Approximation and Heuristic Algorithms for Minimum Delay Application-Layer Multicast Trees

Transcription

1 THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERING Approximation and Heuristic Algorithms for Minimum Delay Application-Layer Multicast Trees A thesis submitted toward the degree of Master of Science in Electrical and Electronic Engineering by Eli Brosh October 2003

2 THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERING Approximation and Heuristic Algorithms for Minimum Delay Application-Layer Multicast Trees A thesis submitted toward the degree of Master of Science in Electrical and Electronic Engineering by Eli Brosh This research was carried out in the Department of Electrical Engineering - Systems, under the supervision of Dr. Yuval Shavitt October 2003

3 Abstract In this thesis we investigate the problem of finding minimum delay applicationlayer multicast trees, such as the trees constructed in overlay networks. It is widely accepted that shortest path trees are not a good solution for this problem since such trees can have nodes with very large degree, termed high load nodes. The load on these nodes makes them a bottleneck in the distribution tree, due to computation load and access link bandwidth constrains. Many previous solutions limited the maximal degree of the nodes by introducing arbitrary constraints. In this work, we show how to directly map the node load to delay penalty at the application host, and create a new model that captures the trade offs between the desire to select shortest path trees and the need to constrain the load on the hosts. Our model enables the evaluation of multicast performance using a single delay cost. In this model the problem is shown to be NP-hard. Therefore, we present a logarithmic approximation algorithm and a heuristic solution with a Ω( n) approximation ratio where n is the size of the multicast group. Both algorithms generate minimum delay trees that intrinsically balance short latency with small degree, and thus avoid an external trial-and-error type of balancing between the two, i.e., our solutions do not impose a maximum degree on the multicast trees. For overlay networks with homogenous costs, we show that the heuristic algorithm is optimal and analyze the convergence rate of a family of non-lazy multicast algorithm. Our work addressed the issue of minimum delay multicasting in the context of fully connected networks and structured topologies used by many peer-2- peer systems. We show that the heuristic scheme achieves an optimal solution for grid graphs, and provide performance bounds for multicasting in grid and tree topologies. Finally, we use a simulation study to evaluate the average performance of the suggested algorithms. The simulations show that the heuristic solution is scalable for large group sizes, and produces results that are very close to optimal. The proposed solution is most useful for delay-sensitive applications, such as real-time media streaming, and can be used to provide an efficient routing infrastructure for group communication applications. i

4 Contents 1 Preface Introduction Background Overlay Communication Network Model 6 3 The Optimal Multicast Tree Problem Problem formulation The optimal recursive computation NP-Completeness Multicast Algorithms Approximation algorithms The postal approximation algorithm The general postal approximation algorithm The MDM approximation algorithm The heuristic algorithm The homogenous case Analysis of Specific Topologies Motivation Trees Spiders Grids A Simulation Study Schemes compared Simulation results Clique topologies Power-law topologies Conclusions Summary 40 ii

5 List of Symbols c(u, v) Communication delay of an edge (u, v) p(v) Processing delay of host v l(p) The cost of a path p T Multicast tree V [T ] Set of nodes in tree T M DM Minimal Delay Multicast M Multicast group set s Source host t T (v) Reception delay of host v using a multicast scheme T M(v) Optimal cost of multicasting from a subtree rooted at v λ uv Communication latency of edge (u, v) in the Postal model s v Switching delay of node v in the Postal model U Terminal set of multicast recivers T Optimal multicast tree for the postal multicasting problem T Maximal generalized degree of tree T L T Weighted diameter of tree T γ Switching to communication ratio γ 0 Maximal value between γ and 1 U i Core subset of terminals computed in the ith iteration OP T Cost of the optimal solution for the MDM problem n Multicast group size GHP Generalized heterogeneous postal model I GHP GHP configuration instance OP T GHP Cost of the optimal solution for the GHP multicasting problem t GHP T Reception delay of host v using a GHP multicast scheme T t[v] Ready time attribute for host v m[v] The mate host of v w u;v Weight of edge (u, v) d u;v Weight of the shortest path from u to v π u;v Predecessor of v on the shortest path from i N(t) Maximal number of hosts that can be reached in the interval (0, t] R (t) Maximal number of messages received in the interval (t τ, t] G m;n Grid graph with size m n D Maximal distance in a grid graph b(v) Optimal broadcast time from v in a grid graph d(v) Additional delay of host v iii

6 List of Figures 1.1 Examples of multicast trees Example of an ordered multicast tree Approx-MDM cost transformation example Greedy tree construction for the MDM problem n approximation ratio of the heuristic tree SPT examples for grid topology Heuristic tree examples for grid topology Examples of B 1 and B 2 structures The multicast delay for a clique topology with random network costs from [1, 10] The multicast delay for a clique topology with random processing costs from [1, 10] and unit communication costs The multicast delay for a clique topology with random communication costs from [1, 10] and unit processing costs The multicast delay for a clique topology with near homogenous random costs selected from [0.75, 1.25] The multicast delay for a power-law topology with random network costs from [1, 10] iv

