On Topology Construction in Layered P2P Live Streaming Networks

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1 On Topology Construction in Layered P2P Live Streaming Networks Runzhi Li College of Info Engineering Zhengzhou University Zhengzhou, Henan, , China Qishi Wu, Yunyue Lin, Xukang Lu Dept of Computer Science University of Memphis Memphis, TN 38152, USA Zongmin Wang Henan Prov. Key Lab on Info Network Zhengzhou University Zhengzhou, Henan, , China Abstract Peer-to-peer (P2P) overlay networks provide a highly effective and scalable solution to live media streaming systems that require the collective use of massively distributed network resources. A P2P media streaming architecture is typically built completely or partially upon a tree-structured network topology and the process of tree construction has a significant impact on the overall system performance. We build network cost models and formulate a specific type of topology construction problem, Maximum Average Bandwidth Spanning Tree (MABST), which aims at optimizing the system s average stream rate. We prove that MABST is NP-complete by reducing from Hamiltonian Path problem and propose an efficient heuristic algorithm. The performance superiority of the proposed algorithm is justified by experimental results using a live media streaming system deployed in real networks and is also illustrated by an extensive set of simulations on simulated networks of various sizes in comparison with other methods based on a degree constraint or a greedy strategy. Index Terms Overlay networks, P2P, NP-complete, spanning tree I. INTRODUCTION Live streaming networks have found widespread applications in various fields especially for media delivery. These applications often require the collective use of massively distributed network resources and therefore are not adequately supported by the traditional client-server architecture in the Internet. Peer-to-peer (P2P) overlay networks enable efficient resource sharing in distributed environments and provide a highly effective and scalable solution to this problem. Such P2P live streaming systems (P2PLSS) include DONet/CoolStreaming [1], PPlive [2], PPStream [3] and many others. The performance measure of a live streaming system concerns several aspects such as jitter, latency, and uplink utilization. Jitter is caused by the unavailability of stream contents at play time. In general, a high level of system throughput can maintain a stream rate that ensures high-quality media playback and continuous supply of stream contents, and therefore is considered as a critical performance requirement. Latency is the time that a media stream transfers from the data source to a client peer. A live event, for example, a sports game, may lose its real-timeness at critical moments because of a high latency. Uplink utilization is another performance metric related to traffic distribution and load balance. To a large degree, these performance metrics rely on the overlay network topology, upon which the P2P streaming system is built. Therefore, constructing an efficient overlay network topology has become a fundamental task in P2P systems to achieve and sustain an acceptable level of quality of service (QoS). There are three widely adopted network architectures in P2P-based live streaming systems: (i) Tree-push mechanism that deploys application-level multicast, including ZigZag [4] and NICE [5]. In this architecture, peers are organized to form one or more peer trees. Each peer in the tree receives media streaming data from its parent peer and forwards the data to its own children peers. The tree-push mechanism requires the live streaming system to maintain the topology of the peer tree in response to changes in the tree structure caused by peers churn. (ii) Mesh-pull mechanism that allows peers to pull video from each other, including CoolStreaming. In this architecture, the video is divided into data chunks which are stored in a local buffer. Each peer makes an independent request for available data chunks to view the video of interest. Compared to the tree-push mechanism, the mesh-pull mechanism reduces the cost of maintaining the tree structure at the expense of increasing redundant traffic among peers requesting for Buffer-Mapping [2]. (iii) Hybrid mechanism that combines both aforementioned approaches, including Anysee [6] and AHLSS [7]. In this architecture, peers are divided into Super Peers (SPs) and Normal Peers (NPs) based on their service capacity such as upload bandwidth, online status, and CPU speed. Since SPs are generally more stable with higher upload bandwidths than NPs, the tree-push mechanism is usually applied to SPs and the mesh-pull mechanism is employed among NPs to handle peers frequent joining and leaving requests. A P2P streaming architecture is built completely, as in the case of tree-push mechanism, or partially, as in the case of hybrid mechanism, upon a tree-structured network topology and therefore the process of tree construction has a significant impact on the overall system performance. There exist a number of tree construction algorithms in the literature including random algorithm, high-bandwidth-first algorithm, and short-

