Embedding Service Function Tree with Minimum Cost for NFV Enabled Multicast

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1 1 Embedding Service Function Tree with Minimum ot for NFV Enabled Multicat angbang Ren, Student Member, IEEE, eke Guo, Senior Member, IEEE, Yulong Shen, Member, IEEE, Guoming Tang, Member, IEEE, Xu Lin, Student Member, IEEE btract Uually, a data flow need to travere a erie of network function, which i called a ervice function chain (SF), before reaching it detination. The emergence of network function virtualization (NFV) make the embedding olution of SF flexible a far a the deployment location i concerned. When provider embed the SF into a ubtrate network, they will hope to minimize the etup cot of the SF and link connection cot toward client. For unicat, ince there i one path connecting the ource node to the detination node, all function of the SF are jut needed to be deployed along the path. However, when embedding the SF for a multicat tak, the topology of the SF may change becaue the function deployment location have impact on the traffic delivery cot. Thu, a ervice function tree (SFT) may be a better choice. Given the huge pace of SFT embedding olution, however, it i extremely hard to find the optimal one uch that the total traffic delivery cot i minimized. In thi paper, we tackle the optimal SFT embedding problem in NFV enabled multicat tak. Specifically, we formally define the problem and formulate it with an integer linear programming (ILP), which turn out to be NP-hard. Then, a two-tage method i propoed to deal with the problem with an approximation ratio of 1+ρ, where ρ i the bet approximation ratio of Steiner tree and can be a mall a 1.9. With extenive experimental evaluation, we demontrate that by applying our SFT embedding olution, the delivery cot of multicat traffic can be reduced by.05% at mot againt three benchmark. Index Term network function virtualization, ervice function chain, minimum cot, multicat I. INTROUTION Multicat i deigned to deliver the ame content from a ingle ource node to a group of detination node []. ompared with unicat, multicat can ignificantly ave the bandwidth conumption and relieve the load of the ource Manucript received May 5, 018; revied January 0, 019, accepted March 1, 019. Thi work wa upported in part by the National Natural Science Foundation of hina under Grant No.61801, No.61775, U1560, U17616, the National Key R& Program of hina No.017YF100700, the Hunan Provincial Natural Science Fund for itinguihed Young Scholar under Grant No.016JJ100 and the NUT Reearch Plan under Grant No.ZK orreponding author: eke Guo, Yulong Shen. Ren,. Guo and G. Tang are with the Science and Technology on Information Sytem Engineering Laboratory, National Univerity of efene Technology, hangha Hunan 1007, hina.. Guo i alo with the ollege of Intelligence and omputing, Tianjin Univerity, Tianjin, 0050, hina. {renbangbang11,dekeguo, gmtang.}@nudt.edu.cn. Y. Shen and X. Lin are with the School of omputer Science and Technology, Xidian Univerity, Shaanxi, hina. {ylhen@mail.xidian.edu.cn,xulin xidian@16.com.} preliminary verion of thi paper i accepted in IS 018 [1]. In thi verion, we modified and added ome content into abtract, introduction, problem formulation and experiment analye. erver a it avoid duplicated tranmiion among independent unicat path, epecially by multiplexing a hared multicat tree. However, the traditional ditributed IP network cannot upport centralized optimal multicat routing (e.g., the Steiner tree []) well. The emergence of oftware defined network (SN) change thi ituation. SN eparate the data plane and control plane of network [], [5]. Thi architecture implifie traffic engineering by uing the controller to manage the routing of flow directly. In SN, the centralized control plane can calculate the optimal multicat routing and then intall relevant routing table into network device [6]. Uually, a requet flow from the ource node to the detination node may be proceed by variou network function in a particular order, which form the ervice function chain (SF) [7], [8]. For example, in the ervice, the data flow will go through viru detection, pam identification, and phihing detection in turn [9]. Thee network function uually are proprietary hardware appliance. Though SN can help to teer traffic to go through thee boxe, bringing new ervice into today network i becoming increaingly difficult due to the proprietary nature of exiting hardware appliance. With the growth of variou new network function need, hardwarebaed appliance rapidly reach the end of life, requiring much of the deign-integrate-deploy cycle to be repeated with little or no revenue benefit. Network function virtualization (NFV) i propoed in 01 to olve the above problem along with SN [10], [11]. The emergence of NFV enable network operator to virtualize their network ervice a virtualized network function (VNF). Thee VNF ubtitute dedicated hardware middleboxe and bring high flexibility and elaticity. Thee VNF not only reduce the cot and time of deploying new network ervice but alo bring new challenge to traffic engineering. Many work invetigate NFV enabled traffic engineering for both unicat and multicat (ee the related work in Section II).. Motivation: from SF to SFT For NFV enabled unicat, there are only one ource and one detination. The SF i traightforward to deploy, e.g., by equentially chooing embedding node and deploying VNF along the unicat routing path [1]. However, embedding the SF for multicat become tricky, epecially when the cot of deployment and connection are enitive to the function location. For example, in the video treaming ervice, ISP trategically deploy network function (e.g., intruion detection, load balance, and format trancoding) among the

2 f f 15 d 1 11 E 1 d (a) Network d 1 15 f 1 (b) S-1 E 1 d f f f f f 1 d 1 11 (c) S- d f f d 1 f 1 E 1 d (d) S- (OPT) Fig. 1. Three SFT embedding trategie for the ame multicat tak in a target network, where i the ource, d 1 and d are the detination, and E are erver node. The link connection cot i labeled beide each edge and the VNF etup cot equal one. network node, a the cot of connecting the ource erver to geographically ditributed uer (travering all the network function) can be much different [1] [15]. onidering the NFV enabled multicat for uch tak, we need to embed the SF for the requet flow carefully, which lead to embedding a ervice function tree (SFT) in the multicat tree. Furthermore, with VNF that have been deployed in a target network, there are multiple choice to deploy the new VNF and contruct the SFT for a particular multicat tak. Neverthele, it i not eay to find the optimal SFT embedding olution with the leat cot under thi ituation. Fig. 1 provide a imple example for SFT embedding over a network with deployed VNF. Fig. 1(a) how i) the target network with eight node and two deployed VNF f and f, ii) the multicat tak with ource node and two detination node {d 1,d }, and iii) link connection cot labeled on the link. uming that the SF requirement for the multicat tak i ( f 1 f f ) and the etup cot of each VNF equal to one, Fig. 1(b) Fig. 1(d) give three SFT embedding olution with traffic delivery cot 1 of 6, and 19, repectively. We can ee that different SFT embedding trategie can reult in divere traffic delivery cot, and our tak i to find the optimal one (a hown by Fig. 1(d) in thi example). The detail of thi example will be further explained in Section III-.. hallenge & ontribution The example given in Fig. 1 i imple and may be eay to olve. In reality, however, the network topology and ize of multicat tak can be much larger and more complex, which lead to a huge pace of feaible SFT embedding olution. Hence, for a pecific NFV enabled multicat