Multicast Flow Aggregation in IP over Optical Networks

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 5, JUNE 2007 1011 Multicast Flow Aggregation in IP over Optical Networks Yi Zhu, Student Member, IEEE, Yaohui Jin, Member, IEEE, Weiqiang Sun, Member, IEEE, Wei Guo, Weisheng Hu, Member, IEEE, Wen-De Zhong, Senior Member, IEEE, and Min-You Wu, Senior Member, IEEE Abstract— It is widely believed that IP over optical networks will be a major component of the next generation Internet. However, it is not efficient to map a single multicast IP flow into one light-tree, since the bandwidth of an IP flow required is usually much less than that of a light-tree. In this paper, we study the problem of multicast flow aggregation (MFA) in the IP over optical two-layered networks under the overlay model, which can be defined as follows: given a set of head ends (i.e. optical multicasting sources), each of which can provide a set of contents (i.e. multicast IP flows) with different required transmission bandwidth, and a set of requested content at the access routers (i.e. optical multicasting destinations), find a set of light-trees as well as the optimal aggregation of multicast IP flows in each light-tree. We model MFA by a tri-partite graph with multiple criteria and show that the problem is NP-complete. Optimal solutions are designed by exploiting MFA to formulate an integer linear programming (ILP), with two parameters: the multicast receiving index α and the redundant transmitting index β. We also propose a heuristic algorithm. Finally, we compare the performance of MFA for different combination of α and β via experiments and show our heuristic algorithm is effective for large-scale network in numerical results. Index Terms— Multicast flow aggregation, IP over optical network, overlay model, tri-partite graph, integer linear programming (ILP), heuristics I. I NTRODUCTION M ULTICASTING technology [1] has become increasingly important since many new applications such as content delivery, IP-TV, video conferences, and multipleplayer gaming require the transmission of real-time multimedia from one source to many destinations. However, multicasting in IP layer [2] is not widely deployed in today’s Internet, partially because there are still many technical issues such as scalability, reliability, security, and QoS control in Manuscript received April 15, 2006; revised December 3, 2006. This work was supported by Natural Science Foundation of China under Contract 60502004, 60602010, 60632010, and in part by Chinese 863 Program. Portions of this paper have appeared in IEEE GLOBECOM’05, St. Louis, MO, Nov. 2005 Yi Zhu was with Shanghai Jiao Tong University. He is now with the University of Texas at Dalls, Richardson, TX 75083 USA (e-mail: yxz053100@utdalls.edu). Yaohui Jin, Weiqiang Sun, Wei Guo, and Weisheng Hu are with State Key Lab of Advanced Optical Communication System and Network, Shanghai Jiao Tong University, China (e-mail: {jinyh, wguo,sunwq, wshu}@sjtu.edu.cn). Wen-De Zhong is with Network Technology Research Centre, School of EEE, Nanyang Technological University, Singapore (e-mail: ewdzhong@ntu.edu.sg). Min-You Wu is with Computer Science Department, Shanghai Jiao Tong University, China (e-mail: mwu@sjtu.edu.cn). Digital Object Identifier 10.1109/JSAC.2007.070613. current IP multicasting [3]–[5]. Recently, optical layer multicasting by using light-trees has been proposed to support point-to-multipoint communication in wide area networks [6]. The use of light-tree can better support bandwidth-intensive applications with guaranteed QoS, but there are also many challenging issues in optical layer multicasting [7]. In the data plane, work has focused on the design of multicast-capable optical crossconnects (MC-OXC) [8], [9]. In the network area, a lot of efficient algorithms have been developed for multicast routing and wavelength assignment [10], [11]. The extension to generalized multi-protocol label switching (GMPLS) has been presented for dynamic control of point-to-multipoint connections in optical networks [12]–[14]. Another critical and important issue is how to interwork the optical multicasting and the existing IP multicasting. In our previous efforts, we have proposed and demonstrated a new multicast-IP over light-tree network model [14]–[16]. The key idea is that a new optical network, with the capability of providing dynamic point-to-multipoint connections, replaces the conventional IP multicasting network in the core, while the edge remains an IP multicasting network. This hierarchical multicast architecture offers several attractive features. Firstly, it is compatible with the existing IP multicasting architecture since it is still IP based at the network edge. Therefore it is not necessary to change or upgrade existing client-side equipment. Secondly, it can support large-scale multicasting because the aggregation of IP sessions significantly reduces the burden of group management and multicast routing protocols. Thirdly, light-trees can provide better survivability for their inside aggregated IP flows [17], [18]. Finally, it can provide multicasting with improved QoS due to the fact that optical point-to-multipoint connections in the core network are circuit-switched with negligible delay and jitter. Generally, the bandwidth of a light-tree is much higher than that required by many typical applications. It is not efficient to map a single multicast IP flow into one lighttree. In this paper, we study the problem of IP multicast flow aggregation (MFA) in the multicast capable optical networks, which can be defined as follows: given a set of head ends (i.e. optical multicasting sources), each of which can provide a set of contents (i.e. multicast IP flows) with different required transmission bandwidth, and a set of requested contents at the access routers (i.e. optical multicasting destinations), find a set of light-trees to accommodate the optimally aggregated multicast IP flows. We would like to point out that it is more consistent with the practical scenarios that the same content can be redundantly distributed in several head ends. Therefore, c 2007 IEEE 0733-8716/07/$25.00