2 tree algorithm with a different focus on simpleness, tree depth, or reliability [8], [9], [10], [11], [12]. These algorithms can be directly applied to both tree-push and hybrid architectures. We formulate and investigate a specific type of topology construction problem, Maximum Average Bandwidth Spanning Tree (MABST), which aims at optimizing the system s overall throughput or stream rate. Being aware of several different performance metrics for evaluating live streaming systems, in this problem we focus on the aspect of system throughput maximization mainly because (i) considering multiple conflictive optimization objectives may lead to an undecidable performance evaluation, and (ii) system throughput affects the performance of live streaming systems more significantly than other factors in small- to medium-sized networks consisting of high-end hosts with low forwarding overhead. In most localarea networks, the typical round trip time (RTT) is on the order of microseconds, and even in wide-area networks across a nation such as United States, the RTT over a long-haul IP path is merely on the order of milliseconds. The effect of latency on the system response time may be negligible if the number of hops from the source to an end peer does not exceed several dozens. We prove that MABST is NP-complete by reducing from Hamiltonian Path problem and propose an efficient heuristic algorithm. The performance superiority of the proposed algorithm is justified by experimental results collected by a live media streaming system deployed in real networks and is also illustrated by extensive simulations performed on a large set of simulated networks of various sizes from small to large scales in comparison with other methods based on a degree constraint or a greedy strategy. The rest of the paper is organized as follows. We provide a survey of related work in Section II. We give a brief introduction to our P2P media streaming system in Section III. We formulate the Maximum Average Bandwidth Spanning Tree problem in Section IV, and prove its NP-completeness in Section V. We present the design of heuristic algorithms in Section VI and evaluate their performances in Section VII. We conclude our work in Section VIII. II. RELATED WORK The tree-based overlay network topology construction algorithm is referred to as a parent selection strategy in tree management, which provides a parent the privilege to select a peer to transmit stream data. In [8], Sripanidkulchai et al. proposed an efficient random algorithm that does not require global topological information. Guo et al. [9] proposed a high bandwidth first algorithm, in which peers are laid out according to their outgoing bandwidth capacities. The tree-based topology construction problem may also be considered as an optimal path selection problem for each newly joining peer with the requirement that the maximal reservable bandwidth of a feasible path is no less than the requested bandwidth. Dijkstra s shortest path algorithm or Bellman-Ford shortest path algorithm can be used to select a feasible path. Several other route computing algorithms were also proposed, such as the widest shortest path [10], the shortest widest path [11], and a utilization based shortest path [12]. In these algorithms, a different weight factor is assigned to limit hop counts and balance network load. Jarvis et al. [13] proposed a heap algorithm to construct overlay topology by moving high-bandwidth and long-lived peers upward in the logical tree to provide better service quality. Closely related to our work is the research efforts on topology construction using a spanning tree-based approach. Peter et al. [14] showed that finding the best topology is a combinatorial optimization problem and computed spanning trees for P2P overlay networks. They proposed a distributed algorithm (TreeOpt) for spanning tree optimization to reduce the overall communication time between any pair of nodes in the graph. Small et al. [15] formulated the optimal multicast tree problem as a minimal delay multicast (MDM) problem and the network topology optimization problem as a minimization problem of server bandwidth cost. They proved that MDM is NP-complete and provided two solutions to MDM. Zhu et al. [16] [17] introduced overlay networks with linear capacity constraints (LCC) and investigated two problems for widest path and maximum flow. Our work differs from the aforementioned ones in that we consider a specific goal of optimizing the system throughput. III. A LAYERED P2P LIVE STREAMING SYSTEM Fig. 1. The architecture of LStream: layer 1 is of a tree topology and layer 2 consists of a number of clusters. We propose a layered P2P live streaming architecture named LStream. As shown in Fig. 1, LStream consists of the following components: (i) BootStrap Server, (ii) Channel Source Peers (CSPs) such as nodes ch1 3, (iii) Super Peers (SPs) such as nodes s1 5, and (iv) Normal Peers (NPs) such as nodes n1 12. The BootStrap Server is responsible for centralized admission and enrollment when new peers join and maintain the peers assignment list. CSPs provide and transmit encoded source media streams. An NP receives streaming data directly or indirectly from CSPs and sends data chunks to its neighbors. Some NPs may communicate with an SP to form a cluster with the SP being the cluster head and the NPs being the cluster members. The entire set of SPs constitutes a data forwarding layer with a tree-based topology structure.