tak, it i a challenging problem to find the mot efficient multicat tree where the embedded SFT i proved optimal, conidering the link connection cot, VNF etup cot, and node capacitie. Indeed, even though we aume that the required SF can be deployed on one node to free the VNF order contraint, the problem i till NP-hard [16]. In thi paper, aiming to olve the optimal SFT embedding problem for NFV enabled multicat, we make the following contribution: We formally define the problem of optimal SFT embedding in NFV enabled multicat tak and formulate the 1 The traffic delivery cot of a multicat tak i computed by the um of all VNF etup cot and link connection cot over the target network [16]. f f Without order contraint Single requet (e.g., MILOM'1, GLOEOM'17 WN'1, I'15) VNF Placement Multiple requet (e.g., INFOOM'15,JS'18) Place the SF in a node (e.g., IS'17) With order contraint Multicat Unicat (e.g., loudnet'1,inp'16, INFOOM'16, SIGOMM'17) Place the SF in multiple node (e.g., IS'17, Our SFT embedding problem) Fig.. laification of the related work for VNF placement. problem by an integer linear programming (ILP), aiming to minimize the traffic delivery cot. We prove that the optimal SFT embedding problem i NP-hard. We propoe a two-tage algorithm to tackle the NP-hard problem. In the firt tage, an initial feaible olution to the ILP problem i produced by embedding an SF together with a Steiner tree. In the econd tage, the initial olution i optimized by adding new VNF intance to contruct an SFT. With ufficient node capacitie, we prove that the approximation ratio of our algorithm i 1 + ρ, where ρ i the bet approximation ratio of Steiner tree and can be a mall a 1.9 [17]. We extend our algorithm to be compatible to the common ituation where a certain number of (virtual) network function have been deployed, like ome public cloud handle bae load by phyical hardware and pillover load by virtual ervice intance [18], and our algorithm can be aware and reue thee deployed function in olving the SFT embedding problem. We invetigate the performance of our algorithm under different parameter etting. With extenive experiment, we demontrate that our SFT embedding method can reduce the total traffic delivery cot by.05% at mot compared to three benchmark. The remainder of thi paper i organized a follow. In Section II, we review the related work. We formulate the optimal SFT embedding problem by an integer linear programming in Section III. We deign a two-tage algorithm to tackle the poed problem and provide the algorithm analyi in Section IV. Section V preent performance evaluation on the propoed algorithm and enitivity analyi to the parameter etting. Section VI conclude the paper.. VNF placement II. RELTE WORK Reearche on NFV enabled multicat and relevant to thi work can be roughly grouped into two categorie: VNF placement with or without order contraint, a hown in Fig.. VNF Placement without order contraint. Thi type of work focu on placing independent VNF. Literature [19] and [0] tudied how to deploy VNF for ingle requet cae. ouet et al. propoed a heuritic algorithm to optimize the cot of VNF deployment which uually concern licene fee and reource conumption [19]. Miloud et al. tudied the optimal VNF placement with the conideration of the QoS requirement [0]. for multiple cae, Rami et al. concentrated on atifying multiple client requet that need variou

3 VNF [1]. In [1], the author invetigated the problem of finding the optimal placement of multiple independent VNF within a phyical network, by minimizing the um of the traffic delivery ditance and the VNF etup cot. Specially, [] ue conformal mapping to give an univeral placement olution conidering demand imbalance. Literature [] [5] invetigate VNF placement problem and network licing in the 5G acce network. We lit the above repreentative work to give a direct comparion to our work, and more related work are urveyed in [8]. VNF Placement with order contraint. The requirement of travering VNF in order (i.e., SF) make VNF placement problem more complicated. In [7], the SF wa embedded into the target network for the unicat tak with multiple objective uch a maximizing remaining data rate, minimizing the number of applied node and minimizing traffic delivery latency. Kuo et al. [6] tudied the joint optimization problem of VNF placement and path election with the conideration of link and erver uage. Lin et al. [7] devoted to minimizing the embedding cot of SF with parallel VNF. The dynamic SF embedding problem wa tudied in [1], [8], where the equence of function could change during the life of a eion. ll the above reearche eentially focu on SF embedding in the unicat requet. However, embedding an SF in a multicat requet i different. Xu et al. [16] devoted to finding a minimum cot peudo-multicat with an aumption that the SF i placed on one node (to neglect the order contraint). The aumption i not practical, epecially under the multi-cloud ervice function chaining architecture [9]. Kuo et al. invetigated to embedding multiple SF intance for the many-to-many flow requet, and a ervice overlay foret wa introduced in [0]. Yi et al. [1] eparated traffic forwarding graph and function delivery graph, which cannot exploit the benefit of multicat multiplexing. the mot relevant work to our, heng et al. [] formulated the SF embedding problem for multicat. They firt contruct a Steiner tree that covering all detination and find a minimum cot path connecting the ource to the Steiner tree. Then they embed the SF along the minimum cot path. However, they embed an SF rather than an SFT. In thi work, we embed an optimal SFT in the hared multicat tree that help to deliver the deired traffic flow from the ource to multiple detination with the minimum cot.. Multicat Routing under SN. Ratnaamy et al. [], [] reviited the multicat deign and gave example of multicat application, including multiplayer game, IPTV and file haring. Li and Freedman [] leveraged the unique topological tructure of datacenter network to build multicat tree. oth work above indicate that the ize of a multicat tree can be ignificantly large, while the emergence of SN technology help alleviate thi problem with convenient way to deal with multicat routing. Multicat traffic engineering ha been widely invetigated under the SN diagram. Shen et al. [6] looked into the packet lo recovery problem for reliable multicat routing and applied an approximation algorithm to minimize both tree and recovery cot. Iyer et al. [5] applied SN aided multicat trategy in data center and preented an SN baed multicat ytem named valanche that wa ued to minimize the ize of the routing tree. III. OPTIML SFT EMEING PROLEM In thi ection, we firt give an illutrative example of the SFT embedding problem for the NFV enabled multicat tak. Then we formally define and formulate the optimal SFT embedding problem, which i the focu of thi work.. Problem Illutration mentioned in Section I, Fig. 1(a) depict a network model with link connection cot labeled on each edge. Five erver node ( E) are in the network upon which VNF can be intalled, and two type of VNF ( f and f ) have been already deployed a illutrated in the network. For implicity, we do not how the witch node between erver node. Indeed, the number of witch node and other factor (e.g., link bandwidth) can be modeled in the link cot. Fig. 1(a), without lo of generality, we aume that the etup cot of VNF intance upon different node are the ame and et the value a one [0]. Here we denote the multicat tak a δ={, ˆ,l}, in which i ource, ˆ i the detination et with ˆ={d 1,d } and l repreent the SF requirement l=( f 1 f f ). For thi pecific multicat requet, Fig. 1(b), Fig. 1(c) and Fig. 1(d) provide three different olution, repectively. In Fig. 1(b), we