1012 Fig. 1. IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 5, JUNE 2007 Multicast flow aggregation in IP over optical two-layered network the source of IP multicast flow is not given before optimal aggregation in the MFA problem, which is different from other work on aggregated multicasting and multicast grooming in the literatures. We model the problem with a tri-partite graph, and prove the NP-Completeness. We then develop an integer linear programming (ILP) for the MFA problem. We also propose a heuristic named least trees first (LTF) algorithm to solve the problem for large-scale networks with hundreds of contents. Numerical and simulation experiments are carried out to verify the effectiveness of our proposed model. The rest of the paper is organized as follows. In section II, we introduce the concept of MFA and discuss its implementation in IP over optical networks. Then we present a tri-partite graph for the MFA problem and provide an overview of the related work. In section III, we develop the ILP for the MFA problem. In section IV, we propose a heuristic algorithm for the MFA problem. In section V, we present a few experimental results. Finally, section VI concludes the paper. II. M ULTICAST F LOW AGGREGATION A. Network model Fig. 1 shows the network model considered in this paper. The head ends are multicasting sources which encode multimedia contents with multicast IP flows and then send such flows to the core network. The required bandwidth of a single multicast IP flow may vary, depending on its image quality and compression standard. For example, by using MPEG-2 technology, standard definition TV (SDTV) and high definition TV (HDTV) require nearly 6 Mb/s and 25 Mb/s transmission bandwidth respectively [19]. Note that the same content may be redundantly provided by several head ends in the network considered in this study, similar to the scenario of today’s TV delivery networks where one channel may be distributed by several stations. At the edges of the core optical network, there are two kinds of aggregation routers. At the head end side, the routers aggregate several multicast IP flows into one light-tree. At the access network side, the routers aggregate a large number of end users’ requests originated at one residential area. After receiving the multicast contents via a light tree, each access network delivers the multicast contents through IP multicasting to the end users. The thin dashed lines in each access network indicate IP multicast flows, where the numbers on each dashed line denote the contents being delivered to a particular end user. In the rest of this paper, for simplicity, we refer to the aggregation routers in the head ends and in the access networks as the heads and the tails respectively. There may be more than one transponders attached to the core optical network at one aggregation router. The transponders in the heads and the tails are the roots and the leaves of the light trees. The bit rate of one transponder is usually much higher than the required bandwidth of a single multicast IP flow. For example, the typical bit rates of Ethernet based interfaces are 100 Mb/s, 1 Gb/s and 10 Gb/s, while that of SONET/SDH based interfaces are 155 Mb/s, 622 Mb/s, 2.5 Gb/s and 10 Gb/s. Therefore, several multicast IP flows can be aggregated into one light-tree. In the network of figure 1, the bit rate of one transponder is assumed to be 3 units of bandwidth. The value in the parentheses following the content number denotes the required bandwidth for that content. If we employ the IP multicasting scheme only, each content corresponds to at least one IP multicast flow. However, if we use the IP over optical two-layer network model described above, only 3 light-trees are needed for 5 multicast IP flows in optical core network, which significantly reduces the number of multicasting trees. B. Implementation In our network model, we require that: 1) the light-tree is constructed by using the shortest path tree, which means that every content is transmitted from head to one tail in the lighttree along the shortest path. 2) Both multicast IP flows and light-trees are one-to-many unidirectional connections. 3) A multicast IP flow cannot be split into two or more light-trees originated at a single head. 4) The transmitting and receiving transponders of a light-tree must be the same type with the same bit rate. 5) The intermediate OXC do not have grooming capability so that the maximum bandwidth of a light-tree is the bit rate of transponders at the aggregation routers. Fig. 2 shows the implementation of multicasting flow aggregation in the “3TNet” project funded by Chinese “863” program, which employs IP over automatically switched optical networks (ASON) over dense wavelength division multiplexing (DWDM) 3-layer architecture. A field-trial has been carried out in Yangtze River Delta [20]. The interconnection of control plane between IP and ASON is based on the overlay model [21]. In the core optical networks, the OXC’s in the data plane and GMPLS as well as user network interface (UNI) in the control plane have been extended to support pointto-multipoint (P2MP) connections [14], [15]. In the access networks, a tail control unit (TCU) collects the user requests information from each router via simple network management protocol (SNMP) and reports to a central scheduler. The scheduler calculates the optimal aggregation based on the content distribution and the user requests, and then sends the aggregation schemes to the corresponding head control units (HCU). The HCU is responsible for creating light- ZHU et al.: MULTICAST FLOW AGGREGATION IN IP OVER OPTICAL NETWORKS 1013 TABLE II T HE RESOURCE UTILIZATION OF LIGHT- TREES IN F IG . 1 Light-tree Bandwidth source destinations 1 2 3 Sum 3 3 3 x1 x2 x2 z1 , z2 , z3 z2 , z3 z3 Resource utilization 12 9 3 24 Fig. 2. Implementation of Multicast Flow Aggregation in IP over Optical networks TABLE I T HE RESOURCE UTILIZATION OF MULTICAST IP FLOWS IN F IG . 