Optimal Distributed P2P Streaming under Node Degree Bounds

Transcription

1 Optimal Distribute P2P Streaming uner Noe Degree Bouns Shaoquan Zhang, Ziyu Shao, Minghua Chen, an Libin Jiang Department of Information Engineering, The Chinese University of Hong Kong Department of EECS, University of California, Berkeley {zsq8, zyshao6, arxiv:8.5248v8 [cs.ni] 9 Jun 22 Abstract We stuy the problem of maximizing the broacast rate in peer-to-peer (P2P) systems uner noe egree bouns, i.e., the number of neighbors a noe can simultaneously connect to is upper-boune. The problem is critical for supporting highquality vieo streaming in P2P systems, an is challenging ue to its combinatorial nature. In this paper, we aress this problem by proviing the first istribute solution that achieves nearoptimal broacast rate uner arbitrary noe egree bouns, an over arbitrary overlay graph. It runs on iniviual noes an utilizes only the measurement from their one-hop neighbors, making the solution easy to implement an aaptable to peer churn an network ynamics. Our solution consists of two istribute algorithms propose in this paper that can be of inepenent interests: a network-coing base broacasting algorithm that optimizes the broacast rate given a topology, an a Markov-chain guie topology hopping algorithm that optimizes the topology. Our istribute broacasting algorithm achieves the optimal broacast rate over arbitrary P2P topology, while previously propose istribute algorithms obtain optimality only for P2P complete graphs. We prove the optimality of our solution an its convergence to a neighborhoo aroun the optimal equilibrium uner noisy measurements or without timescale separation assumptions. We emonstrate the effectiveness of our solution in simulations using uplink banwith statistics of Internet hosts. I. Introuction Peer-to-peer (P2P) systems have provie a scalable an cost effective way for streaming vieo in the past ecae. Recent stuies [] [4], however, inicate that the practical performance of P2P streaming systems can be far from their theoretical optimal. There have been work stuying the performance limit of P2P systems to unerstan an unleash their potential. One focus is on the streaming capacity problem [5] in P2P live streaming systems, i.e., maximizing the streaming rate subject to the peering an overlay topology constraints. The problem is critical for supporting high-quality vieo, which is etermine by the streaming rate, in P2P live streaming systems. In this paper, we focus on the broacast scenario where all peers in the system are receivers. The case of unconstraine peering on top of a complete graph is well stuie, where the maximum broacast rate is erive in several papers [] [3], [6], [7]. The case of unconstraine peering over general graph can also be aresse by using a centralize solution [5]. The streaming capacity problem becomes NP-Complete over general graph with noe egree bouns []. Noe egree is efine as the number of simultaneous active connections that a noe maintains with its neighbors. Due to connection overhea costs, it is necessary to limit the number of simultaneous connections a peer can maintain. This naturally bouns the noe egrees in P2P systems. For instance, in practical systems such as PPLive [8], the total number of neighbors of a noe is usually boune aroun 2, an the number of active neighbors of a noe is usually boune by -5 [5]. In such large P2P systems with hunres of thousans of peers, the system topology is not a complete graph. There has been work stuying this challenging problem of maximizing streaming rate uner noe egree bouns an over general P2P graph. SplitStream/CoopNet [6], [7], ZIGZAG [8], PRIME [9] an most practical systems (such as PPLive [8] an UUSee [9]) boun noe egree but o not provie rate optimality guarantee. Recently, the authors in [] propose a centralize Cluster-Tree algorithm that achieves near-optimal broacast rate with high probability over complete graph, uner the assumption that the noe egree boun is at least logarithmic in the size of the network. A summary an comparison of previous work an this work are in Table I. Despite of these exciting results, the following two important questions remain open: What is the maximum broacast rate uner arbitrary noe egree bouns, an over general P2P overlay graph? How to achieve the maximum broacast rate in a istribute manner? Systems running istribute algorithms, compare with those running centralize algorithms, are more aaptable to peer churn an network ynamics. In this paper, we answer the above two questions an make the following contributions: We provie the first istribute solution that achieves a broacast rate arbitrarily close to the optimal uner arbitrary noe egree bouns, an over arbitrary overlay graph. Our solution runs on iniviual noes an utilizes only the informatiorom their one-hop neighbors. Our solution consists of the following two algorithms that can be of inepenent interests. We propose a istribute broacasting algorithm that achieves the optimal broacast rate over arbitrary overlay graph. Previous istribute P2P broacasting algorithms

2 TABLE I Summary an comparison of previous work an this work for maximizing P2P broacast rate. References General Arbitrary Noe Exact or ǫ Distribute Overlay Graph? Degree Boun? Optimality? Solution? Mutualcast [] an the algorithms in [2], [3] Iterative in [4], [5] CoopNet/SplitStream [6], [7] ZIGZAG [8], PRIME [9] Cluster-tree [] conitionally optimal This paper The Cluster-Tree algorithm is ( ǫ)-optimal with high probability if the noe egree boun is O ( log N ). are optimal only for complete overlay graph [] [3]. Our algorithm is base on network coing an utilizes backpressure arguments. We also propose a istribute algorithm that optimizes the topology. In this algorithm, each noe hops among their possible set of neighbors towars the best peering configuration. Our algorithm is inspire by a set of log-sumexp approximation an Markov chain base arguments expoune in [2]. We prove the optimality of the overall solution. We also prove its convergence to a neighborhoo aroun the optimal equilibrium in the presence of noisy measurements or without time-scale separation assumptions. We emonstrate the effectiveness of our solution in simulations using uplink banwith statistics of Internet hosts. A. Settings an Notations II. Problem Formulation We moel the P2P overlay network as a general irecte graph G=(V, E), where V enotes the set of noes an E enotes the set of links. Each link in the graph correspons to a TCP/UDP connection between two noes. Let N v enote the neighbor set of noe v V in the graph. Each noe v V is associate with an uploa capacity C v. We assume there is no constraint on the ownloaing rate for each noe v V. This assumption can be partly justifie by the empirical observation that as resiential broaban connections with asymmetric uploa an ownloa rates become increasingly ominant, bottlenecks typically are at the uplinks of the access networks rather than in the mile of the Internet. As such, P2P networks