Multicast with Network Coding in Application-Layer Overlay Networks

Transcription

1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY Multicat with Network Coding in Application-Layer Overlay Network Ying Zhu, Baochun Li, Member, IEEE, and Jiang Guo Abtract All of the advantage of application-layer overlay network arie from two fundamental propertie: (1) The network node in an overlay network, a oppoed to lower-layer network element uch a router and witche, are end ytem and have capabilitie far beyond baic operation of toring and forwarding; and (2) The overlay topology, reiding above a denely connected IP-layer wide-area network, can be contructed and manipulated to uit one purpoe. In thi paper, we eek to ignificantly improve end-to-end throughput in application-layer multicat by taking full advantage of thee unique characteritic. Thi objective i achieved with two novel inight. Firt, we depart from the conventional view that data can only be replicated and forwarded by overlay node. Rather, a end ytem, thee overlay node alo have the full capability of encoding and decoding data at the meage level uing efficient linear code. Second, we depart from traditional widom that the multicat topology from ource to receiver need to be a tree, and propoe a novel and ditributed algorithm to contruct a 2-redundant multicat graph (a directed acyclic graph) a the multicat topology, on which network coding i applied. We deign our algorithm uch that the cot of link tre and tretch are explicitly conidered a contraint and minimized. We extenively evaluate our algorithm by provable analytical and experimental reult, which how that the introduction of 2-redundant multicat graph and network coding may indeed bring ignificant benefit, eentially doubling the end-toend throughput in mot cae. Index Term Application-layer overlay network, network coding, application-layer multicat. I. INTRODUCTION Due to the lack of a widely available IP multicat ervice at the network layer in backbone network, recent reearch (e.g., [1], [2], [3], [4]) ha examined the feaibility and trade-off of implementing multicat ervice in the application layer. The general approach common to all exiting propoal i to have application elf organize into a logical overlay network, and to tranfer data along the edge of uch an overlay network uing unicat tranport ervice. Each application-layer node communicate only with it neighbor in the overlay network. Multicating i implemented by forwarding meage along overlay multicat tree that are contructed and embedded in the virtual overlay network. Application-layer multicat, in general, enjoy two attractive advantage over traditional IP multicat: (1) Multicat upport in the network layer i not required; (2) Data i tranmitted between node via unicat, effectively exploiting all exiting ecurity, flow control and reliable delivery mechanim that are Ying Zhu, Baochun Li and Jiang Guo are with the Department of Electrical and Computer Engineering, Univerity of Toronto. Their addree are {yz, bli, jguo}@eecg.toronto.edu. readily available and mature. However, an overlay multicat approach, however efficient, cannot perform a well a IP multicat. It i impoible to completely prevent multiple overlay edge from travering the ame phyical link, cauing unavoidable redundant traffic (identical copie of applicationlayer meage) on the ame link, referred to a link tre [1]. Further, unicat communication between end ytem involve travering other end ytem, potentially increaing latency. It i therefore critical to evaluate and eek to minimize both the relative increae of end-to-end latencie (caued by link tretch 1 ) and the increae in per-link bandwidth requirement a compared with network-layer multicat. Beyond what ha been extenively tudied in previou work, we emphaize that the advantage of deploying applicationlayer overlay network arie from two fundamental propertie. (1) Network node in an overlay network, a oppoed to lowerlayer network element uch a router, are end ytem and have capabilitie far beyond baic operation of toring and forwarding. (2) The topology of an overlay network can be manipulated willfully to uit one purpoe ince it reide on top of a denely connected IP-layer network. The link between node can be dynamically created or torn down to contruct topologie that are conducive to better network performance. Recent reearch in application layer multicat (e.g., [1], [2]) ha hown that it may well be worth the incurred cot of topology contruction and maintenance to profit in increaed robutne, flexibility and efficiency. In thi paper, we eek to improve end-to-end eion throughput in an application-layer overlay multicat topology by taking full advantage of both of thee unique characteritic. We deviate from the conventional view that data can only be replicated and forwarded by overlay routing node. Rather, a end ytem, thee overlay node alo have the full capability of encoding and decoding data. We apply the mechanim of network coding [5], [6], [7] on intermediate overlay node. In addition, we alo depart from the traditional widom that the multicat topology need to be a tree from ource to receiver; rather, we eek to contruct a 2-redundant multicat graph (a directed acyclic graph to be defined in Sec. III) a the multicat topology, on which network coding i applied. Baed on thee inight, our main contribution i to propoe a et of ditributed algorithm to contruct uch multicat graph and to ubequently aign linear code and apply network coding, uch that a a provable property, the endto-end throughput may be ignificantly increaed (doubled in 1 Formally, tretch i defined a the ratio of path length from the ource to the multicat group member along the overlay to the length of the direct unicat path [3].