1 Multicast IP flow y1 y2 y3 y4 y5 Sum Bandwidth Source Destinations 1 1 2 2 2 x1 x2 x2 x1 x2 z1 , z3 z2 z3 z1 , z2 , z3 z3 Resource utilization 3 2 2 8 2 17 tree in optical networks via UNI signaling and configuring aggregation in the routers of head ends via SNMP. C. Resource overhead Although the MFA strategy provides many benefits for large-scale streaming media delivery, it may lead to extra resource overhead. We define the resource utilization of a tree is equal to the product of its bandwidth and all the links along the tree. For the case without MFA strategy, we consider that the core network is based on IP routers with the same topology as the optical network and the IP flows are transmitted along the same shortest path as the light-tree. For example, Tables I and II show the resource utilization of multicast IP flows and light-trees respectively in network of figure 1. 5 multicast IP flows would consume 17 units of bandwidth while 3 light-trees will do 24 units. The resource overhead of MFA is 7 units of bandwidth. This overhead comes from mismatch both in the heads and in the tails. In the heads, if the total bandwidth of aggregated IP flows is less than the transponder bandwidth, it will lead to head wastage, e.g. light-tree 3 wastes 1 unit in head x2 . In the tails, if one access router receives unwanted contents, it will lead to tail wastage, e.g. tails z2 and z3 received unwanted contents y3 and y2 respectively in light-tree 2. The tail wastage is also called leaky match in other literatures [22] because the multicast IP flow cannot be perfectly matched into the aggregated tree. The objective of optimal MFA problem is to minimize the overall resource overhead. D. Tri-partite graph formulation The MFA problem can be abstracted as a tri-partite graph G(X ∪ Y ∪ Z, EH ∪ ET ∪ ER ), which is composed of three bipartite sub-graphs GH (X ∪ Y, EH ), GT (Y ∪ Z, ET ), and GR (X ∪ Z, ER ), where the vertices X, Y and Z are the sets of heads, contents and tails, respectively, and the edges EH , Fig. 3. Tri-partite graph for multicasting flow aggregation ET and ER are the relationship among them. As an example, figure 3 shows the tri-partite graph representation of figure 1. In order to illustrate the graph on the plane more clearly, we repeat the set X with dashed line on the right side of Z to show the subgragh GR . For simplicity, we consider homogenous transponders whose bandwidth is t. The weight bm on the vertex y is its required bandwidth. The weight cx,z on ER represents the length of the shortest path from head x to tail z. Note that here we just use the length of the shortest path instead of network topology for the core optical network because our implementation is the overlay IPover-Optical model, in which the detailed network topology information of optical layer should not be exposed to upper IP layer [21]. Such linear approximation partially reflects to the network routing due to the following reasons: 1) we do not know which head will be the root and which tails are the leaves before aggregating IP flows to light-tree; 2) given a light-tree, the sum of the shortest-path length from its root to all the leaves is the upper bound of its total links since the shortest-path light-tree may share some of links of the shortest paths. For example in figure 1, light-tree 1 consumes 4 links in the optical networks, while the sum of the shortest-path length is 5 links because link A-D is shared by z1 and z2 . Let P ={P1 , P2 , . . . , Pk , . . .}, where Pk is a bipartite graph with {xk ∈ X, Zk ⊆ Z}, representing a root-leaves set of one light-tree, and Q = {Q1 , Q2 , . . . , Qk , . . .}, where Qk ⊆ Y represents an aggregation group of contents in a light-tree Pk . Three complete bipartite graphs GHk with {xk , Qk }, GT k with {Qk , Zk } and GRk with {xk , Zk } are induced from Pk and Qk as shown in figure 4. We define the edge difference for the combination Pk and Qk . d(Qk ) = |Qk | · |Zk | − |E(GT ∩ GT k )| (1) 1014 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 5, JUNE 2007 b) c) (a) GH1 , GT 1 and GR1 induced from P1 , Q1 means all elements in Qk must connect to the same node xk in GH ; the neighbor set of Qk must be equal to Zk in GT , which means that the contents in Qk are the aggregated multicast flows that tails Zk intend to receive;
ym ∈Qk bm ≤ t, which means the total bandwidth of aggregated IP flows in one light-tree must be not greater than the effective bandwidth of a transponder. Definition 1: The multicast flow aggregation (MFA) problem can be stated as follows: Given a tri-partite graph G(X ∪ Y ∪ Z, EH ∪ ET ∪ ER ) and t Find an optimal combination of P and Q satisfies with the above constrains such that: 

 d) The total head wastages k t − ym ∈Qk bm must be as small as possible;
e) The total edge difference k d(Qk )must be as small as possible; f) The
total length of light-trees
c ) must be as small as ( x∈Pk ,z∈Zk x,z k possible. In appendix, we prove that the MFA problem is NP-complete. E. Related work (b) GH2 , GT 2 and GR2 induced from P2 , Q2 (c) GH2 , GT 2 and GR2 induced from P2 , Q2 Fig. 4. Induced graph from P and Q If d(Qk ) = 0, it is called perfect match; otherwise it is leaky match indicated by dashed lines in figure 4, which corresponds to the unwanted contents in the tails. P and Q satisfy the following constraints: a) in GH , ∃xk ∈ X, ∀y ∈ Qk , (x, y) ∈ EH , which In IP networking research area, the MFA problem is related to aggregated multicast (AM) model [22]–[24]. The motivation of AM is to solve scalability and reliability issues in traditional IP multicast. The key idea is that, multiple IP multicast sessions is “forced” to share a single distribution tree in the core network so as to reduce the number of multicast states. Compared to the AM problem, the MFA problem considers the bandwidth constraint of a light-tree as well as the redundant distribution of contents in different head ends. Furthermore, instead of a dynamic scenario in the AM problem, the MFA problem is a static optimization