Noname manuscript No.(will be inserted by the editor)

Optimal Mapping of Virtual Networks with Hidden Hops

Juan Felipe Botero · Xavier Hesselbach · Andreas Fischer · Hermannde Meer

Received: date / Accepted: date

Abstract Network Virtualization has emerged as a so-lution for the Internet inability to address the requiredchallenges caused by the lack of coordination amongInternet service providers for the deployment of newservices. The allocation of resources is one of the mainproblems in network virtualization, mainly in the map-ping of virtual nodes and links to specific substratenodes and paths, also known as the virtual networkembedding problem. This paper proposes an algorithmbased on optimization theory, to map the virtual linksand nodes requiring a specific demand, looking for themaximization of the spare bandwidth and spare CPUin the substrate network, taking into account the band-width demanded by the hidden hops when a virtuallink is mapped. The components of the virtual networks(nodes and links) that do not ask for an specific demandare then allocated following a fairness criteria.

Keywords Network Virtualization · Virtual NetworkEmbedding · Virtual Network Mapping · OptimizationTheory

Juan Felipe Botero

Jordi Girona Street, 1 and 3, Barcelona, SpainTel.: +34-93-4017041E-mail: jfbotero@entel.upc.edu

Xavier HesselbachJordi Girona Street, 1 and 3, Barcelona, SpainTel.: +34-93-40159871E-mail: xavierh@entel.upc.edu

Andreas FischerInnstr. 43, 94032 Passau, GermanyTel: +49-851-509-3057

E-mail: andreas.fischer@uni-passau.de

Hermann de MeerInnstr. 43, 94032 Passau, Germany

Tel: +49-851-509-3050E-mail: demeer@fim.uni-passau.de

1 Introduction

The deployment of new Internet services is nowadaysbeing more and more difficult, the lack of cooperationamong stakeholders does not allow radical changes tothe Internet architecture [3,12,11,14]. This tendency iscalled ossification.

Network virtualization has been proposed as the al-ternative to face up ossification [12]. It allows multipleheterogeneous networks to cohabit on a shared physicalsubstrate (SN)1.

A Virtual Network (VN) - sometimes also called“Overlay Network” - consists of active and passive net-work elements realized on top of a substrate network.The active elements are called virtual nodes whereasthe passive elements are called virtual links. Virtualnodes are interconnected through virtual links, forminga network that can be represented by a graph, where thevirtual nodes are represented by the nodes in the graphand the virtual links are represented by the edges.

An instance of such a VN then is realized through amapping of its elements to the substrate network. Thismapping defines the relationship of virtual network ele-ments to their respective counterparts in the substratenetwork. Several virtual nodes in the VN can corre-spond to a single node in the SN (i.e. the mapping isn : 1). Likewise, several virtual links (VL) can corre-spond to a single link in the substrate network. Thisn : 1 mapping is accomplished through resource shar-ing, with the resource being CPU time and memory fornodes and bandwidth for links.

This concept of virtual networks is realized in sev-eral different implementations. Examples are Virtual

1 This paper will use indifferently the terms substrate network

and physical network

Copyright and Reference Information: This material (preprint, accepted manuscript, or other author-distributable version) is provided to ensure timely dissemination of scholarly work.Copyright and all rights therein are retained by the author(s) and/or other copyright holders. All persons copying this work are expected to adhere to the terms and constraints invoked bythese copyrights. This work is for personal use only and may not be redistributed without the explicit permission of the copyright holder. The definite version of this work is published as

[·] Juan F. Botero, Xavier Hesselbach, Andreas Fischer and Hermann De Meer. Optimal mapping of virtual networks with hidden hops. Telecommunication Systems, Volume 52, Number 3,pages [NA], 2013. The original publication is available at www.springerlink.com, to appear.

See http://www.net.fim.uni-passau.de/papers/Botero2013a for full reference details (BibTeX, XML).

2

Private Networks, Peer-to-Peer networks and networksvirtualized with System Virtualization.

In Virtual Private Networks (VPNs), a dedicated(usually encrypted) Virtual Link is set up between tworouters, possibly spanning several physical links. VPNscan be set up on different layers in the network stack,with the IPSec implementation on the network layerbeing one of the more popular ones [8].

In Peer-to-peer (P2P) overlay networks [13] an en-tire network structure is created on top of an existingnetwork. P2P nodes correspond to Virtual Routers, set-ting up Virtual Links between them, that do not reflectthe underlying network infrastructure, but rather thelogical grouping of nodes within the P2P network.

In networks virtualized with System Virtualization[2], core routers host several operating systems withrouting functionality. The routers are interconnectedwith Virtual Links that may again span several physicallinks. This approach allows for easy management andeven mobility of Virtual Routers. Moreover, it becomespossible to deploy different network protocols alongsideeach other with the concept of system virtualizationproviding a clear compartmentalization. Besides, net-work virtualization provides reusable topology and anenergy efficient scheme. Fig. 1 is an example of thisimplementation.