2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY many cae) for all member of the multicat group. Since overlay multicat come with the cot of link tre and tretch, we deign our algorithm uch that thee cot are explicitly conidered a contraint and optimized. Achieving the objective of increaing end-to-end multicat throughput i particularly important when application uch a content ditribution ervice demand the highet capacity poible in overlay network. We extenively evaluate our algorithm by uing both analytical and imulation-baed experimental tool. We how that our algorithm i indeed able to bring ignificant benefit with repect to increaing end-to-end eion throughput. The remainder of thi paper i organized a follow. Sec. II motivate the cae for application-layer coded multicat, by preenting concept, advantage and requirement of network coding. Sec. III preent formal definition of term, leading to the main theorem with repect to the maximally achievable throughput with network coding. Our algorithm i formally preented in Sec. IV, along with it provable propertie and relevant dicuion. Sec. V preent experimental reult uing imulation. Finally, Sec. VI evaluate our propoal in the context of related work, and Sec. VII conclude the paper. II. A CASE FOR APPLICATION-LAYER CODED MULTICAST The main contribution of thi paper are: (1) contructing multiple data path from ource to multicat group member, and (2) applying the concept of network coding in applicationlayer multicat, motivating the cae for application-layer coded multicat. The objective i to take advantage of alternate path and exce capacity in the IP-layer network topology, and to ignificantly increae end-to-end multicat capacity. The information-theoretic apect of network coding wa firt propoed and tudied by Ahlwede et al. [5]. With network coding, node have the capability of encoding and decoding data at the per-meage level uing efficient linear code; the aim i to ue bandwidth more efficiently and thereby increae network capacity. We briefly review the concept of network coding with an example hown in Fig. 1(c). The example how how the eion throughput of an 1-to-2 multicat eion may be improved. In the figure, a and b repreent two independent information flow originating from the ource S. Node tranmit the coded flow a b along the bottleneck link to node, which then forward the coded flow to both detination t 1 and t 2. Receiver t 1 and t 2 can recover {a, b} from {a, a b} and {b, a b}, repectively. The eion achieve a throughput of 2C, auming each link ha capacity C. Without network coding, it can be verified that the achievable throughput i only 3C/2. However, network coding i not the panacea when it come to increaing multicat eion throughput. There exit many topologie including all form of multicat tree where network coding fail to be more effective with repect to improving throughput. It help to increae throughput only in network graph that conform to pecial pattern. It i extremely difficult to manipulate node in the IP layer to contruct multicat graph that conform to pecific pattern, and it i infeaible to modify all IP router and witche to upport coding. Overlay network, on the other hand, have exactly the propertie that could be leveraged to employ network coding for higher throughput in application-layer multicat: flexibility in topology contruction, and capability of encoding and decoding. Traditionally, the fundamental topological tructure of multicat, whether it be IP-layer or application-layer, i a tree. Hence, every multicat group member in the tree ha only one path from the ource root; it throughput i limited by thi path. To increae throughput by adding another path from the ource to each receiver, one i faced with two problem: (a) The gain may be overhadowed by the cot 2 of the additional link and node in the alternative path. (b) One mut enure that the alternative path do not conflict with the original path in order to avoid throughput-limiting bottleneck. Fig. 1. W u 2 t 1 t 2 (a) Example with two receiver, t 1, t 2. (b) Shortet path multicat tree. a b a b u 2 a a+b t 1 t 2 (c) Throughput i doubled with network coding and alternative path (dark edge form econd path for t 1, econd path for t 2 i the mirror image). b u 2 t 1 t 2 The effect of network coding: an example. W u 2 t 1 t 2 (d) Throughput i doubled uing alternative path, but not uing network coding. We propoe a new application-layer multicat trategy that, by appropriate ue of network coding, will reolve thee problem and achieve the higher throughput without commenurate cot or complexity. We ue the previou example to illutrate how we apply network coding advantageouly. The overlay network i repreented by the graph in Fig. 1(a), in which t 1 and t 2 are the two receiver in the multicat group, and i the ource. Each edge ha the ame bandwidth of 1 except that the bandwidth available on edge (, ) i w 1. Thi i the cae when, for example, can not utain an outgoing bandwidth of much more than 2. The uual all-widet-path multicat tree i hown in Fig. 1(b); the widet alternative path are added in Fig. 1(c), while the other choice of (narrower) alternative path are hown in Fig. 1(d). Without network coding, it i impoible to double throughput in Fig. 1(c), ince the alternative path to t 1 and t 2 interfere with each other widet path uch that they cannot 2 The cot of a link embodie critical metric of concern, uch a bandwidth, latency and lo rate.

3 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY both double their throughput. The conflict can be eliminated by chooing the path in Fig. 1(d), but the bandwidth of the alternative path i much le (narrower) than that of the original tree, again making it not feaible to double the throughput. Thi example exemplifie the two concern pecified previouly. With network coding, however, the graph in Fig. 1(a) can be afely ued in the multicat to double the throughput to both receiver, a hown in Fig. 1(c). The tremendou power of network coding lie in the fact that any conflict reulting from interfering path in the multicat graph can be reolved to obtain the ame throughput for each receiver a if it wa the only receiver. III. PRELIMINARIES We conider a network graph with a ource node and a et of multicat group member a receiver node (or imply receiver). We define two notion of maximum flow and k- redundant multicat graph. Definition 1 (individual maxflow). Given any ingle receiver t, we ay that the individual maximum flow (or imply individual maxflow ) of t i the maximum flow from to t when data flow from only to t, and the other receiver are not conidered except a part of the ubgraph erving the data flow from to t. Definition 2 (imultaneou maxflow). When all multicat group member receive data flowing from imultaneouly, i.e., a multicat, the maximum flow that a receiver t achieve i the imultaneou maximum flow of t. Definition 3 (k-redundant multicat graph). A k- redundant multicat graph for ingle-ource multicat i a directed