have capacity limits on the noes instea of links. This is ifferent from traitional unerlay networks where the capacity limits are on the links. We focus on the single-source streaming scenario, i.e., a source s broacasts a continuous stream of contents to the entire network; we enote its receiver set as RV {s}. We consier the peering constraints that each noe has a egree boun B v, i.e., it can only exchange streaming content with up to a B v number of neighbors simultaneously ue to connection overhea cost. We allow ifferent noes to have ifferent egree bouns. Fig. shows four sample peering configurations of a 5-noe network with noe egree boun 3 for each noe. (a) (c) Fig.. Peering configuration examples for a 5-noe network with noe egree boun 3 for each noe. Let F enote the set of all feasible peering configurations over graph G uner noe egree bouns. Given a configuration f F, we obtain a connecte sub-graph of G that satisfies the noe egree boun constraints. We enote this sub-graph as G f = ( ) V, E f, where E f represents the set of links in this sub-graph. We enote N v, f as the set of noe v s neighbors in this sub-graph. We have N v, f B v where represents the size of a set. (b) () B. Problem Formulation an Our Approach For a configuratio F, let x f be the maximum achievable broacast rate uner f, i.e., the highest rate at which every noe in the system can receive the streaming content simultaneously. The problem of maximizing broacast rate uner noe egree bouns can be formulate as follows: MRC : max x f. () This problem is combinatorial in nature which is known to be NP-complete [], an there is no efficient approximate solution to the problem even in a centralize manner. In this paper, we aress this problem by proviing a istribute solution. In particular, we first evelop a istribute broacasting algorithm that can achieve x f uner arbitrary f F. We then esign a istribute algorithm that optimizes towars the best peering configurations. They operate

3 in tanem to achieve a close-to-optimal broacast rate uner arbitrary noe egree bouns, an over arbitrary overlay graph. We elaborate on these two algorithms in the following two sections. III. The Propose Distribute Broacasting Algorithm By exploiting network coing [2], we esign a backpressure base istribute broacasting algorithm. Backpressure type algorithm is propose initially in [22]. This type of algorithms select a subset of queues in the system with the maximum back-pressures an serve these queues subject to resource constraints, where back-pressure is efine as the ifference between the queue at the local noe an that of its ownstream noes. Back-pressure algorithm esign has foun applications in many network resource allocation omains [23], [24], [25]. In this paper, we apply this metho for the first time to esign istribute P2P broacasting algorithm. Our algorithm can achieve the maximum broacast rate over arbitrary P2P topology. A. Routing vs. Network Coing In P2P systems, there are two approaches for broacasting contents: one is base on routing [26], in which noes only store an forwar packets; an the other is base on network coing [2], [26], in which a noe is also allowe to mix information an output ata as functions of the ata it receive. Some commercial P2P systems are built upon routing-base approach (e.g., PPLive [8]), an some are base on network coing (e.g., UUSee [9], [27]). It is known that both routing an network coing approaches can achieve optimal broacast rate over arbitrary P2P graph [2], [7]. Compare to routing-base approach, the network-coing base approach introuces aitional packet heaer overhea for carrying coing coefficients (e.g., 3% extra overhea accoring to [29]) an computation complexity for encoing an ecoing (e.g., [3], [27] iscuss how to keep the complexity low). However, the network-coing base approach is robust to peer ynamics since there is no nee for constructing an maintaining the spanning trees. In this section, we esign a istribute broacasting algorithm base on network coing that is robust to ynamics. In Section VII, we will iscuss how the overall problem can be solve by using centralize solutions when only routing is allowe. B. Network Coing Base Formulation Accoring to the Max-Flow-Min-Cut theorem, a ata transmission of rate z between source s an a receiver is feasible if an only if there exists a flow, enote as f, satisfying the We refer intereste reaers to [27], [28] for more etails on performance of routing-base an network-coing-base practical P2P systems. We focus on optimal istribute P2P broacasting algorithm esign base on network coing in this paper. following flow conservation constraints:, v R {}, (2) f uv z u out(s) f su, (3) f, (4) where in(v){u (u, v) E f } is the set of noes sening content to v uner configuratio, an out(v){u (v, u) E f, u s} is the set of noes receiving content from v. A powerful theorem establishe in [2] states that a multicast or broacast rate z from s to a set of receivers is achievable if an only if z is feasible for s an any receiver. This is a strong result as it says that if the network can support a unicast rate of z between s an any receiver assuming other receivers traffic is absent, then it can support a multicast rate of z to all the receivers simultaneously. Such rate z can be achieve by every noe in the network performing network coing [2]. Further, authors in [29], [3] show that it is sufficient to perform ranom linear network coing. In ranom linear network coing, by inepenently an ranomly choosing a set of coing coefficients from a finite fiel, each noe sens out the coe packet as a linear combination of the noe s receive packets. The combination information is specifie by a coefficient vector in the packet heaer, which is upate by applying the same linear transformations as to the ata. When one noe receives a full set of linearly inepenent coe packets, it can ecoe an recover the original packets. In this paper, we focus on the istribute algorithm esign. The iscussions of ecoing probability an implementation etails can be foun in [29], [3]. Uner the setting of network coing, we can consier f as a virtual informatiolow between s an. Multiple informatiolows piggyback together to transmit over the physical links. The actual physical rate over a physical link is only the maximum rate of iniviual informatiolows passing over it. Let g uv be the physical flow rate over a link (u, v) E f, then we have f uv g uv for all R. With the above unerstaning, we formulate the problem of maximizing broacast rate uner configuratio as follows: MP : s.t. max U(z) (5) z, f,g fuv + z½ {v=s}, v V {}, R,(6) g vu, v V,, R, (7) g vu C v, v V, (8) where U (z) is a twice-ifferentiable strictly concave utility function 2, ½ { } enotes the inicator function. The constraints 2 It might seem unnecessary to involve a strictly concave utility function in this formulation. The reason is that we later esign a primal-ual algorithm to solve the problem, an using a strictly concave utility function can avoi its potential instability problem [7].