problem that can help us to find a theoretical lower bound. Recently, the multicast grooming (MG) problem has begun to attract attention in WDM networks, which is defined as follows: given a set of multicast sessions with various capacity requirements, satisfy all of the multicast sessions, and minimize the network cost at the same time [25]. To support grooming of multicast traffic in an optical network, the switch architecture must be enhanced with a grooming fabric [6]. The related research works of the MG problem can be categorized into: static optimization [26]–[29] and dynamic scenario [30], [31]. We note that the MG problem is different from the MFA problem in the following aspects: 1) the MG problem may have multi-hops in the optical network while the light-tree is transparent in the MFA network model; 2) the MG problem does not have tail wastage issue since multicast IP flows can be groomed at the intermediate nodes in the core network; 3) the roots of multicast sessions are given in the MG problem while they are not given before aggregation in the MFA problem since the contents are redundantly distributed in the different head ends. ZHU et al.: MULTICAST FLOW AGGREGATION IN IP OVER OPTICAL NETWORKS N = |X| M = |Y | I = |Z| S := [sn,m ]N×M R := [ri,m ]I×M C := [cn,i ]N×I B := [bm ]M t K W V the total number of heads the total number of contents the total number of tails the adjacency matrix represent GH (X ∪Y, EH ) the adjacency matrix represent GT (Y ∪Z, ET ) the shortest path matrix, whose element cn,i represents the length of the shortest path from head n to tail i the bandwidth vector whose element bm is the required bandwidth of content m the bandwidth of a transponder the maximum possible number of light-trees to be set up P the upper bound for k d(Qk ) A very large integer number, is greater than max (N , M , I) H := [hn,k ]N,K F := [fk,m ]K,M L := [ln,i,k,m ]N,I,K,M A := [an,i,k ]N,I,K the matrix represents the relationship between the head and the light-trees, whose element hn,k is 1 if head n is the root of light-tree k, otherwise 0; the matrix denotes the relationship between the multicasting contents and the light-trees, whose element fk,m is 1 if content m is aggregated in light-tree k, otherwise 0; the matrix represents the relationship between the set P, Q and the light-trees, whose element ln,i,k,m is 1 if content m is aggregated in light-tree k which originates at head n and terminates at tail i, otherwise 0; the matrix indicates the relationship between the set P and the light-trees, whose element an,i,k is 1 if the light-tree k originates at head n and terminates at tail i, otherwise 0. III. ILP F ORMULATION In this section, we present the ILP formulation for the MFA problem. First, we define some notations and variables. • Input parameters: The multicast receiving index α is defined in equation (2), which measures the multicasting degree of the receiving matrix:  RB/B−1 I > 1, I−1 (2) α= 1 I = 1.
I
M
M where RB = i=1 m=1 ri,m bm and B = m=1 bm . α varies from 0 to 1. If α is 0, the light-tree is a unicast one, while if it equals to 1, the light-tree is a broadcast one. The redundant transmitting index β is defined in equation (3), which measures the redundancy of transmitting matrix:  SB/B−1 N > 1, N −1 (3) β= 1 N = 1.
M
where SB = N n=1 m=1 sn,m bm . β varies from 0 to 1. • Variables of the ILP: A. Constraints We now discuss constrains for the MFA problem. • Head constraints

k i ln,i,k,m ≤ sn,m (4) V It guarantees the constraint a) in the tri-partite graph model. • Tail constraints 1015  ln,i,k,m ≥ ri.m (5) n k

i ln,i,k,m ≤ fk,m V  ln,i,k,m ≥ fk,m n n (6) (7) i fk,m1 − fk,m2 + 1 (8) V Equation (5) ensures that the results will not contradict to the constraints b) given in tri-partite graph modeling. Equations (6)-(8) states how to generate GT k . Equation (6) ensures that fk,m = 1 if at least one tail receives content m through the light-tree k. If there is no tail receiving content m through such light-tree, Equation ( 7) ensures that such content is not aggregated in light-tree k. We now explain equaiton (8) in detail with the following three cases: 1) fk,m1 = fk,m2 = 0: in this case, neither m1 nor m2 is aggregated into light-tree k. We will find the left hand side of Equation (8) is 1, and the right hand side is 1/V , so the constraints are always satisfied. 2) fk,m1 = fk,m2 = 1: in this case, both m1 and m2 are aggregated into the light-tree k. The constraint given by Equation (8) guarantee that if tail i receives one content, it should receives the other; otherwise tail i receives neither of them. 3) fk,m1 = 1, fk,m2 = 0 or fk,m1 = 0, fk,m2 = 1: in this case, only one content is aggregated into the lighttree. m1 and m2 can be chosen from 1 to M arbitrarily. Consider that m1 is aggregated into the tree and then exchange the position of m1 and m2 that is just the second situation in this case. If we change the position of m1 and m2 , i.e., only m2 is aggregated, now ln,i,k,m2 can take either 0 or 1 while ln,i,k,m1 = 0. In order to keep the unequal relationship, we add 1 and 1/V to the left hand side and the right hand side, respectively. • Bandwidth constraints  ln,i,k,m bm ≤ t (9) ln,i,k,m1 − ln,i,k,m2 + 1 ≥ m It guarantees the constraint c) in the tri-partroot-leaves setite graph model. • Tree constraints
m ln,i,k,m ≤ an,i,k (10) V  ln,i,k,m ≥ an,i,k (11) m