acyclic graph (DAG) which ha the following two propertie: 1) The et of all node, A, i the union of three dijoint ubet {} A I A T : {}, the ource, indegree() =0and outdegree() > 0; A I, the intermediate node (who are not member of the multicat group), denoted by u i, 1 i n I, 1 indegree(u i ) k and outdegree(u i ) > 0; A T, the receiver node (i.e., multicat group member), denoted by t i, 1 i n T, indegree(t i )=k and outdegree(t i ) 0; 2) If each edge in the graph ha unit bandwidth, then for any node whoe indegree i k, it individual maxflow i k. In particular, ince the indegree of all receiver i k, the individual maxflow of each receiver t i k, if the graph ha unit-bandwidth edge. If edge have different bandwidth, then the individual maxflow of t i clearly k time the bandwidth of the bottleneck edge between and t. Without lo of generality, we can aume that all the edge have the ame bandwidth (that of the bottleneck). Thi aumption will be made throughout thi ection to implify preentation of proof. Let n =1+n I + n T be the total number of node. In thi paper, we only conider the cae of 2-redundant multicat graph. The jutification are two-fold: (1) A k increae, both the utainable phyical link tre leading to overlay node and the limited number of intermediate node and receiver ignificantly decreae the probability of finding multiple good path from the ource to each receiver. In the example of Fig. 1(a), when the utainable tre on all node i no greater than 3 incoming and outgoing flow, it i infeaible to contruct a k-redundant multicat graph if k>2. (2) A k increae, the code aignment algorithm (Sec. IV-F) become more complex and avere to the dynamic of node join and departure. We now etablih the ufficiency of a maximum indegree of 2 in a 2-redundant multicat graph. It i not immediately clear why no node in the graph need more than two incoming edge to enure individual maxflow of 2 to each receiver. We prove that a maximum indegree of 2 i ufficient by firt proving the following obervation (Propoition 1 and 2) about dijoint path. Definition 4 (dijoint path). We ay two path from to t are dijoint if they do not hare any common edge. Propoition 1: Given a node t with indegree 2 in a 2- redundant multicat graph with ource and unit-bandwidth edge, t ha two dijoint path from if and only if t ha individual maxflow of 2. Proof: ( ) Thi direction i traightforward. It i obviou that if there are two dijoint path to t, then it ha maxflow of 2. ( ) A maxflow of 2 implie (in thi cae where edge have unit bandwidth) that there are two flow, f 1,f 2, each of bandwidth 1. Each flow clearly mut define a path from to t, let p 1,p 2 denote the path for f 1,f 2, repectively. If p 1 and p 2 hare a common edge e, then f 1 and f 2 mut hare the bandwidth of e, which i only 1. So the value of each flow i 1/2, thi i a contradiction. Therefore, p 1 and p 2 do not hare any common edge and are two dijoint path to t from. Propoition 2: It i not neceary for any node in a 2- redundant multicat graph to have indegree of greater than 2 to obtain two dijoint path for each receiver from. Proof: We only need to how that by contructing two dijoint data path for a receiver t, it i not neceary for adding a third incoming edge to any node in the exiting multicat graph. When contructing the firt path p 1, uppoe, by contradiction, that a third incoming edge i to be added to a node u. Thi clearly i not neceary, ince a path mut exit from u to t, u t, and a path mut already exit from to u (due to connectedne), u, and o p 1 can be imply a concatenation of u and that from u t. Thi contradict the neceity of adding the third incoming edge to u. When contructing the econd path p 2, uppoe, again by contradiction, that a third incoming edge i to be added to a node u. Since u ha an indegree of 2, by property 2 in Definition 3 and Propoition 1, u mut have two dijoint path from, denote them by,u 2, repectively. There are two cae. Cae 1: Firt path for t, p 1, i dijoint from at leat one of and u 2. Without lo of generality, uppoe p 1 i dijoint from, then p 2 can be contructed by concatenating u 2 and u t. It i clear that p 2 thu formed i dijoint from p 1. Cae 2: Both and u 2 hare common edge with p 1. Without lo of generality, uppoe the common edge cloet to t i hared by and p 1. Let v w denote thi common

4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY edge, and alo let v denote the egment of until v and w t denote the egment of p 1 from w to t. A new firt path for t, p 1, can then be contructed from concatenating v and w t; and p 2 i formed by u 2 followed by u t. It i eay to ee that p 1 and p 2 are dijoint. The property of 2-redundant multicat graph that the maximum indegree of a node i 2 not only reduce complexity in the algorithm to contruct the graph, but moreover contribute ignificantly toward minimizing tre. Toward the objective of contructing a 2-redundant multicat graph, Propoition 1 how the importance of contructing two dijoint path from the ource to a receiver t. Further, directed acyclicity mut be preerved in the contruction of the multicat graph, i.e., there mut be no directed cycle in the graph. Directed cycle in the network introduce great complexity and difficulty in determining the linear code ued for multicat [5], [6]. In fact, Koetter et al. [6] have not preented any olution for the cae of general network with cycle. We next how that, in order to contruct a 2-redundant acyclic multicat graph, the et of intermediate node, A I, need to be non-empty. Propoition 3: A 2-redundant multicat graph with only receiver, i.e., every node beide ha two dijoint directed path from, contain a directed cycle. Proof: Let t 1 be any node that ha a directed edge from. Since it ha two dijoint path from, there i a econd directed edge pointing to it from another node, ay, t 2, denoted by t 2 t 1. Similarly, t 2 ha at leat one incident directed edge from a node that i not, ay, t 3. A chain of node may be built with t 2 t 1, t 3 t 2, and o forth. Since there exit a finite number of node, eventually t i become the ame node a t j with j<i, i.e., t j i in the chain before t i, thu forming a directed cycle. Obviouly, there may not exit two dijoint path from to each u i, which lead to the concluion that multicat group member may not erve a intermediate node. We need to recruit dedicated high-degree relay node in the overlay network a intermediate node, who do not belong to the multicat group. For thi purpoe, we may deploy a pool of end hot or proxy erver connected to high-bandwidth phyical link. Naturally, the number of required intermediate node mut be minimized, and calable to large-cale multicat group. Such a pool of dedicated node i the price we pay to exploit the power of network coding to ignificantly increae throughput. Thee intermediate node do not place much additional tre on the network, ince they do not require multiple data delivery path from the ource. Facilitated by intermediate node, we eek to contruct 2-redundant multicat