7 Chapter 1 Preface 1.1 Introduction Multicast is a key component in the design of group communication applications which require efficient data delivery to multiple destinations. However, IP multicast which implements multicast functionality at the network layer is still not widely deployed in current IP networks. To alleviate this problem, several recent proposals [1] have advocated an alternative approach, termed application layer multicast or end-host multicast, which implements multicast functionality at the application layer using unicast network level services only, forming an overlay network between end hosts. The goal of application layer multicast [2] is to construct and maintain efficient distribution trees between the multicast session participants, minimizing the performance penalty involved with application-layer processing. Many proposals attempt to optimize the cost of the multicast delivery tree using application level performance metrics such as delay or throughput. The systems which aim at reducing the overall delay [2, 3, 4, 5, 6, 7] construct a minimum height (or minimum diameter) tree with constrained degrees. The degree constraints are used to control the network resource usage, i.e., available bandwidth or stress on the physical links. However, this solution stipulates the usage of a dual cost optimization objective which mixes network level and application level costs to characterize applications performance. In this thesis we advocate an application-centric approach which quantifies system performance using application level costs only. We claim that the conventional overlay network model and its corresponding delay measure are designed to characterize multicast systems which assume network-level data distribution capabilities. Unfortunately, message processing by end-hosts involves an additional delay penalty which is not captured by such models and is related to application-layer implementations of packet duplication and routing. In particular, the shift of multicast functionality to the upper level influences the simultaneous distribution capabilities of end-hosts, implying a communi- 1

8 CHAPTER 1. PREFACE 2 cation model with sequential message distribution. This constraint stems from the fundamental change in the characteristics of the routing infrastructure assumed by the overlay network, attributed to the difference between message distribution speeds of routing nodes (i.e., end-hosts) in overlay networks and packet distribution speeds of routers in conventional physical networks. For example, consider the simple network of Figure 1.1A, composed of three hosts H 1, H 2, and H 3 and two routers R 1 and R 2 connected using a high speed backbone, where host H 1 uses a low-bandwidth access link for network connectivity, e.g., modem access, and H 2, H 3 use high-bandwidth LAN access connectivity. Assume that the goal of the overlay system is to devise a multicast tree that provides minimal distribution delay from H 1 to H 2 and H 3. Clearly, a multicast system must be careful to avoid delegating large degree to the low bandwidth host H 1 in order to eliminate unnecessary bottleneck due to its low-speed data distribution capabilities. Figure 1.1B depicts the corresponding optimal multicast tree. Now, consider the conventional routing algorithm used by many application-layer multicast architectures that optimize tree delay, namely the shortest path tree algorithm. In this case the shortest path multicast tree reduces to a star topology (Figure 1.1C), which ignores the performance penalty at the star center. Hence, serialized message distribution which is irrelevant to IP multicast schemes must be accounted for in the evaluation of overlay multicast architectures. Surprisingly, however, many application-layer architectures which optimize tree delay have neglected these implications on the overall performance of group communication applications. Another factor which constrains parallel message distributions in overlay networks is the processing capacity of end-host machines. For instance, consider a server which implements router like functionality at the application layer and therefore may not have enough CPU power to handle message processing at the full speed of its network interfaces. Hence, the effective message distribution rate of an end-host is shaped by two factors, the bandwidth of the access link connecting the host or its local area network to the physical network, and the processing power and the computational load on the host machine. A recent study [8] that measured the actual end-host heterogeneity of popular p2p overly systems showed that the bandwidth and latency parameters can vary by several orders of magnitude across different hosts in the system. In this paper, we present an overlay network model which captures the realistic costs involved with application-layer multicast. The model is a mathematical generalization of a communication model developed by Cidon et al. [18] for high-speed networks, and similarly it incorporates two separate delay measures. The processing delay measure, which is a reciprocal of the effective message distribution speed of an end-host application, and the communication delay measure, which represents the delay of traversing an overlay link. This framework enables us to characterize the performance of multicast trees using

9 CHAPTER 1. PREFACE 3 H1 H1 H1 R1 H2 H 2 H 3 R 2 H3 H2 H 3 Low-Speed Access Link High-Speed Access Link Backbone Link (A) (B) (C) Figure 1.1: Comparison between application-layer multicast and network-layer multicast in a simple heterogeneous overlay network a single cost, the overall delay of message distribution. We use the proposed framework to develop heuristic and approximation algorithms for the problem of constructing delay optimized multicast trees. Both the heuristic and the approximation generate minimum delay trees that intrinsically balance short latency with small degree, and thus avoid an external trial-and-error type of balancing between the two, i.e., we do not impose a maximum degree on our trees. Our heuristic algorithm constructs such trees efficiently and thus can scale to large multicast groups, which is a known problem [2]. Note that the suggested solution works both for fully connected topologies, and for structured topologies, used in some p2p overlay networks [9]. Therefore, we address the issue of multicasting in partially connected networks, and provide performance bounds for tree and grid graphs. The presented algorithmic solutions can be effectively used to implement centralized overlay systems, such as p2p and server based systems. The heuristic algorithm is particularly useful in the context of two-tier server based architectures [5, 10, 3] which construct a virtual tree among the servers to provide an efficient content and data delivery services to end-users. Each end-user registers to a server in order to receive multicast services, and the server handles the dissemination of the aggregated traffic. Such semi-static architectures employ reliable servers to provide high-availability service, stipulating a simple implementation with low computational overhead due to minor topology changes. Furthermore, a centralized approach is capable of providing quick