4 in (6) escribe the flow conservation requirements. The constraints in (7) come from the piggybacking property of information flows. The noe uploa capacity constraints are in (8). The problem MP is a convex problem. All feasible broacast rates must satisfy the constraints in (6)-(8) an are achievable by using ranom linear network coing. C. Algorithm Design via Lagrange Decomposition To procee, we first relax the first set of constraints in (6) in problem MP to obtain a partial Lagrangian as follows: L(z, f, g,λ) =U(z) v V {} R λ v, =U(z) λ v, f uv + z½ {v=s} f uv + z½ {v=s}, (9) whereλ v,, v V {}, R are Lagrange multipliers,λ, =, R, an u in(s) fus=. The strong uality hols for problem MP since the Slater conitions are satisfie [3]. Therefore, we can solve problem MP by fining the sale points of L(z, f, g,λ). Noticing that λ v, z½ {v=s} = z λ s, () Given λ an z, we consier the following scheuling subproblem o, g: SSP : max f,g (λ v, λ u, ) (3) s.t. R v V (7) (8). The above linear programming problem has a structure that allows us to solve it istributely. The first observation is that if an optimal g is given, then an optimal f can be obtaine as follows: u, v V, R, ( f vu ) =, ifλ v, λ u,, g vu, otherwise. (4) As such, it is sufficient to stuy the following problem in g: g vu w vu (5) where max g s.t. v V g vu C v, v V, w vu [λ v, λ u, ] +, (u, v) E f. (6) R enotes the aggregate back-pressure between two neighboring noes u an v, an [ ] + max(, ). For any v V, let u (v)arg max w vu (7) be one of its neighbors with the maximum back-pressure (breaking ties arbitrarily). Then one optimal solutioor problem SSP is as follows: an ( g vu ) = C v, if u=u (v),, otherwise, ( ) f = ifλ v, λ u,, vu, g vu, otherwise. Give an g, primal-ual algorithms can be esigne to aapt z anλto pursue the esire optimal solution. R We summarize the above analysis into a istribute algorithm incluing the following components: an Primal-ual Rate Control: we pursue the sale point in λ v, fuv = (λ z an λ simultaneously as follows: u, λ v, ), R v V ż=α[u (z) Rλ s, ] + z, () [ ( ) λ v, = k v, f + uv z½v=s we cain the sale points of L(z, f, g,λ) by solving the ( ) f ] + vu, v V {}, R, λ v, λ following problem successively in z, f, g,λ: v, λ, =λ, =, R, (2) min λ max(u(z) λ s, )+ max (λ v, λ u, ) z f,g whereαan k v, are positive step sizes, an the function R R v V (2) [b] + max(, b), a, a= s.t. (7) (8). b, a>. (8) (9) Neighbor Scheuling, Content Scheuling, an Network Coing: Every noe v V maintains a queue storing packets that are intene for. Whenever a transmission opportunity arises, noe v chooses one neighbor u (v) with the maximum back-pressure accoring to (7). If w vu (v)>, noe v sens packets to u (v) at rate C v. Every output packet is constructe as follows. Noe v chooses one packet from the hea of each queue of ifλ v, λ u (v),>, an output one ranom linear combination of these hear-ofqueue packets. If otherwise w vu (v) or there is no hea-ofline packets to coe, noe v oes nothing. We have the following observations. The Lagrangian variableλ v, is proportional to the length of queue storing packets that are intene for receiver.