3 The proposed LStream system supports multi-channel live streaming (from nodes ch1-3 in Fig. 1). Since one channel results in one logical tree rooted at the corresponding CSP, there exists a multi-tree structure in layer 1 where different logical trees might share one physical SP that receives and transmits multi-channel streaming contents. Our objective is to construct a logical tree topology among SPs for each channel to maximize the system stream rate. IV. TOPOLOGY CONSTRUCTION PROBLEM FORMULATION An overlay computer network generally consists of a number of heterogeneous computer nodes interconnected by disparate overlay network links (IP paths). Depending on the underlying network infrastructure, the topology of an overlay network may be complete as in the case of the Internet based on layer-3 IP routing, or not as in the case of most dedicated research testbed networks using layer-1 or 2 circuit/lambda switching or MPLS/GMPLS techniques. Note that even in Internet environments, the overlay network topology may not be always complete because the network connectivity and resource accessability or availability could be largely affected by system dynamics and firewall settings on either routers or end hosts. In most local- and wide-area computer networks, the overlay nodes are of disparate system resources and the overlay links are of different transport properties. We represent an overlay network of arbitrary topology as a directed weighted graph G = (V, E), where each node v V has an incoming (downloading) bandwidth, an outgoing (uploading) bandwidth, a splitting outgoing bandwidth, and a throughput, which are defined as follows: Definition 1: incoming bandwidth is the maximum downloading speed of a node. As evidenced in many commercial production networks, a node s incoming bandwidth is typically much higher than its outgoing bandwidth and the data rate, and hence is assumed to be unconstrained in our problem. Definition 2: outgoing bandwidth is the maximum uploading speed of a node denoted by a finite constant value, which is much smaller than the incoming bandwidth 1 and usually causes a bottleneck of data transfer. Definition 3: splitting outgoing bandwidth is the bandwidth share over an outgoing overlay link of a node. In a tree-structured graph, if the outgoing bandwidth of a node v is b and it has p child nodes, each of which is connected via one overlay link, then the splitting outgoing bandwidth of v is b/p for each overlay link, assuming that the outgoing bandwidth of v is fairly shared by its p child nodes. This cost model is based on the fact that the transport bottleneck is often located on or close to the end node (last mile) in the underlying physical network and the fair share of bandwidth is well supported by the wide deployment and use of TCP or TCP-friendly protocols in the Internet. Definition 4: node throughput is the data receiving rate of a node, which is equivalent to the bandwidth of the path from 1 In the current Internet access market, most Internet Service Providers offer very limited uploading speed compared to downloading speed. the root node to the current node in a tree-structured graph, assuming a sufficiently large source data rate 2. Note that the path bandwidth is only determined by the narrowest link of the entire path. Based on the above network models, we formulate the Maximum Average Bandwidth Spanning Tree (MABST) problem as follows: given a directed weighted graph G = (V, E), outgoing bandwidth b v for each node v V, and a specified root node v 0 V, find a spanning tree rooted at node v 0 in G such that the average throughput of all the nodes except the root node in the tree is maximized. In MABST problem, we assume that the overlay link capacity and the node incoming bandwidth are much larger than the node outgoing bandwidth. Therefore, the bandwidth of a path from the root node to any other node is determined by the minimal splitting outgoing bandwidth of all the nodes on that path. V. COMPLEXITY ANALYSIS OF MABST PROBLEM We prove that MABST is NP-complete by reducing from Hamiltonian path problem, whose NP-completeness is well known [18]. We first formulate a decision version of MABST problem as follows: given a directed weighted graph G = (V, E), outgoing bandwidth b v for each node v V, a specified root node v 0 V, and a bound β, does there exist a spanning tree rooted