graph with no directed cycle (a DAG). A an example, the multicat graph in Fig. 1(c) i 2- redundant, with receiver t 1 and t 2 both having two dijoint path from (a explained in Sec. II). We now proceed to illutrate the eence of linear code. Definition 5 (linear coding multicat). Linear coding multicat view a block of data flowing over an edge a a vector and aign a linear tranformation for each node u in the multicat graph uch that: for each outgoing edge e of u, the vector ent out on e i a linear combination of the vector of the incoming edge; for ource, any vector can be ent out on it outgoing edge; all the vector are in the ame infinite-dimenional vector pace over a bae field. Linear code are the coefficient that determine the linear tranformation. For example, in Fig. 1(c), the data ent by on the outgoing edge i a linear combination of {a, b}: 1 a +1 b = a + b, where + i defined in a finite field, e.g., GF(256). Such a linear combination i repreented a (1, 1). The main reult of network coding, firt propoed in [7] (a lightly weaker verion wa propoed earlier in [5]), i tated in the following theorem. It eentially tate that in an acyclic network with a ource and multiple receiver, the maximum individual throughput of each receiver can alway be achieved a if there wa no interference at all from data flowing in the network to the other receiver, by uing only linear coding. Theorem 1 (Li and Koetter). For every multicat graph, there exit a et of linear code that could be ued for multicat (linear coding multicat) uch that imultaneou maxflow of t i i equal to individual maxflow of t i. Proof: The intereted reader i referred to [6] or [7] for detailed proof uing different methodologie. IV. ALGORITHM AND ANALYSIS The ultimate goal of our algorithm i to build and maintain a 2-redundant multicat graph a defined in Sec. III. There are everal non-trivial challenge. Our algorithm addree each of thee challenge, and much of the complexity lie in tackling all of them in conjunction. (1) In order to ubequently apply network coding, we need to correctly contruct a 2-redundant acyclic multicat graph from the ource to all member of the multicat group 3. During the contruction proce, data delivery path hould be optimized in the multicat graph to the receiver. Each receiver eentially ha two path from the ource; both path hould be carefully choen to maximize the aggregated throughput to the receiver. (2) We need to minimize the number of intermediate node with a given number of receiver while preerving good performance. (3) Minimizing tre i paramount ince it directly determine how much actual bandwidth a virtual link ha and high tre can everely diminih end-to-end throughput. Thee problem embody the fundamental objective of maximizing multicat performance (end-to-end throughput and latency) while minimizing the penalty incurred by elevating the functionality of multicat from the IP layer to the application layer and by uing a multicat graph intead of a multicat tree. In particular, minimizing tretch i an integral part of optimizing the path and minimizing tre i covered by impoing a maximum node degree in the multicat graph. The algorithm we developed for our multicat cheme are fully ditributed. Our cheme mainly conit of three tep. The firt tep i building a relatively denely connected graph of the et of all node in the group, A, referred to 3 Henceforth, the term multicat group member and receiver will be ued interchangeably.

5 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY a the rudimentary graph. Fig. 2(a) how a imple example of a rudimentary graph of a mall multicat group with 7 node; the intermediate node, A I, conit of {,u 2, }. The econd and third tep are carried out for data delivery. In the econd tep, a panning tree of only the intermediate node with ource a the root i contructed, referred to a rudimentary tree. With node r 1 being the ource, the rudimentary tree i hown, a darkened edge with incident node, in Fig. 2(a). Uing the rudimentary graph and tree, the third tep contruct the 2-redundant multicat graph by carefully electing two path through intermediate node from the ource for each leaf receiver. In our imple example, the leaf receiver A T = {r 2,r 3,r 4 } each ha two dijoint path from, a hown in Fig. 2(b). Both the rudimentary tree and the multicat graph have the degree contraint: every node ha degree ; i.e., i the maximum node degree. r 2 r 1 u 2 r 3 (a) rudimentary graph; u i ' are intermediate node; darkened edge form rudimentary tree, r 1 i ource. r 4 u2 r 2 a a+b b r 3 (b) 2-redundant multicat graph; r i' are leaf receiver Fig. 2. A imple example of a multicat group, with rudimentary graph, rudimentary tree, and the reulting 2-redundant multicat graph. A. Rudimentary graph When a node join the group, it i given a et of node already in the group, thee are it initial neighbor in the rudimentary graph. The new node contact it neighbor o they are made aware of it. Every node maintain a et of neighbor with which it periodically exchange group information. That i, each node tore a lit of the addree of all the node it know about in the group and periodically exchange it with it neighbor lit, and update accordingly. After a node join, the information will eventually propagate through the rudimentary graph. Each edge (or link), e, in the rudimentary graph ha aociated with it a 2-tuple weight, w(e) = (β,λ), where β i the link bandwidth and λ i the link latency. Each path, p =(e 1,e 2,...,e m ), alo ha an aociated weight, w(p) = (β,λ), and β = min(β i,i =1,...,m),λ = m i=1 λ i, where w(e i )=(β i,λ i ). A uual, path bandwidth i the minimum of the link bandwidth and path latency i the um of the link latencie. Let w(p 1 )=(β 1,λ 1 ),w(p 2 )=(β 2,λ 2 ) be the weight of p 1,p 2, repectively. p 1 i better than p 2 if β 1 >β 2, or β 1 = β 2 and λ 1 <λ 2. Alo, periodically, each node u chooe randomly another node v in the group that i not a neighbor and by ending a probing meage, etimate the bandwidth and latency of the direct overlay link, (u, v). If the direct link i better than mot of it (direct) link to it current neighbor, then v i added a a neighbor of u and the edge (u, v) i added to the rudimentary graph. The goal i to have overlay link (edge) r 4 that have good performance in the rudimentary graph. Let x be a current neighbor of u. From the ame motivation, if (u, x) i much wore than the link u ha to it other neighbor, and both u and x ue thi link rarely (i.e., ue it to reach very few node), then u drop x a it neighbor and (u, x) i removed from the rudimentary graph. The dynamic of adding high-quality edge and dropping poor-quality edge i vital to the performance of the entire multicat cheme. Becaue, ultimately, the edge in the rudimentary graph are ued to contruct the data delivery path, whoe performance depend directly on the property (i.e., weight) of thee edge. The graph reemble the Narada meh [1], with the following important difference: (1) For every intermediate node, the