5 The back-pressure w vu measures the aggregate ifference in the queues of all Rbetween v an u. The larger the back-pressure is, the more esperate noe u wants to receive ata from v. Our algorithm can be implemente in a istribute manner. It only requires noes to exchange information with its one-hop neighbors, an thus is robust to peer churn an system ynamics. When a new peer arrives, it connects to a set of neighbors, assigne by the streaming server or trackers. Then the peer starts exchanging streaming ata with them following the strategy efine by our algorithm. When a peer leaves, its neighbors are informe an then close the connections. For the network coing operation, theoretically we nee to ajust the size of fiel where the coing coefficients are chosen to make sure of the ecoing probability when the number of noes changes [32], [33]. While [29] an [3] show that in practice the finite fielf 2 8 orf 2 6 is enough to have a sufficiently high ecoing probability. Therefore, only local configuration changes corresponing to ynamics, which is easy to implement compare to centralize algorithms where typically global information is neee for whole configuration change (e.g., spanning trees reconstruction in spanning tree base solutions). Although our algorithm is esigne for P2P broacast scenarios, it also works for P2P multicast scenarios where helper noes exist. The helper noes simply also perform the operations escribe in (8)-(2). Our algorithm can be consiere as the extension of the algorithm in [3] from link-capacity-limite unerlay networks to noecapacity-limite overlay networks. The following theorem characterize the convergence of the propose algorithm. Theorem : The algorithm in (8)-(2) converges to the optimal solution of problem MP globally asymptotically in time. The proof utilizes stanar Lyapunov arguments an a Lyapunov functioor primal-ual algorithm, similar to those use in [7], [34]. The proof is relegate to Appenix -A. Remark: We erive our algorithm an prove its convergence base on a flui moel formulation. It is also possible to obtain a similar back-pressure base istribute algorithm with packet-level ynamics taken into account an prove its stability, following a set of Lyapunov rift arguments elaborate in [35]. IV. The Propose Distribute Topology Hopping Algorithm We recently propose in [2] to use Markov chain as a principle approach in esigning istribute algorithms for solving combinatorial network problems approximately. In particular, we show one can esign istribute algorithms for a combinatorial network optimization problem in the following way. First, construct a special class of Markov chains with problem-specific steay-state istribution. Secon, search for a Markov chain in this class that allows istribute implementation. If such Markov chain can be foun, which is usually challenging an problem-specific, the istribute implementation irectly yiels a istribute algorithm for the problem. In this paper, we follow the framework from [2] an esign a istribute topology hopping algorithm for our problem (). There are two steps in esigning our algorithm uner the Markov approximatioramework [2]: log-sum-exp approximation an constructing problem-specific Markov chains that allows istribute implementation. A. Log-Sum-Exp Approximation First, the maximum broacast rate can be approximate by a log-sum-exp function as follows: max x f β log exp ( ) βx f, (2) whereβis a positive constant. Let F enote the size of the setf, then the approximation accuracy is known as follows [2]: β log exp ( ) βx f max x f log F. (22) β As β approaches infinity, the approximation gap approaches zero. As iscusse in [2], however, usually β shoul not take too large values as there are practical constraints or convergence rate concerns in the algorithm esign afterwars. To better unerstan the log-sum-exp approximation, we associate with each configuratio F a probability p f. Consier the following problem MRC EQ : max p s.t. p f x f (23) p f =. (24) Its optimal value is max x f an is obtaine by setting the probability corresponing to one of the best configurations to be one an the rest probabilities to be zero. Hence, problem MRC EQ is equivalent to the original problem MRC. On the other han, accoring to [2] we have the following observations. Theorem 2 (cf. [2]): The optimal value of the following optimization problem MRC β : max p f x f p f log p f (25) p β s.t. p f = (26) is given by β log[ exp ( )] βx f. The optimal solution of problem MRC β is given by p f (x)= exp ( ) βx f exp ( ),. (27) βx f

6 As such, by the log-sum-exp approximation in (2), we obtain an approximate version of the maximum broacast rate problem MRC, off by an entropy term β p f log p f. If we can time-share among ifferent configurations accoring to the optimal solution p f (x) in (27), then we can solve the problem MRC approximately an obtain a close-to-optimal broacast rate. B. Markov Chain Guie Algorithm Design We esign a Markov chain with a state space being the set of all feasible peering configurations F an has a stationary istribution as p f (x) in (27). We implement the Markov chain to guie the system to optimize the configuration. As the system hops among configurations, the Markov chain converges an the configurations are time-share accoring to the esire istribution p f (x). The key lies in esigning such Markov chain that allows istribute implementation. Since p f (x) in (27) is prouctform, it suffices to focus on esigning time-reversible Markov chains [2]. Let f, be two states of Markov chain, an enote q f, f as the transition rate from state f to f. We have two egrees of freeom in esigning a time-reversible Markov chain: The state space structure: we can a or cut irect transitions between any two states, given that the state space remains connecte an any two states are reachable from each other. The transition rates: we can explore various options in esigning q f, f, given that the etaile balance equation is satisfie, i.e., p f (x)q f, f = p f (x)q f, f, f,. (28) Satisfying the above equations guarantees the esigne Markov chain has the esire stationary istribution as in (27). Recall that for a noe v V, the set of its neighbors uner configuratio is enote by N v, f. We call noe in N v, f v s in-use neighbor an noe in N v \N v, f v s not-in-use neighbor. For the ease of explanation, we further efinen f as the set of all the noe-pairs uner f, i.e.,n f ={{v, u}, v V, u N v, f }. Note we o not ifferentiate noe pairs{u, v} an{v, u}. As an example, for the peering configuratio shown in Fig. (b), N f is given by{{s, },{s, 2},{s, 4},{, 2},{, 4},{2, 