at node v 0 in G such that the average throughput of all the nodes except the root node in the tree is at least β? G Fig. 2. An illustration of instance transformation from Hamiltonian path to MABST. Theorem 1: MABST is NP-complete. Proof: We first show that MABST NP. Given a solution (a spanning tree rooted at node v 0 ) to MABST, one can verify in O( V ) time the validity of the solution by checking whether or not the average throughput of all the nodes except the root node is greater than or equal to β. We now reduce the Hamiltonian path problem [18] to MABST. The Hamiltonian path problem is defined as follows: given graph G = (V, E), does there exist a Hamiltonian path that visits each vertex exactly once? Let G be an arbitrary instance of Hamiltonian path problem. We construct an instance (G, B, v 0, β) of MABST from the instance G in polynomial time such that G has a spanning tree rooted at node v 0 and the average throughput of all the nodes except the root node is greater than or equal to β, if 2 If the bottleneck is limited by the source data rate, the tree construction problem becomes trivial. v 0 G'

4 and only if there exists a Hamiltonian path in G. Here, B represents the outgoing bandwidth list for all nodes. G is simply a duplicate of G with an additional node v 0 and V number of new links (v 0, v), v V. Therefore, V = V v 0, and E = E {(v 0, v), v V }. An example of the instance transformation from Hamiltonian path to MABST is illustrated in Fig. 2. We set b v = 1, v V, and β = 1. Obviously, this instance construction can be done in polynomial time. Suppose that there exists a Hamiltonian path P in G. We can find a corresponding spanning tree T in G, where T is simply a duplicate of P with an additional node v 0 and a new link from node v 0 to one of the end nodes in P. Since each node except the leaf node in T has only one child node and its outgoing bandwidth is 1, the throughput of all the nodes except root node v 0 in T is 1, which is equal to β. Hence, T composes a solution to MABST. Conversely, let T be the spanning tree rooted at node v 0 in G such that the average throughput of all the nodes except the root node is at least β = 1. Since the outgoing bandwidth of node v is 1, v T, v v 0, and the average throughput of all the nodes except the root node v 0 in T is 1, the throughput of each node except the root node is exactly 1. Therefore, any node except the leaf node in T has only one child node, and T is Hamiltonian path in G. We can find a corresponding Hamiltonian path P in G, where P is simply a duplicate of T by deleting node v 0 and links from node v 0 to other nodes. Hence, P is the Hamiltonian path in G, which composes a solution to Hamiltonian path problem. Proof ends. VI. ALGORITHM DESIGN We propose a heuristic topology construction algorithm, Maximum Average Bandwidth (MAB), to solve MABST problem, and also design a greedy algorithm for performance comparison. We define the following notations to facilitate our explanation on the algorithm design: B: the list of outgoing bandwidths of all nodes. g(v): the throughput of the computed path from v 0 to v. Q(v): the set of all neighbor nodes of v. f(v): the total throughput of all nodes along a path starting from node v. MAB computes a spanning tree rooted at a specified root node to maximize the average throughput of all the nodes except the root node in the tree. We first present a heuristic algorithm based on an iterative greedy strategy, which takes as input a graph G, outgoing bandwidth list B for all nodes, and root node v 0, and computes a spanning tree rooted at v 0. As shown at lines 1-6 in Algorithm 1, we initialize the states of all nodes and links to be undetermined. Starting from the root node v 0, at each iteration, we select the neighbor node with the largest outgoing bandwidth of the current code as the child node in the spanning tree. Once a child node is selected, the state of the link between the current node and the selected child node is marked and the next iteration of search starts from the selected child node. If there does not exist any neighbor node which has not been selected for the current node, the algorithm rolls back to the parent node of the current node. This iterative search process terminates when all nodes have been selected, and the final spanning tree is constructed by using only those marked links. Since