i an,i,k ≤ hn,k (12) an,i,k ≥ hn,k (13) V  i  hn,k ≤ 1 (14) n As a tree, we have to consider some constraints to guarantee the content-tree relationship, leaf-tree relationship and roottree relationship, respectively. 1016 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 5, JUNE 2007 1) Content-tree relationship: The contents must be aggregated into the corresponding existing trees. Equation (10) ensures that an,i,k should be 1 if there exists one content to deliver. If no content will be delivered from the heads to the tails through the tree, Equation (11) ensures that the tree will not be set up. 2) Leaf-tree relationship: The tail must be a leaf of the corresponding existing tree. Equation (12) guarantees that when there exists a leaf of the tree that is an,i,k > 1, the head should set up such tree hn,k = 1. On the other hand, if hn,k = 0, no tail should add to the light-tree k since head n will not set up the tree. This is guaranteed by Equation (13). 3) Root-tree relationship: One tree must only have a single root since we consider a point-to-multipoint unidirectional connection. Equation (14) meets such condition by searching all the possible heads. B. Objective of ILP formulation We can use the unified criteria of bandwidth to formulate objectives (d), (e) and (f) The head wastage is   hn,k − fk,m bm (15) Wh = t n k k m The tail wastage is        Wt = ln,i,k,m − ri,m × bm i m k The total path bandwidth for all the light-trees is  cn,i × an,i,k Wr = t n (16) n i (17) k There are many approaches [32] to solving multi-criteria problem and here we use an easy one - weighted sum model. The overall objective of the MFA problem is to minimize α1 × Wh + α2 × Wt + α3 × Wr (18) As we discussed before, the overall objective of the MFA problem has to reflect minimization of the resource overhead. However, the topology information of optical network is not known by upper layer under the overlay model. So the objective function 18 should be a linear approximation of the resource overhead. We consider the tri-partite graph. When GH and GR are dense, many choices for Pk should be considered for Qk . In this case, Wr will affect the objective more than the other two. When GT is sparse, Wt will affect the objective more than the other two. As for Wh , it is just related to Qk . when α is larger and βis smaller, Wh will dominate the objective. The following assignments capture such ideas: α3 α2 α1 =β =1−α =1+α−β IV. A H EURISTIC A PPROACH While the ILP formulation is useful in providing insights into the nature of the problem, it may be hard to solve for large networks with hundreds of contents because of the NPcompleteness of the original problem. In this section, we propose a heuristic algorithm, named least trees first (LTF), for large-scale problem. The heuristic has three phases. Phase 1 chooses the first content for the empty new lighttree, and we use three rules for preferable content selection: • Rule 1. Least number of heads contained first, which means that we choose the content that can be sent from the least head ends. • Rule 2. Least number of tails received first, which means that we choose the content that minimum tails required to receive. • Rule 3. Shortest path first, which means that we calculate the shortest path for all possible content sending from one head to all requested tails and then choose the shortest one. Algorithm 1 shows how we use these three rules. It ensures that we choose the content most likely to aggregate with other contents, that is to say, phase 1 meets the demand of first criterion (d). Phase 2 finds the candidate content to be aggregated into the tree. Note that phase 2 will do recursively. Algorithm 2 shows how we find such candidate. In every step, algorithm 2 chooses the content that minimizes the edge difference, so phase 2 guarantees the second criterion (e). Phase 3 optimally aggregates candidate content into the light-tree. There are two cases for this aggregation: • Case 1 (total aggregation): if d(Qk ) is smaller than |Qk |· |Zk |, then we just aggregate this contents. Otherwise, • Case 2 (partial aggregation): in this case, we will recursively remove the tail z in Zy which does not receive Qk . Algorithm 3 will give more details about aggregation with these two cases. We consider the routing has the effect on our problem, so when we decrease the number of tails which only receive content yn , we should also know that those tails will receive yn from some other trees. Therefore, Phase 3 guarantees the third criterion (f). A. LTF algorithm Based on the three phases discussed above, we can get our LTF algorithm. Before giving the algorithm, it is worth to pointing out why we name this algorithm least trees first (LTF). Note that this algorithm tries to combine more contents into one tree. By doing so, the head wastage will be least. That is to say, the total number of light-trees finally set up in the network will be least. Now we propose the whole LTF algorithm as follows. B. Complexity of the heuristic For step one, we need O(|Y ||X|) time to check Rule 1, O(|Y ||Z|) time to check Rule 2 and O(|X||Y ||Z|) time to check Rule 3, so at the worst case, we need O(|X||Y ||Z|) time for step 1. As for the second step, we need O(|Y ||Z|) ZHU et al.: MULTICAST FLOW AGGREGATION IN IP OVER OPTICAL NETWORKS Algorithm 1 Find the first content for the light-tree k Input: a tri-partite graph G(X ∪ Y ∪ Z, EH ∪ ET ∪ ER ) that are represent by S, R and C. All reminder content set A, Pk = ∅, Qk = ∅. Output: the combination of Pk and Qk where there is one element in Qk . Begin for each content is in set A do Calculate the number of heads contains such content in S end for Choose the content with minimum number of heads to join the set Qk if (|Qk | = 1) then Call this content ym , Qk = {ym }, Zk contains all tails which receive Qk , xk is the head which contains Qk and have total shortest paths form xk to Zk else for (contents in Qk ) do Calculate the number of tails receive such content end for end if Drop all contents from Qk and choose the content with minimum number of tails to join the set Qk if (|Qk | = 1) then Call this content ym , Qk = {ym }, Zk contains all tails which receive Qk , xk is the head which contains Qk and have total shortest paths form xk to Zk else for (contents in Qk ) do Calculate shortest path for transmitting such content from one head to all tails request it end for Choose one content with minimal value from the set Qk and drop the others, call this content ym . Zk contains all tails which receive Qk , xk is the head which contains Qk and have total shortest paths form xk to Zk end if A = A − Qk End time. It will take O(|Z|) time when we consider step three. When we consider the complexity of the total algorithm, there are