deploy new intance of f 1, f, f on,,e, repectively, o the traffic delivery cot i =6. In Fig. 1(c), we teer flow from to detination through and. Since f and f have already been deployed on and, their etup cot i zero, and then the traffic delivery cot i =. Particularly, the olution in Fig. 1(d) build a ervice function tree by i) reuing the deployed VNF (on the branch) and ii) etablihing a new flow path (on the E branch). With the newly builted SFT, the multicat flow avoid expenive link e Ed1 and e d1 d, which further cut down the traffic delivery cot and reult in the minimum traffic delivery cot, i.e., = 19. With the example hown in Fig. 1(d), we formally define the ervice function tree (SFT) a follow: efinition 1. Service Function Tree. Given a target network with or without deployed VNF and a multicat tak, we need to intall ome new VNF intance among the network node to fulfill the SF requirement. Particularly, the goal of minimizing the total traffic delivery cot will enforce a tree tructure of ervice function embedded in the hared multicat tree, which we name ervice function tree (SFT). Note that the number of VNF i certain for an SF, while the number of VNF for an SFT i not determinitic given the target network and a multicat tak, i.e., multiple verion of SFT can fulfill the multicat tak (e.g., Fig. 1 give three different SFT). Under thi ituation, conidering the VNF etup cot, link connection cot and node capacity

4 S d Source Intermediate VNF etination Fig.. Four location of node u in contraint (). for VNF deployment, it i meaningful yet challenging to find the optimal SFT embedding olution that lead to the leat traffic delivery cot for the multicat tak.. Problem efinition Target Network. We conider a network G=(V, E) with V =V M VS, in which V M and V S denote the et of erver node and witch node, repectively. For a erver node v V M, cap(v) repreent it capacity for VNF deployment and i meaured by it reource. Thi capacity can be evaluated by the bottleneck reource, uually i PU. For an edge e uv, c uv denote the link connection cot of tranferring the flow. It i important to emphaize that c uv i the input data of our problem and can be caled down/up according to the flow ize. VNF eployment ontraint. ny VNF will have a etup cot when it i deployed upon the erver node in V M. binary number π f j u i ued to denote whether f j ha been deployed on node u; a real number γ f j u i ued to denote the etup cot of a new VNF intance f j upon node u. For thoe VNF intance that have already been deployed, we treat their etup cot a zero; a real number µ f j i applied to repreent the reource amount that deploying f j i needed. For implicity, we aume that each intance of VNF can erve traffic flow with any ize by caling vertically o that the ame VNF intance will not be deployed repeatedly due to flow plitting [6]. efinition. Multicat Tak. Given the target network G with or without deployed VNF, a multicat tak can be repreented by a three-element tuple δ =(, ˆ, l), where, ˆ, l are the ource node, detination node, and the SF requirement for the multicat flow, repectively. Specifically, the SF i denoted by l=( f 1, f,..., f n ). efinition. Optimal SFT Embedding Problem. For a multicat tak δ given by efinition, under the VNF deployment contraint, the optimal SFT embedding problem i to build an embedded ervice function tree upon the target network uch that i) the multicat requet i atified and ii) the traffic delivery cot i minimized.. Problem Formulation For eae of reference, we lit all the notation in Table I. To decribe the optimal SFT embedding problem eaily, we can treat a f 0 and d a f n+1. Then the SF requirement l can be expanded a ˆl = ( f 0, f 1,..., f n, f n+1 ). Though the multicat can ave reource by multiplexing the hared tree, the flow entering into each detination mut travere a complete SF. In Fig. 1(d), the flow entering into d 1 travere ( ) to get the ervice of ( f 1 f f ). For each detination node d, the flow entering it mut go f j TLE I SUMMRY OF NOTTIONS Notation ecription SFT ervice function tree MO the mutlilevel overlay directed network T S our two-tage algorithm ST the Steiner tree baed algorithm S the minimum et cover baed algorithm RS the randomly electing baed algorithm G graph of network topology V the et of all node in G V M the et of all erver node in G V S the et of all witch node in G E the et of all edge (link) in G u,v a node in G N u the neighbor node of u e uv an edge of G cap(v) the capacity of node v δ the multicat tak the multicat ource ˆ the et of all multicat detination d i the i th detination of multicat l the SF requirement ˆl the extended SF requirement f i the i th VNF of l γ f j u the etup cot when f j i deployed in node u c uv the link cot of edge c uv π f j u denote whether f n ha been deployed in node u µ f j the amount of reource needed to deploy f j ω f j u indicate whether or not a new VNF intance f j i deployedon node u τ d, f j,u,v denote if the edge e uv i located between f j and f j+1 for the traffic flow heading for detination d φ f j repreent whether the flow heading for detination d i du erved by VNF f j on node u ψ f j,u,v denote if edge e uv i located between f j and f j+1 through all VNF of ˆl only once. Thi characteritic mean that the VNF will never proce the flow and the phyical erver node jut forward the flow when the flow have been proceed by the ame VNF deployed on the other erver node viit the VNF again. We ue φ f j du to repreent whether the flow entering to d get the ervice of f j at the node u. Then, we have φ f j du =1, f j ˆl,d (1) u V M Remark 1. When we ay that each detination need to be erved by a SF mean that all VNF of the SF proce the flow in order before it reache to the detination. Some detination may get ervice by the ame VNF intance due to the multiplexing characteritic of the multicat. Though the ource can be een a f 0, each detination cannot get ervice of f 0 except from ource. Thu, we have φ f 0 d =1, d () Moreover, we need to enure that the new deployed VNF intance never violate the capacity contraint. We ue a binary variable ω f j u to denote whether f j i deployed in u. Then (π f j u+ω f j u)µ f j cap(u), u V M () f j ˆl The mot challenging part in our problem formulation i how to model the link connection cot. The requirement of

5 5 travering the SF in correct order may enforce ome edge be viited multiple time. However, the content of package in thi flow are different at each time viiting the ame edge. The reaon behind thi characteritic i that each VNF doe effect toward the package. For a detination d, there i a walk Walk(,d) connecting it to in the multicat routing. Thi path can be divided into n + 1 egment by ˆl. Take Fig. 1(d) for example, the path from to d can be divided into four part, i.e., {( f 0 ) ( f 1 ) ( f ) E( f ) d ( f )}. For the edge e uv, we ue a binary variable τ d, f j,u,v to denote whether it i included in the final multicat routing. If e uv lie in the egment between f j and f j+1, then τ d, f j,u,v = 1. It need to emphaize that τ d, f j,u,v i different from τ d, f j,v,u ince the flow ha direction from to d. Embedding an SFT i not only to decide on which node VNF intance hould be deployed, it hould alo find the proper routing to enure that the flow received by the detination completely goe through all the required VNF of ˆl in order. There are four type node (i.e, ource node, intermediate node, VNF node and detination node) in Walk(,d) a hown in Fig.. We ue N u to denote the neighbor node of u. onidering both the conervation of flow and the SF ordering requet, we have τ d, f j,u,v τ d, f j,v,u φ f j du φ f j+1 du, v N u v N u f j ˆl,u V,d Remark. node u ha four kind of poible poition: ource node, intermediate node, VNF node and detination