10 CHAPTER 1. PREFACE 4 and efficient session management services by sharing the computational load among several overlay servers [4]. The main applicability of our algorithms is in the context of delay-sensitive multicast applications, which require tight bounds on the end-to-end delays due to jitter and timing constrains. Applications which belong to this category include audio conferencing, real-time media streaming, content distribution services, and multi-player distributed games. The rest of the thesis is organized as follows. The next chapter formulates the overlay communication model. In Chapter 3 we discuss the problem of optimal multicast tree construction and show that this problem is NP-Complete. In Chapter 4 we develop approximation and heuristic algorithms for solving this problem. Chapter 5 deals with performance analysis of the heuristic algorithm for several overlay topologies. An experimental evaluation of our solutions is presented at Chapter 6. Finally, Chapter 7 concludes our work. 1.2 Background Broadcast and multicast are important communication primitives which have many applications in distributed and parallel systems. The problem of designing efficient broadcast and multicast algorithms under the assumption of sequential message distribution have been extensively studied in the context of several communication models. One model which was widely investigated is the telephone model [11], which assumes a synchronous communication model where each node can either send or receive a single message per communication round. Some telephone model studies have focused on the problem of designing optimal broadcast schemes for specific classes of graphs (see [12] for a comprehensive survey), while others have suggested approximations which support efficient broadcasting in general graphs ([13, 14, 20, 15]). Cidon et al. [18] presented a homogenous communication model for highspeed networks which captures communication costs using two parameters transmission delay and computation delay. In this model, the network is represented by an undirected graph G = (V, E). Each node is associated with a processing delay cost P, and each edge is associated with a communication delay cost C. The model assumes sequential processing of messages, such that the time needed for a node to handle the transmission of i messages is ip. In addition, they proposed an optimal tree-based broadcast algorithm [18] for complete graphs, and showed that such trees achieve a broadcast delay which is logarithmic in the size of V. Raz and Shavitt [19] presented a general version of this model which supports IP-like routing, and considered efficient multicasting algorithm (based on balanced trees) for line topologies. The postal model, introduced by Bar-Noy et al. [11], is a similar homogenous model which captures network communication costs by incorporating a latency

11 CHAPTER 1. PREFACE 5 parameter λ. It assumes a fully connected communication model where each host can either transmit or receive a single message per time unit. More specifically, this model assumes that a message originator, u, spends one unit of time preparing and sending a message to its destination. After the transmission u is free to perform other functions including sending other messages. The destination completes the reception λ units of time after the initial preparation of the message by the originator. LogP [16] is a similar homogenous model which uses an extended set of parameters to characterize communications costs. Optimal broadcast schemes for these homogenous models can be found in [11, 17]. The heterogeneous postal model [20] extends the postal model by assuming that the communication costs are not uniform. In addition the model incorporates a switching time measure which represents the minimal gap between message transmissions. This model represents the communication network using an undirected graph G = (V, E), a switching time function which associates a sending time s v with each node v V, and a communication latency function which associates a length λ uv with each pair of nodes (u, v) E. A log k approximation algorithm is given in [20] for the problem of optimal multicast where k is the size of the multicast group.

12 Chapter 2 Overlay Communication Network Model In this chapter we define the overlay communication model and the corresponding delay measures which characterize the performance of application-layer multicast solutions. An overlay network is a fully connected virtual network formed by hosts which communicate with each other using a physical network, such as the Internet. The physical network is composed of routing nodes (i.e., routers) and communication links. Hosts are connected to the routing nodes using access links (for example, see Figure 1.1A). The underlying assumption is that backbone links have abundant bandwidth, while access links may impose system bottlenecks due to potential bandwidth limitations. The overlay network utilizes the regular unicast services of the physical network to provide communication among hosts, and do not require any special support at the network level. The delay experienced by a message that travels between hosts is composed of two elements: I. Communication delay, which represents the delay of traversing the unicast path between the hosts. This component includes the accumulated propagation and queuing delays of the physical links on the unicast path, and the message reception overhead at the receiver host. II. Processing delay, which represents the delay of processing a message at the sender host. This element includes the overhead of preparing a message for transmission and the transmission delay through the physical access link. Although current operating systems and their communication services have mechanisms which allow applications to perform simultaneous (or near simultaneous) message transmissions, the simultaneous effect is overridden by the 6

13 CHAPTER 2. OVERLAY COMMUNICATION NETWORK MODEL 7 inherent serialization involved with message transmission through a physical access link. This type of serialization is typically performed at the hardware level by the access equipment. Furthermore, the sequential distribution prohibits the usage of unrealistic application design schemes which relies on simultaneous message transmissions. We define an overlay network model based on a generalization of a communication model developed by Cidon et al. [18]. The overlay network is modeled by a directed complete graph G = (V, E), where V is a set of vertices representing hosts, and E is the set of edges representing the communication paths in the overlay topology. We use the terms host and link to refer to the vertexes and edges in the overlay graph. Each overlay edge (u, v) E is associated with a communication delay cost, c(u, v), and each host v V is associated with a bounded and finite processing delay cost, p(v). The original model of Cidon et al. [18] assumes homogenous processing and communication costs, i.e., p(v) = P, v V, and c(u, v) = C, (u, v) E The direct communication between hosts is characterized as follows. Assume that at time t, host u initiates processing of a message targeted to host v. Then host u is busy processing this message during the time interval [t, t+p(u)], and the message arrives at host v at time t + p(u) + c(u, v). After this processing period (i.e., [t, t + p(u)]), host u is free to handle the following messages. Therefore, the processing delay represents the minimum time interval between consecutive message transmissions. It is important to note that in our model, the delay costs between pairs of hosts do not necessarily satisfy the triangle inequality. This is a known phenomenon in the Internet, stemming in part from policy routing. For example, Jamin et al. [21, Figures 2 and 3] show that about 30-50% of the triangles in the Internet do not obey the triangle inequality. Given a pair of hosts v 1 and v k which are connected by a path p v1,v k =< v 1,..., v k > of length k 1, the cost of the path from v 1 to v k, denoted by l(p v1,v k ), is defined as the sum of the communication delays of the overlay links in this path and the processing delays of the traversed hosts (the processing delay of v k is excluded since it doesn t consume processing resources for further delivery). Therefore k 1 l(p v1,v k ) = p(v i ) + c(v i, v i+1 ) (2.1) i=1 where v i, 1 i k denotes the ith host on the path p v1,v k. One may view this cost as a measure of the minimum distribution delay (using the specified path) from v 1 to v k. We believe that this model captures the heterogeneity of current overlay networks, and provides a realistic framework for evaluating the performance of multicast algorithms.