Substrate Network

Virtual Network

1

A

2

5

3

4

B

C

D

Virtual Nodes Mapping:

Virtual Node Substrate NodeA 1B 3C 2D 5

Virtual Links Mapping:

Virtual Link List of PossibleSubstrate Paths

(A, B) (1,2),(2,3)(1,4),(4,5),(5,2)(2,3)

(A,C) (1,2)(1,4),(4,5),(5,2)

(B,D) (3,2),(2,5)(3,2),(2,1),(1,4),(4,5)

(C,D) (2,5)(2,1),(1,4),(4,5)

Fig. 1 Example of a Virtual Network Mapped Over a SubstrateNetwork

The network embedding problem consists on the ef-ficient mapping of a set of VN requests to substratenodes and links. We define a VN request as a set of vir-tual nodes (with or without CPU requirements) thatmust be mapped to a set of SN nodes (i.e. nodes from

the SN) with enough CPU resource to accomplish therequirements, and a set of virtual links (with or with-out bandwidth requirements) to be mapped to a set ofSN paths (i.e. paths from the SN) accomplishing the re-quired bandwidth demands and the hidden hops (inter-mediate substrate nodes) CPU demands. The networkembedding problem is computationally hard to solve,as it will be shown in section 3, so the developed algo-rithm should follow a heuristic to try to obtain a nearoptimum value.

This problem turns out to be a form of the wellknown Multicommodity-Flow problem (MFP). This pa-per proposes an algorithm to solve the virtual networkembedding problem based in optimization theory. Theaim is to maximize the remaining SN resources (CPUand Bandwidth) while mapping the virtual nodes andlinks with specific demands. Apart from these demandsconstraints, the virtual nodes and links might imposemore constraints, e.g. specific location in virtual nodesand delay constraints in virtual links. When the virtualnodes and links with specific demands are assigned, thealgorithm assigns, in an equal way, the remaining sub-strate network resources to the virtual nodes and linkswithout specific demand requests.

The paper is organized as follows: Next section de-scribes the optimization model proposed to efficientlysolve the embedding VN problem accomplishing theconstraints and looking for the maximization of thespare bandwidth and spare CPU. Section 3 shows thecomplexity of the raised problem. Section 4 presents theheuristics used to solve the problem. Section 5 presentsan example of the algorithm and finally, conclusions andfuture work are presented in section 6.

2 Optimization Model

2.1 Hypothesis

To accommodate a demand between two virtual nodesinside a virtual network, up to now, only one pathis taken into account. This is often a realistic restric-tion due to the used routing protocol, or simply anexplicit management requirement. However, multi-pathapproaches have been proposed. In [16] the virtual linkdemand is split among the possible paths, reducing, inthis way, the computational complexity of the problem.Although this approach is computationally better, thedifficulty of its implementation is higher [7].

In our model, the relationship among the elementsof SN and VNs is as follows: One virtual router can bemapped to only one physical router, while a virtual linkis represented by a path (group of consecutive links) inthe SN. We consider that a physical node is a hidden

3

hop if it is part of a SN path mapped to a virtual link(i.e. if it is an intermediate node in the SN path).

Besides the CPU demand of some virtual nodes, weconsider that each substrate node’s, acting as a hid-den hop (intermediate node) in a substrate path, thatrepresents a virtual link, will have a CPU expenditurebecause it has to be configured and it will have to cor-rectly forward the packets passing through this virtuallink. The CPU resource that must be assigned to anintermediate node is a function of the virtual link de-mand. This resource expense had not been consideredin previous studies [16,17,10].

This model assumes that the virtual nodes are as-signed (mapped) to specific substrate nodes before theoptimization process starts. This hypothesis is consid-ered because we think that, for each virtual node, nodemapping might not be done taking into account all thesubstrate nodes as potential candidates. At least, a vir-tual node must be subject to location constraints, so thepotential candidates would be the nodes accomplishingthese constraints. This aspect has not been taken intoaccount in [10,17]. In future works, location constraintswill be studied to map the most suitable substrate node,among a set of candidates, to a specific virtual node.The first attempt to include location constraints in nodemapping can be found in [4].

The optimization model assumes virtual network re-quest where there can be bandwidth and CPU fixeddemands for some virtual links and nodes, but also thepossibility that some other nodes and links do not askfor any resources is contemplated. After the demandedresources (bandwidth and CPU) in virtual links andnodes are assigned; a fair allocation is made, with theremaining resources, among the virtual links and nodeswith no resources demanded. This fair allocation lies inallocates the shortest paths for the virtual links.