node. Fig. depict thee four cae. For cae 1, f 0 denote. Given f j = f 0,u=,v=, if ha other VNF, like f 1, then v Nu τ d, f0,u,v=0 a the flow leaving from i proceed by f 1. v Nu τ d, f0,v,u=0, φ f 0 du =1 and φ f 1 du =1. Thu, we have If ha no VNF, then τ d, f0,u,v=1, v Nu τ d, f0,v,u=0, φ f 0 du =1 and φ f 1 du =0, then we have Similarly, The ituation for cae, cae and cae can alo be proved to atify contraint (). The content of multicat flow will be changed after being proceed by different VNF. Thu, the content of thi flow in different egment of Walk(,d) are different. Multicat avoid duplicated tranmiion and only copy itelf at the branch node. For different detination (e.g. {d 1,d }), the cae with τ d1, f j,u,v = τ d, f j,u,v = 1 mean that there i only one flow copy that i proceed by f j on edge e uv. Thi mean that we can merge variable τ d, f j,u,v for different detination. We ue a binary variable ψ f j,u,v to denote whether edge e uv i ued to tranfer flow that ha been proceed by f j but not f j+1. For intance, in Fig. 1(d), τ d1, f 0,,=1 and τ d, f 0,,=1, and we jut need to tranfer one flow copy, ince the flow content from i the ame. Thi multicat flow contraint can be modeled a ψ f j,u,v τ d, f j,u,v f j ˆl,u V,v V,u v,d ontraint (5) indicate that ψ f j,u,v=1 if τ d, f j,u,v=1 for any d. With the definition of ω f j u and ψ f j,u,v, we can eaily depict () (5) the total cot of tranferring uch NFV enabled multicat a follow. ω f j uγ f j u+ u V M f j ˆl f j ˆl e uv E ψ f j,u,vc uv (6) The goal of embedding optimal SFT i to minimize the cot of embedding SFT, i.e., Min (6) (7a).t. (1),(),(),()and(5) (7b) Theorem 1. The optimal SFT embedding problem formulated in (7) i NP-hard. Proof: We prove the theorem by a polynomial-time reduction from the Steiner tree problem. ume there i a graph G=(V,E) where V and E are the node et and edge et, repectively. For each edge e E, it ha an edge cot of c e. Given a ubet of the node et V, Steiner tree problem i to find a minimum cot tree OPT G panning all node in. Then, we contruct an intance of embedding SFT a follow. We firt replicate G into G, into. eide G, conider another connected graph with the node et P={p 0, p 1,...p n } and each edge in the graph ha a cot. Then we connect d j in G to p i P and aign the edge a random cot. lo, p 0 can act a the ource node, and other node in P can act a erver where VNF can be deployed. ll node in G can only act a intermediate node or detination node. We aume that ome VNF have already been deployed in P. Each node in P ha a capacity contraint and deploying new VNF intance in P ha a cot. onider a multicat tak δ=(=p 0,=,l) in G P. ume that we can find the optimal olution OPT G for SFT embedding. Since all VNF can be deployed only in P, then we can delete P and the edge between P and G. The remainder ubgraph of OPT G in G i the OPT G for G. Otherwie, OPT G i not the optimal olution. Thi mean that we can find the optimal olution for the Steiner tree problem. Neverthele, the Steiner tree problem i NP-hard []. Thu, the aumption i not true, and Theorem 1 i proved. IV. METHOOLOGY: TWO-STGE LGORITHM In thi ection, we propoe a two-tage approximation algorithm to olve the optimal SFT embedding problem. the preparation of the algorithm, we firt contruct a multilevel overlay directed (MO) network which include all information of the original network.. MO Network ontruction the preliminary, we tranform the target network to a multilevel overlay directed (MO) network, which contain all information of the original network including the link connection cot and VNF etup cot. Fig. (a) how an original network with four node. The weight attached to each edge denote the link connection cot, and the number near the node repreent the node capacity. The deployment cot

6 6 (a) Original network G with link cot and node capacity. 5 f 1 f f f 1 5 (b) Multilevel overlay directed network G Fig.. n example of MO network with l=( f 1 f f f ). of thee four function on the four node are different, and a matrix can be ued to denote them a hown in Equation (8). f 1 f f f 1 deployment cot = (8) Fig. (b) illutrate a MO network, which i tranformed from the target network coniting of node a hown in Fig. (a). The MO network can be partitioned by column and row, in which each row denote one node and each column denote one VNF. Specifically, the order of column i conitent with the order of the SF. The weight attached to each node repreent the deployment cot of the correponding VNF on the node. For intance, the weight attached to the node lie in the firt column, and the firt row repreent that the cot of deploying f 1 on i 1. ll node in the left column connect to all node in it right neighbor column with directed edge, and the cot of thee edge are the ame with the total link cot of the correponding hortet path in the original network [7]. There are three tep to get a MO network a follow. Step 1: Replicate all node of the network k time, where k denote the length of the SF. Place thee node in a matrix form, in which column denote VNF and row denote phyical erver node. lgorithm 1 The contruction of multilevel overlay directed network Input network G=(V, E, c(e)) and a ervice function chain l={ f 1, f,..., f k }. Output Multilevel overlay directed network G. 1: alculate all hortet path between each pair node in G : Replicate all V =n node k time and place thee n k node in a form of n k matrix. Each node can be denoted by v nk : for i=1; i + +; i k 1 do : onnect all node in the i th column to all node in the i+1 th column with directed edge 5: Set the edge cot between two node a the correponding hortet path cot in G 6: Set the node weight equal to the correponding VNF deployment cot in G. Step : For each node in the i th column, connect it to all the node in the i + 1 th column with directed edge. Step : Set the node weight equal to the correponding VNF etup cot. Set the edge weight equal to the total link connection cot of the correponding hortet path in G. The above three tep can tranform any target network with a certain SF requirement into a MO network. The procedure for contructing a MO network enure that the original network i a ubgraph of the overlay network, which guarantee that no information will be mied. lgorithm 1 depict the procedure in detail.. Stage One: Find a Feaible Solution In thi tage, we firt deign an algorithm for the problem under the condition that each node ha ufficient reource, and then generalize it to common ituation. To find a feaible olution to the optimal SFT embedding problem, we firt embed the SF into the tranformed MO network and then connect the lat node of the SF to all the detination. Fig. 5 how an expanded MO network baed on the network in Fig. (b). hown in the figure, all node in G are plit into two identical node connected by a virtual link with connection cot equal to the VNF etup cot. aed on the expanded MO network, the detailed operation to find a feaible olution are a follow. Step 1: dd the ource node into the expanded MO network; connect it to all node in the firt column; et each link connection cot a that of the correponding hortet path in original network G. Step : From the ource node to one node in the lat column (where the lat VNF i deployed), find a path to embed the required SF, uing the hortet (weighted) path earch algorithm (e.g., ijktra algorithm). We will prove that uch a path ha the leat traffic delivery cot. Step : For the node deployed the lat VNF, find a Steiner tree connecting it to all detination node. Thu, the Steiner tree along with the path reulting from Step yield a feaible olution for the SFT embedding problem. Theorem. In Step, for the elected node in the lat column of the expanded MO network, the (weighted) hortet path i optimal for SF embedding.