number of it neighbor that are intermediate node mut be no larger than. Thi i neceary to enure that the maximum degree of node in the rudimentary tree (of intermediate node) contructed from thi graph i limited by. (2) For every node (intermediate or not), the total number of it neighbor may be larger than. The algorithm for building the multicat graph later explicitly enforce the degree contraint, o node in the rudimentary graph can have more than neighbor. Since thee extra link are for control meage and not for data tranmiion, more of them can be allowed without raiing concern about performance degradation. (3) The ubgraph of intermediate node with their incident edge mut be connected. Definition 6 (core graph). The core graph i the ubgraph of the rudimentary graph with the et of vertice A I and all the incident edge. The core graph i kept connected by the ame heuritic ued for keeping the entire graph connected. B. Rudimentary tree The rudimentary tree i built from the ubgraph coniting of the core graph and (with it edge incident with the core). We adopt the ditributed algorithm propoed by Wang et al. [8] baed on ditance vector that find the hortet widet path. The widet path, or the path with the highet end-to-end bandwidth, i elected; and if there i more than one widet path, the hortet, one with the lowet end-to-end latency, i elected. C. Multicat graph The baic idea of building the multicat graph i to ue the rudimentary tree a a bai and add edge, when neceary, from the rudimentary graph. Contruction of the 2-redundant multicat graph G 2r adhere to the following rule for ource node and every intermediate node u: 1) ha k intermediate node a children and 2 k 1 (outdegree() = k); 2) total degree of u = indegree(u) + outdegree(u) ; 3) 1 indegree(u) 2; 4) number of children of u that are leaf receiver 1 indegree(u).

6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY The fourth rule force an intermediate node to leave at leat one outdegree available for adding an edge to another intermediate node. Thi enure that the earch for a path i alway ucceful a long a there are enough intermediate node in the rudimentary graph, without reorting to expenive exhautive earch. We hall ue a running example throughout the decription of the algorithm to help illutrate how the algorithm work. The core graph of the rudimentary graph and the rudimentary tree of the example are hown in Fig. 3. Note that the maximum degree,, i 4 in our example. u2 u 5 u 6 u 7 u u 9 u (a) core graph of rudimentary graph u 2 u 5 u 6 u 8 u (b) rudimentary tree Fig. 3. The core graph of rudimentary graph and the rudimentary tree for an example et of intermediate node, with =4. We now need to define two additional term. Definition 7 (leaf intermediate). An intermediate node v i called a leaf intermediate if v doe not have any downtream intermediate node in G 2r, i.e., none of v children in G 2r i an intermediate node. Definition 8 (aturation). We ay v i aturated if either degree(v) =, orv i a leaf intermediate and degree(v) = 1. For intance, in the G 2r in Fig. 4(b), u 7 i a leaf intermediate node while i not. If we aume = 4 for the graph in Fig. 4(b), then will be aturated if one leaf receiver i connected to it. While u 7 will be aturated if two leaf receiver are connected to it. Even when a leaf intermediate node i aturated, there i an outdegree preerved from leaf receiver to allow poibility of adding at leat one child intermediate node. The 2-redundant multicat graph G 2r i initialized to the rudimentary tree, o initially it ha no leaf receiver. Adding each leaf receiver t entail contructing two dijoint data delivery path for t. Edge not in the tree (but in the rudimentary graph) may be added to G 2r in the proce. We now decribe procedure for obtaining two dijoint path, p f and p, for a leaf receiver t. The ditributed algorithm are formulated in Table I and Table II. We eek to create, for t, two data delivery path from, denoted by p f and p, denoting the firt path and the econd path, repectively. Moreover, p f and p do not hare any edge in common. To find p f, t firt contact all it neighbor in the rudimentary graph that are intermediate node and find out which are unaturated. If t ha unaturated neighbor, then t compare their tree path appended with the edge from them to t, and elect node u with the bet path. Let u f denote the node that i parent of t in p f, then u f i aigned u. If all of t neighbor are aturated, t initiate a breadth-firt earch of the u 7 TABLE I PROCEDURE FOR CONSTRUCTING p f FOR t Leaf receiver t: if not all t intermediate node neighbor are aturated (u i ) u = Find bet path(t, {u i }) u f = u and p f = p (u, t) ele if all neighbor of t are aturated Breadth-firt earch of tree, halt when k unaturated node {u i } found u = Find bet path(t, {u i }) u f = u and p f = p (u, t) Find bet path(t, {,...,u m}): requet u i for weight {w(p i )=(β i,λ i )} of their path in tree for each (unaturated) u i β = bandwidth of edge (u i,t) compute w(p i (u i,t)) = (min(β i,β),α+ α i ) chooe the bet (hortet widet) path, p return u that correpond to p TABLE II PROCEDURE FOR CONSTRUCTING p FOR t Leaf receiver t: find k unaturated intermediate node {u i }, which are not in p f, from it neighbor and/or random probing. end p f to each u i and requet bet path p i from to u i that doe not interect with p f and it weight w(p i ) u = Find bet path(t, {u i }, {p i }, {w(p i )}) Find bet path(t, {u i }, {p i }, {w(p i )}): for each u i β = bandwidth of edge (u i,t) compute w(p i (u i,t)) = (min(β i,β),α+ α i ) chooe the bet (hortet widet) path, p return u that correpond to p Intermediate node u i : Upon receiving p f and requet for path from to u i dijoint from p f if u i tree path P (, u i ) doe not interect with p f then return p = P (, u i ) and w(p) ele if u i tree path interect p f then if u i ha an alternative path p from then return p and w(p) ele if u i ha indegree 1 then contact a different child c of than the one whoe ubtree u i i in. c conduct breadth-firt earch of it ubtree and return to u i firt unaturated or leaf intermediate node v. return p = P (, v) (v, u) and w(p) tree to find the firt k unaturated node. Comparion of the k tree path appended by repective edge from thee node to t yield node u with the bet path. A above, u f i et to u. In either cae, p f = P (, u f ) (u f,t), where P (, u f ) i the tree path from to u f. The primitive of the breadth-firt earch of the tree ued in the procedure will recur in later procedure. The detail of it implementation are preented in Sec. IV-D. The number k repreent a trade-off between efficiency of the algorithm and optimality of the path contructed. It hould not be too high to avoid near-exhautive earch of the tree for an unaturated node, when there are many leaf receiver aturating many node. Suppoe we want to contruct p f,p for t with the rudimentary graph and tree in Fig. 3, and the exiting 2-redundant multicat graph i a hown in Fig. 4(a). Node,u 2, are aturated.

