A 2-Layers Virtual Network Mapping Algorithm Based on Node ...staff.ustc.edu.cn/~qzheng/li.pdf · This work was supported in part by ZTE’s collaborative project under Grant Research

Received November 2, 2018, accepted November 14, 2018, date of publication November 26, 2018,date of current version December 31, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2883251

A 2-Layers Virtual Network MappingAlgorithm Based on Node Attributeand Network SimplexQUAN ZHENG1,2, JUN LI 1,2, HONGLIANG TIAN3, ZHIZHEN WANG1,2, AND SONG WANG21Laboratory for Future Networks, University of Science and Technology of China, Hefei 230000, China2Department of Automation, University of Science and Technology of China, Hefei 230000, China3ZTE Corporation, Shanghai 518057, China

Corresponding author: Quan Zheng ([email protected])

This work was supported in part by ZTE’s collaborative project under Grant Research on Access Network Oriented ContentService Network and in part by the ‘‘Internet Plus’’ major projects for the ‘‘Internet Plus’’ Coordinated ManufacturingCloud Service Support Platform.

ABSTRACT Network virtualization is one of the most important techniques to overcome Internet ossifica-tion, which can enable multiple networks to run on a common infrastructure. Virtual network mapping is akey process of network virtualization to efficiently map virtual network onto the substrate network withinconstraints of node and link resource. However, previous VNM heuristic algorithms completely separate thenode mapping stage from link mapping stage and ignore the effect of link mapping results on physical noderanking, which may lead to higher mapping cost and distributed distribution of virtual nodes on the substratenetwork. The shortest path algorithm is used at most to resolve virtual link mapping which results in longermapping time. To address these issues, a two-layer virtual network mapping algorithm named simplex-VNMis proposed based on node attribute and network simplex in this paper. In node mapping layer, a new noderanking approach which considers topology attributes, global network resources, and mapped physical nodeinformation provides candidate nodes and node mapping cost for link mapping layer. Then, in link mappinglayer, network simplex algorithm which has smaller time complexity compared to traditional shortest pathalgorithms has been introduced to find the optimal link mapping solution under the result of node mappinglayer. Finally, according to the results of two layers, the virtual node and relevant links will be mappedwith minimal total mapping cost. Extensive simulation results have demonstrated that our proposed methodbehaves better than four other algorithms that consider node attribute and use shortest path algorithm interms of total mapping cost and mapping time.

INDEX TERMS Virtual network mapping, network simplex, node attribute, node-ranking approach.

I. INTRODUCTIONWith the development of the Internet, the current networkarchitecture is difficult to meet the elastic needs of the net-work due to the network ossification [1], [2]. Network vir-tualization [3] has been proposed to allow multiple networkarchitecture to run on a common substrate, which is an impor-tant supporting technology for building a new generation ofnetwork architecture [4]. Virtual network mapping is oneof the basic contents in network virtualization research [5].Its main goal is to map virtual networks to common physicalinfrastructure with the constraints of node and link resourcesand make more efficient use of the substrate resources.

Virtual network mapping problem currently faces severalmajor challenges, such as resource constraints, diversetopologies, and online requests [6]. These challenges makeit difficult to find the optimal solution in polynomial time.At present, many researches aim at improving physical infras-tructure revenue and reducing the mapping time of virtualnetwork mapping problem. In fact, virtual network mappingis an NP-Hard [7] problem which is difficult to find theoptimal solution.

Currently, heuristic algorithm, the typical method for solv-ing virtual network mapping, is composed of three stage.In the node ranking stage, the virtual nodes and physical

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VOLUME 6, 2018

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Q. Zheng et al.: 2-Layers Virtual Network Mapping Algorithm

nodes are ranked based on certain rules. Then in node map-ping stage, virtual nodes are mapped to physical node usinggreedy mapping strategy meanwhile satisfying the resourceconstraints of node. Finally, virtual links are mapped ontosubstrate network by shortest path algorithm under the con-straints of link bandwidth. However, previous VNM heuristicalgorithms [7]–[13] completely separate the node mappingstage from link mapping stage and ignore the effect of linkmapping results on physical node ranking, which may leadto higher mapping cost and distributed distribution of virtualnodes on the substrate network. Latest work [12] takes globalresources into consideration in the node ranking stage, butthe proposed algorithm only explores part of information ofnodes such as CPU capacity or degree. Previous link mappingapproaches use the shortest path algorithm at most to resolvevirtual link which result in longer mapping time.