7 7 f 1 f f f Fig. 5. The expanded MO network. Proof. We aume that the elected node deploying the lat VNF i v t. Indeed, any one olution of embedding the SF in original network G can be mapped into a path in the expanded MO network. The cot of embedding an SF can be treated a the um of two part, i.e., the VNF etup cot and the link connection cot. The VNF etup cot can be mapped to the virtual edge weight, and the link connection cot between neighbor VNF can be mapped to the real edge weight between neighbor column. The condition that node have ufficient reource enure that any path from to v t can be ued to embed the SF, and the hortet path computed by our algorithm in the expanded MO network provide the optimal SF embedding. Thu, Theorem i proved. Neverthele, if the aumption that each node ha ufficient reource i not held, the path generated by Step with the leat traffic delivery cot may be infeaible in practice. The reaon i that ome node along the path may be overloaded. Therefore, for the obtained olution, we further check if there exit the overloading problem. If o, correponding node adjutment will be performed a follow. We check all the VNF of the SF in order. If one VNF ha been deployed in an overloaded node, then we hould find another node to deploy it. Without lo of generality, we aume VNF f i ha been deployed on node v j that i overloaded, and it neighbor VNF f i 1 and f i+1 have been deployed on node v k and v m, repectively. To find a new node to deploy VNF f i, we check all the other node in the ame column which v j lie. For each node with enough capacity, we compute the um of i) it link connection cot to v k, ii) it link connection cot to v m, and iii) it VNF etup cot. Thu, we chooe the node with the minimum cot a the new deployment place for f i. In Step, the node with the lat VNF make a big impact on the final olution, a the different location of the node will lead to a different Steiner tree. Therefore, we conider all the cae for the choice of the node with the lat VNF and elect the olution with the minimum cot a the output of the firt tage. Theorem. The olution reulting from tage one i a feaible olution to the SFT embedding problem. Proof. In the problem of embedding SFT for NFV enabled multicat δ =(,, l), the feaible olution require that the flow travere all VNF of l in order before reaching de- lgorithm The firt tage of our two-tage algorithm Input n overlay network G, a multicat tak δ=(, ˆ,l) and VNF deployment cot on all node. Output olution with an SF and a Steiner tree. 1: dd the ource node into G. : onnect to all node of the firt column in G and et the cot a correponding hortet path. : ivide each node in G into two part, and connect thee two part with cot γ fk u. : for each node v of the lat column in G do 5: Find the hortet path ϒ from to v in G and get [ω fn u]. 6: uild a Steiner tree to cover v and all detination. 7: for j=0; j + +; j n do 8: heck whether f j i deployed in an overloaded node, if o, find a new node with the minimum um of etup cot and connection cot to deploy f j. 9: Update [ω fn u],ϒ and get the total cot of the olution. 10: Output the olution with the minimum cot. tination. In tage one, we firt embed the SF and then connect the node deploying the lat VNF to all detination. Thi tep enure that flow i teered from the firt VNF to the lat VNF in order and then goe to the detination. Thu, each detination receive the flow proceed by the entire l, theorem i proved. Overall, the firt tage of our two-tage algorithm i hown in lgorithm.. Stage Two: Optimize the Feaible Solution For the feaible olution in tage one, it only contain an embedded SF. we have mentioned in Section III-, embedding an SFT for the multicat tak can outperform embedding an SF. Therefore, in the econd tage, we tranform the obtained SF in the firt tage to an SFT by adding new VNF intance and eliminating link with expenive cot. Fig. 6 depict an example of SFT. In uch an SFT, we call f i the predeceor VNF of f j if f i take effect before f j, and correpondingly, f j i the ucceor VNF of f i. In Fig. 6, it can be oberved that the number of predeceor VNF i maller than that of ucceor VNF, and we can prove that thi i neceary for an SFT with Theorem. Theorem. In an SFT, the number of predeceor VNF i alway maller than that of ucceor VNF.