The required demand of the hidden hops and themapping of undemanded request had not been takeninto account in previous work [16,17,10,4]. These twoimprovements helps to carry out a more realistic virtualnetwork mapping, because more virtual networks (eventhose without explicit demand requests) can be mappedtaking into account the CPU demand of hidden hops.

2.2 Variables Definition

SN is represented by an undirected graph G(V, E), whereV is a set of elements called vertices and E is a set of el-ements called arcs or edges, with one or more numbersassociated with each arc. Considering a ordered set ofvertices V1, V2, ..., Vn, Vn+1; a free of cycles directed path(just path from now on) is any sequence of arcs ∈ E ofthe following type: {(V1, V2), (V2, V3), ..., (Vn, Vn+1))}.

A virtual network is represented by the undirectedgraph Gk(V k, Ek), where k is the identifier of the VN.Each virtual network request is represented by the graphGk(V k, Ek), a group of variables h(ik, jk) representingthe bandwidth demand of the virtual link (ik, jk) ∈ Ek,a variable cpu(ik) representing the CPU requirement ofthe virtual node ik ∈ V k and a variable I(ik, jk) whichrepresents the CPU demand of every intermediate nodecomposing the substrate path that maps the virtual link(ik, jk); this value can be given by the virtual networkrequester, or can be calculated based on the bandwidthdemand h(ik, jk), Fig. 2 is an example of these requestsover a substrate network with the NSF topology.

0

1

2

34

6 89

1110

13

12

5 7

100

120100

120

80

100

100

80

120

80 80

120

100

100100

1000

100

101000

10100

100 1000

1000

10010

100 100

1000100

100

10

100

CPU Capacity of a substrate node

Bandwidth Capacity of a substrate link

Substrate Network

Virtual Network Requests

A

E

BC

D

25170

102

50

10

15

5

32

A

D

BC

15378

100

200

80

92

20300

A

E

BC

D

20170

102

50

60

85

45

12 100

CPU demand (c(ik)) of a virtual node

Bandwidth demand(h(ik,jk)) of a virtual link

VN 1 VN 2

VN 3

Fig. 2 Example of a Set of VN requests over a NSF SubstrateNetwork

If h(ik, jk) = 0 the virtual link (ik, jk) is not ask-ing for a specific bandwidth demand, in the same wayif cpu(ik) = 0 the virtual node ik is not asking for aspecific CPU demand. The notation of the substratenetwork variables is shown in Table 1, while variablesof each specific virtual network are deeply detailed inTable 2.

4

Table 1 Definition of Substrate Network’s Variables

Terms Definition

G(V, E) Undirected graph representing the substrate network

V Set of physical nodes (routers) belonging to the sub-strate network

E Set of links belonging to the substrate network

(i, j) (i, j) ∈ E is the link from substrate’s node i to sub-strate’s node j

V N Set of virtual networks, virtualized from the substratenetwork

V Nk V Nk ∈ V N represents the virtual network number k

C(i) CPU capacity of the substrate network’s node iB(i, j) Bandwidth capacity of the substrate network’s link

(i, j)

Table 2 Definition of Variables Specific of Virtual Network k

Terms Definition

Gk(V k, Ek) Undirected graph representing the virtual networkk

V k Set of virtual nodes (routers) belonging to the vir-tual network k

a virtual node mapping is given by the followingfunction f(ik) ∈ V and ik ∈ V k, f(ik) has to fulfillthe CPU demand and location constraints

cpu(ik) CPU that must be assigned to the node ik ∈ V k ofthe virtual network k

Ek Set of links belonging to the virtual network k

P (ik, jk) Set of cycle-free paths in SN connecting the sub-strate nodes f(ik) and f(jk)

I(ik, jk) CPU requirement of a substrate node acting as in-termediate (hidden hop) in a substrate path repre-

senting the virtual link (ik, jk)

dp(i) Assigned bandwidth to substrate node i ∈ V actingas an intermediate node in path p ∈ P (ik, jk)

(ik, jk) (ik, jk) ∈ Ek is the link from virtual node ik to

node jk in the virtual network k.