In this paper, we focus on the problem of high mappingcost and long mapping time in virtual network mapping,and propose Simplex-VNM algorithm based on nodeattribute and network simplex. First, we introduce net-work simplex algorithm to deal with link mapping problem.Kelly and O’Neill [14] and Muramatsu [15] implement thenetwork simplex method based on the original-dual rule,and Wang and Lin [16] improve the initial feasible solutionand the iterative update sequence of the original-dual rule.We integrate the above studies in Simplex-VNM algorithmto implement the network simplex algorithm. Then, we intro-duce a utility function to calculate the total mapping costincluding node mapping cost and link mapping cost. For anyvirtual node and relevant links, we will complete the mappingprocess through the 2-layers mapping algorithm. In nodemapping layer, there are many different mapping optionsfrom virtual node to physical node, and we select the set ofphysical nodes that may be mapped according to topologyattribute, global network resources and the information ofmapped physical node. In link mapping layer, we introducenetwork simplex algorithm to find the optimal mapping costof the relevant link under the result of node layer. Finally,according to the results of the above two layers, the virtualnode will be mapped to physical node which has minimaltotal mapping cost. The two layers depend on each other untilthe mapping of virtual network is completed. To further provethe efficiency of Simplex-VNM, a comprehensive simulationis conducted in this paper. Simulation results have demon-strated that our proposed method behaves better than fourother algorithms that consider node attribute and use shortestpath algorithm in terms of total mapping cost and mappingtime.

The main contributions of this paper are listed below:1. We propose a new node ranking approach by consid-

ering topology attribute, global network resources and theinformation of mapped physical node. This node rankingapproach measures global network resources from a differentperspective, and the information ofmapped physical node canensure the distribution of virtual node more concentrated inthe substrate network.

2. In the link mapping layer, we implement net-work simplex algorithm which has lower time complex-ity than that of traditional shortest path algorithms, andit makes the Simplex-VNM achieve better performancein mapping time.

3. A 2-layers virtual network mapping algorithm is pro-posed to deal with virtual network mapping problem. In thenode mapping layer, physical nodes are ranked by the newnode ranking approach to provided candidate nodes of linkmapping layer. In the link mapping, the relevant virtual linksaremapped by network simplex algorithm. Every virtual nodeand relevant links are mapped according to the results ofthe node mapping layer and the link mapping layer, and theinterdependence of mapping results per layer can reduce thetotal mapping cost.

4. Under the experimental scenario of small-scale topologyand large-scale topology respectively, it is verified that thealgorithm has better results in terms of mapping cost andmapping time.

The remainder of this paper is organized as follows.Related work is presented in Section II. A brief descriptionof virtual network mapping problem is given in Section III.Section IV presents our algorithm Simplex-VNM. Section Vverifies the effectiveness of Simplex-VNM through sim-ulation experiments. Finally, we conclude the paper inSection VI.

II. RELATED WORKIn [7]–[25], some of algorithms are put forward to solvethe virtual network mapping problem. A traditional methodis to convert the virtual network mapping problem into aninteger linear programming problem and then uses an optimalalgorithm to solve it. The advanced algorithms such as meta-heuristics algorithms are also used in VNM. And the mostcommonmethod is to complete themapping of nodes throughheuristic algorithm and complete the link mapping throughthe shortest path algorithm. We will introduce some studiesusing the above methods separately.

In terms of optimal algorithm, Chowdhury et al. [17] con-structing the virtual network mapping problem as the mixedinteger programming (MIP) model [18] and then modifying itinto a linear programming (LP) model, the link mapping canbe solved by Dijkstra’s algorithm [19] if the LP model has afeasible solution. Melo et al. [20] formulated an integer linearprogramming model to solve the virtual network mappingproblem. Three cost functions are presented to minimize theresource consumption and balance the network load. How-ever, due to the limitation of mapping time, complex linearprogramming models are difficult to applied to medium-size or large-size substrate networks.

Meta-heuristic algorithms such as genetic algorithm [21],simulated annealing [22] and particle ant colony optimiza-tion [23] can find near optimal solutions by iterativelyimproving a candidate solution, and it can achieve goodresults if the initial values are selected properly and iterativeprocess is designed reasonably. Zhang et al. [24] proposed

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a unified enhanced particle swarm optimization-based algo-rithm, called VNE-UEPSO, to solve virtual network mappingproblem which support path splitting. Infhr and Raidl [25]introduced a memetic algorithm to solve virtual networkmapping problem and they also offered an analysis of theinfluence of different problem representations and in par-ticular the implementation of a uniform crossover for thegrouping genetic algorithm. The drawback of the meta-heuristic algorithm is that it requires a lot of computing time,so it is difficult to use in practice.

Virtual network mapping can be decomposed into twotasks: node mapping and link mapping [8]. In general,a heuristic algorithm is used to solve the node mapping prob-lem, and then the linkmapping is converted into theminimumcost flow problem [7] based on the pre-determined sourcenode and destination node, and is solved by the shortestpath algorithm or cancellation algorithm finally. Yu et al. [6]advocated a different approach which rethink the designof the substrate network to enable simpler mapping algo-rithms and more efficient utilization of resources withoutrestricting the problem space. The algorithm simplifies vir-tual link mapping in two ways: (i) allowing the substratenetwork to split a virtual link over multiple substrate pathsand (ii) employing path migration to periodically re-optimizethe utilization of the substrate network. Cui et al. [9] pro-vided a virtual network mapping algorithm based on theconnectional characteristics of virtual network topologies andintroduced a node connection-degree to measure the priorityof the alternative physical nodes. The algorithm can make themapped substrate nodes closer to each other by using the nodeconnection-degree based on the virtual topologies feature.Cheng et al. [10] proposed a topology-aware node mappingalgorithm that applies theMarkov RandomWalk (RW)modelto rank nodes based on its resource and topological attributes.The virtual node mapping sequence will depend on node-ranking values, and the virtual links, whether split or un-split, will be mapped by k-shortest path or multi-commodityflow method. In the recent research, Zhang et al. [12] haveproposed a novel five-node ranking metric to quantify theimportance of substrate node, but the global resources in thispaper are not comprehensive enough. The definition of theglobal resources, ignoring the resource information of theneighbor nodes, only considers the ratio of the link bandwidthof neighbor nodes.