3},{3, 4}}. In our Markov chain esign, we first specify its state space structure as follows: we set the transition rate q f, f to be zero, unless f an f satisfy that N f \N f =or N f \N f =. In other wors, we only allow irect transitions between two configurations if such transitions correspon to a single noe aing a new noe in its in-use neighbor set or removing one in-use neighbor from its in-use neighbor set. Secon, given the state space structure of Markov chain, we esign the transition rates to favor istribute implementation while satisfying the etaile balance equation in (28). One possible option is to set q f, f to be exp (βx f ). One way to implement this option is for every noe to generate a timer accoring to its measure receiving rate an counts own accoringly. When the timer expires, the eicate noe performs the neighbor swapping an resets its timer. As simple as the implementation may soun, this option is expensive to implement. Once the peering configuration changes, the system nees to notify all the noes to measure the new receiving rate an reset their timers accoringly. It is not clear how to implement such system-wie notification in a lowoverhea manner. In this paper, we esign q f, f an q f, f as follows: an q f, f = exp(τ) (29) ) q f, f= exp(τ), (3) where τ is a constant. It is straightforwar to verify that etaile balance equation is satisfie. As will be clear in the next subsection, our choices of transition rates o not require coorination or notification among peers in its implementation. C. Distribute Implementation One istribute implementation of our esigne Markov chain is briefly escribe as follows. Initialization: Each peer v V ranomly selects neighbors from its neighbor list N v uner the noe egree boun an buils connections with these selecte neighbors. Step : Let f enote the current configuration. Each noe v V generates an exponentially istribute ranom number inepenently with mean 2 exp(τ), an counts own accoring to this number. Step 2: When the count-own expires, noe v measures its current receiving rate as an estimate of the broacast rate x f. Then with probability N v, f noe v goes to the Step 2a; with probability N v N v, f, noe v goes to the Step 2b; Step 2a: Noe v ranomly selects one in-use neighbor in N v, f an removes it from N v, f. Uner the new peering configuratio, noe v measures its receiving rate as an estimate of x f. With the estimates of x f an x f, peer v stays in the new configuratio with probability f with probability )+exp(βx f ) exp(βx f ), an switches back to. Noe v then repeats Step. Step 2b: Noe v ranomly selects one not-in-use neighbor in N v \N v, f. If the noe egree of the selecte not-in-use noe is equal to the boun or v s noe egree is equal to the boun, noe v jumps back to Step immeiately. Otherwise, noe v as this selecte noe into N v, f. Uner the new peering configuratio, noe v measures its receiving rate as an estimate of x f. With the estimates of x f an x f, peer v stays in the new configuratio with

7 probability with probability )+exp(βx f ) exp(βx f ), an switches back to f. Noe v then repeats Step. It is straightforwar to summarize the above implementation into a istribute algorithm that runs on iniviual noes an utilizes only the measurement from their one-hop neighbors. The correctness of the implementation is shown as follows: Proposition : The implementation iact realizes a timereversible Markov chain with stationary istribution in (27). The proof is relegate to Appenix -B. Remarks: a) In Step, the generation of count-own timers oes not epen on the receiving rate, thus the system oes not nee to notify the noes about changes of peering configurations. b) With the above implementation, the system hops towars configurations with better broacast rate probabilistically. For example, if x f > x f, then the system will be more likely to stay in configuratio than i, an vice versa. c) With large values of β, the system hops towars better configurations more greeily. However, this may as well lea to the system getting trappe in locally optimal configurations. Hence there is a trae-off to consier when setting the value ofβ. Moreover, the value ofβalso affects the convergence rate of the time-reversible Markov chain to the esire stationary istribution. It is worth future investigation to further unerstan the impact ofβ. ) In the presence of peer ynamics, our algorithm incurs only simple actions base on local information. When a new peer arrives, a neighbor set an a neighbor list are assigne to it. The peer buils connections with the noes in the neighbor set. Then the peer starts counting own as Step an follows the strategy of our algorithm. When a peer leaves, we just eliminate it from the neighbor list of its previous neighbors an en up connections. V. Convergence Properties of Overall Solution We have esigne the istribute broacasting algorithm in Section III an the Markov chain guie topology hopping algorithm in Section IV. The pseuocoes of each algorithm are shown in Algorithm an Algorithm 2 respectively. Both algorithms are simple to implement, run on each iniviual noe, an only require noes to exchange information with their neighbors. If the broacasting algorithm converges instantaneously, i.e., time-scale separation assumption hols, then we can obtain the accurate value of x f for any configuratio F. Transiting base on the accurate x f, the esigne Markov chain will converges to the esire stationary istribution in (27). Hence by operating these two algorithms in tanem, we obtain a close-to-optimal broacast rate uner arbitrary noe egree bouns, an over arbitrary overlay graph. The optimality gap is characterize in (22). In practice, however, it is possible to obtain only an inaccurate measurement or estimate of x f. These inaccuracies root in two sources. One is the noisy measurements of the maximum broacast rates given the configuration. The other is the fast state transition of Markov chain, i.e., the Markov Algorithm Broacasting Algorithm : The following proceure runs on each iniviual noe inepenently. 