This virtual link is given by the following functiong(ik, jk) ∈ P (ik, jk) and (ik, jk) ∈ Ek, g(ik, jk)has to fulfill the bandwidth demand

h(ik, jk) Bandwidth that must be assigned (demand) to the

virtual link (ik, jk) of the virtual network k

bp(ik, jk) Allocated Bandwidth to the virtual link (ik, jk) in

the path p ∈ P (ik, jk) of the virtual network k

ρp(i, j, k) Binary variableIt is 0 → if the substrate link (i, j) in the virtualnetwork k, is not part of the path p for the virtuallink (ik, jk). It is 1 elsewhere

αp(ik, jk, l) Binary variableIt is 0 → if the substrate node l ∈ V in the virtualnetwork k, is not an intermediate node of the pathp for the virtual link (ik, jk). It is 1 elsewhere

2.3 Optimization Model Definition

The model aims to maximize the spare bandwidth andCPU in the physical network, while the demands forvirtual nodes and links are fulfilled. Formally the opti-mization model is the following:

Maximize:

F =X

(i,j)∈E

0

B

@

B(i, j) −|V N|X

k=1

X

(ik,jk)∈Ek

|P (ik,jk)|X

p=1

ρp(i, j, k)b

p(i

k, j

k)

1

C

A

+X

l∈V

C(l) −|V N|X

k=1

X

ik∈V k|f(ik)=l

cpu(ik)+

+X

(ik,jk)∈Ek

|P (ik,jk)|X

p=1

αp(i

k, j

k, l)d

p(l)

!!

(1)

Subject to:

B(i, j)−|V N|X

k=1

X

(ik,jk)∈Ek

|P (ik,jk)|X

p=1

ρp(i, j, k)b

p(i

k, j

k) ≥ 0, ∀(i, j) ∈ E

(2)

C(l) −|V N|X

k=1

X

ik∈V k|f(ik)=l

cpu(ik)+

+X

(ik,jk)∈Ek

|P (ik,jk)|X

p=1

αp(i

k, j

k, l)d

p(l)

!

≥ 0, ∀l ∈ V

(3)

|P (ik,jk)|X

p=1

ρp(i, j, k)b

p(i

k, j

k) = h(i

k, j

k)

for 1 ≤ k ≤ |V N |, ∀(ik, j

k) ∈ E

k

(4)

|P (ik,jk)|X

p=1

αp(i

k, j

k, l)d

p(l) = I(i

k, j

k)

∀l ∈ V, for 1 ≤ k ≤ |V N | and ∀(ik, j

k) ∈ E

k

(5)

∃!bp(i

k, j

k) 6= 0, ∀p ∈ P (i

k, j

k) (6)

∃!dp(n) 6= 0, ∀p|p ∈ P (i

k, j

k) (7)

|V N | ≥ 0 |E| ≥ 0

|P (ik, j

k)| ≥ 0 for 1 ≤ k ≤ |V N |, 1 ≤ l ≤ |C(k)|

ρp(i, j, k) ≥ 0 for (i, j) ∈ E, 1 ≤ k ≤ |V N |, 1 ≤ p ≤ |Pl(k)|

αp(i

k, j

k, l) ≥ 0 for (i

k, j

k) ∈ E

k, l ∈ V, 1 ≤ k ≤ |V N |,

1 ≤ p ≤ |Pl(k)|(8)

The objective is to find, for each virtual network,the paths accomplishing the constraints that maximizethe objective function F (Equation (1)), representingthe sum of the spare bandwidth and spare CPU in thesubstrate network. First group of constraints (2), (3)are of capacity, they assure the sum of the bandwidthsand CPU, assigned to each virtual link and node, doesnot exceed substrate’s link nor node capacity. Second

5

group of constraints are related with demand (4), (5);they assure that each virtual link obeys its demandedbandwidth and each intermediate virtual node reachesits CPU demand. Third set of constraints (6), (7) as-sures the uniqueness of the chosen substrate path foreach virtual link. Last set of constraints (9) force vari-ables to be greater or equal than zero.

3 Problem Complexity

The model exposed in the previous section may havedifferent degrees of complexity depending on the waythe allocation of the virtual link demands is approached.To consider unique substrate path to transport a vir-tual link demand is often a realistic restriction due tothe used routing protocol, or simply an explicit man-agement requirement stipulating avoidance of packetre-sequencing in receiving nodes. In the right side’s topof Fig. 2 an example of an allocation using just one pathis shown. If this restriction is present in the problem,it can be formulated as the unsplittable flow decisionproblem (UFP) [9]:

INSTANCE: Graph G = (V,E), and edge capacityB(i, j) for each (i, j) ∈ E, a set of demands T , whereh(s, t)) ∈ T is the demand between nodes s and t ∈ V .We say that T is realizable in G if there exist a set ofpaths P (s, t) such that s and t are the endpoints of eachpath, and the following capacity constraint is met forevery edge:

∑(s,t)∈T

∑p∈P (s,t)

ρp(i, j)h(s, t) ≤ B(i, j) ∀(i, j) ∈ E (9)

where ρp(i, j) is a binary variable that is 1 if thelink (i, j) ∈ E is part of the path p between the nodes(s, t), and 0 otherwise.

QUESTION: Is T realizable in G?This problem is a generalization of the well known

Edge Disjoint Paths (EDP) decision problem, that wasshown to be NP-complete [6], that is why UFP is NP-complete.