The aforementioned heuristic algorithms mostly deal withnode and link mapping separately in virtual network map-ping. In the process of node mapping, nodes are rankedfirst according to some factors such as node resources andtopological attributes, but the factors considered are relativelyone-sided. And such ranking method ignores the fact thatthe ranking values of remainder physical nodes will changewhen some physical node have been mapped. The traditionalshortest path algorithm which has high complexity will beapplied in link mapping. It is difficult for such a heuristicalgorithm to guarantee overall optimization although eachstage is the optimal algorithm.

Our work has some similarities with these heuristic algo-rithms, and the major difference is that our method couplesthe mapping of nodes to links through node mapping layerand link mapping layer. In node mapping layer, a new noderanking approach, combining the topology attribute in [13]and improving calculation of global resource in [12], is usedto calculate the rank value of physical node. In link mappinglayer, the network simplex algorithm, rather than the shortestpath algorithm, is used tomap the virtual links associatedwiththe virtual node.

III. VIRTUAL NETWORK MAPPINGIn this section, we make a description of virtual networkmapping, including network model, algorithm objectives andconstraint condition.

A. NETWORK MODELThe main research goal of the virtual network mapping isto maximize the final network revenue when satisfying thevirtual node and virtual link resource requirements.

In the virtual network mapping model, the substrate net-work is denoted by a weighted undirected graph Gs =(Ns,Es,Cs,Bs), where Ns and Es refer to the set of physicalnodes and links, respectively. The resources of nodes andlinks are denoted byCs andBs. The virtual network is denotedby the weighted undirected graph Gv = (Nv,Ev,Cv,Bv),where Nv and Ev are the set of virtual nodes and links,Cv and Bv refer to the constraint of virtual nodes and links.In this paper, we consider the resources requirement of virtualnode and link as the constraint. Virtual network mappingis defined as a mapping M(Gv,Gs) from virtual networkrequests Gv to substrate networks Gs, i.e.,

M (Gv,Gs) : (Nv,Ev,Cv,Bv)→ (Ns,Es,Cs,Bs) (1)

The virtual network mapping M(Gv,Gs) can be decom-posed into node map Mn(Gv,Gs) and link map Me(Gv,Gs)as follows:

Mn (Gv,Gs) : (Nv,Cv) → (Ns,Cs) (2)

Me (Gv,Gs) : (Ev,Bv) → (Es,Bs) (3)

B. OBJECTIVESThe virtual network mapping problem is a multi-objectiveoptimization problem under constraints of node and linkresources. The main evaluation indicators include the vir-tual network’s request acceptance ratio [26], revenue to costratio [6], and mapping time [27]. This paper evaluates theperformance of the algorithm by mapping time and the rev-enue to cost ratio of substrate network. The research goal is tomaximize the revenue to cost ratio and minimize the mappingtime while meeting the requirements of virtual network nodesand links. In the case of multiple virtual network requests,it is difficult to accurately count the mapping time due to thetime interval between virtual network requests and the failureof virtual network mapping. So we approximately multiply

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virtual network mapping problem to a single network map-ping problem. When not considering the operational effi-ciency of the substrate network after mapping, the revenueto cost ratio of substrate network can be represented by thecost of mapping.

The total cost of virtual network mapping includes nodecost and link cost, and we can define C (Gv) as follows:

C (Gv) = α ∗∑

ev∈EvCe (ev)+ β ∗

∑nv∈Nv

Cn(nv) (4)

where Ce (ev) represents the link mapping cost and Cn(nv)represents the node mapping cost, α and β are the weightcoefficients.

C. CONSTRAINT CONDITIONVirtual network mapping has many constraints, and we willfocus some of them as follows:

1) NODE CONSTRAINTAssuming that there is a virtual node nv and the correspondingmapped physical node is ns, then the available resource of nsshould be greater than or equal to the requirement of nv:

Cs (M (nv)) ≥ Cv (nv) ,∀nv ∈ Vv (5)

where Cv (nv) is the requirement of nv and Cs (M (nv)) is theresource of ns that can be assigned to nv.

2) MAPPING CONSTRAINTMost of the existing studies apply the mapping method inwhich the virtual nodes cannot be separated but the virtuallinks can be split. That is, a virtual node can only be mappedto a single and different physical node, and a virtual link canbe mapped to multiple physical links. So this constrain isformulated as:

Mn (i) 6= Mn (j) , ∀i, j ∈ Vv, i 6= j (6)

As for virtual link lv = (i, j), we suppose the set of pathsthat lv maps to is PM (lv), and then |PM (lv) | ≥ 1.