8 8 f 1 f 1 f f f f f f f f f f f f f Logical link Fig. 6. n example of SFT with the SF requirement ( f 1 f f f ). Note that the link connecting VNF are logical. d 7 f f 1 f d 7 f f 1 f lgorithm The Second tage of our two-tage algorithm Input feaible olution X 0 produced by lgorithm, ervice function chain l={ f 1, f,..., f k }, network G, all node capacity and VNF deployment cot γ f j u. Output Optimized olution X opt. 1: Find all connection node in X 0. : for j=k; j ; j>0 do : if f j can be deployed in other node according to the rule in Section IV- then : eploy new intance of f j in correponding node, update connection node and [ω fn u],ϒ 5: ele 6: break 7: return X opt ={[ω fn u],ϒ}. lgorithm The overview of our two-tage algorithm (TS). Input network G=(V,E,c(E)), a multicat tak δ=(, ˆ,l) and VNF deployment cot γ f j u on all node. Output minimum cot SFT embedding olution to deliver the multicat tak δ. 1: ontruct the MO G according to lgorithm 1. : Embed the SF to get the feaible olution X 0 according to lgorithm. : Find the optimal olution X opt by tranforming the SF into the SFT according to lgorithm. f E f f F f d 1 d d d d 5 d 6 (a)the feaible olution of the firt tage d 1 d d d (b)the optimized olution of the econd tage Fig. 7. n example of the econd tage where all link denote the correponding hortet path. Proof. In the SFT, predeceor VNF are parent node of ucceor VNF. ll leaf node in the SFT mut be the intance of lat VNF. Thi property enure that the predeceor VNF in the SFT mut have children node. Otherwie, thee intance will not be contained by the SFT due to the minimum cot contraint. Since each parent node ha at leat one children node, predeceor VNF have fewer intance than ucceor VNF. Thu, Theorem i proved. To decribe tage two clearly, we provide an example to demontrate the procedure. Suppoe that there i a multicat tak δ={, ˆ,( f 1 f f f )}, and = {d 1,d,d,d,d 5,d 6,d 7 }. Fig. 7(a) how a feaible olution produced in tage one. In tage two, we optimize the feaible olution additionally by tranforming the SF into an SFT. In the lat tep of the firt tage, a Steiner tree i found to connect all detination with the lat VNF. hown in Fig. 7(a), the Steiner tree connect the detination {d 1,d,d,d,d 5,d 6,d 7 } and the lat VNF f. From node that deploy the lat VNF, there are four path for the multicat flow to all detination (i.e., leave of the Steiner tree {d,d,d 6,d 7 }). Thee four path can be divided into two categorie baed on whether they have common edge with the embedded SF. In Fig. 7(a), the path ( d 7 ) d 5 d 6 overlap with the SF, and we call uch a path dependent path. Otherwie, like path ( d ), d and ( d 6 ), which have no common edge with the SF, we call them independent path. For an independent path, it may contain multiple detination and we define the one nearet to the ource a it connection node. In Fig. 7(b), the connection node are d 1,d and d 5. fter identifying the dependent/independent path a well a connection node, we can further optimize the olution. Since Theorem how that predeceor VNF cannot have more intance than ucceor VNF, we increae the number of VNF intance in an inverted order. Thu, a illutrated by Fig. 7(b), we further deploy new intance of f in different node. Rule of dding New VNF Intance: In Fig. 7(a), each independent path get ervice of f and f from and. While in Fig. 7(b), f can be deployed on {,E,F}. enote the cot of the hortet path between two node v i and v j a c vi v j. If we could find a node E uch that c d1 E+c E + γ f E<c d1, we can deploy a new intance of f on E. Similarly, if c d5 F+c F + γ f F<c d1, we can deploy a new intance of f on F. fter adding intance of f, all node with f become the new connection node of each dependent path. Repeat the above procedure until one VNF cannot be deployed on multiple node. Theorem enure that thi terminate condition i valid. The detail of the econd tage can be formally depicted by lgorithm. Overall, our two-tage algorithm (TS) i ummarized in lgorithm.. lgorithm for Network with eployed VNF For a target network, ome VNF may have already been deployed. We modify lgorithm 1 to tackle the SFT embedding problem in uch a network.

9 9 Indeed, when contructing the MO network, thoe VNF that have already been deployed in the network can be divided into two categorie. For thoe VNF in the SF l, we treat their etup cot a zero, ince there i no need to deploy them again for a ingle multicat tak. For thoe VNF not in the SF l, they will not occupy a column in the MO network according to lgorithm 1. Thu, we only need to modify the capacitie of thoe node where VNF have already been deployed. Then lgorithm and lgorithm can be applied to the modified MO network to get the olution. Target Network Multicat Tak TLE II PRMETER SETTINGS. Parameter Setting Network ize [50, 50] eployed VNF eploy randomly Node apacity [1, 5] Link onnection ot The Euclidean ditance VNF eployment ot Relate to the connection cot S Select randomly Select randomly SF length [5, 5] E. lgorithm nalyi Theorem 5. The computational complexity of the two-tage algorithm i O((k + ) V ), in which V i the number of node in the target network, i the number of detination and k i the length of the required SF for each traffic flow in the multicat tak. Proof. In the procedure of contructing the MO network, we firt need to find the hortet path between each pair of node in the original network. Floyd algorithm can find all hortet path in O( V ) [7]. The overlay network ha k V edge and k V, thu the contruction of overlay network need O( V )+O(k V )=O( V ) tep. In the firt tage, we firt need to find the hortet path between S and T, and thi path can be found by ijktra algorithm in O(k V ) [8]. for the node that violate the capacity contraint, correponding adjutment procedure need O(k V ) comparion. The time complexity of finding the Steiner tree in the original network i O( V ) []. One iteration in the firt tage need O(k V )+O(k V )+O( V )=O((k + ) V ). Thu, the computational complexity of the firt tage i O((k + ) V ). In the econd tage, the computational complexity of the feaible olution optimizing depend on the number of dependent path and the length of SF, which need O(k ) comparion. Therefore, the computational complexity of all the above operation i O( V )+O((k + ) V )+O(k )=O((k + ) V ). Thu, Theorem 5 i proved. Theorem 6. With ufficient reource on each node, the twotage algorithm guarantee an approximation ratio maller than (1+ρ), where ρ i the bet approximation ratio of Steiner tree and can be a mall a 1.9 [17]. Proof. X opt repreent the optimal olution of delivering the multicat tak and there will be an SFT in X opt. X alg i the olution generated by our algorithm, and it alo ha an SFT. Indeed, X alg i generated after executing the econd tage algorithm. In the firt tage of our algorithm, a feaible olution X alg ha been generated which ha embed an SF. Since X alg i the olution optimized on the bai of X alg, then c(x alg )<c(x alg ). X alg i compoed of an SF P alg and a Steiner tree T alg. X opt ha an SFT, and we can find an SF P opt in the SFT and a tree T opt connecting all detination. ume the lat node of P opt i W, and the optimal Steiner tree covering all node in W i T W. There i no doubt that c(p opt )<c(x opt ) and c(t W )<c(t opt )<c(x opt ). In Fig. 8. The Palmetto network [9]. lgorithm, node W ha been conidered, and the olution can be denoted a the SF P W alg add the T W. ccording to lgorithm, c(p alg )+c(t alg )<c(p W alg )+c(t W ). Theorem ha proved that P i the optimal SF when all node have ufficient reource, thu c(p W alg ) < c(p opt ) < c(x opt ). The bet approximation ratio of the Steiner tree can be a mall a 1.9 [17]. However, in thi proof, we do not concentrate the concrete value, and we ue ρ denote the approximation ratio. nd we can get, c(t W ) < ρc(t opt ) < ρc(x opt ), ince T opt i a feaible tree to cover all detination and W. Then c(p W alg )+c(t W ) < c(x opt ) + ρc(x opt ), c(x alg ) < c(x alg ) = c(p alg) + c(t alg ) < (1 + ρ)c(x opt ). Thu, Theorem 6 i proved. V. EXPERIMENTL EVLUTION In thi ection, we firt introduce the evaluation environment and methodology and then evaluate the performance of the algorithm. Our evaluation focu on quantifying the benefit of our method at delivery cot and running time reduction.. Experiment eign Experiment onfiguration. To evaluate our method, we firt generate ynthetic target network uing the ER random graph [0]. Then, we make ue of the real-world network of PalmettoNet hown in Fig. 8, which i a backbone network with 5 node in the U.S. [9]. With the ynthetic and realworld network, we evaluate the impact of the multicat ize (i.e., the number of detination), the average VNF etup cot and the length of SF (the number of VNF). It i worth to mention that the link connection cot and VNF etup cot are prior knowledge and can be et according to the network dynamic. For example, if one link i congeted or