Fig. 3 shows a virtual network on top of a sub-strate. There is a virtual network demand between vir-tual nodes A and D. In the right side of the figure;two methods to allocate the traffic are shown, in parta, the bandwidth to fulfill the virtual link demand isassigned to just one of the possible paths between thispair of nodes. In the bottom of the figure (part b), theallocation of bandwidth is made using all the possi-ble paths between the substrate nodes. This approachcan be solved using the multi-commodity-flow problem

(MFP). Algorithms running in polynomial time thatsolve MFP are available [1].

Substrate Network

Virtual Network

A

B

C

D

Just one path allocated for the

virtual link demand between virtual nodes A

and D

A DVirtual Link between virtual nodes A and D

Virtual link (A,D) demand allocated

following two paths

a)

b)

Fig. 3 Different Strategies to Allocate Virtual Link Demands

4 Heuristics

In this section, we propose an algorithm to map a setof virtual network requests (offline algorithm) trying tomaximize the spare bandwidth and spare CPU in thesubstrate network. The algorithm takes into accountthat each VL request mapped to a SN path uses cer-tain CPU capacity in the intermediate path nodes; thiscapacity could be calculated as a function of the virtuallink BW demand as well as a given parameter enclosedin this demand.

The algorithm is divided in two different steps: firstly,the algorithm maps the requests of each VN, both forthe virtual nodes and virtual links, that explicitly asktheir demands. In second place, the remaining resourcesare distributed equally among the remaining virtualnodes and links.

The second step of the algorithm, to allocate theresources equally among the request without demand,is divided in two parts; the virtual node mapping is easybecause we assume that each virtual node is alreadymapped (f(ik) is known for ik ∈ V k∀k), the mappingof a virtual link is performed by allocating the resourcesto the shortest SN path.

The optimum solution to the first step of the al-gorithm is computationally hard to solve as shown inthe previous section, so a heuristic solution is proposed

6

to find a near-optimum result. This heuristic is mainlybased in an approximated greedy algorithm [15] pro-posed to solve the Unsplittable-Flow problem, we makefew modification to adapt the algorithm to the mappingproblem.

4.1 Mapping of Demanded Virtual Nodes and Links

The algorithm to map the virtual nodes and links, ofall the virtual network requests asking for explicit de-mand, uses a sub-algorithm to realize just one virtuallink mapping. For the mapping of virtual nodes, theassumption that virtual nodes are already mapped isdone, that is, each VN k request provides f(ik) ∀ik ∈V k.

We employ a small modification of the known greedyalgorithm (GA) [9] to look for the best substrate virtualpath for a single virtual link obeying the bandwidthand CPU constraints. This algorithm uses the short-est path, with hops number as metric, as the preferredsubstrate path to map each virtual link. This decision ismade because when the shortest path is used, the sparebandwidth and CPU of the SN is maximized, that is,as the shortest path maps the required demands to lessresources (less physical links and nodes in the short-est path), the remaining resources in the network aremaximized each time a shortest path is used.

Before presenting the algorithm that allocates a sub-strate path for each virtual link, some temporary vari-ables and functions are explained in detail.

The algorithm needs some entry parameters: thegraph G(V, E) representing the substrate network, aswell as the virtual nodes of the virtual link (ik, jk) thathave associated the bandwidth demand h(ik, jk) andCPU demand I(ik, jk) ∀i ∈ V acting as an intermedi-ate node belonging to the path between f(ik) and f(jk).The function FIND-SP (n, ik, jk, G(V, E)) returns theshortest path (in number of hops) number n betweenthe nodes f(ik) and f(jk) ∈ V accomplishing the CPUand Bandwidth restrictions, this function is based onthe algorithm to find the k-shortest paths proposed in[5]. HOPS(p) is a function that returns the number ofhops of a path p. MAX(Paths, bin) returns the max-imum from a set of paths taking into account the binvariable that can be BW or CPU, that is, if bin=BWthe chosen path will be the path with more remainingbandwidth (sum of the remaining BW in all links). Iftwo (or more) paths have the same remaining band-width, the one with higher spare CPU (sum of the re-maining CPU in all nodes of the path) will be chosen.If bin=CPU the opposite process is made. INTER(p)returns a set containing the intermediate nodes of thepath p.