3) LINK CONSTRAINTSupposing physical path p ∈ PM (lv) allocates bandwidthsB(lv, p) to virtual link lv. Then there must be enough band-width resources BS (ls) on the physical link to meet the fol-lowing constraint:

BS (ls) ≥∑

p:p∈PM (lv),ls∈pB (lv, p), ∀ls ∈ ES (7)

IV. 2-LAYERS ALGORITHM BASED ON NODEATTRIBUTE AND NETWORK SIMPLEXThis section describes the Simplex-VNM algorithm in detail.

First, virtual nodes are ranked by new approach, and thegreedy virtual node mapping is implemented based on therank result. Then two mapping layers will complete the vir-tual node mapping and virtual link mapping. In node map-ping layer, algorithm will select the set of physical nodesbased on topology attribute, global network resource and the

information of mapped physical node. In link mapping layer,simplex network algorithmwill find the optimal linkmappingcost under the result of node mapping layer. And finally,the virtual node will be mapped to physical node which hasminimal total mapping cost.

A. VIRTUAL NODE RANKINGIn the node mapping stage, virtual nodes are ranked byheuristic algorithms and then mapped in order. Yu et al. [6]proposed a classical heuristic algorithm, which is mainlybased on node resources and link bandwidth. The value ofnode ranking can be formulated as Equation (8).

C (n) = C (n) ∗∑

l∈L(n)B (l) (8)

The algorithm simply uses the local resources of the node,ignoring the impact of the resources of neighbor nodes, whichmay result in unbalanced link load after mapping. In addi-tion, the algorithm does not consider the network topologyattribute such as degree of node.

In this paper, the nodes will be ranked by global resourceand topology attribute. First, the resources of a single nodecan be represented by the Equation (8). According to topol-ogy attribute, the node degree indicates the number of adja-cent links of node. As for a virtual node, the greater the degreeof node, the harder it is to be mapped. As for a physicalnode, the greater the degree of node, the more it needs to bemapped. That is to say, the node with the greater degree hashigher priority. So the attribute value of a single node can beexpressed by Equation (9)

Hv (ni) = C (ni) ∗ De (ni) ∗∑

nj∈nbr(ni)B(lij) (9)

where Hv (ni) represents the attribute value of the node ni,C (ni) represents the CPU capacity of the node ni, De (ni) rep-resents the degree of node ni, B(lij) represents the bandwidthresource of the link lij, and nbr (ni) represents the set of nodewhich directly connect to node ni.In order to consider the influence of neighbor nodes on

the node rank value, we define a new node attribute rankingformula:

RC (ni) = Hv (ni)+∑

nj∈nbr(ni)

Hv(nj)∗

B(lij)∑nk∈nbr(ni) B(lik )

(10)

where RC (ni) represents the global attribute value. Note thatwe take bandwidth resource into account for the proportionof total bandwidth resource as probability, and use the pro-portion to measure the relative importance of neighbor nodes.In summary, our ranking method of virtual node considers theglobal resource and topology attribute.

In the node mapping stage, the virtual nodes will be rankedaccording to RC, then map according to the order of ranking.

B. PHYSICAL NODE RANKINGIn the traditional node mapping algorithm, the virtual nodeand physical node perform node ranking according to the

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samemanner, and then the virtual nodes will bemapped to thephysical node that has the same level in order. This mappingmethod simply maps according to node rank order, whichmay cause the mapped nodes to be particularly scattered onthe substrate network because of ignoring the influence of themapped nodes on the unmapped nodes. In addition, completeseparation of node mapping from link mapping may causevirtual network mapping to fail.

For the ranking of physical nodes, we will consider theimpact of the already mapped nodes on the physical nodes.If virtual node nv will select physical node ni to mapping,then the attribute value of physical node ni can be expressedby Equation (11):

Hs (ni) = C (ni) ∗ De (ni) ∗∑

nj∈MF(lij) (11)

where Hs (ni) represents the attribute value of the physicalnode ni, C (ni) represents the CPU capacity of the node ni,De (ni) represents the degree of node ni, M represent the set ofmapped nodes in the substrate network associatedwith virtualnode nv, F

(lij)represents the maximum flow from physical

node ni to physical node nj.The same as virtual node rank, we also consider the impact

of neighbor nodes, and Equation (11) can be converted to:

RC (ni) = Hs (ni)+∑

nj∈nbr(ni)

Hs(nj)∗

B(lij)∑nk∈nbr(ni) B(lik )

(12)

Internet service provider (ISP) wants the physical nodesmapped by the virtual nodes to be as close together as pos-sible, because it can reduce operating expenses and decreasenetwork propagation delay. Then the physical node rankingformula should be added to the distance factor and can beexpressed by Equation (13).

Nrank (ni) =RC (ni)∑

ns∈Z D(ns, ni)(13)

where Nrank (ni) is the global attribute value of physicalnode ni, Z is a subset of M where the maximum flow betweenthe physical node and ni is greater than zero, D(ns, ni) isthe distance between two nodes. In summary, our rankingmethod of physical node considers the global resource, topol-ogy attribute and the information of mapped physical nodeincluding feasible flow and link distance of node.