7 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY u u 7 t 1 t 2 t 3 (a) Exiting 2-redundant multicat graph t u 8 (b),u 8 are unaturated neighbour of t; t will chooe the bet path from thee two path t Returning to our example, we let k =2. For contructing p, t firt find two unaturated node that are not in p f : u 7 and u 8. t then end requet to u 7,u 8 for their bet path from. The tree path of u 7 from doe not interect with p f,ou 7 will return it tree path p =(,,u 7 ).Ift chooe p (u 7,t) a the bet path, then p i hown in Fig. 5(a). Since u 8 i in the ame ubtree a, it will need to find an alternative path from, p =(, u 2,u 6,u 8 ).Ifp (u 8,t) i a better path than {(,,u 7 )} (u 7,t), then p i hown in Fig. 5(b). u f = u u 7 u 2 u 8 t t 1 t 2 t 3 (c) Adding the firt data path p f = (,,,t) for new leaf receiver t; u f =, dark edge are p f. Fig. 4. Example of 2-redundant multicat graph. Ellipi indicate part of graph there i not hown.... u 2... u 7... t (a) Second data path p = (,,u 7,t) if t chooe u = u 7 ; dark edge form p. u5... u 6... u 7 u 8 t (b) Second data path p = (,u 2,u 6,u 8,t) if t chooe u = u 8 ; dark edge form p. In our example, the unaturated node t conider are,u 8. t chooe the bet path from the directed path (,,,t) and (,,,u 8,t), a hown in Fig. 4(b). If the better path i the firt one, then p f for t i hown in Fig. 4(c). To keep the figure clean, the direction on the edge in the multicat graph in later figure are omitted. The direction of the edge are uniformly downward. For reference, the procedure for determining p i preented in Table II. The leaf receiver t firt find k unaturated node {u i }, which are not in p f, from it neighbor. If there are fewer than k uch neighbor, t randomly probe intermediate node. Now each u i i requeted by t to give the bet path, p i, from to u i. After t receive p i and w(p i ) from all the u i, t imply elect the bet path among {p i (u i,t)}. Let u denote the parent intermediate node of the bet path p that t elect. The econd data delivery path to t i p = p (u,t). Each u i, when requeted by t, firt check if it tree path P (, u i ) interect with p f. If not, then P (, u i ) i returned. If it doe interect, then u i find an alternative path from. An alternative path may already exit (if u i ha indegree 2), in which cae, u i replie with that. Otherwie, u i end a meage to a child node c of that i different from the child who i uptream from u i. It i a requet for c to do a breadth-firt earch of it own ubtree and reply to u i the firt unaturated or leaf intermediate node found. Let v denote thi node. u i replie to t with P (, v) (v, u). Note that the earch will alway be ucceful, ince a breadth-firt earch will alway eventually find a leaf intermediate node. Thi i not true if the earch wa only for unaturated node. Moreover, the algorithm require for each leaf intermediate node to leave one outdegree free for an edge to another intermediate node (recall the definition of node aturation). Thi i the reaon that we allow adding a econd incoming edge to u i. The rationale behind the allowance i quite intuitive: we are trying to find two bet path from to t which mut be dijoint, it i entirely poible that they do not both belong to the rudimentary tree. Fig. 5. Two example of adding a econd data path, p ; darkened line repreent p. D. Breadth-firt earch primitive Breadth-firt earch of a tree to find unaturated node can be made much more efficient than blind and exhautive earch. Keeping record of the tate of aturation of the ubtree of children i not hard. When a node firt become aturated, it imply end that information uptream in the multicat graph. Recurively, a node, which i the root of ubtree R, know R i aturated when all it children have ent aturation notification to it. In thi cae, the breadth-firt earch take guidance from the indicator of ubtree aturation at the root of ubtree. If the ubtree i aturated, then no node in the ubtree i earched henceforth. E. Analyi We prove the correctne of the algorithm, how ome bound on the number of intermediate node required, and dicu calability. Theorem 2. The graph contructed by the algorithm i indeed a 2-redundant multicat graph. Proof: The graph i 2-redundant by contruction, i.e., it i contructed to be uch that each leaf receiver ha two dijoint path from the ource. We only need to how that the graph contructed i acyclic. We how by induction that thi hold. The initialization of the graph i the rudimentary tree with directed edge from parent to child, which i certainly acyclic. Suppoe an exiting multicat graph i acyclic. We now prove that after adding a new leaf receiver t, the multicat graph remain acyclic. Let p f,p be the two path choen for t. If p f,p were already in the multicat graph, then only two directed edge to t are added, and the reulting graph i clearly till acyclic.