14 Chapter 3 The Optimal Multicast Tree Problem 3.1 Problem formulation In this chapter we state our design objective formally and show that the optimal multicast tree problem is NP-Complete. We use the term multicast scheme to refer to the task of distributing a message from a source host to a subset of hosts M in the overlay network. Since one cannot relay on the cooperation of non-participating hosts (i.e., hosts which do not belong to the multicast group M), we assume that only the hosts in M are allowed to participate in the distribution. Thus, a multicast scheme in the graph G = (V, E) can be viewed as a broadcast scheme, i.e., the task of distributing a message to the entire network, in the subgraph induced by the host set M V. We formulate the optimal multicast tree problem, also denoted as minimal delay multicast (MDM) problem, as follows. Definition 3.1. The optimal multicast tree problem (MDM): Given a directed complete graph G = (V, E), a multicast group M V, a source host s M, a non-negative real processing delay p(v) for each vertex v V, and a non-negative real communication cost c(u, v) for each edge (u, v) E, find a multicast scheme which minimizes the delay by which all the hosts in M receive a message from s, namely a scheme which minimizes the arrival time of the last message. Our goal is to devise a multicast scheme which minimizes the distribution delay. Therefore, we restrict this study to non-lazy multicast schemes, in which a host that has already received a message does not delay message distribution by becoming idle (this term was introduced in [20]). Such schemes correspond to an ordered directed tree T, rooted at s and spanning M. The root of the tree corresponds to the source host s, and the 8

15 CHAPTER 3. THE OPTIMAL MULTICAST TREE PROBLEM s 1 1 v1 v v3 v4 v5 v6 Figure 3.1: Example of an ordered multicast tree rest of the nodes corresponds to the rest of the hosts in the multicast group M. Each non-leaf node u has an ordered list of k children u 1,..., u k which correspond to the ordered set of k hosts u 1,..., u k to which host u sends the multicast message. That is, the ith outgoing edge of node u corresponds to the ith transmission from u. The reception delay of host v M, denoted by t T (v), is defined to be the time at which v receives a message from the source host, s. The reception delay of s is by definition 0. The reception delay of node v at depth k which is connected to the tree root using a unique directed path, p v0,v k =< v 0,..., v k > such that v 0 = s and v k = v, is computed by summing the communication delays of the path edges and the processing delays of the traversed hosts. Therefore we have k 1 t T (v) = (c(v i, v i+1 ) + p(v i ) g(v i, v i+1 )) (3.1) i=0 where g(v i, v i+1 ) denotes the index of the transmission from v i to v i+1, 0 i k 1 (i.e., the processing round number used to prepare and deliver the message from v i to v i+1 ). The cost of a multicast tree T is defined as the earliest time at which all the hosts have been notified, i.e., this cost is max v M t T (v). For example, consider the ordered tree in Figure 3.1. The communications costs are located near the edges, and the processing costs near the nodes. The edges of this tree are depicted from left to right according to increasing transmission order. By applying equation (3.1) we get that the reception delay of v 1 is = 3. Observe that s distributes the message to v 2 after two processing rounds, and therefore the reception delay of v 2 is = 3. Similarly, the reception delays of v 3, v 4, v 5, v 6 are 7, 8, 9, 11, respectively, and the cost of the tree is 11. Given a multicast tree we can easily calculate the optimal ordering using

16 CHAPTER 3. THE OPTIMAL MULTICAST TREE PROBLEM 10 a recursive computation, working bottom-up. This recursion is presented in the following section. In the rest of the thesis we neglect the ordering and concentrate on finding the optimal tree. 3.2 The optimal recursive computation Given a tree T = (V, E) rooted at s with associated processing and communication costs p and c, respectively, we need to compute the optimal cost and optimal ordering of T. We employ a bottom-up recursive computation approach. For each node v V, v s with parent u we compute the quantity M(v). This quantity represents optimal cost of the subtree rooted at v, i.e., the minimal multicast delay from v to the nodes in its subtree. We use the quantity m(v) to incorporate the communication cost from u to v: m(v) = M(v) + c(u, v) (3.2) Given a non leaf node v with k children v 1,..., v k, we denote by r(v, i) a rank function that returns a child node of v with the ith largest m quantity. Therefore, the recursive formulation of the optimal cost of a tree rooted at a non-leaf node v is M(v) = max {m(r(v, i)) + i p(v)} (3.3) 1 i k The optimal cost of a tree rooted at a leaf node v is by definition m(v) = 0. The optimal multicast delay of tree T is simply M(s). The corresponding optimal ordering is computed according to the rank function, such that the ith child of a non-leaf node v is r(v, i). The time complexity of this computation is Θ( V ). The correctness follows from the fact that in a tree graph any multicast scheme is characterized only by the message distribution order of non-leaf hosts. The following lemma provides a simple proof for the optimality of the recursive computation. Lemma 3.1. The recursive computation provides an optimal solution for minimum delay multicast in a tree graph T = (V, E). Proof. Assume to the contrary, that there exists an optimal multicast scheme in which host u V distributes a message to child hosts u i and u j in processing rounds k and l, k < l, respectively, whereas m(u i ) < m(u j ). By swapping the transmissions order to these child nodes, the multicast delay may be reduced by at most (l k) p(u), which contradicts the assumption that this multicast scheme outperforms the distribution order of the recursion.