C. SIMPLEX-VNMSimplex-VNM is mainly composed of node mapping layerand link mapping layer. The two layers attempt to map thenodes and links to obtain the optimal node mapping costand link mapping cost. Based on the results of two mappinglayers, the mapping cost of current node and all relevant linksare used as a utility function Ctotal(ns) to evaluate the finalutility of the physical node. So Ctotal(ns) can be expressed as:

Ctotal (ns) = α ∗∑

nm∈MCLink (ns, nm)+ β ∗ CNode(ns)

(14)

where CLink (ns, nm) is link mapping cost of mapping avirtual node to physical node ns,CNode(ns) is node mappingcost and Ctotal (ns) is total mapping cost, α and β are theweight coefficients. Selecting the mapping node with thisutility function can effectively reduce the total mapping cost.

In the node mapping layer, physical node will be rankedin descending order based on Equation (13) first, then thetop N physical nodes are selected to mapping virtual node,and finally the node mapping cost of top N nodes will becalculated for link mapping layer.

For example, as illustrated in Figure 1. For virtual network,we assume that the CPU capacity requirement of all virtualnodes is 1. For substrate network, we also assume that theCPU capacity of all physical node is 1 except nodeD and theirunit resource price is 1. Supposing virtual nodes a and b havebeen mapped to physical node A and B, and virtual link (a, b)has been mapped to physical link (A, B). Virtual node c isthe next node to be mapped, and we suppose N is 2. ThenEquation (13) is used to calculate the Nrank value of physicalnode C, D and E, and the two nodes with largest Nrank valueare selected for link mapping layer. We know that E and D isthe candidate node and their node mapping cost is 1.

FIGURE 1. An example of node mapping layer.

In the link mapping layer, the relevant links will be mappedaccording to the result of node mapping layer to get optimallink mapping cost. Based on the results of two mappinglayers, we can get the final mapping cost including nodemap-ping cost and all relevant links mapping cost, and completethe mapping with minimum cost.

For example, as illustrated in Figure 2. For virtual network,we assume that the bandwidth requirement of all virtual linksis 1. For substrate network, we also assume that the bandwidth

FIGURE 2. An example of link mapping layer.

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of all physical links is 1 and their unit resource price is 1.Physical node D and E is the result of node mapping layer.As for physical node E, supposing the best link mappingresult is that (a, c) maps to (A, E) and (b, c) maps to (B, E),the relevant links mapping cost is 2 and the total mapping costis 3. As for physical node D, supposing the best link mappingresult is that (a, c) maps to (A, D) and (b, c) maps to (B, E, D),the relevant links mapping cost is 3 and the total mapping costis 4. According to the total mapping cost, physical node E isselected to mapping virtual node c and the relevant virtuallinks should mapping to (A, E) and (B, E).

The following describes the specific steps of our 2-layersalgorithm, as shown in algorithm 1:

Algorithm 1 Simplex-VNMInput: (Vv,Ev) (Vs,Es)Output: {M (nv) |∀nv ∈ Vv}, {M (lv) |∀lv ∈ Ev}1: M = ∅,Di = ∅, i = 0, step = 0,N2: for all nv ∈ Vv3: count the RC (nv) of nv4: end for5: arrange nv in descending according to RC (nv)6: for all niv ∈ Vv7: if (M = ∅) then8: select nis ∈ Vs which has biggest RC(nis)9: else10: rank nis ∈ Vs by Nrank(nis)11: for all nis ∈ Vs and step < N12: Mapping relevant links and count the

Ctotal(nis), add the step13: nis→ Di14: end for15: select nis ∈ Di which has minimum Ctotal(nis)16: nis→ M17: end if18: end for

Step (1) is initialization, where M and Di represent theset of physical nodes that have been mapped to and the setof preselected physical nodes of virtual nodes. Steps (2)-(5)are the process of ranking virtual nodes. Steps (7)(8) are tomap the first virtual node. Steps (9)-(16) are other virtualnode mapping process, where step (10) represents physicalnode ranking process, steps (11)-(14) represent the process ofselecting the first N physical nodes ranked by node attributevalue, and step (15) represents the final selection process ofthe physical nodes. Specifically, steps (6)-(10) is the processof node mapping layer, and link mapping layer is reflected instep (12).

The final selection basis Ctotal(nis) in steps (10)-(15) istotal mapping cost including node mapping cost and relevantlinksmapping cost. By considering the cost of the each step ofnode mapping and relevant links mapping comprehensively,our 2-layers algorithm can make the total mapping the costsmaller and avoid the result of the local optimal cost in

traditional two-stage algorithm. In Algorithm 1, we introducethe framework of Simplex-VNM, especially the node map-ping of node mapping layer. In order to obtain link mappingcost in step (15), we need mapping relevant links first in linkmapping layer and we will introduce it in the next section.