8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY Suppoe new edge are added to the graph for p f and p. For p f, only an edge directed to t i added, o no cycle i introduced. For p, there are two cae. The firt cae i when the parent node of t in p, u, i not in the ame ubtree a the node in p f. p i the path P (, u ) in the rudimentary tree plu the added edge from u to t, which i the ame cenario a for p f above, hence no cycle exit. The econd cae i when u i in the ame child-ubtree of a node in p f. So a directed edge (w, u ) not in the rudimentary tree i added (in addition to (u,t)), where w i in a different ubtree. We prove by contradiction that the directed edge (w, u ) cannot be part of a cycle. Suppoe that a cycle containing (w, u ) indeed exit. Since w and u are in different child-ubtree of, there mut be a directed edge (u, v) in the cycle uch that u i in the ame child-ubtree of a u, while v i in a different child-ubtree. Moreover, u i a decendant of u. The edge (u, v) wa added before and when it wa added, u wa unaturated. But (u, v) exit only if a breadth-firt earch from the top yielded that u wa aturated, otherwie, (u,v) would have been choen intead. Thi i a contradiction. With the retriction of maximum degree,, on the node in the multicat graph, the number of intermediate node needed increae with the number of leaf receiver. For n T leaf receiver, we would like to determine the maximum and minimum number of intermediate node poible. It may be eaier to conider the contrapoitive of thee: (1) the maximum number of leaf receiver poible, in a 2-redundant multicat graph, n T max, for a given number of intermediate node, and (2) the minimum number of leaf receiver that can aturate a 2-redundant multicat graph uing the above algorithm, n T min. Theorem 3. Given n I intermediate node, n T max = (( 2)n I +1)/2, (1) n T min = (( 4)n I +1)/2. (2) Proof: It i eay to ee that to prove Equation (1) i equivalent to finding the number of leave in a ( 1)-regular tree. (An m-regular tree i a tree in which every non-leaf (i.e., intermediate) node ha exactly m children.) The total number of children in tree =( 1)n I = number of non-leaf node + number of leaf node + root = n I + n T +1, and the claim follow directly. Now we prove Equation (2). A above, conider the ( 1)- regular tree. Starting with thi tree, the number of leaf node decreae every time the algorithm add a econd incoming edge to a non-leaf (intermediate) node. There are two incident node of thi additional edge and the degree of each decreae by 1, which mean that the number of children of each decreae by 1. So each uch econd incoming edge added decreae the total number of children by 2. The wort cae i when every non-leaf node ha a econd incoming edge. Therefore, the leat number of total children i ( 1)n I 2n I = n I + n T +1, and the claim follow. Now we dicu the iue of calability, by firt noting that each node only exchange control meage with a contant number of neighbor. The tep of building the rudimentary graph and the rudimentary tree i a variant of the ditancevector algorithm (alo known a the ditributed Bellman-Ford algorithm), which ha been proven to converge and ha time complexity of O(VE), where V i the number of node and E i the number of link. Since the number of neighbor i contant for each node, the complexity for n node O(n 2 ). Similarly, the overhead of control meage for contructing the rudimentary graph and tree i the ame a that for the ditance-vector algorithm, variant of which are commonly ued in realitic network (e.g., Border Gateway Protocol). Hence, our protocol i clearly calable, in term of both time complexity and control overhead, to high number of group node. It i eay to ee that all the procedure in the lat tep are dominated, in time complexity and control overhead, by the procedure of finding k unaturated node. All other operation are contant time with repect to n (the number of node). To find k unaturated node, it take contant time if a contant number of random probe are ucceful; otherwie, a pecial verion of a breadth-firt earch, decribed in Sec. IV- D, i executed. A node need to exchange control meage with at wort O(log n) other node, ince our breadth-firt earch primitive include record-keeping at the node. Thu the control overhead in the wort cae i O(log n). Overall, it i clear that our graph-contruction algorithm i calable to large multicat group ize. To handle dicrepancy in bandwidth or rate of two incoming flow, we reort to exiting flow control mechanim (e.g., TCP) to ynchronize the incoming flow rate, a it i traditionally done for matching incoming and outgoing rate in a flow-controlled reliable connection. We know that the end-to-end throughput in a multicat tree i determined by the minimum bandwidth link in the tree. Since two incoming flow ynchronize rate in a 2-redundant multicat graph, the end-to-end throughput i twice the minimum bandwidth in the two (dijoint) path. To reolve a difference in latency of two incoming ignal, buffer i needed; the buffer ize i proportional to the latency difference and i finite. Since the node in the overlay network are end ytem (with abundant memory pace), the iue of available memory for buffering i not likely to be ignificant. It i alo poible and not hard to add optimization technique to the exiting algorithm to minimize the latency difference when finding a pair of dijoint path during graph contruction. F. Linear coding multicat Once a multicat graph i contructed, a et of linear code mut be found to realize linear coding multicat. Both Koetter et al. [6] and Li et al. [7] give algorithm for contructing the linear code. However, becaue both paper are theoretical in nature, Li et al. [7] with an information-theoretic perpective while Koetter et al. [6] ha an algebraic-geometric formulation, their briefly decribed algorithm have been included mainly for completene. The algorithm are moreover centralized, difficult to implement in a ditributed manner, and intended for general multicat cenario. Since the graph contructed by our algorithm are of a pecific tructure, a more light-weight