two circles in this algorithm. So the total complexity is O(|Y |(|X||Y ||Z| + |Y |(|Y ||Z| + |Z|))) that is equal to O(|X||Z||Y |2 + |Z||Y |3 ). V. N UMERICAL R ESULTS AND D ISCUSSION In this section, we present some numerical examples to show that our ILP model can solve the small-scale problem very well while our heuristic algorithm can achieve good performance dealing with large-scale network with hundreds of contents. The ILP model was solved using CPLEX 7.0 [33]. A. Effects of the transponder bandwidth We first examine the effects of the different transponder bandwidth t. Let N = 3, M = 16 and I = 5, and matrices 1017 Algorithm 2 Find the Candidate Content for the Light-tree k Input: a tri-partite graph G(X ∪ Y ∪ Z, EH ∪ ET ∪ ER ) that are represent by S, R and C, bandwidth vector B and effective bandwidth of the transponder t. The reminder content set A, Qk and Pk Output: candidate content yn , Zy , and d(Qk ∪ yn ) Begin Yc = ∅ for all
y is in set A do if ym ∈Qk bm + by ≤ t and y ∈ Xk then Yc = Yc ∪ {y} end if end for for all y is in set Yc do Calculate d(Qk ∪ y) = (|Qc | + 1) × |Zk ∪ Zy | − |E(GT ∩ (GTk ) ∪ Gy )|, Gy is the subgraph of GT with all edges from y to Zy end for Choose one content with the minimal d(Qk ∪ y), called as yn , as the candidate content. End TABLE III E FFECTS OF THE TRANSPONDER BANDWIDTH t(Mb/s) 100 155 622 1000 KO 7 5 5 5 RH 0.69 0.78 0.93 0.95 Wt (Mb/s) 100 180 180 180 S, R, B and C are given as follows: ⎡ 0 1 0 0 0 1 1 1 1 0 0 S=⎣ 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 1 1 ⎡ 0 1 0 0 0 1 1 1 1 1 1 ⎢ 1 0 0 0 1 0 1 0 1 0 0 ⎢ R=⎢ ⎢ 0 1 1 1 1 0 0 0 1 0 0 ⎣ 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1  B = 6 25 6 6 6 25 25 6 25 25 6 ⎡ ⎤ 2 2 4 4 4 C=⎣ 5 3 3 3 2 ⎦ 1 4 2 3 2 PB 5.75 8.0 8.0 8.0 ⎤ 1 0 1 1 0 0 1 0 0 0⎦ 0 0 0 0 1 ⎤ 1 1 1 1 1 1 1 0 0 0⎥ ⎥ 0 1 0 0 0⎥ ⎥ 1 0 1 1 1⎦ 1 1 1 0 0  6 6 25 6 6 Table III shows the results of the optimal number of lighttrees KO , the average head wastage ratio RH , the tail wastage Wt and the average sum of path bandwidth PB for all the light-trees against the transponder bandwidth t = 100Mb/s, 155Mb/s, 622Mb/s and 1000Mb/s, respectively. The average head wastage ratio is defined as the total head wastage over the product of KO and t. The average sum of path bandwidth ratio PB is defined as the total sum of path bandwidth for all light-trees over the product of KO and t. Obviously, the comparison of the average head wastage ratio and the average sum of path bandwidth ratio is more valid than that of the head wastage and the sum of path bandwidth. The results of KO are straight-forward since more multicast IP 1018 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 5, JUNE 2007 Algorithm 3 Optimally Aggregate Candidate Content into the Light-tree k Input: a tri-partite graph G(X ∪ Y ∪ Z, EH ∪ ET ∪ ER ) that are represent by S, R and C, Qk , Pk , yn , Zy and d(Qk ∪ yn ) which comes from Algorithm 2. Output: the combination of Pk and Qk Begin j=0 if (d(Qk ) < |Qk | × |Zk |) then Qk = Qk ∪ yn , Zk = Zk ∪ Zy and Pk = {xk , Zk }; A = A − yn else for every z in Zy which only receives yn do Drop z from Zy Calculate if (d(Qk )∪yn ) < |Qk ||Zk | and |Qk |+2 > cxk ,Z then Qk = Qk ∪ yn , Zk = Zk ∪ Zy and Pk = {xk , Zk } Break; else if Zy ⊆ Zk then Break; end if end if end for end if End Algorithm 4 LTF Input: a tri-partite graph G(X ∪ Y ∪ Z, EH ∪ ET ∪ ER ) that are represent by S, R and C, bandwidth vector B and effective bandwidth of the transponder t. Output: the combination of P and Q Begin A = Y, k = 1; repeat Qk = ∅; Call Algorithm 1 to find the first content for the light-tree k repeat Call Algorithm 2 to find the candidate content Call Algorithm 3 to aggregate candidate content until Yc = ∅; until A = ∅; End flows can be aggregated in one light-tree as the transponder bandwidth increases. Thus the total number of trees decreases. The head wastage ratio partially reflects that the network load decreases as the transponder bandwidth increases because the total bandwidth of the aggregated multicast IP flows increases much slower than that of the transponder. The tail wastage and the average sum of path bandwidth ratio remain unchanged when the transponder bandwidth t is greater than 155Mb/s, which shows that the aggregation of multicast IP flows for each light-tree are the same. This is because the bandwidth of the transponder is greater than the total bandwidth needed by the multicast IP flows in every head. Consequently the Fig. 5. Two specific multicasting trees from different two heads to all tails aggregation of multicast IP flows and the trees to deliver such aggregated multicast IP flows are the same. The head wastage increases with the bandwidth of the transponder. B. Verification of the heuristic In this subsection, we will check the different combination of the parameter α and β on the performance. By comparing the results obtained by the ILP model and the heuristic algorithm with the same inputs, we can verify the effectiveness of the heuristic. Let N = 2, M = 6, and I = 6. The bit rate of transponder is 155Mb/s. We consider 3 typical types of contents whose encoded bandwidth are 2, 6, 25 Mb/s respectively. The content bandwidth vector B is given as follows:   B = 2 25 6 25 25 25 Fig. 5 shows the broadcast shortest path trees from two heads to all the tails, respectively. For simplicity, the length between any two neighboring nodes is assumed to be unity. Then we can set up the shortest path matrix C as follows:   2 2 3 3 3 3 C= 5 3 3 3 2 2 We randomly generate the transmission matrix S and the receiving matrix R for different values of the parameter α and β . Table IV shows the number of light trees, the head wastage, the tail wastage obtained by the ILP model and the heuristic with the same inputs. The last columns of Table III give the objective function 18. In the table, we also compare the sum of path bandwidth for all the light-trees Wr and the actual used link bandwidth Bt of light-trees along with the given broadcast shortest path trees. From the results in Table IV gained by ILP model, we