the Greedy algorithm always picks up the node with the largest outgoing bandwidth, in general nodes with larger outgoing bandwidths are deployed at higher tiers of the spanning tree. The computational complexity of this Greedy algorithm is O( V 2 ) in the worst case. Algorithm 1 Greedy Algorithm Input: network G(V, E), outgoing bandwidth list B, and root node v 0 Output: spanning tree T 1: for all (u, v) E do 2: state((u, v)) = 0; 3: end for 4: for all v V do 5: state(v) = 0; 6: end for 7: v s = v 0 ; 8: state(v s ) = 1; 9: while v V, state(v) 0 do 10: if v Q(v s ), state(v) 0 then 11: u = argmax v Q(v s),state(v) 0 {b v }; 12: state(u) = 1; 13: state((v s, u)) = 1; 14: parent(u) = v s ; 15: v s = u; 16: else 17: v s = parent(v s ); 18: end if 19: end while 20: Construct the spanning tree T in G using links l with state(l) 1; 21: return T. The Greedy algorithm does not consider the bandwidths of those nodes that are more than one hop way from the current node in selecting the child node. Note again that the bandwidth of a path from the root node to any other node is determined by the bottleneck splitting outgoing bandwidth. Therefore, a node with a relatively small outgoing bandwidth degrades the performance of all its downstream nodes. We further propose a heuristic algorithm, referred to as Maximum Average Bandwidth (MAB), to address this issue by taking the bandwidths of downstream child nodes into consideration. The pseudo-code of the MAB algorithm is provided in Algorithm 2. Lines 13 to 23 compute the total node throughput of the greedy path starting from each neighbor node, which is used for determining the child node. The greedy path is computed in a similar way to the Greedy algorithm shown in Algorithm 1. Since the splitting outgoing bandwidth of a node also depends on the number of its child nodes, we keep track of the number of child nodes and use this information to adaptively adjust the splitting outgoing bandwidth in the algorithm. At each iteration of the search process, the neighbor

5 node with the largest path throughput is selected for spanning the tree. The computational complexity of the MAB algorithm is O( V 4 ) in the worst case. Algorithm 2 Maximum Average Bandwidth (MAB) Algorithm Input: network G(V, E), outgoing bandwidth list B, and root node v 0 Output: spanning tree T 1: for all (u, v) E do 2: state((u, v)) = 0; 3: end for 4: for all v V do 5: state(v) = 0; 6: numchildren(v) = 0; 7: end for 8: v s = v 0 ; 9: state(v s ) = 1; 10: g(v s ) = b v0 ; 11: while v V, state(v) 0 do 12: if v Q(v s ), state(v) 0 then 13: for all v Q(v s ), state(v) 0 do 14: x 0 = v; b vs nchild(v s)+1 ); 15: g(x 0 ) = min(g(v s ), 16: f(v) = g(x 0 ); 17: while x Q(x 0 ), state(x) 0 do 18: y = argmax {b x }; x Q(x 0),state(x) 0 19: g(y) = min(g(x 0 ), b x0 ); 20: f(v) = f(v) + g(y); 21: x 0 = y; 22: end while 23: end for 24: u = argmax {f(v)}; v Q(v s),state(v) 0 25: state(u) = 1; 26: state((v s, u)) = 1; 27: parent(u) = v s ; 28: numchildren(v s ) = numchildren(v s ) + 1; b vs numchildren(v s) ); 29: g(u) = min(g(v s ), 30: v s = u; 31: else 32: v s = parent(v s ); 33: end if 34: end while 35: Construct the spanning tree T in G using links l with state(l) 1; 36: return T. We illustrate in Fig. 3 an example of the spanning trees generated by the Greedy and MAB algorithms, where Fig. 3(a) is the original network graph with root node v 0, and Fig. 3(b) and (c) are the spanning trees generated by the Greedy and MAB algorithms, respectively. The number enclosed in a pair of parentheses is the node s outgoing bandwidth. The procedure of the Greedy algorithm is as follows: (i) starting from v 0, the Greedy algorithm compares its two neighbor nodes v 1 and v 2, and chooses v 2 with a larger outgoing bandwidth to be its child node; (ii) the Greedy algorithm moves to the child node and repeats step (i) until arriving at node v 1 ; (iii) since the neighbor nodes of v 1 are all determined, the algorithm returns to the parent node of v 1, i.e. v 3, and checks if any neighbor nodes of v 3 have not been determined yet; (iv) the algorithm repeats step (iii) and arrives at node v 2, and chooses the neighbor node v 4 as its second child node; (v) the algorithm terminates when all the nodes are determined. The procedure of MAB algorithm is similar to that of the Greedy algorithm. The main difference is that at each iteration, MAB chooses