10 10 one erver node cannot upport VNF deployment, we can coniderably increae their cot uch that the embedding of an SFT would never elect uch kind of link. We alway report the average reult of 50 round in the ame parameter etting. Table II ummarize the parameter etting with the following conideration: The network ize: Following the previou work [16], we et the network ize in the range of [50,50], which cover mall network and large one. The deployed VNF: There are many value-added (VNF) ervice a the NFV market i developing [1]. We arbitrarily elect thirty different VNF and deploy ome of them randomly among the erver node (a the exited VNF). The node capacity: To eae the evaluation, we et the node capacity randomly within [1,5], which mean at mot 1 5 VNF can be deployed on the node. The link connection cot: To eae the evaluation, we et the link connection cot of any pair of node a the Euclidean ditance between them. The VNF deployment cot: we et the VNF deployment cot upon any erver node following the Normal ditribution N(µl G,σ ), where µ {1,}, σ = l G /, and l G i the average hortet path cot of the network (to balance the cot of link connection and VNF deployment). It i vital to emphaize that the VNF deployment cot can follow any ditribution. The ize of a multicat tak: For each imulated multicat tak, the ource node and the detination node are elected randomly from the target network. Furthermore, we et the value of / V a 0.1 or 0. to evaluate the impact of multicat ize [16]. The SF length: the NFV market develop, the SF length would be increaed. Thu, we et the SF length from 5 to 5, which i twice larger than it in real deployment []. enchmark lgorithm. To the bet of our knowledge, thi i the firt work tackling the SFT embedding problem (refer to the related work). In [], heng et al. have propoed a method which we call ST (Steiner tree-baed). ST can olve NFV enabled multicat, but it embed an SF rather than an SFT. In detail, the ST method firt contruct a Steiner tree covering all detination of the multicat and find the minimum cot path to connect the ource to thi tree. Then ST embed the required SF along the minimum cot path. We treat thi ST method a one of the benchmark that evaluate the cot of tranferring NFV enabled multicat. eide, we compare our algorithm (i.e., TS) with another two benchmark we deigned that can embed SFT: the minimum et cover algorithm (S) and the randomly electing algorithm (RS). Thee three algorithm are different in generating the feaible olution at the firt tage, while the optimization procedure at the econd tage i the ame. S trie to occupy a few node a poible when embedding the SF in the firt tage. It chooe the minimum number of node to cover a many VNF a poible. If ome VNF ha no exited intance in the network, S will deploy a new intance upon the nearet Total ot TS S RS ST Network ize (a) The traffic delivery cot Running Time(m) TS S RS ST Network ize Fig. 9. The changing trend of the two metric when / V =0.1 Total ot TS S RS ST Network ize (a) The traffic delivery cot Running Time(m) TS S RS ST Network ize Fig. 10. The changing trend of the two metric when / V =0. node to the predeceor VNF. for RS, it randomly elect VNF that have been deployed. While for thoe VNF that have not been deployed, RS randomly elect node with ufficient capacitie to deploy them. fter all requeted VNF having been deployed, RS connect them in order with the hortet path. With variou parameter etting, we compare our algorithm with thee three benchmark concerning the traffic delivery cot and the method running time. Furthermore, to evaluate the accuracy of approximated olution, we perform experiment over the Palmetto network, a it network ize i uitable to obtain the optimal olution.. Evaluation with Synthetic Network 1) The impact of detination ratio: We define the detination ratio a the number of detination in the network to the total number of network node, i.e., / V. To concentrate on the impact of detination ratio under different network ize, we fix the SF length with 5 and the average VNF deployment cot with µ =. Fig. 9 and Fig. 10 how the variation of traffic delivery cot and running time a the growth of network ize under different detination ratio. In average, the traffic delivery cot reulting from TS are 11.01% and 16.6% lower than thoe from ST hown in Fig. 9(a) and Fig. 10(a), repectively. We can ee that the traffic delivery cot of the olution increae a the growth of network ize. The reaon for thi phenomenon i that the final olution include more link (incurring more link connection cot) a the network ize increae. lo, the number of detination increae a the network ize increae, which lead to the increae of both connection link and VNF intance. Thu, comparing Fig. 9(a) with Fig. 10(a), we can find that the olution in the latter figure poe higher traffic delivery cot. The curve that repreent the method running time are illutrated in Fig. 9(b) and Fig. 10(b). The running time of TS can be larger than them of ST by 7.% and 7.9%,

11 11 Total ot TS S RS ST Running Time(m) TS S RS ST Total ot TS S RS ST Running Time(m) TS S RS ST Network ize Network ize Length of SF Network ize (a) The traffic delivery cot (a) The traffic delivery cot Fig. 11. The changing trend of the two metric when the average etup cot i 1 the average hortet path cot. Total ot TS S RS ST Network ize (a) The traffic delivery cot Running Time(m) TS S RS ST Network ize Fig. 1. The changing trend of the two metric when the average etup cot i the average hortet path cot repectively. The extra time conumption reult from the multiple comparion in the tage one of TS. Thee two figure ee the ame change trend, both of which increae gradually. the number of detination and network ize increae, the time conumption increae nonlinearly. The reaon behind thi phenomenon i that the computation complexity in contructing the Steiner tree i nonlinear. ) The impact of etup cot: To evaluate the impact of VNF etup cot under different network ize, we retrict the value of detination ratio to 0. (i.e., / V =0.) and the length of SF to 5. The curve change trend hown in thee two figure are almot the ame a thoe in Fig. 9 and Fig. 10. ompared with ST, TS reduce the traffic delivery cot by 16.7% and 17.9% (under different VNF etup cot) in Fig. 11(a) and Fig. 1(a), repectively. Particularly, the cot reduction i up to.05% in Fig. 1(a). We can ee that the traffic delivery cot in Fig. 1(a) i higher than that in Fig. 11(a), a higher VNF etup cot contribute more to the traffic delivery cot. Fig. 11(b) and Fig. 1(b) are almot ame, which illutrate that the average VNF etup cot doe no effect in the method running time. Thi phenomenon agree with what i indicated in Theorem 5. ) The impact of SF length: To evaluate the impact of SF length, we retrict the value of detination ratio to / V =0., the average deployment cot to µ=, and the network ize to V =00. The reult are hown in Fig. 1(a). We can ee that the olution of thee four method poe different traffic delivery cot. The performance gap between the curve of ST and TS increae a the SF length grow up, and the average cot reduction can be up to 1.91%. Thi i becaue longer SF provide more opportunitie to inherit deployed VNF and more optimization pace to embed SFT (rather than SF) with our TS. Fig. 1. The changing trend of the two metric with different SF length. Total cot TS S RS PLEX ST # of etination (a) The traffic delivery cot Running Time(m) TS S RS ST Network ize Fig. 1. The changing trend of the two metric in PalmettoNet when the number of detination are different. Fig. 1(b) depict the curve of the running time. Specifically, the curve of TS, S and ST rie along with the growth of SF length, while the curve of RS i table. Thi i becaue RS randomly elect erver node for VNF deployment without comparion, while the other three method perform more comparion between erver node when deciding the VNF location due to higher SF length.. Evaluation with Real-world Network In thi experiment, we apply the real-world network topology of Palmetto and evaluate our method by changing the number of detination and the SF length. Furthermore, with thi mall-cale network, we can obtain the optimal olution uing PLEX [], and then compare it with the deigned method. onidering the ize of thi real network, we et [5,5] to evaluate the impact of the multicat ize. We retrict the SF length to 10 and et the average deployment cot with µ =. Fig. 1 illutrate the impact of the number of detination. In Fig. 1(a), the traffic delivery cot grow up a the number of detination increae. Our TS can reduce the traffic delivery cot by 5.1% and 1.86% in average againt ST and RS, repectively. ompared with the optimal olution (illutrated by the yellow curve in Fig. 1(a)), our ST method can achieve an approximation ratio of 1.51 a far a the average algorithm performance concerned in thi experiment, which i larger than the given theoretical reult in Theorem 6. Thi gap reult from the SFT embedding olution and the approximate Steiner tree. We have ued the fat algorithm in [] to contruct the Steiner tree. ccording to Fig. 1(b), the running time of all four method rie a the number of detination increae. Note that the running time of PLEX are not able to be illutrated in the figure ince they are much (>0 time) higher than that of the other four method.