17 CHAPTER 3. THE OPTIMAL MULTICAST TREE PROBLEM 11 Consider the example depicted in Figure 3.1. Using the recursive computation we get that M(v 1 ) = max{1 + 3, } = 5 and M(v 2 ) = max{2 + 4, } = 8. The optimal cost of the tree is achieved when s distributes the first message to v 2, such that the overall multicast delay, M(s), is max{1 + 9, 2 + 7} = 10. We further discuss tree graphs in Section 5.2, where we develop worst case performance bounds for multicasting in such graphs. 3.3 NP-Completeness We now show that the MDM problem is NP-complete using a simple reduction from the telephone broadcast (TB) problem. In the Telephone model (see [11]) communication is synchronous, i.e., at each time unit a node may send a message to at most one other node. The TB problem seeks an optimal broadcast scheme which distributes a message from a source node to all the nodes in a given graph in a minimal number of time units. The TB problem is known to be NP-Hard [22, ND49] for arbitrary graphs. Theorem 3.1. The decision version of the MDM problem, finding a multicast tree with a delay bound K, is NP-complete. Proof. The problem is in NP since we can verify in polynomial time whether a certificate multicast scheme satisfies the delay bound. We prove that it is NP-Hard by a reduction from TB. Let G = (V, E) be an instance of the TB problem, and let n denote the number of nodes in V. We construct an instance of the MDM problem as follows. We form a complete graph G = (V, E ), where E = {(u, v), u, v V }, and we define the processing cost function as p(v) = 1, v V, and the communication cost function as { 0 if (u, v) E, c(u, v) = n if (u, v) / E. We let M = V. It is easy to verify that the TB instance, represented by G, has a broadcast scheme which completes within K time units iff the MDM instance, represented by (G,p,c), has a multicast scheme with a delay of K time units. Note that the problem is shown to be NP-Hard when a uniform set of processing costs is considered. That is, the problem still remains hard even when using homogenous processing costs, implying that the processing cost measure is a true bottleneck for designing efficient multicast algorithms.

18 Chapter 4 Multicast Algorithms Since the problem of finding the optimal multicast tree is NP-complete, we seek to devise approximations and heuristics described as follows. We begin with developing approximation algorithm based on a modified version of the postal approximation algorithm. This algorithm accepts an undirected overlay graph input, implying that its domain is limited to overlay networks with symmetric links. This restriction is in many cases unrealistic due to the widespread deployment of asymmetric access links, such as ADSL and cable-modem connections. Furthermore, the approximation needs to solve multiple linear programs and therefore its running time is at least Θ(n 7 ) (see [15]). We therefore devise an alternative cost-effective heuristic tree construction algorithm that supports directed overlay networks, and evaluate its performance. Finally, we analyze homogenous overlay networks where the processing and communication costs are uniform, and show that non-lazy trees achieve logarithmic multicast delay. We also discuss the extensions of these algorithms to support shared tree solutions. In the shared tree approach [23] a single tree is constructed for the purpose of multi-source multicast, such that the shared tree is used by all the group members. Our analysis show that the presented algorithms can be easily modified to support shared trees without major impact on the performance. Of course, using multiple single source multicast trees will always achieve lower delay, but at the expense of the management and resource usage overhead. 4.1 Approximation algorithms We base our approximation on the algorithm of Bar-Noy et al. [20] originally designed for the heterogeneous postal model. The postal model represents the communication network using an undirected graph G = (V, E), a switching time function which associates a sending time parameter s v with each node v V, and a communication latency function which associates a length parameter λ uv with each pair of nodes (u, v) E. In this model, the direct communication between nodes is characterized as follows [20]: assume node u sends a message 12

19 CHAPTER 4. MULTICAST ALGORITHMS 13 to node v at time 0 and the message arrives at v at time λ u,v. The assumption is that u is free to send a new message at time s u (i.e., u is considered busy engaged only with the current transmission in the first s u time units following the previous transmission time). By definition, s u is smaller than λ u,v, i.e., the delay λ u,v takes into account the sending time at u (and the receiving time at v). Although both models (postal and overlay) have common properties, the postal model differs from the overlay model in the following aspects. In the postal model the communication latency of a link incorporates the sending time, while in the overlay model the sending time is incorporated in the processing delay of the sender host. Thus, the cost (i.e., delay) of delivering the ith message from u to v is the sum of the cost of (u, v) and i 1 times the cost of u in the case of the postal model, and the sum of the cost of (u, v) and i times the cost of u in the case of the overlay model. The postal model requires that s u < λ uv, (u, v) E. Therefore, we need to adapt the postal approximation algorithm before applying it to the overlay model. We do this in three steps: 1. We define the generalized heterogeneous postal (GHP) model which eliminates the restriction on the values of the communication and switching (sending) measures. 2. The original postal approximation algorithm is adapted to support GHP models. This adaptation, termed general postal approximation, increases the approximation ratio by a multiplicative factor γ = max (v,w) E { s v λ vw } 3. We construct a cost preserving transformation from the overlay model to the GHP model and apply the general postal approximation to compute the multicast tree. This process results in an approximation algorithm, Approx-MDM, which increases the original approximation ratio by an additive factor. We now proceed to define the GHP model. Definition 4.1. The GHP model is a heterogeneous postal model which excludes the restriction on the network costs, such that the edge length parameter in the GHP model is finite and positive, i.e., λ uv > 0, (u, v) E. Therefore, in the GHP model the communication latency between nodes can have any finite positive real value, providing a framework that includes nodes with switching time which is larger than the communication latency to the neighbors. The following measure captures the proportion between switching and communication times.