D. VIRTUAL LINK MAPPING BASE ON NETWORK SIMPLEXIn the link mapping layer, the cost solution of link mappingcan be transformed into the minimum cost flow problem,which is to find a scheme with minimal cost under the condi-tion of constant flow size between node i and node j. In thissection, we make an implementation of the network simplexalgorithm based on the algorithms of [14] and [16] and findthe optimal solution of the link mapping in shorter time.

As for a topology G = (N,A), N and A are the set of nodesand arcs, respectively. The weight cij of the arc (i, j) ∈ Arepresents the cost of the unit flow, uij represents the capacitylimitation of arc (i, j), and b(i) respresents the supply anddemand function of node i.

For any feasible solution x of the minimum cost flowproblem, arc (i, j) is a free arc with 0 < xij < uij and arestricted arc with xij = 0 or xij = uij. Then the arc set Acan be divided into three subsets:

(1) T is the set of free arc.(2) L is the set of restricted arc with flow being 0.(3) U is the set of restricted arc with flow being upper

bound.

In the network simplex problem, we should define thepotential function π (i) of node i ∈ N, and the check num-ber cπij of arc (i, j):

cπij = cij − π (i)+ π (j) (15)

Supposing (T, L, U) is a feasible solution of minimum costflow problem, algorithm will stop when the check number cπijmeets the following three conditions:

cπij = 0, ∀(i, j) ∈ T (16)

cπij ≥ 0, ∀(i, j) ∈ L (17)

cπij ≤ 0, ∀(i, j) ∈ U (18)

then (T, L, U) is an optimal solution for the minimum costflow problem. The minimal cost flow steps of the networksimplex algorithm are shown as follows:

The network simplex algorithm adjusts the arc flowiteratively based on the flow circle. We will introduce a singleadjustment of the specific process in detail, and the numberin parentheses is the corresponding label of the algorithm inalgorithm 2:Step 1 (1-2): Generate an initial feasible solution. It con-

sists of the supply and demand functions b(i) for each node i,and the corresponding flow xij for each arc (i, j).Step 2 (3-8): Let the initial potential function of the source

node be 0,and the check number of arc in set T is 0, then thecheck number of other arcs and the potential function of othernodes can be calculated by formula (11).

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Algorithm 2 Link Mapping Algorithm Base on NetworkSimplexInput: G = (N,A), s, t, fOutput: Minimum cost flow f of graph G1: b (s) = f, b (t) = −f, {b (i) = 0|∀(i! = s&&i! = t)}2: find a feasible (T,L,U)→ A, π (s) = 03: for node i ∈ N and i! = s4: count π (i) of node i5: end for6: for arc (i, j) ∈ A7: count cπij of arc (i, j)8: end for9: while exist arc (ik, jk) not satisfy(17)or(18)10: (ik, jk)→ T11: Whilecπik ,jk ! = 012: for node ip associate with (ik, jk)13: Adjust π (ip) of node ip14: end for15: for arc (i, j) ∈ A16: Adjust cπij of arc (i, j)17: end for18: end while19: end while

Step 3 (9-10): Find the arc that does not satisfy the optimalsolution condition and add it to the spanning tree, then adjustthe flow distribution according to the circle flow conservationprinciple.Step 4 (11-18): Adjust the potential function and check

number of relevant nodes and arcs under the new principleof flow distribution, then ∀ (i, j) ∈ T, cπij = 0.

All links associated with the node nis can be mappedaccording to the network simplex algorithm. Then the utilityfunction Ctotal(nis) can be calculated in algorithm 1 and theentire virtual network mapping will be completed finally.

E. A EXAMPLE FOR SIMPLEX-VNMConsider the following topologies of the virtual network andthe substrate network as illustrated in Figure 3. For the virtualnetwork, the CPU capacity requirements of node and thebandwidth requirement of link are marked in parentheses.For the substrate network, the first number in parentheses

FIGURE 3. An illustration demo for Simplex-VNM.

indicates the number of available resources, and the secondnumber indicates the unit price of the resource. Supposing Nis 3, α and β are 1.Initialization: Both the virtual node and physical node will

be ranked according to Equation (10), and the virtual nodewith the highest rank value will be mapped to the physicalnode with the highest rank value. Table 1 shows the rankresults. As illustrated in Figure 4, virtual node a is mappedto physical node A, and virtual node b is the next node to bemapped.

TABLE 1. Rank value of node.

FIGURE 4. Mapping result of the first node by Simplex-VNM.

For virtual node b and relevant links, we will map themthrough 2-layer algorithm. In node mapping layer, physicalnodes that are not mapped will be ranked according toEquation (12), then the top 3 nodes will be selected tocalculate the node mapping cost. As illustrated in table 2,the top 3 nodes are B, C and D, and the CPU capacity ofnode E does not satisfy the constraint.

TABLE 2. Mapping result of node b in node mapping layer.

In link mapping layer, we will map the relevant linksaccording to the results of the node mapping layer to obtainthe link mapping cost. Table 3 shows the result of link map-ping layer. Based on the results of node mapping layer andlink mapping layer, virtual node b is mapped to physicalnode B, and virtual link (a, b) is mapped to physical link(A, B), as illustrated in Figure 5.

