Zhag ad Y.C. Hu, Borg: A Hybrid Protocol for Scalable Applicatio-Level Multicast i Peer-to-Peer Networks, Proc. 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Turer, Routig i Overlay Multicast Networks, Proc. INFOCOM 02, Jue [22] B.Y. Zhao, J.D. Kubiatowicz, ad A.D. Joseph, Tapestry: A Ifrastrucre for Fault-Tolerat Wide-Area Locatio ad Routig, Uiv. of Califoria Berkeley Techical Report UCB/CSD-0-4, Apr [23] A. Rowstro ad P. Druschel, Pastry: Scalable, Decetralized Object Locatio, ad Routig for Large-Scale Peer-to-Peer Systems, Proc. Middleware 0, Nov [24] S. Ratasamy, P. Fracis, M. Hadley, R. Karp, ad S. Sheker, A Scalable Cotet Addressable Network, Proc. ACM SIG- COMM 0, Aug [25] S. El-Asary, L.O. Alima, P. Brad, ad S. Haridi, Efficiet Broadcast i Strucred P2P Networks, Proc. It l Workshop Peerto-Peer Systems (IPTPS 03), Feb [26] I. Stoica, R. Morris, D. Karger, F. Kaashoek, ad H. Balakrisha, Chord: A Scalable Peer-to-Peer Lookup Service for Iteret Applicatios, Proc. ACM SIGCOMM 0, pp , Aug [27] M. Kaashoek ad D. Karger, Koorde: A Simple Degree-Optimal Distributed Hash Table, Proc. Secod It l Workshop Peer-to-Peer Systems (IPTPS 03), Feb [28] T.S.E. Ng ad H. Zhag, Predictig Iteret Network Distace with Coordiates-Based Approaches, Proc. IEEE INFOCOM 02, Jue [29] A. Mukherjee, O the Dyamics ad Sigificace of Low Frequecy Compoets of Iteret Load, Iteretworkig: Research ad Experiece, vol. 5, o. 4, pp , 994. Shipig Che received the BS degree i electrical egieerig from JiagXi Uiversity of Chia i 984. He received the MS ad PhD degrees i computer sciece from the Istite of Computig Techology of the Chiese Academy of Scieces ad Fuda Uiversity i 990 ad 2006, respectively. He joied the Uiversity of Shaghai for Sciece ad Techology i 990 ad is curretly a full professor i the Computer Sciece Departmet. He is also the director of the etwork ceter of the uiversity. His research iterests iclude peerto-peer etworks, etwork commuicatios, ad database systems. Baile Shi joied the Departmet of Computer ad Iformatio Techology of Fuda Uiversity i 975. He was promoted to a associate ad the a full professor i 980 ad 985, respectively. He was the departmet chair from 985 to 996. His research field is database theories ad applicatios. He has published more tha 70 papers i the top Chiese jourals ad writte more tha 0 textbooks. He has wo umerous awards, icludig oe atioal sciece ad techology advacemet award, oe Guaghua award, ie Shaghai sciece ad techology advacemet awards, ad four textbook awards. Shigag Che received the BS degree i computer sciece from the Uiversity of Sciece ad Techology of Chia i 993. He received the MS ad PhD degrees i computer sciece from the Uiversity of Illiois at Urbaa-Champaig i 996 ad 999, respectively. After graduatio, he worked with Cisco Systems for three years before joiig the Uiversity of Florida as a assistat professor i His research iterests iclude etwork security, peer-to-peer etworks, ad sesor etworks. He received a IEEE Commuicatios Society Best Tutorial Paper Award i 999. He was a guest editor of the ACM/Baltzer Joural of Wireless Networks (WINET) ad the IEEE Trasactios o Vehicle Techologies. He served as a techical program committee cochair for the Computer ad Network Security Symposium of IEEE IWCCC 2006, a techical program committe vice chair for IEEE MASS 2005, a vice geeral chair for QShie 2005, a techical program committee cochair for QShie 2004, ad a techical program committee member for may cofereces icludig IEEE ICNP, IEEE INFOCOM, IEEE SANS, IEEE ISCC, IEEE Globecom, etc. Ye Xia received the PhD degree from the Uiversity of Califoria, Berkeley, i 2003, the MS degree i 995 from Columbia Uiversity, ad the BA degree i 993 from Harvard Uiversity, all i electrical egieerig. He became a assistat professor i the Computer ad Iformatio Sciece ad Egieerig departmet at the Uiversity of Florida i August Betwee Jue 994 ad August 996, he was a member of the techical staff at Bell Laboratories, Lucet Techologies i New Jersey. His research iterests are i the computer etworkig area, icludig performace evaluatio of etwork protocols ad algorithms, cogestio cotrol, resource allocatio, ad load balacig o peer-to-peer etworks. He is also iterested i probability theory, stochastic processes, ad queuig theory.. For more iformatio o this or ay other computig topic, please visit our Digital Library at