Algorithm 1 Greedy Algorithm to Map a Single Vir-tual Link (GA)Require: G(V, E), ik, jk, bin1: Condition=TRUE, FT=TRUE, SP={∅}, paths={∅},

Counter=1, prevhops=0, Path

2: while Condition do3: SP=FIND-SP(Counter,ik,jk,G(V, E))4: if HOPS(SP) 6=0 then5: if HOPS(SP) = prevhops or FT then6: FT=FALSE, paths=paths+{SP}, pre-

vhops=HOPS(SP)7: else

8: Condition=FALSE9: end if

10: else11: Condition=FALSE

12: end if13: end while14: if HOPS(SP) 6= {∅} then15: Path=MAX(paths,bin)

16: for i ∈ INTER(Path) do17: C(i) = C(i) − I(ik, jk)18: end for19: return Path

20: else21: return 022: end if

The algorithm GA (Alg. 1) works by finding theshortest path, with number of hops as the metric, ac-complishing the bandwidth and CPU constraints. Ifmore than one shortest path is found, a path with moreremaining bandwidth or remaining CPU, based on thebin variable, is chosen. When the path is chosen, thealgorithm proceeds to subtract the intermediate nodedemand associated with the virtual link from the re-maining CPU demand of each intermediate node.

The general algorithm (GAP-M) maps the virtuallinks and nodes with explicit demands of one virtualnetwork request making use of the previous procedure(GA) and uses the following temporary variables andfunctions.

H and A are sets of unmapped and mapped (re-spectively) connections of a specific virtual network.SORT(B) returns the group of links ∈ B ordered indecreasing order of demands h(ik, jk). FIRST(H) re-turns the first item (ik, jk) ∈ Ek of H. SORTP(A) re-turns the connections of A in decreasing order, tak-ing into account the number of hops of each corre-sponding SN path. REALLOCATE-RESOURCES(k)updates the capacities of the substrate network (B(i, j),∀(i, j) ∈ E and C(i),∀i ∈ V ) by reallocating them withthe mapped virtual links and nodes resources of thevirtual network k. The variable TOL (tolerance) is themaximum number of times that the largest path is elim-inated to allow the mapping of a request, that does notfit in the physical network, due to the lack of resources.

7

Algorithm 2 Algorithm to Map a Virtual Network(GAP-M)Require: G(V k, Ek), G(V, E), TOL, bin1: for Counter ∈ V k do2: if C(f(counter)) ≥ cpu(counter) then

3: C(f(counter))= C(f(counter)) - cpu(counter)4: else5: GOTO 396: end if7: end for8: Path=0, A={∅}, B=Ek

9: H=SORT(B), (ik, jk)=FIRST(H), Path=GA(G(V,E),

f(ik),f(jk), bin)10: if Path6= 0 then11: g(ik, jk)=Path12: for (i, j) ∈ g(ik, jk) do

13: B(i, j) = B(i, j) − h(ik, jk)14: end for15: H=H-{(ik, jk)}, A=A+{(ik, jk), Path}, GOTO 916: else

17: if H={∅} then18: Mapping completed, Stop Algorithm19: else20: GOTO 22

21: end if22: end if23: for Counter=1 to TOL do

24: D= SORTP(A), (gk, hk)=FIRST(D)25: H=H+{(gk, hk)}, A=A-{(gk, hk), Path}26: for (i, j) ∈ g(gk, hk) do27: B(i, j) = B(i, j) + h(ik, jk)

28: end for29: Path=GA(G(V,E), f(ik),f(jk), bin)30: if Path 6= 0 then31: g(ik, jk)=Path

32: for (i, j) ∈ g(ik, jk) do33: B(i, j) = B(i, j) − h(ik, jk)34: end for35: H=H-{(ik, jk)}, A=A+{(ik, jk), Path}, B=H,

GOTO 936: end if37: end for38: if Path=0 then

39: It is not possible to map Virtual Network k40: REALLOCATE-RESOURCES(k)41: return 0

42: end if

Algorithm GAP-M works by mapping each virtuallink, belonging to the virtual network. Virtual links aremapped, one by one, in decreasing demand order. Tomap each link, the algorithm GA (Alg. 1) is used. Ifthe mapping of a virtual link is not possible (GA re-turns 0), the algorithm tries to look for the resourcesneeded to assign this requirement; the longest of themapped paths (already assigned to another virtual link)is eliminated of the mapping because this path is morelikely to share substrate link and nodes with the cur-rent request, and so, the resources are returned to thesubstrate network variable and another try to map vir-tual link is done. If even in this way the request can

not be mapped, the following mapped longest path iseliminated and the same process is performed until thevirtual link is assigned or until a tolerance (TOL) valueof times is reached. This value is an algorithm input andis thought to limit the number of mapping attempts ofa virtual link; if after TOL attempts, the mapping hasnot been done, we consider that this virtual networkcan not be mapped.

To map the set of VNs into the SN, the previousalgorithm (Alg. 2) must be called for each VN.

4.2 Mapping of Undemanded Virtual Nodes and Links

After the mapping of the request asking for a specificdemand, the second step in the algorithm is to assignthe remaining unconstrained demand in an equal wayamong the requests that did not demand any resource.To do the mapping, it is necessary to know, in firstplace, if the physical nodes that correspond to the vir-tual nodes (without demand), have non-zero CPU re-maining capacity. To map a virtual link, the shortestpath with non-zero remaining capacity, both in virtuallinks and nodes, is chosen.