Finally, virtual node c and the relevant links (a, c) and(b, c) will also be mapped by 2-layer algorithm. The mapping

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TABLE 3. Mapping result of link relevant with node b in link mappinglayer.

FIGURE 5. Mapping result of the second node by Simplex-VNM.

process is illustrated in Table 4, Table 5 and Figure 6 showsthe final mapping result. Virtual node C is mapped to physicalnode C, and virtual link (a, c) and (b, c) are mapped tophysical link (A, C) and (B, C) respectively.

TABLE 4. Mapping result of node c in node mapping layer.

TABLE 5. Mapping result of link relevant with node c in link mappinglayer.

V. PERFORMANCE EVALUATION AND ANALYSISIn this section, we perform simulation experiments to evalu-ate the Simplex-VNM. The substrate network and the virtualnetwork topology is generated by the GT-ITM [28] topologygenerator randomly.

In this paper, our main work is to use network simplexalgorithm which has lower time complexity to complete vir-tual link mapping in link mapping layer. So mapping time isthe first metrics to verify the advantages of network simplex

FIGURE 6. Mapping result of virtual network request by Simplex-VNM.

algorithm in link mapping. In order to eliminate the impactof time interval between virtual network requests and thefailure of virtual network mapping on the mapping time,we focus on the mapping efficiency of single virtual net-work. Theoretically speaking, compared with the traditionalheuristic algorithm, Simplex-VNM can find a better mappingsolution by coupling the mapping between nodes and links.So the mapping cost is the second metric.

Five VNM algorithms make up the simulation part totally.Apart from Simplex-VNM, the other four algorithms areGreedy-VNM [6], RW-VNM [10], DCC-VNM [13] andTAGRD-VNM [11]. The Greedy-VNM algorithm is the clas-sical VN mapping algorithm where the virtual nodes aremapped by greedy algorithm and the virtual links are mappedby shortest path algorithm. The RW-VNM algorithm is atopology-aware node ranking algorithm where the virtualnodes are mapped based on random walk algorithm and thevirtual links are mapped by shortest path algorithm. TheDCC-VNM is a topology-based algorithm where the virtualnodes are mapped based on degree and clustering coefficientinformation and the virtual links are mapped by shortestpath algorithm. The TAGRE-VNM algorithm is a novel noderanking algorithm where the virtual nodes are mapped bytopology attributes and global network resources and thevirtual links are mapped by shortest path algorithm. Thesefour algorithms are typical, latest in VNM area. We willexperiment in both small-scale substrate topology and large-scale substrate topology and compare the performance of fivealgorithms in total mapping cost and mapping time.

A. SMALL-SCALE TOPOLOGYIn the simulation of small-scale topology, we use CERNET(China Education and Research Network) as substratenetwork topology. Table 6 summarizes the parameters ofsubstrate network and virtual network.

The five algorithms are run on the small-scale substratetopology. Figure 7 and Figure 8 represent the mapping costand mapping time of the five algorithms, respectively.

Figure 7 shows the mapping cost of five mapping algo-rithms with different number of virtual nodes, and the map-ping cost of the five algorithms increase with the number ofvirtual nodes. But it is evident that Simplex-VNM performs

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TABLE 6. Network parameters in small-scale topology.

FIGURE 7. Mapping cost with the number of virtual nodes in small-scaletopology.

FIGURE 8. Mapping time with the number of virtual nodes in small-scaletopology.

significantly better than four other algorithms for all virtualnodes numbers, and the Greedy-VNM has the maximummapping cost. The reason is that, in the mapping stage,RW-VNM only selects physical node that has best connec-tivity to map, without considering those physical nodes with

poor connectivity and lower mapping cost, and Greedy-VNMadjusts the compactness of the node based on the distancelimitation of the mapping, without considering the connec-tivity of nodes. DCC-VNM and TAGRD-VNM reduce themapping cost to a certain extent because of consideringtopology attributes and global network resource. But in theSimplex-VNM algorithm, not only global network resourceand topology attribute are considered in the node mappinglayer, but also the result of link mapping layer makes thetotal mapping cost smaller. The more nodes virtual networktopology has, the better Simplex-VNM will be.

Figure 8 shows the mapping time of five mapping algo-rithms with different number of virtual nodes, the mappingtime of the five algorithms become larger as the number ofvirtual nodes increases, and the Simplex-VNM has a betterperformance. We can make an analysis of it, Greedy-VNM,RW-VNM, DCC-VNM and TAGRD-VNM all perform linkmapping through the shortest path algorithm, but it takesmore time for DCC-VNM and TAGRD-VNM to computenode ranking value in the node mapping stage. Comparedwith Greedy-VNM and RW-VNM, Simplex-VNM spendsmore time in the node mapping stage due to consideringglobal network resource, topology attribute and the informa-tion of mapped physical node, but the Simplex-VNM haslower mapping time, because the network simplex algorithm,which is more efficient than other traditional shortest pathalgorithms, is used in link mapping stage and the time spenton link mapping occupies a larger proportion in the wholeprocess of virtual network mapping.