20 CHAPTER 4. MULTICAST ALGORITHMS 14 Definition 4.2. Given a GHP model with graph G = (V, E), switching time function s, and a communication latency function λ, define γ = max (v,w) E { s v λ vw } as the switching to communication ratio of the graph G. Before proceeding to the general postal approximation we provide an outline of the postal approximation algorithm. The interested reader is directed to [20] for the full details. In the context of the postal model the multicast problem is defined as follows. Given a configuration of an undirected graph with associated communication and switching cost functions (G = (V, E), s, λ), a set of terminals U V, and a source node r U, find the minimal time (i.e., delay) scheme that distributes a message from r to the terminal set U, where all the nodes in V may participate in the distribution. Observe that we preserve the notations of [20] which denote the multicast group by U The postal approximation algorithm The basic idea of the algorithm is to find a multicast tree T which minimizes the quantity T + L T, where T denotes the maximum generalized degree (the actual degree multiplied by the corresponding switching time) of T, and L T denotes the maximum distance (in T ) from r to any of the nodes in U The algorithm computes a multicast tree, which approximates the cost of the optimal tree, T, iteratively using l rounds. Let U i denote the terminal set in the ith round. The algorithm starts with the initial set U 0 = U and terminates when U l = {r}. In the ith round the algorithm uses the core procedure to compute the following, for any i l: 1. a core subset U i U i 1 of size at most 3 4 U i 1 where r U i 2. a multicast scheme from U i to U i 1, such that the obtained multicast time is linear in the optimal multicast time from r to U i 1. The computation of core(u i ) involves the following steps: 1. Solve a linear program, variant of a multicommodity flow. The resulting set of fractional paths is rounded [20, Theorem 4] producing a set of U i integral paths, one for each terminal. The length and congestion properties of the integral paths are linear in T + L T. The congestion property is the generalized degree of the graph spanned by a set of paths. 2. Transform the set of paths into a set of spider graphs, which are graphs where at most one node has a degree larger than two. The resulting spiders span at least half of the nodes in U i. Each spider contains at least two terminals, and its generalized degree and diameter is linear in T + L T.

21 CHAPTER 4. MULTICAST ALGORITHMS Select an arbitrary terminal from each spider together with nodes which are not spanned by any spider to be included within the resulting core. This selection ensures that the chosen spider terminal is able to distribute a message to the remaining spider nodes in O( T + L T ). In [20] it is shown that the resulting tree has a O(log U ) multiplicative approximation factor. The postal approximation algorithm cannot be directly applied to the GHP model due to the particular selection of the coefficients of the rounding matrix. The following section describes the new rounding matrix used to derive the general postal approximation algorithm The general postal approximation algorithm We now describe the GHP rounding matrix which enables the support of networks with γ 1, i.e., GHP models. We preserve the notations of [20], P 1, P 2,... denotes the length bounded fractional flow paths, and V (P i ) and E(P i ) denotes the set of nodes and edges in a path P i, respectively; f(p i ) denotes the amount of flow pushed on path P i, and P j denotes the set of all paths that carry flow of the jth commodity. To simplify the presentation of the results we define γ = max{1, γ}. In the general postal approximation, the following rounding matrix (termed GHP rounding matrix) is used for rounding the fractional solution of step (1) in the core computation. for each v s v f(p i ) 6 T for all j 4L T γ i: v V (P i ) i: P i P j f(p i ) = 4L T γ The sum of positive entries in the ith column is: s v λ vw γ + s tj 4L T γ. v V (P i ) (v,w) E(p i ) where the second part of the equation follows from the definition of γ. The sum of the negative entries at each column is at most 4L T γ. By invoking the rounding theorem [20, Theorem 4] we get a set of integral paths such that their congestion is at most 6 T + 4L T γ and the length of each path is at most 4L T γ. The general postal approximation algorithm. The general postal approximation algorithm is a postal approximation algorithm which employees a GHP rounding matrix instead of the original one.