2: For the source s an each time slot, 3: x [ x+α(u (x) Rλ s, ) ] + 4: For each noe v V an each time slot, 5: w 6: for o 7: for for R o 8: w vu w vu + max(λ v, λ u,, ) 9: en for : if w vu > w then : w w vu 2: u u 3: en if 4: en for 5: if w vu > then 6: for R o 7: ifλ v, λ u,> then 8: f vu C v 9: en if 2: en for 2: en if 22: for R o 23: λ v, [ λ v + k v, ( f uv f vu) ] + 24: en for chain transits before the unerlying broacasting algorithm converges an thus it transits base on inaccurate observations of the broacast rates. Consequently, the topology hopping Markov chain may not converge to the esire stationary istribution p f (x). This observation motivates our following stuy on the convergence of Markov chain in the presence of inaccurate transition rates. For each configuratio F with broacast rate x f, we assume its corresponing inaccurate observe rate belongs to the boune region [ ] f, f. f is the inaccuracy boun an can be ifferent for ifferent f. For easy explanation of our approach, we further assume the observe broacast rate for configuratio only takes one of the following 2 + iscrete values: [x f f,..., x f n f, x f, x f + n ] f,..., x f + f, f f where is a positive constant. Further, with probability η j, f, the observe broacast rate takes value x f + j f, j {,..., } an j= η j, f =. With the inaccurate observe broacast rates, the topology hopping behaves as follows. Suppose the current configuration is f an the observe broacast rate is x f + j f, where j {,..., }. After some count-own process, the system hops to a new configuratio an probes its broacast rate. In configuratio, the broacast rate is observe as x f + j n f f, j {,..., }. The system stays in the new

8 Algorithm 2 Topology Hopping Algorithm : The following proceure runs on each iniviual noe inepenently. We focus on a particular noe v V. 2: proceure Initialization Initialize N v, B v ; ranomly connects to peers from N v uner the egree boun. Generate a timer that follows exponential istribution with mean equal to 2 exp(τ)/() an begin counting own. 3: en proceure 4: 5: When the timer expires, invoke the proceure Transition. 6: proceure Transition 7: With probability N v, f, 8: N o N v, f ; 9: ranomly remove one in-use neighbor from N v, f ; : x f f v uv; : N v, f N o with probability / ( ) ; 2: refresh the timer an begin counting own; 3: With probability N v, f, 4: N o N v, f ; 5: ranomly a one not-in-use neighbor v in N v \N v, f to N v, f ; 6: if N v, f = B v or N v, f = B v 7: refresh the timer an begin counting own; 8: en if 9: x f f v uv ; 2: N v, f N o with probability / ( ) ; 2: refresh the timer an begin counting own; 22: en proceure configuratio with probability exp(β(x f + j n f f )) exp(β(x f + j n f f ))+exp(β(x f+ j f )), an switches back to configuratio with probability exp(β(x f + j n f f )) exp(β(x f + j n f f ))+exp(β(x f+ j f )). By arguments similar to the proof of Proposition, the transition rate from configuratio with broacast rate x f + j f to configuratio with broacast rate x f + j n f f is given by η j, f exp(τ) exp(β(x f + j f )) exp(β(x f + j f ))+ exp(β(x f+ j f )). (3) We construct a Markov chain to capture an stuy the above topology hopping behavior. In this Markov chain, a state is associate with a configuration an an observe broacast rate. Given any configuratio F an its corresponing x f, there are 2 + states in the extene Markov q OriginalTopologyHoppingMarkovChainM withexactbroacastrates, x 2, x 2 3, x3 CorresponingExteneMarkovChainM withinaccuratelyobservebroacastrates, x 2, x2 2 3, x3 3 (, x ),(2, x2 ) q(, x ),(2, x2 2 ) q(, x ),(2, x2 2 ), x 2, x 2 3, x3, x 2, x2 2 3, x3 3 Fig. 2. An example of the original three-state topology hopping Markov chain an the extene Markov chain. M is the original topology hopping Markov chain with accurate broacast rates. M is the corresponing extene Markov chain with inaccurate broacast rate observations. For each configuratio {, 2, 3}, the observe broacast rate takes values x f f, x f, x f + f with probabilityη, f,η, f anη, f respectively. The transition rates are assigne accoring to (32) an (33). ( ) chain: f, x f + j f, j {,..., }. Further, Given irect transitions between configuratio an f in the original topology hopping Markov chain, there are irect transitions between states ( f, x f + j f ) an ( f, x f + j n f f ) ( j {,..., }, j {,..., }) in the corresponing new Markov chain. The corresponing transition rates are shown as follows: an q ( f,x f + j f ),( f,x f + j f = η j, f exp(τ) exp(β(x f + j f )) exp(β(x f + j f ))+exp(β(x f+ j f )) q ( f,x f + j f,( f,x f+ j f ) = η j, f exp(τ) exp(β(x f + j f )) (32) exp(β(x f + j f ))+exp(β(x f + j n (33) f f )), where j= η j, f = an j = n η f j, f =. This new Markov chain can be thought as an extene version of the original topology hopping Markov chain. As an example, an extene Markov chain is shown an explaine in Fig. 2. The extene Markov chain has a unique stationary istribution since it is irreucible an only has a finite number of states. We can stuy the impact of inaccurate broacast rates by comparing the stationary configuration istribution of the new Markov chain an that of the original topology hopping Markov chain. We enote the stationary istribution of the states in the new Markov chain by p [ p f,x f + j f, j {,..., }, ]. (34)

9 We also enote p : [ p f (x), ] as the stationary istribution of the configurations in the extene Markov chain. Given a configuratio F, there are 2 + states associate with f in the extene Markov chain. We have p f (x)= j {,..., } p f,x f + j f,. (35) Recall that the stationary istribution of the configurations for the original topology hopping Markov chain is p : [p f (x), f F ]. We use the total variance istance [36] to quantify the ifference between p an p, as TV (p, p) p f 2 p f. (36) We have the following result: Theorem 3: Let max = max f, an x max = max x f. The TV (p, p) are boune as follows: TV (p, p) exp ( 2β max ). (37) Further, the optimality gap in broacast rates p x T px T is boune as below: p x T px T 2x max ( exp( 2β max )). (38) The proof is relegate to Appenix-C. Remarks: a) The upper boun on TV (p, p) shown in (37) is general, as it is inepenent of the number of configurations F, the values of(, an the istributions of inaccurate observe ratesη j, f j, ). b) The upper boun on TV (p, p) shown in (37) ecreases exponentially with the worst inaccuracy boun max ecreasing. c) It woul be interesting to explore a tighter upper boun on TV (p, p) than the one in (37). TABLE II Peer uploa capacity istribution Uploa Capacity (kbps) Fraction (%) respectively. These parameters are empirically chosen to obtain smooth algorithm upating an small errors. In our simulations, we assign noe egree bouns in the following two ways. The first is to set ientical boun on each noe s noe egree. The secon is to set egree boun proportional to the noe s uploa capacity. This is base on the empirical observations that noes with high uploa capacities usually have more system resource (e.g., memory an CPU power) than noes with low uploa capacities. With more system resource, noes can maintain more concurrent connections, thus have larger noe egree