can find that as α increases from 0 to 1, the average of head wastage Wh decreases from 720 Mb/s to 150 Mb/s, the difference is nearly equal to the total bandwidth of 4 lighttrees. At the same time, the optimal number of trees in the network also decreases as the same pace as the head wastage when α increases from 0 to 1. Secondly, the average tail wastage Wt varies from 0 Mb/s at α = 1 to 184 Mb/s at α = 0.5. When α= 0, the tails receive IP multicast flows quite different from each other, so the heads should deliver contents individually to the tails, resulting in the reduction in the tail wastage. When α= 1, the tails receive almost all the multicast IP flows, so the heads should first guarantee that all required ZHU et al.: MULTICAST FLOW AGGREGATION IN IP OVER OPTICAL NETWORKS 1019 TABLE IV R ESULTS OBTAINED BY THE ILP MODEL (I) AND THE HEURISTIC ALGORITHM (H) α 0 0 0 0.5 0.5 0.5 1 1 1 Fig. 6. β 0 0.5 1 0 0.5 1 0 0.5 1 I 4 6 6 3 3 4 2 2 1 KO H 4 4 4 3 3 2 2 2 1 Wh(Mb/s) I H 512 512 822 512 822 512 357 357 357 357 512 202 202 202 202 202 47 47 Wt(Mb/s)/d(Q) I H 100/4 154/8 0/0 139/8 0/0 81/4 141/9 222/13 175/7 241/13 175/7 254/12 0/0 0/0 0/0 0/0 0/0 0/0 24-node mesh network contents are delivered to the tails by aggregating and then sending them to the tails. When α= 0.5, there is counterbalance between aggregating the contents and sending one content per tree, and consequently the average tail wastage is the largest at α= 0.5. Thirdly, we can find that, for a given α, the average Wr decreases as β increases, which partially reflects the actual resource consumed by the light-trees in optical network. As for the results obtained by the heuristic algorithm in Table III, we can find they have the similar trends on Wt and Wr comparing with that done by the ILP model. The major difference between them is the number of trees and Wh that is just because we try to combine as many contents into one tree as possible in LFT algorithm. When α is equal to 1, the ILP model will aggregate the contents as the same manner. That is why we obtain the same results at that point. C. Large-scale design using the heuristic We adopt a 24-node mesh network, shown in Fig. 6, for the network-level simulation in which the maximum hop-distance is 6. We randomly choose 5 nodes and 10 nodes as the heads and tails respectively. 500 contents will be delivered from heads to tails. Among these contents, 60% require 25Mb/s, 30% require 6Mb/s and 2Mb/s for the others. We randomly generate the transmission matrix S and the receiving matrix R for α = 0.5 and β = 0.5. For simplicity, we just assign length 1 to each link in the graph, that is to say, the shortest path is just based on the hop counts. We use two common bandwidth values 1Gb/s and 2.5Gb/s as t for the light-tree and do each experiment three times. Table V shows the results obtained by the heuristic algorithm. In the network without the MFA strategy, it requires at least 500 multicast IP sessions for 500 contents. By introducing the MFA strategy, we can find that the number Wr(Mb/s)/Bt(Mb/s) I H 2480/2170 2480/2170 2170/2170 2325/2325 2170/2170 2790/2170 6510/3565 5115/3565 4650/2945 5115/3565 2170/1860 2790/2170 5890/3100 5890/3100 5890/3100 5890/3100 2170/1550 2170/1550 Objective I 100 1446 2170 601 2769.5 2436 404 3248 2217 (Mb/s) H 154 1457.5 2871 646.5 2880 3118 404 3248 2217 of light-trees is significantly reduced. It is consistent with the conclusion drawn by ILP that the number of light-trees decreases as transponder bandwidth increases. In addition, we find that content distribution and user requests have a strong impact on the results even for the same αand β . For example, when t = 2.5Gb/s, the maximum number of tree is 17 that is almost twice of the minimum number of trees. We also compare the resource utilization of multicast IP flow without the MFA strategy and light-tree the MFA strategy. The overhead ratio is defined as the difference of resource utilization over the resource utilization of light-tree. It does not change greatly over the experiments with the same α and β . Note that, the transponder bandwidth does not affect the resource utilization of multicast IP flow. However, the resource utilization of light-tree increases as the transponder bandwidth increases. Consequently, the overhead ratio also increases. In the practical scenario, we have to choose proper transponder bandwidth to balance the number of light-trees and the average overhead ratio. VI. C ONCLUSIONS We proposed and investigated the multicast traffic aggregation (MFA) problem in IP over optical networks which is NP-complete and can be modeled by a tri-partite graph with multiple criteria. We have also developed an integer linear programming (ILP) model and a simple approach to dealing with multiple criteria. Although ILP model can do solve the smallscale problem perfectly, we propose the heuristic algorithm for the large-scale network with hundreds of contents. We have carried out numerical studies to verify the effectiveness of our model and heuristic algorithm. The experimental results show that 1) as the objective ruled, the model meets the main criteria very well at the extreme cases; 2) as the bandwidth of a transponder increases, the heads intend to aggregate more multicast IP flows into one light-tree while the tails still want to receive fewer unwanted flows, and there is a tradeoff between them; 3) heuristic algorithm gets the close results to the ILP model and can solve the large scale problem delivering 500 contents in the 24-node mesh network. In our research, we use a linear approximation to deal with multicast routing issue in optical network and a simple weighted-sum model for multiple criteria optimization since we consider our IP over Optical networks is based on the overlay model. However, more work needs to be done such as taking the network topology into consideration under the 1020 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 5, JUNE 2007 TABLE V L ARGE -S CALE NETWORK RESULTS WITH α = 0.5 AND β = 0.5 Transponder Bandwidth (Mb/s) Experiment