22 CHAPTER 4. MULTICAST ALGORITHMS 16 The correctness of the modified algorithm follows from the fact the algorithm structure and its underling theorems and lemmas are not related to the specific switching and communication cost values, except of the constrained selection of the rounding coefficient which we handle appropriately. Therefore it remains to show the approximation ratio. The transformation performed on the rounded paths, step (2) in the core procedure, yields a set of spiders, such that the diameter of each spider is at most 4 γ ( T + L T ) and the generalized degree (of the center) of a spider is at most 6 γ ( T + L T ). Since the algorithm invokes O(log U ) iterations of the core procedure and the cost of the optimal tree T is at least 0.5 ( T + L T ) [20, Lemma 1], we have that the multicast time from the root r to a set of terminals U is at most O(log U max{1, γ}) times the optimal multicast time The MDM approximation algorithm The following is a polynomial algorithm that approximates the minimum multicast delay. The algorithm accepts as an input an overlay network configuration, (G, c, p), which consists of an undirected graph G = (M, E ) with associated processing and communication cost functions, p and c, respectively, and a source host s M. Given an overlay network graph (V, E), the input graph, G, is the subgraph induced by the multicast group set M V (see Section 3.1). Algorithm Approx-MDM( s, G, p, c) 1. Construct a GHP configuration instance I GHP = (G, s, λ), from the graph G, switching time function s v = p(v), v M and communication latency function λ u,v = c(u, v) + (p(u) + p(v))/2, (u, v) E. 2. Invoke the general postal approximation to compute a multicast tree using I GHP, source host s, and multicast group U which spans M, i.e., U = M. 3. Return the computed multicast tree An example of the cost transformation performed in step (1) of the Approx- MDM algorithm is presented in Figure 4.1. The input instance (see Figure 4.1A) is a simple overlay network with three hosts. The corresponding GHP network is presented in Figure 4.1B. The algorithm runs in polynomial time due to the polynomial time construction performed is step (1) and the polynomial execution time of the approximation algorithm in step (2). The correctness of the algorithm follows immediately from the correctness of the general postal approximation algorithm.

23 CHAPTER 4. MULTICAST ALGORITHMS 17 p 1 p 1 = s 1 c 1 c 2 p 1 + p 3 l 1 = + c 2 1 c 2 + p 1 + p 2 2 = l 2 p 3 p 2 c 3 p 3 = s 3 p 2 = s 2 p c p 2 = l 2 3 (A) (B) Figure 4.1: Example that illustrates the cost transformation of the Approx- MDM algorithm (A) The overlay instance (B) The resulting GHP instance The following lemma shows the approximation ratio of the algorithm. Let OP T denote the multicast delay of the optimal tree and let n denote the size of M. Let p max = max v M p(v) and p min = min v M p(v) be the maximal and minimal processing costs of the hosts in M. Theorem 4.1. The multicast delay of the Approx-MDM algorithm is at most (OP T + (p max p min )) O(log n) Proof. Given a multicast tree T which spans M and a host v M, let t GHP (v) T be the reception delay of v assuming GHP timing. Similarly to equation (3.1), the GHP reception delay of v is k 1 t GHP (v) = (λ T vi,v i+1 + s vi (g(v i, v i+1 ) 1)) (4.1) i=0 where v i, 0 i k 1, v 0 = s and v k = v, denotes the ith host on the unique directed path in T from s to v, and g(v i, v i+1 ) denotes the index of the transmission from v i to v i+1. Observe that, due to the definition of the postal model (see Section 4.1), there is a delay gap of a single processing round (per each traversed host) between the message delivery delays in the postal and overlay models. By substituting the computed switching and communications costs of I GHP with the corresponding overlay input costs, i.e., s v = p(v), λ u,v = c(u, v) + (p(u) + p(v))/2, we get the following relationship between the reception delay costs. p(s) p(v) t GHP (v) = + t T 2 T (v) (4.2) Now, consider the following quantities which are computed assuming GHP timing model. Let OP T GHP be the multicast delay of an optimal tree TGHP for

24 CHAPTER 4. MULTICAST ALGORITHMS 18 the I GHP configuration. Let u M be the node with the maximal reception delay in TGHP. Therefore we have that OP T GHP OP T + p(u) p(s) 2 OP T + p max p min 2 (4.3) where the first inequality follows from Equation (4.2) and the second one follows from the substitution of p(u), p(s) in equation(4.3) with p max, p min. The constructed I GHP instance satisfies γ < 2, since p(v) 0.5 (p(v) + p(w)) + c(v, w) < 2, (v, w) E (4.4) The constant bound on γ implies that the multicast delay of the resulting tree is at most OP T GHP O(log n). Substituting OP T GHP according to equation (4.3) gives the requested upper bound for the multicast delay. When the processing costs are all equal, it improves our approximation for the MDM problem to O(log n). We do not restrict the communication costs to be homogeneous. The following theorem handles this case. Theorem 4.2. Consider an overlay model with homogenous processing costs, i.e., p(v) = p, v V. The multicast delay of Approx-MDM algorithm for this case is at most OP T O(log n) Theorem 4.2 can be obtained by substituting p max = p min = p in Theorem 4.1. Given a network with symmetric communication costs, a multicast tree T rooted at s can be easily adapted to support multicasting from multiple sources. To enable the multicast from a host v M, v s, we reverse the direction of the edges on the path from s to v. This modification results in a multicast scheme with a delay which at most p(v) p(s) + 2 C (4.5) where C denotes the cost of T. Therefore, the undirected version of T can be used as a shared tree such that the multicast delay of any host v s is at most 2 (OP T s + (p max p min )) O(log n) (4.6) where OP T s denotes the optimal multicast delay from s.