bouns. In our secon egree bouns assignment, noes set their noe egree bouns proportional to the ratio between their uploa capacities an 64 kbps. In particular, noes with 64 kbps have a egree boun of 2, an noes with 28 kbps have a egree boun of 4, etc. We carry out two sets of simulations. First, we evaluate the performance of our istribute broacasting algorithm uner Setting I an II. Secon, we evaluate the overall performance when we combine the topology hopping algorithm an the broacasting algorithm uner Setting I an III. In these two sets of simulations, we also compare the performance uner the two egree bouns assignments explaine in the previous paragraph. VI. Performance Evaluation We implement a packet-level simulator to our propose solutions an use this simulator to evaluate the performance of our solutions. A. Settings In our simulations, time is choppe into slots of equal length, an we aopt three ifferent settings. In Setting I, we set the total number of noes to be, an assign the noe uploa capacities ranomly accoring to the istribution in Table II, which is obtaine from the uplink banwith statistics of Internet hosts [37]. We set the source s uploa capacity to be 768 kbps; with this uploa capacity, source is not the broacast bottleneck [], [3]. Setting II is the same as Setting I, except we set the total number of noes to be. In Setting III, there are 4 ifferent peering configurations as shown in Fig. 3. Every noe has a unit capacity. Uner configuratio, f 2 an f 3 the maximum broacast rate is, an uner configuratio 4 the maximum broacast rate is.5. When running our network coing base broacasting algorithm, we set the upating step size of z anλto be. an Fig. 3. Peering configurations uner Setting III. For the ease of illustration, we only allow noe to a or remove neighbors between noes 2 an 4. The rest noes keep their neighbors fixe. B. Evaluation of the Propose Broacasting Algorithm In this simulation, we evaluate our istribute broacasting algorithm propose in Section III. We ranomly choose a sub-graph that satisfies the noe egree bouns constraints, an run our algorithm over it. We evaluate three aspects of the propose algorithm: ) oes it converge to optimal broacast rate as expecte from theoretical analysis? 2) How fast oes it converge? 3) How woul ifferent values of egree

10 rate(kbps) source streaming rate peer receiving rate rate(kbps) source streaming rate peer receiving rate rate(kbps) source streaming rate peer receiving rate CDF boun= proportional boun full mesh rate boun= time slot 5 5 time slot 5,5 time slot peer receiving rate(kbps) (a) (b) (c) () Fig. 4. Broacasting algorithm evaluations. a) The source broacast rate an average peer receiving rate uner Setting II when egree boun is set to 3; b) The source broacast rate an average peer receiving rate uner Setting I when egree boun is set to 4; c) The source broacast rate an average peer receiving rate uner Setting I when egree boun is set to. ) This figure shows the impact of egree boun on the peer receiving rate uner Setting I. The full-mesh rate is the maximum broacast rate when the noe egrees are unboune [] rate(kbps) rate uner our overall algorithm rate uner our broacasting algorithm 22% rate uner the simple algorithm 55% rate(kbps) rate uner our overall algorithm 55% rate uner the simple algorithm 25% rate uner our broacasting algorithm rate(kbps) rate uner our broacasting algorithm rate uner the simple algorithm 7% rate uner our overall algorithm 25 5 time slot 25 5 time slot time slot (a) (b) (c) Fig. 6. Evaluation of our overall solution which combines the topology hopping algorithm an the broacasting algorithm. a) The average peer receiving rate when the noe egree boun is 3 anβis 2; b) The average peer receiving rate when the noe egree boun is 3 anβis 5; c) The average peer receiving rate when peer egree boun is proportional to its uploa capacity an β is 2. The percentage of average receiving rate improvement of our overall algorithm against our broacasting algorithm an the simple heuristic algorithm are shown in these three figures. For example, in (a), 22% means that the average receiving rate of our overall algorithm is.22 times of that of our broacasting algorithm, an 55% means that the average receiving rate of our overall algorithm is 6.5 times of that of the simple heuristic algorithm. fraction(%) p(f ) p(f 2 ) p(f 3 ) p(f 4 ) β= β=5 β= (a) fraction(%) β= p(f ) p(f 2 ) p(f 3 ) p(f 4 ) β=5 β= Fig. 5. a) Optimal configuration istributioor ifferent values of β uner Setting III; b) Configuration istribution obtaine by our algorithm for ifferent values of β uner Setting III. bouns affect the maximum broacast rate? The results are summarize in Fig. 4. From Fig. 4(a) an Fig. 4(b), we see that our broacasting algorithm converges. It converges faster in the small size network as shown in Fig. 4(a) than in the large size network as shown in Fig. 4(b). From Fig. 4(), we also see the converge rate when the noe egree bon is is very close to a theoretical upper boun the optimal broacast rate uner no egree bouns compute accoring to [], [7], [3]. This suggests that our algorithm converges to the optimal broacast rate. Uner ifferent egree bouns, the optimal broacast rate varies. Fig. 4() shows that the optimal broacast rate increases (b) when we increase the noe egree bouns. We plot the CDF of peer receiving rates (after the broacasting algorithm converges) for the case where egree boun is 4,, an proportional to the peer s uploa capacity. It s seen that when the boun is, the obtaine rate is close to the full-mesh rate, which suggests that we o not nee a large egree boun to achieve close to the full-mesh rate. The obtaine rate is also close to the full-mesh rate when egree boun is proportional to the peer s uploa capacity. C. Evaluation of the Overall Solution Our overall solution, which combines the Markov chain guie topology hopping algorithm an the back-pressure an network coing base broacasting algorithm, achieves the near optimal broacast rate uner arbitrary noe egree boun an over arbitrary overlay graph. To evaluate its performance, we generate a sub-graph ranomly, run our algorithms on every noe, an evaluate the achieve broacast rate. The topology hopping algorithm runs on top of the broacasting algorithm. Uner given topology, the broacasting algorithm achieves the optimal broacast rate. Noes swap neighbors base on their observe receiving rate, thus changing the topology from time to time. In the simulation, we run the broacasting algorithm long enough so that it converges before the topology transits accoring to the Markov chain.