The time complexity of the minimum cost flow algorithmbased on the shortest path is O(F ∗ E ∗ logV), where F repre-sents the link traffic demand, E represents the total number ofphysical links, and V represents the total number of physicalnodes. The time complexity of the network simplex algorithmimplemented in this paper is O((m+ n) ∗m ∗ n∗C2

∗ F),which is proved in [18], where C represents the maximumbandwidth cost of the physical link, n represents the numberof nodes whose traffic demand is not zero, and m representsthe number of links whose traffic is not zero. In this paper,the network simplex algorithm has better time complexitythan other minimal cost flow algorithms.

B. LARGE-SCALE TOPOLOGYIn the simulation of large-scale topology, the parameters ofsubstrate network and virtual network are showed in Table 7.The probability of connection between physical nodes insubstrate network is P = β ∗ e ∗ [−d

5

α∗L ], where d is thedistance of the node, L is the generation scope of the node,α and β is the network characteristic parameters. The resultsof simulation based on the above environment are shownin Figure 9 and Figure 10.

The experimental results under large-scale topology showthat Simplex-VNM also has better performance in the map-ping cost and mapping time. As shown in Figure 9, in thecase of fewer virtual nodes, the mapping cost of various

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TABLE 7. Network parameters in large-scale topology.

FIGURE 9. Mapping cost with the number of virtual nodes in large-scaletopology.

FIGURE 10. Mapping time with the number of virtual nodes in large-scaletopology.

algorithms are similar since global network resources andtopology attribute have less impact on mapping result. As thenumber of virtual nodes increases, the number of virtuallinks will increase at a faster rate, and the proportion of linkmapping cost to total cost will continue to increase. Simplex-VNM achieves better performance in scenarios with morevirtual nodes since the link mapping layer makes the totalmapping cost smaller.

According to the result of mapping time in Figure 10,the larger the virtual network size is, the more obvious theSimplex-VNM advantage shows. The reason is that the linkmapping has a higher proportion of time in the entire mappingprocess with the increase of the virtual network topology size.At the same time, the time complexity of network simplexalgorithm is smaller than other shortest path algorithms inlink mapping process.

VI. CONCLUSIONIn this paper, we propose a 2-layers virtual network mappingalgorithm named Simplex-VNM based on node attribute andnetwork simplex. Compared with the traditional heuristicalgorithm, Simplex-VNM reduces the total mapping cost byconsidering topology attributes, global network resources andmapped physical node information in node mapping layer,and using final mapping cost as the utility function. Thenetwork simplex algorithm has been implemented in the linkmapping layer, which reduces the time complexity of themapping compared to the traditional shortest path algorithm.Simulation results show that the mapping cost and mappingtime of Simplex-VNM are better than four other algorithms(Greedy-VNM, RW-VNM, DCC-VNM and TAGRD-VNM)on small-scale and large-scale substrate networks. In thefuture, we plan to perform virtual network mapping inNDN [29]. If the physical node has the cache capacity, thelink mapping will be no longer a narrow minimum cost flowproblem. How to solve this problem is an interesting researchdirection.

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QUAN ZHENG received the B.S. degree in pro-duction process automation from the Dalian Uni-versity of Technology, Dalian, China, in 1992,and the M.S. degree in automatic control theoryand application and the Ph.D. degree in computersoftware and theory from the University of Scienceand Technology of China (USTC), Hefei, China,in 1995 and 2003, respectively. He is currentlyan Associate Professor with the Department ofAutomation and the Deputy Director of the Labo-

ratory for Future Networks, University of Science and Technology of China.His research interests includemedia content distribution and future networks.

JUN LI received the B.S. degree from the Depart-ment of Automation, Anhui University, Hefei,in 2016. He is currently pursuing the M.S. degreewith the University of Science and Technologyof China. His current main research direction isvirtual network mapping and NDN.

HONGLIANG TIAN received the B.S. degreefrom the Department of Computer, HuazhongUni-versity of Science and Technology, Wuhan, China,in 1988, and the M.S. degree and the Ph.D. degreein information science and electronic engineer-ing from the Department of Computer, ZhejiangUniversity, Hangzhou, China, in 1990 and 2000,respectively. He is currently with ZTE Corpora-tion. His research interests include optical access,multimedia, edge computing, and SDN and NFV.

ZHIZHEN WANG received the B.S. degree fromthe Department of Automation, University ofScience and Technology of China (USTC), Hefei,in 2015, and theM.S. degree in control science andengineering from USTC in 2018. His current mainresearch direction is virtual network mapping.

SONG WANG received the B.S. degree from theDepartment of Automation, University of Scienceand Technology of China (USTC), Hefei, in 1997,and the M.S. degree in control theory and appli-cation and the Ph.D. degree in control scienceand engineering from USTC in 2003 and 2010,respectively. He is currently a Lecturer with theDepartment of Automation, USTC. His researchinterests include future networks, media services,and P2P technology.

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A 2-Layers Virtual Network Mapping Algorithm Based on Node ...staff.ustc.edu.cn/~qzheng/li.pdf · This work was supported in part by ZTE’s collaborative project under Grant Research