The algorithm uses a new function SP-WR (G(V, E),i, j) that returns the shortest path between i and j in thegraph G(V,E), this path must have non-zero remainingbandwidth and CPU in each physical link (composingthe virtual one) and physical intermediate node, if thiscondition does not apply the function returns 0.

Algorithm 3 Algorithm to Map Request Without De-mand in a Virtual Network (UVL)Require: G(V k, Ek), G(V, E)1: for Counter ∈ V k do2: if C(f(counter)) = 0) then

3: GOTO 164: end if5: end for6: U=Ek

7: (ik, jk)=FIRST(U), Path=SP-WR(G(V, E), f(ik), f(jk))8: if Path 6= 0 then9: g(ik, jk)=Path, U=U-{(ik, jk)}, GOTO 7

10: else

11: if H={∅} then12: Mapping completed, Stop Algorithm13: end if14: end if

15: if Path=0 then16: It is not possible to map Virtual Network k17: return 018: end if

The mapping process finishes with the execution ofthe UVL algorithm (Alg. 3) in each virtual networkrequest. In first place, the algorithm assures that each

8

physical node corresponding with a virtual node of thisrequest, has remaining CPU capacity, if it is not thecase, the virtual network is not mapped. After that step,the virtual link mapping is performed, the result of themapping is located in g(ik, jk), these variables indicatesthe path each virtual link has been mapped to.

5 Example of Application

Fig. 4 shows an example with a simple SN topologyand three virtual network requests, two of them withdemands in each virtual node and link, the third re-quest does not ask for any demand. This example ispresented to show how a network would behave follow-ing the proposed algorithm. It will be easier for thereader to understand how the mapping is carried out.

Substrate Node i ik

jk

i

C(i)

i jB(i,j)

Substrate Link

(i,j)

cpu(ik) Virtual Node

ik, where f(ik)=

i

Substrate Linkik h(ik,jk)

I(ik,jk)

3

100

1 4

3 4

1 5

15

5

Substrate Network

12

10

8

20 25

VN

Request #1

VN

Request #2

After VN 1

mapping

3 4

1 5

15

5

20 25

After VN 2

mapping

1 5

VN

Request #3

(without

demand)

VN

Request #2

VN

Request #3

(without

demand)

VN

Request #3

(without

demand)

1

2

3 5

4

250

100

2040 7010

1

2

3 5

4

88

234

92

2030 7010

83

1

2

3 5

4

229

67

2010 70

10

g(11,31)= {(1,2),(2,3)}

g(31,41)= {(3,4)}

g(32,42)= {(2,3),(1,2),(1,4)}

After VN 3

mapping

83

1

2

3 5

4

229

67

2010 70

10

g(13,53)= {(1,3),(3,5)}

Fig. 4 Example of a Virtual Network Mapping Scenario

In this example, three virtual network request aredone to a substrate network that counts on 5 nodesconnected by 6 links. The information of each physical

node (node id i, remaining bandwidth capacity C(i))and physical link (link remaining capacity, B(i, j)) isshown in the figure. The virtual network requests con-tains also all the needed information in each VN. Itis possible to see, for each virtual node being part ofV Nk, in Fig. 4; the node i to which each virtual nodeik is mapped (to facilitate the reading, f(ik) = i) andthe CPU demand cpu(ik), and for each virtual link, thebandwidth demand h(ik, jk) and the CPU requirementof the intermediate physical nodes I(ik, jk) that willcompose the SN path fulfilling the virtual link require-ments.

Following the proposed algorithms, the process tomap the VN requests is as follows: Firstly, the 1st re-quest is processed by GAP-M (Alg. 2), lets supposearbitrary values to TOL = 2 and bin=BW. The firststep of the algorithm is to subtract of the remainingCPU in physical nodes, the amount of demand madeby the corresponding virtual nodes. The result is dis-played in the drew SN after the VN 1 mapping. Nodes1, 3 and 4 (nodes of the 1st VN request) new CPU ca-pacity is the result of the subtraction of the virtual nodedemand from the old capacity. Node 2 remaining CPUdecreases as well. In second place, the algorithm looksfor the virtual link request with the higher bandwidthdemand (h(ik, jk)), the result is the virtual link (1, 3)with 20 units of bandwidth demand. In third place, aproper path fulfilling the virtual link requirements issearched, to find this path the GA (1) is invoked, thisalgorithm returns a path accomplishing the bandwidthand intermediate node CPU requirements. In this case,the chosen path is g(11, 31) = {(1, 2), (2, 3)}, althoughthe shortest path between nodes 1 and 3 is direct (nohops), it does not obey the BW requirements. So thedemanded bandwidth is subtracted from the remainingbandwidth capacity of each physical link contained inthe substrate path. As this path contains an interme-diate node (node 2), the CPU demand associated withthe bandwidth request is subtracted from the remain-ing CPU capacity in this physical node. The case ofthe second virtual link in the first request is easier tosolve. The shortest path between physical nodes 3 and 4can accomplish the bandwidth and intermediate nodesCPU’ demand, so the bandwidth demand h(31, 41) issubtracted from the remaining bandwidth in this linkB(3, 4) and it is clear that g(31, 41) = {(3, 4)}. The re-sult of this mapping is the substrate network shown inFig. 4 after “VN 1 mapping” arrow.