11 This way, the overall algorithm converges to the close-tooptimal broacast rate. In all simulations, we compare our overall algorithm with our back-pressure an network coing base broacasting algorithm to illustrate the benefit of topology hopping, an with a simple heuristic algorithm introuce below to illustrate the benefit of our overall solution. Remin that no existing works solve the problem of streaming-rate maximization uner general noe egree bouns an over arbitrary topology we stuie in this paper. The simple heuristic algorithm we compare our overall algorithm against is also compose of two parts: routingbase broacasting algorithm an ranom topology hopping algorithm. In routing-base broacasting algorithm, each peer evenly allocates its uploa capacity to its neighbors. Given the topology an capacity allocation, a centralize routing strategy (e.g. spanning trees base solution) is use to achieve the best broacast rate the system can support. Similarly, the ranom topology hopping algorithm runs on the top of the broacasting algorithm. Every peer maintains a timer. When the timer of one peer expires, the peer ranomly rops one active neighbor which is exchanging ata with it, an then selects one ranom caniate from its feasible neighbor list an starts to exchange ata with it. By oing so, we actually allow noes running the simple scheme to have a noe egree beyon the bouns. This relaxation gives the simple scheme more egree of freeom to optimize its performance. Overall, the topology changes ranomly on the top uner which peers use routing to exchange streaming ata. Our first observation is that our overall scheme converges to the solution that theory preicts. We carry out simulations uner Setting III. Uner this setting the optimal broacast rate is. The optimal configuration solution to problem MRC β is calculate an shown in Fig. 5(a) for ifferent values ofβ. We run the overall scheme for this specific case an show the empirical configuration istribution in Fig. 5(b). Comparing the istributions in Fig. 5(a) an Fig. 5(b), we can see that the istribution obtaine by our overall solution is very close to the optimal one. We also calculate the achieve broacast rate uner ifferent values ofβ. Forβ=, 5 an, the broacast rate is.97,.987, an.998 respectively. We see that with largeβ, the achieve broacast rate is close to the optimal value, as preicte by our analysis in Section IV. Next, we evaluate our overall solution uner Setting I. In Fig. 6(a) an Fig. 6(b), the broacast rates obtaine are 35 kbps an 32 kbps respectively. They are about 22% an 25% higher respectively than the broacast rate 25 kbps achieve by running the broacasting algorithm over a ranomly chosen topology. This emonstrates the avantage of performing topology hopping to optimize the configuration, as compare to only ranomly choosing topology. By setting noe egree bouns proportional to peers uploa capacity, noes with higher uploa capacity maintain more connections. From Fig. 6(c), we observe that this flexibility offers a broacast rate of 475 kbps. Although the aitional gain of topology hopping is small uner the specific P2P simulation settings (e.g., noe uplink capacity istribution), we remark that our topology-hopping base algorithm is theoretically guarantee to achieve close-to-optimal streaming rate uner arbitrary noe egree bouns an P2P settings, while the broacasting algorithm with ranom topology selection has no performance guarantee. Moreover, in practical P2P streaming systems, the noe egree bouns are typically small. For example, in PPLive, the noe egree bouns are 5-2 [5], while the size of the system (i.e., total number of peers that are simultaneously watching the same channel) is usually hunres of thousans. Thus, we suspect we can see substantial gain of topology hopping if our algorithm is implemente in such system with small noe egree bouns, as suggeste by our simulation results uner small noe egree bouns. From Fig. 6(a), Fig. 6(b) an Fig. 6(c), we observe that the average receiving rate of our overall algorithm is about times higher than that of the simple algorithm respectively. An also we can see from Fig. 6(a) an Fig. 6(b), our algorithm can achieve smoother streaming rate than the simple algorithm because our algorithm optimizes the topology hopping an stays in the optimal topology while the simple algorithm hops among topologies ranomly an arbitrarily. VII. Discussions an Future Work In this paper, we propose a istribute solution to achieve a near-optimal broacast rate uner arbitrary noe egree bouns, an over arbitrary overlay graph. Our solution is istribute an consists of two algorithms that can be of inepenent interests. The first is a istribute broacasting algorithm that optimizes the broacast rate given a P2P topology. It is erive from a network coing base problem formulation an utilizes back-pressure arguments. It can be consiere as the extension of the algorithm in [3] from linkcapacity-limite unerlay networks to noe-capacity-limite overlay networks. The secon algorithm is a Markov chain guie hopping algorithm that optimizes the topology, inspire by the Markov Approximatioramework introuce in [2]. Assuming the unerlying broacasting algorithm converges instantaneously, the topology hopping algorithm converges to the optimal configuration istribution. When the broacasting algorithm oes not converge fast enough, the topology hopping Markov chain transits base on inaccurate observations of the maximum broacast rates associate with the configurations. We show that the topology hopping algorithm still converges, but to a sub-optimal configuration istribution. We characterize an upper boun on the total variance istance between the optimal an sub-optimal configuration istributions, as well as an upper boun on the gap between the achieve an the optimal broacast rates. We show that both bouns ecreases exponentially as the boun on inaccuracy ecreases. Using uplink banwith statistics of Internet hosts, our simulations valiate the effectiveness of the propose solutions, an emonstrate the avantage of allowing noe egree bouns to scale linearly with their uploa capacities.