25 CHAPTER 4. MULTICAST ALGORITHMS The heuristic algorithm We introduce a heuristic tree construction algorithm which solves the directed variant of the MDM problem. The proposed algorithm computes the multicast tree incrementally using a greedy approach; for each host not yet included in the tree, the algorithm computes its minimum reception delay, and the host with the maximal delay quantity is selected. The tree is extended with the hosts on a minimum delay path between the selected host and a notified host. Figure 4.2 shows the steps of the algorithm. The input to this algorithm is the same as the input to the Approx-MDM, except for the source host which is denoted by s. The algorithm maintains a ready time attribute t[v] for each host v M which records the minimal time at which the host is free to initiate processing of a new message. The ready time is set to infinity to indicate non notified host. The constructed tree is denoted by T and the corresponding set of notified hosts by V [T ]. In each iteration, the algorithm determines for each host u M V [T ] its mate host m[u] V [T ] by selecting a path which minimizes the ready time attribute of u, setting v to indicate the host with the maximal reception delay. Then, it updates the ready time of the hosts on the path from m[v] to v to reflect the processing time involved with delivering a message to a newly notified host v, and it adds the path hosts to the constructed tree T. The variable w indicates the current updated host. The algorithm terminates when all the hosts are notified. To be able to calculate the connection cost between a non notified host and a notified host, a preprocessing phase of computing all pairs shortest path using the Floyd-Warshall algorithm [24] is implemented. Given a pair of hosts v 1 and v k connected by a path < v 1,..., v k > of length k 1, the cost of this path is defined as k 1 i=1 p(v i) + c(v i, v i+1 ), where v i, 1 i k denotes the ith host on this path, i.e., this cost represent the minimal distribution delay (along the specified path) from v 1 to v k. A shortest path from host u to host v is defined as any path between these hosts with minimum cost. Therefore, the input to the Floyd-Warshall computation is an n n weight matrix W = (w vi,v j ) defined as: { p(vi ) + c(v w vi,v j = i, v j ) if v i v j, (4.7) 0 otherwise. where n denotes the size of M. The output of the all pairs shortest path computation is composed of two n n matrices; all pairs distance matrix D = (d vi,v j ) and predecessor matrix Π = (π vi,v j ); where d vi,v j stores the weight of the shortest path from v i to v j, and π vi,v j stores the predecessor of v j on its shortest path to v i assuming that such a path exists, otherwise π vi,v j stores a null pointer. Observe that the shortest path from the source s to any host v is a lower bound on the cost of the optimal tree. This algorithm can be extended to support a shared tree solution using

26 CHAPTER 4. MULTICAST ALGORITHMS 20 Algorithm Heuristic-MDM(G, p, c, s) 1. t[s] = 0, set s as the root of a tree T 2. for each v M {s} 3. do t[v] 4. for each (u, v) E 5. do w u,v = c(u, v) + p(u) 6. for each (u, v) / E 7. do if v = u then w u,v = 0 else w u,v = 8. D, Π All-Pairs-Shortest-Path(G, W ) 9. while M V [T ] 10. for each host u M V [T ] do 11. m[u] arg min v:v V [T ] {t[v] + d v,u } 12. v arg max u:u M V [T ] {t[m[u]] + d m[u],u } 13. w v 14. while w m[v] do 15. t[w] t[m[v]] + p(w) + d m[v],w 16. add w to T as a child of π m[v],w 17. w π m[v],w 18. t[m[v]] t[m[v]] + p(m[v]), t[v] t[v] p(v) 19. return T Figure 4.2: Greedy tree construction for the MDM problem

27 CHAPTER 4. MULTICAST ALGORITHMS 21 the following modification. At the initialization phase the longest path in the graph G is computed using the weight matrix W, and the hosts on this path are used as the initial set of notified hosts in T. The shared tree variant uses this initial selection instead of the original one and proceeds with normal tree construction as in the original algorithm. The complexity analysis of this algorithm is straightforward. The all pairs shortest path computation requires Θ(n 3 ) time. Each iteration requires O(n) time to find a single mate host, and O(n) time to extend the tree. The total time per iteration is therefore O(n 2 ), and the total running time of the heuristic algorithm is Θ(n 3 ). We conjecture this time complexity cannot be improved since any algorithm should at least calculate the all pair shortest path. We show using an example (see Figure 4.3A) a lower bound on the approximation ratio of the heuristic tree. Consider the following complete undirected graph G = (V, E) with n + 1 hosts denoted by v 0,..., v n, with processing costs defined as p(v) = 1, v V, and communication costs c(v i, v j ) defined as c(v i, v j ) = 0 if i = 0, j = 1,... n, δ if 1 i n 1, j = i + 1, n otherwise. (4.8) where δ 0. For the simplicity of presentation Figure 4.3A omits the edges with cost n. Assume that the source host is v 0 and that M = V. Therefore, the heuristic scheme would have v 0 distribute the message to the rest of the hosts using n processing rounds, such that the tree cost is n (see Figure 4.3B). On the other hand, consider an improved scheme in which v 0 distributes the message to k hosts, and the last host in the graph receives the message in k + mδ time units, where m is a positive integer. Let p i denote the length (i.e., the number of edges) of path p i. Such a scheme can be obtained by a tree composed of k paths p 1,..., p k which share only single host, v 0 (i.e., only v 0 has an out-degree more than two), and the lengths of these paths form the following non increasing sequence: p i 1 = p i+1,1 i k 1, whereas for a single index j in this set we may have p j = p j+1. Figure 4.3C depicts such a tree when n = k (k+1). Assume that v 2 0 distributes the message to these paths (i.e., to its k children in the tree) according to a decreasing path length order. Therefore, the cost of the optimal tree is less than (1 + δ) k. Since the set of paths span all the hosts in V we have that k = O( n), and therefore we get Ω( n) approximation ratio for the multicast delay. We conjecture that this example represents the worst case, namely that our heuristic algorithm is an n-approximation. 4.3 The homogenous case Consider a fully connected overlay network with homogeneous processing and delay costs, i.e., p(v) = p, v V, c(u, v) = c, (u, v) E. We denote this