the neighbor node that has the largest path throughput instead of the largest outgoing bandwidth. Starting from root node v 0, the MAB algorithm computes two greedy paths starting from its neighbor nodes v 1 and v 2, respectively. The path starting from v 1 is v 1 v 3 v 2 v 4, with a total throughput = 27 of all the nodes along the path; while the path starting from v 2 is v 2 v 3 v 1, with a total throughput = 23 of all the nodes along the path. The MAB algorithm chooses v 1 as its child node based on comparison. In this example, the total throughput of the nodes except the root node in the spanning trees generated by the Greedy and MAB algorithms are 19.5 and 27, respectively. v 1 (6) v 0 (9) v 2 (7) v (8) v (5) 3 4 v 1 (6) v 0 (9) v (8) v (5) 3 4 v (7) 2 v (6) 1 v 0 (9) v 2 (7) v (8) v (5) 3 4 ( a ) ( b) ( c) Fig. 3. An example showing the spanning process of the Greedy and MAB algorithm: (a) the original graph; (b) the spanning tree generated by the Greedy algorithm; and (c) the spanning tree generated by the MAB algorithm. VII. PERFORMANCE EVALUATION We conduct a set of experiments based on simulated and real networks of various sizes and topologies for an extensive comparative performance evaluation. The proposed MAB algorithm is compared with the Greedy algorithm shown in Algorithm 1 as well as a simple k Degree Constrained (k- DC) algorithm that was previously implemented in our existing live streaming system. Here, the control parameter k is used to limit the number of child nodes of a node to at most k in a balanced tree. The k-dc algorithm first sorts the nodes by their capacities in a decreasing order, and then adds neighbor nodes to each node (starting from the root node) until the degree

6 Fig. 4. Spanning tree of 14 nodes produced by DC algorithm. Fig. 5. Spanning tree of 14 nodes produced by Greedy algorithm. Fig. 6. Spanning tree of 14 nodes produced by MAB algorithm. constraint is reached. When the nodes at the same tier in the tree have found all their child nodes, the algorithm moves to the next tier and repeats the same tree growing procedure until an entire tree is constructed. A. Experimental Results We build a live streaming testbed by deploying 20 computers (peers) in a local-area network and run LStream, the P2P live streaming system developed by Henan Education and Research Network [19] on each of these nodes. Besides, we install NetLimiter [20], a network measurement tool on each node to control and limit its outgoing bandwidth. The source video streaming rate is set to be 500 Kbps in our experiments. The outgoing bandwidth of each node is configured to be a random quantity in a range from 200 Kbps to 800 Kbps in the granularity of 20 Kbps. We also use NetLimiter to measure the actual receiving data rate of each node. We run three algorithms in comparison to compute three trees in each of 9 overlay networks with different numbers of nodes varying from 4 to 20 at an interval of 2 nodes. We then construct the tree topology accordingly for streaming experiments and collect their average throughput performance measurements as tabulated in Table I, where the corresponding simulation results are also provided in pair for comparison. In particular, for k-dc algorithm, we vary the value of k and choose the one that produces the best throughput performance. We observe that the experimental results match well with the simulation results, which illustrates the accuracy of our cost models and also validates our assumptions on the location of the bandwidth bottleneck. In those two smallest networks, three algorithms obtain the same spanning tree and achieve the same throughput performance. As the network size increases, the performance superiority of the MAB algorithm over the Greedy and k-dc algorithms becomes more substantial and evident. For a visual comparison, in the network of 14 nodes, we plot the topology of three different spanning trees produced by the DC, Greedy and MAB algorithms in Fig. 4, Fig. 5 and Fig. 6, respectively. The number enclosed in a pair of parentheses is the original node ID in the overlay network. The tree computed by the DC algorithm with degree constraint k = 3 has a smaller depth, and therefore the outgoing bandwidth of a node is shared by more child nodes, resulting in a smaller total throughput. The trees generated by the Greedy and MAB algorithms have relatively larger