12 1 Total cot TS S RS PLEX ST Length of SF (a) The traffic delivery cot Running Time(m) TS S RS ST Network ize Fig. 15. The changing trend of the two metric in PalmettoNet with different SF length. Fig. 15 evaluate the impact of the SF length. In thi experiment, we retrict the number of the detination to 15 (i.e., =15) and et the average deployment cot with µ =. Similar to Fig. 1, the curve in Fig. 15 how the ame trend. ompared with RS and ST, our TS can reduce the traffic delivery cot by 18.69% and 19.6% on average, repectively. The curve in Fig. 15(b) alo how the ame trend a Fig. 1(b), i.e., the running time of all the four method rie a the SF length increae. VI. ONLUSION In NFV enabled multicat, a the cot of VNF deployment and link connection are enitive to the VNF location, embedding a ervice function tree (SFT) i proved to be a better choice for cot aving than a ervice function chain (SF). In thi paper, we tudy the optimal SFT embedding problem for NFV enabled multicat. Firtly, we formally defined and formulated the optimal SFT embedding problem with an integer linear programming. Secondly, to olve the problem efficiently, we deigned a two-tage algorithm by i) producing an initial feaible olution with embedding an SF and ii) optimize the initial olution by adding new VNF intance and contructing an SFT. Then we proved that the approximation ratio of our algorithm i 1 + ρ where ρ can be a mall a 1.9, under the condition that the erver node have ufficient reource. Finally, with extenive experimental evaluation, we dicloed the impact of different parameter on embedding an SFT and howed that our SFT embedding method could reduce the traffic delivery cot by.05% at mot againt other three benchmark. REFERENES [1]. Ren,. Guo, G. Tang, Y. Qin, and X. Lin, Optimal ervice function tree embedding for nfv enabled multicat, in Proc.of IS, 018. [] H. L. Hao, H. H. hun, S. S. Hiang, Y.. Nian, and. W. 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13 1 [5] I. akah, K. Praveen, and M. Vijay, valanche: ata center multicat uing oftware defined networking, in Proc.of IEEE OMSNETS, 01. [6] V. Eramo, E. Miucci, M. mmar, and F. G. Lavacca, n approach for ervice function chain routing and virtual function network intance migration in network function virtualization architecture, IEEE/M Tranaction on Networking., vol. 5, no., pp , 017. [7] F. R. W, lgorithm 97: hortet path, ommunication of the M, vol. 5, no. 6, p. 5, 196. [8]. E. W, note on two problem in connexion with graph, Numeriche mathematik, vol. 1, no. 1, pp , [9] The internet topology, [0] P. Erdo S and. Re nyi, On random graph i, Publ. Math. ebrecen, vol. 6, pp , [1] G. rown, S. nalyt, and H. Reading, White paper: Service chaining in carrier network, [] M.. Luizelli,. Raz, and Y. Sa ar, Optimizing NFV chain deployment through minimizing the cot of virtual witching, in Proc. of INFOOM, 018. [] IM ILOG PLEX, ization-tudio/. Guoming Tang (S 1-M 17) i an itant Profeor at the Key Laboratory of Science and Technology on Information Sytem Engineering, National Univerity of efene Technology (NUT), hina. He received the Ph.. degree in omputer Science from the Univerity of Victoria, anada, in 017, and both the achelor and Mater degree from NUT in 010 and 01, repectively. ided by machine learning and optimization technique, hi reearch focue on computational utainability in ditributed and networking ytem. bangbang angang Ren received the.s.degree and M.S. degree in management cience and engineering from National Univerity of efene Technology in 015 and 017, repectively. He i currently working toward Ph. in ollege of Sytem Engineering, National Univerity of efene Technology, hangha, hina. Hi reearch interet include oftware-defined network, network function virtualization, approximation algorithm. eke Guo eke Guo received the.s. degree in indutry engineering from the eijing Univerity of eronautic and tronautic, eijing, hina, in 001, and the Ph.. degree in management cience and engineering from the National Univerity of efene Technology, hangha, hina, in 008. He i currently a Profeor with the ollege of Sytem Engineering, National Univerity of efene Technology, and i alo with the ollege of Intelligence and omputing, Tianjin Univerity. Hi reearch interet include ditributed ytem, oftware-defined networking, data center networking, wirele and mobile ytem, and interconnection network. He i a enior member of the IEEE and a member of the M. Yulong Shen Yulong Shen received the.s. and M.S. degree in computer cience and the Ph.. degree in cryptography from Xidian Univerity, Xian, hina, in 00, 005, and 008, repectively. He i currently a Profeor with the School of omputer Science and Technology, Xidian Univerity, and alo an ociate irector of the Shaanxi Key Laboratory of Network and Sytem Security. He ha alo erved on the technical program committee of everal international conference, including the NN the IEE, the INoS, the IS, and the SOWN. Hi reearch interet include wirele network ecurity and cloud computing ecurity. Xu Lin Xu Lin received the.s. degree in computer cience from Xidian Univerity, Xian, hina, in 016. He i currently a Profeor with the School of omputer Science and Technology, Xidian Univerity. He i currently working toward Ph. in the School of omputer Science and Technology, National Univerity of efene Technology, hangha, hina. Hi reearch interet include oftwaredefined network, network function virtualization, network ecurity.