After the first virtual network is mapped, the samealgorithm is applied to the following VN request. The2nd request is just of one link: (32, 42). So, as in theprevious request, the first step is to update the re-maining capacities of the physical node. In this case,

9

nodes 3 and 4 remaining CPU is updated. The secondstep, with GA algorithm’s help, is to find the most ap-propriate path for the unique virtual link request ofthe request. Possible paths between these nodes are({(3, 4)}, {(1, 3), (1, 4)}, {(2, 3), (1, 2), (1, 4)}), but theonly path obeying the bandwidth and CPU require-ments is the largest g(32, 42) = {(2, 3), (1, 2), (1, 4)}, sothis is the chosen path to map the virtual link (32, 42),and its bandwidth demand (h(32, 42)) is subtracted fromthe bandwidth remaining capacity (B(i, j)) of each sub-strate link composing the path. The CPU requirementassociated with the virtual link I(32, 42) is also sub-tracted from the remaining CPU capacity of each in-termediate node (nodes 2 and 1). After the successfulmapping of the VN 2 request, the resultant substratenetwork is shown in Fig. 4 after “VN 2 mapping” arrow.

The last VN request does not ask for any demandneither of virtual nodes nor of virtual links. So, as it isshown in Fig. 4 the substrate network does not changeafter the mapping of this request, the remaining re-sources are available to be used by this VN. But it is stillinteresting to know which is substrate path that will bechosen to fulfill the demand of the unique virtual linkin the third VN request. This request among the sub-strate nodes 1 and 5 is fulfilled, as it is stated in the al-gorithm Alg. 3, using the shortest substrate path, withnon-zero remaining resources (BW and CPU), amongthese nodes; in this case it is clear that the chosen pathis g(32, 42) = {(2, 3), (1, 2)}.

6 Conclusions and Future Work

The introduction of network virtualization would pro-duce big changes in current network architecture. But itsets out some problems, such as the mapping of virtualnetworks in top of the substrate network.

We have proposed an optimization model contem-plating the possibility of mapping virtual networks withexplicit bandwidth and CPU demands, but also withthe possibility of accept virtual links or nodes, or evenvirtual networks, that do not ask for any demand. Asthe problem is computationally hard to solve, we haveproposed an algorithm (heuristic) that tries to map infirst place, the virtual links and nodes that are ask-ing for an explicit demand in each virtual network, thismapping is done with the objective of maximize thespare bandwidth and CPU resources in the substratenetwork; the aim of this optimization objective is toleave as much resources as possible, but obeying all thedemands, to distribute them among the remaining re-quests (without explicit demands).

Previous studies in the embedding problem havebeen focused in the mapping of VNs taking into ac-

count the demand in both, virtual link bandwidth andvirtual node CPU. CPU demand due the mapping ofa virtual links in the intermediate nodes (hidden hops)of a substrate path, has not been taken into accountas an important parameter in the problem. It is logicalto think that some CPU is consumed by the interme-diate nodes, because they will have to be configured toprocess and forward the packets passing through thatvirtual link. The proposed optimization model (and sothe algorithm) includes this important feature.

The proposed algorithm is based on a modified greedyheuristic that can be improved, if the problem is set outas a Multi-Commodity flow problem, and after solvingit, the application of rounding techniques might driveus to more optimized solution taking into account theconstraint stating that just one SN path must be usedto map each virtual link.

An optimization objective that is energy-awarenesscould be also interesting to obtain an embedding thatreduces the energy expense in the substrate network.

Acknowledgements This work has been partially supported

by national Spanish project CICYT TSI2007-66637-C02, Euro-pean Network of Excellence Euro-NF FP7-ICT-2007-1, Grant No.216366, AutoI: STREP, FP7 Call1 (ICT-2007-1-216404) and Re-sumeNet STREP, FP7 Call2 (ICT-2007-2-224619). Also, it has

been partially supported by the ”Comissionat per a Universitatsi Recerca del DIUE” from the ”Generalitat de Catalunya” andthe Social European Budget (”Fons Social Europeu”).

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