depths and the branches are located at the lower part of the tree, resulting in better throughput performance. B. Simulation Results The size of overlay networks for real streaming experiments is limited due to the lack of network and system resources. To account for this limitation, we further conduct simulationbased performance comparison among these three algorithms in a large set of simulated networks. For a given number of nodes and links, each simulated network is created with a randomly generated network topology and a random outgoing bandwidth within a range from 0 to Kbps is assigned to each node. We perform two sets of simulations on different simulated networks to examine the scalability and robustness of these three algorithms. 1) Scalability: In the first set of simulations, we test the scalability of these three algorithms based on a series of 10 simulated networks, indexed from 1 to 10, with a varying number of nodes from 20 to 200 at an interval of 20 nodes, and a varying number of links from 60 to 600 at an interval of 60 links, respectively. For a convenient reference, we tabulate these network parameters in Table II. The capacity or outgoing bandwidth of each node is randomly assigned. For each given network size, we create 10 instances with different network topologies. We run three algorithms in each of these 10 network instances and compute the average throughputs of three resultant trees. The mean and standard deviation of the average throughput measurements over 10

7 TABLE I THROUGHPUT PERFORMANCE COMPARISON BETWEEN k-dc, GREEDY AND MAB ALGORITHMS. Simulation results / Experimental results Alg 4 nodes 6 nodes 8 nodes 10 nodes 12 nodes 14 nodes 16 nodes 18 nodes 20 nodes k-dc 25.30/ / / / / / / / /10.79 Greedy 25.30/ / / / / / / / /13.63 MAB 25.30/ / / / / / / / /18.84 TABLE II PERFORMANCE COMPARISON BETWEEN k-dc, GREEDY AND MAB ALGORITHMS Index of problem size Number of nodes Number of links Average throughput (Kbps) DC Greedy MAB Average throughput (Kbps) DC Greedy MAB Index of problem size Fig. 7. Average throughput performance comparison (mean and standard deviation) among three algorithms based on a series of 10 simulated networks of various sizes ranging from small to large scales Number of links Fig. 9. Average throughput performance comparison (mean and standard deviation) among three algorithms based on a series of 10 simulated networks of 200 nodes and a varying number of links from 200 to 2000 at an interval of 200 links. Performance speedups DC Greedy Index of problem size Fig. 8. Performance speedups of MAB over DC and Greedy based on a series of 10 simulated networks of various sizes ranging from small to large scales. instances for each network size are plotted in Fig. 7. We observe that the MAB algorithm consistently outperforms the other two algorithms in all the problem cases we studied. We also plot in Fig. 8 the performance speedups of MAB over the MAB (DC or Greedy) (DC or Greedy), DC and Greedy algorithms defined as which shows that the MAB algorithm achieves at least 20% more average throughput than the Greedy algorithm and up to 2 times improvement over the DC algorithm. 2) Robustness: We conduct a second set of simulations to study the robustness of these three algorithms based on 10 simulated networks of 200 nodes and a varying number of links from 200 to 2000 at an interval of 200 links. These networks represent the diversity of network connectivity ranging from sparse to dense. Similarly, for each given number of links (note that the number of nodes is fixed to be 200), we create 10 instances with different network topologies. The mean and standard deviation of the average throughput measurements obtained by each of these three algorithms over 10 instances for each number of links are plotted in Fig. 9 and the corresponding performance speedups of MAB over the other two algorithms are plotted in Fig. 10. As the number of links increases, more bandwidth resources are made available in the network, and all three algorithms produce increasingly higher throughputs. We also observe that the MAB algorithm consistently achieves better throughput performance than the other two algorithms. VIII. CONCLUSION We formulated and investigated a specific type of spanning tree problem, MABST, with the objective to maximize the average network throughput in live streaming applications.