9 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 1, JANUARY algorithm tailored for thi pecific type of multicat graph can be devied. We propoe a ditributed algorithm that i eay to implement for obtaining the linear code for the 2-redundant multicat graph. We oberve that coding i only needed at the ource and the intermediate node, and that an intermediate node ha either one or two incoming edge (by contruction). Furthermore, ince each leaf receiver ha exactly two path from the ource, end the data vector (a, b) and both a and b hould be obtained by each receiver. We aume that there i a function gen(m) that generate a equence of m tranformation vector {(p 1,q 1 ) T, (p 2,q 2 ) T,...,(p m,q m ) T } uch that p i,q i are element in a field; (p i,q i ) and (p j,q j ) are linearly independent, i j; (p i,q i ) T define a linear tranformation of data vector (a, b): c i =(a, b)(p i,q i ) T = p i a + q i b (p i,q i are coefficient of a linear combination of a, b), i.e., (p i,q i ) T determine a vector c i, i. Hence c i and c j are linearly independent, i j. Let C = {c 1,...,c m }. Eentially, gen(m) generate code for m linear tranformation. For example, gen(5) could be {(1, 0) T, (0, 1) T, (1, 1) T, (2, 1) T, (1, 2) T }, then the correponding C would be {a, b, a + b, 2a + b, a +2b}. Any two element from C are linearly independent and therefore, a and b can be obtained from them. It follow that a leaf receiver i able to get both a and b a long a it receive any two ditinct element from the et C. We aign only one linear tranformation to every intermediate node u, ou end out the ame data over all it outgoing edge. u ha 1 incoming edge: the identity tranformation i aigned, i.e., data on the incoming edge i forwarded, with no encoding, on all the outgoing edge; u ha 2 incoming edge: a tranformation vector v u = (v 1,v 2 ) T i aigned by the algorithm, o if x i received on one incoming edge and y on the other, then (x, y)(v 1,v 2 ) T = v 1 x+v 2 y i ent on all of u outgoing edge. We will now decribe how the v u are obtained for thoe u with indegree 2. The ditributed algorithm ha two phae, AignCode and DieminateCode. Due to the pecial topology of the 2-redundant multicat graph, we can aume that the ource i multicating a 2-dimenional data vector (a, b) to every leaf receiver. Every node u A I will determine a vector w u =(p u,q u ) T uch that the data ent on it outgoing edge i (a, b)(p u,q u ) T. Thi way, a node u with indegree 2 can obtain thee from it two parent node and together with it own w u, it can eaily obtain v u, a will be hown later. In the AignCode phae, firt multicat a meage through the rudimentary tree to initiate the AignCode phae. If an intermediate node u ha 2 incoming edge, then it end a meage containing it addre to requeting a code. When enough time ha paed for all the node to have a chance to end requet, uppoe m requet were received and ha j children, then generate m+j linear code uing gen(m+j) and end to each requeting node one of the firt m code. Thi vector will be the w u vector of u. The peudocode i given in Table IV. The w i for node i with indegree 1 are determined in the DieminateCode phae (ummarized in Table V), followed by obtaining v u for every u, The lat j vector generated by gen(m + j) are ent by to it children, one to each child. A child i of aign the received vector to it w i. Each node u with one incoming edge imply et it w u to the vector received and forward it on all it outgoing edge. (Alo, u ha the identity tranformation.) Each node u with two incoming edge already ha w u = (p u,q u ) T from the AignCode phae and pae it onto it outgoing edge. Node u alo receive (p 1,q 1 ), (p 2,q 2 ) on it incoming edge, repectively. Now u need to determine v u =(v 1,v 2 ) T. Let α, β denote the data received on the two incoming edge, repectively, then (α, β)(v 1,v 2 ) T i the data u end out on it outgoing edge. We know u hould end out (a, b)(p u,q u ) T, but we alo know ( ) p1 p ( a b ) 2 q 1 q 2 =( α β ) We have ( a b ) ( ) ( ) pu p1 p 1 ( ) qu =( α β ) 2 pu q 1 q 2 qu So the product of the matrix and vector on the right of (α, β) i the vector v u. Thi i correct only if the matrix in the equation i invertible, i.e., (p 1,q 1 ) and (p 2,q 2 ) are linearly independent. Two edge carry the ame (p, q) only when they come out of the ame intermediate node. But ince the contruction algorithm enure that the path containing one uptream node of u do not interect with any path containing the other uptream node of u, except at. Therefore (p 1,q 1 ) and (p 2,q 2 ) are linearly independent. The ame logic applie to the leaf receiver, o the data on one incoming edge and the data on the other incoming edge are linearly independent. Thi, in fact, prove the correctne of the algorithm. Encoding at any intermediate node u i completely defined by the tranformation vector v u if u ha indegree 2 and no encoding i done at intermediate node with indegree 1. Decoding at the leaf receiver i imple, becaue in the DieminateCode phae, each leaf receiver get the code from it two uptream node and can ue thee to decode the data they receive. It only remain to find gen(m). We define gen(m) to be a et uch that every (p, q) in the et i ditinct and p, q are two prime number with p q. It i clear that any two vector are linearly independent, becaue they are only linearly dependent if one i a multiple of the other, which i impoible when they are not equal and are vector of prime number. A function for generating prime in increaing order tarting from 2 i ued. It i traightforward to code uch a function or find an exiting efficient function. We give the algorithm in Table III. Complexity analyi of linear code algorithm In the AignCode phae, one initiate-eion meage i multicat through a tree, i.e., O(n) tranmiion of the meage occur, where n i the number of node; then O(m) meage are unicat between the ource and intermediate node, where m i the number of intermediate node. The DieminateCode phae involve only each node ending one