Number of light-trees IP LT Overhead Ratio = (LT-IP) / LT Average Overhead Ratio Resource Utilization (Mb/s) 1000 1 23 199550 501700 60.2% peer model, and other approaches to solving multiple criteria optimization. ACKNOWLEDGEMENT We thank anonymous reviewers for their valuable comments and all the colleagues in the 3TNet project. Yi Zhu thanks Guobiao Zhou, Xiaodong Huang, and Jason Jue for their helpful discussion. We are grateful to Guowu Xie for help in the preparation of this manuscript. A PPENDIX The MFA problem can be divided into three sub-problems based on the distribution of the multicasting contents. • Partition distribution sub-problem (MFA-P, β = 0): In this case, one multicast content is only provided by a single head, thus the node degree of Y in GH is equal to 1; • Mirroring distribution sub-problem (MFA-M, β = 1): In this case, any multicast content are provided by all the head end, thus GH is a complete bipartite sub-graph; • Partially redundant distribution sub-problem (MFA-R, 0 < β < 1): In this case, the node degree of Y in GH is not less than 1 while GH is a proper graph of the complete bipartite graph. In order to prove the total MFA problem is NP-complete, we first prove the MFA-1, which refers to MFA problem with a single head, is NP-complete. Lemma 1: MFA-1 is NP-Completeness. Proof: Given one partition for the MFA-1 problem, we first check whether this
partition satisfies with the contraints ∀k = {1, 2, . . . , K}, ym ∈Qk bm ≤ t and d(Qk ) ≤ W . It is easily to find that the check can be done in polynomial time. We now show that the well-known BIN-PACKING problem is the special case of the MFA-1 problem. For simplicity, we set B as the all-1 vector. The decision problem for bin-packing can be stated as follows [34]: Given a set E = e1 , e2 , . . . , eM and an positive integer K and value
t, find the partition of the set E ′ = E1 , E2 , . . . , EK , such that ej ∈Ek ej ≤ t, ∀j = 1, 2, . . . , M.∀k = 1, 2, . . . , K Obviously, if we let E = Y and GT is a complete bipartite graph, bin-packing problem is the special case for MFA-1. It is worth to point that the MFA-P, the MFA-M, and the MFAR problems are degenerated into the MFA-1 problem for the single head. However, when we consider multiple heads, it is more complicated than MFA-1. Theorem 1: The MFA problem is NP-complete. 2 20 178500 463525 61.5% 58.6% 2500 3 26 212500 463700 54.2% 1 17 195500 880250 77.8% 2 9 173500 573250 69.7% 73.9% 3 14 184500 719075 74.3% Proof: In the case of the MFA-P problem, note that the content sets provided by any two heads are disjoint since β = 0, so the MFA-P problem with N heads is equivalent to N MFA-1 problems. In the case of the MFA-M problem, all contents can be provided by all heads since β = 1, MFAM with N heads is equivalent to N ! MFA-1 problems. 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Steuer, “Multiple Criteria Optimization: Theory, Computation and Application,” 1985. [33] [Online]. Available: http://www.ilog.com/products/cplex/ [34] [Online]. Available: http://www.nist.gov/dads/html/binpacking.html Yi Zhu received the B.S and M.S degree in electricity engineering from Shanghai Jiaotong University, China, in 2003 and 2006, respectively. He is currently working toward the Ph.D. degree in computer science at the University of Texas, Dallas. His research is on multicast aggregation, traffic grooming, and complexity of optical network. 1021 Yaohui Jin is a professor in the State Key Laboratory of Advanced Optical Communication Systems and Network, Shanghai Jiao Tong University, China. Prior to joining SJTU, he was a member of technical staff at Bell Labs Research China from 2000 to 2002. He served as the TPC member in many international conferences. He published more than 50 paper in technical journals and conferences. His research interests include optical networking, optical grid and switch scheduling. Weiqiang Sun is currently a Lecturer at the State Key Laboratory on Fiber-Optic Local Area Networks and Advanced Optical Communication Systems, Shanghai Jiao Tong University. His research interests include Automatically Switched Optical Networks (ASON), optical multicast and TV distribution in overlay networks. Guo Wei is an associate professor of state key lab of advanced optical communication system and network in Shanghai Jiao Tong University since 2003. Before she entered in SJTU, she was a senior engineer and a project manager of the Fiberhome Telecommunication Technologies CO., LTD from 2001-2003. She has over 50 publications published in technical journals and conferences. Her research interests include optical grid, network planning, and optimization algorithm. Weisheng Hu is the Professor and Director of the State Key Laboratory on Fiber-Optic Local Area Networks and Advanced Optical Communication Systems, Shanghai Jiao Tong University. His interests are on generalized automatic switched optical network, and optical packet switching. He is the author or co-author of over 100 journal and conference papers. Wen-De Zhong is an associate professor with School of Electrical and Electronic Engineering, Nanyang Technological University (NTU), Singapore. He received his Ph.D degree from the University of Electro-Communications, Tokyo in 1993. He has published more than 100 refereed journal and conference papers and has given several invited presentations at international conferences. He has served on organizing and/or TPC for numerous international conferences. His research interests include optical WDM systems and networks. Min-You Wu is an IBM Chair Professor in the Department of Computer Science and Engineering at Shanghai Jiao Tong University. He serves as the Chief Scientist at Grid Center of Shanghai JiaoTong University. He is a research professor of the University of New Mexico, USA. His research interests include grid computing, wireless networks, sensor networks, overlay networks, multimedia networking, parallel and distributed systems, and compilers for parallel computers. He has published over 140 journal and conference papers in the above areas.