8 Performance speedups DC Greedy Number of links Fig. 10. Performance speedups of MAB over DC and Greedy based on 10 simulated networks of 200 nodes and a varying number of links from 200 to 2000 at an interval of 200 links. We proved that MABST is NP-complete by reducing from Hamiltonian Path problem and proposed a heuristic algorithm. The performance superiority of the proposed algorithm was illustrated by extensive experimental and simulation results in comparison with other methods based on a degree constraint and a greedy strategy. The current work is focused on a single throughput optimization objective. We will revisit the tree construction problem with multiple objectives including latency and reliability in addition to throughput to support live streaming systems in large-scale networks with low-end peers. We will study the tree reconstruction problem that handles new nodes joining and existing nodes departure in a dynamic manner. We plan to refine the system cost models by considering other system resources such as CPU processing power and memory capacities. It is also our future interest to consider new overlay network models where the bandwidth bottleneck is not necessarily located at a node s outgoing port. [5] S. Banerjee, B. Bhattacharjee, and C. Kommareddy, Scalable application layer multicast, in Proc. of the ACM SIGCOMM, 2002, pp [6] C. Zhang, H. Jin, D. Deng, S. Yang, Q. Yuan, and Z. Yin, Anysee: Multicast-based peer-to-peer media streaming service system, in Proc. of Asia-Pacific Conference on Communications, 2005, pp [7] R. Li, C. Guo, M. Fa, and Z. Wang, AHLSS: A hierarchical, adaptive, extendable P2P live streaming system, in Int. Symp. on Advances in Computer and Sensor Networks and Systems, Apr. 2008, pp [8] K. Sripanidkulchai, A. Ganjam, B. Maggs, and H. Zhang, The feasibility of supporting large-scale live streaming applications with dynamic application end-points, in Proc. of the 4th ACM/IEEE Symp. on Architectures for Networking and Communications Systems, 2004, pp [9] M. Guo and M. Ammar, Scalable live video streaming to cooperative clients using time shifting and video patching, in Proc. of IEEE INFOCOM, vol. 3, March 2004, pp [10] R. Guerin, A. Orda, and D. Williams, QoS routing mechanisms and OSPF extensions, in Proc. of Global Telecommunications Conference, 1996, pp [11] Z. Wang and J. Crowcroft, Quality of service routing for supporting multimedia applications, IEEE Journal on Selected Areas in Communications, vol. 14, no. 7, pp , Sep [12] Q. Ma, P. Steenkiste, and H. Zhang, Routing high bandwidth traffic in max min fair share networks, in Proc. of ACM SIGCOMM, 1996, pp [13] S. Jarvis, G. Tan, D. Spooner, and G. Nudd, Constructing reliable and efficient overlays for P2P live media streaming, Int. Journal of Simulation, vol. 7, no. 2, pp , March [14] P. Merz and S. Wolf, TreeOpt: Self-organizing, evolving P2P overlay topologies based on spanning trees, in Proc. of KiVS, Feb. 2007, p. 12. [15] T. Small, B. Li, and B. Liang, On optimal peer-to-peer topology construction with maximum peer bandwidth contributions, in Proc. of the 23rd Biennial Symposium on Communications, 2006, pp [16] Y. Zhu and B. Li, Overlay networks with linear capacity constraints, IEEE Tran. on Parallel and Distributed Systems, vol. 19, no. 2, pp , Feb [17] Y. Zhu, B. Li, and K. Pu, Dynamic multicast in overlay networks with linear capacity constraints, IEEE Tran. on Parallel and Distributed Systems, vol. 20, no. 7, pp , July [18] M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness. New York: W.H. Freeman and Company, [19] HERNET. [20] Netlimiter. ACKNOWLEDGMENT This research is sponsored by U.S. National Science Foundation under Grant No. CNS with University of Memphis and the National High-Tech Research and Development Program of China (863 Program) under Grant No. 2008AA01A315 with Zhengzhou University. We would like to thank the project team in the Henan Provincial Key Laboratory on Information Network for their valuable contributions to the implementation, deployment, and experiments of the live media streaming system in real networks. REFERENCES [1] X. Zhang, J. Liu, B. Li, and T. Yum, Coolstreaming/donet: a datadriven overlay network for peer-to-peer live media streaming, in Proc. of IEEE INFOCOM, vol. 3, March 2005, pp [2] X. Hei, C. Liang, J. Liang, Y. Liu, and K. Ross, A measurement study of a large-scale P2P IPTV system, IEEE Transactions on Multimedia, vol. 2, no. 8, pp , March [3] PPStream. [4] D. Tran, K. Hua, and T. Do, Zigzag: an efficient peer-to-peer scheme for media streaming, in Proc. of INFOCOM, vol. 9, Dec. 2007, pp