Joint Backup Capacity Allocation and Embedding for Survivable Virtual Networks

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1 Jont Backup Capacty Allocaton and Embeddng for Survvable Vrtual Networks Nashd Shahrar, Shhabur Rahman Chowdhury, Reaz Ahmed, Amal Khan, Raouf Boutaba, Jeebak Mtra, and Lu Lu Davd R. Cherton School of Computer Scence, Unversty of Waterloo {nshahra srchowdhury rahmed a73khan Huawe Technologes Canada Research Center Huawe Technologes Abstract A key challenge n Network Vrtualzaton s to effcently map a vrtual network (VN) on a substrate network (SN) whle accountng for possble substrate falures. Ths s known as the Survvable Vrtual Network Embeddng (SVNE) problem. The state-of-the-art lterature has studed the SVNE problem from nfrastructure provders (InPs) perspectve,.e., provsonng backup resources n the SN. A rather unexplored soluton spectrum s to augment the VN wth suffcent spare backup capacty to survve substrate falures and embed the resultng VN accordngly. Such augmentaton enables InPs to offload falure recovery decsons to the VN operator, thus, provdng more flexble VN management. In ths paper, we study the problem of jontly optmzng spare backup capacty allocaton n a VN and embeddng the VN to guarantee full bandwdth n the presence of sngle substrate lnk falure. We formulate the optmal soluton to the jont optmzaton problem as a quadratc nteger program that we transform nto an nteger lnear program. We propose a heurstc algorthm to solve larger nstances of the problem. Smulaton results show that our heurstc allocates % extra resources compared to the optmal, whle executng several orders of magntude faster. I. INTRODUCTION Infrastructure provders (InPs), such as data center network operators, Internet servce provders, and transport network operators are leveragng Network Vrtualzaton (NV) to offer slces of ther networks to servce provders (SPs) [], []. NV enables InPs to better utlze ther nfrastructure,.e., substrate network (SN) and also to open new revenue streams. However, the benefts from NV come wth addtonal resource management challenges, such as effcently mappng the vrtual nodes and lnks of a vrtual network (VN) request onto substrate nodes and paths, respectvely. Ths s known as the VN embeddng (VNE) problem [3]. If a VNE soluton does not take possble substrate falures nto account, then such falures can result n degraded Qualty of Servce (QoS) for VNs, leadng to Servce Level Agreement (SLA) volatons. A VN embeddng that can survve substrate falures s known as the Survvable VNE (SVNE) [], and has receved sgnfcant attenton n the research communty [] [3]. The SVNE research lterature focuses prmarly on proactvely (protecton durng embeddng) or reactvely (restoraton after falure) provsonng backup resources n the SN, ISBN c 07 IFIP possbly dsjont from the VN s prmary resources. A rather unexplored spectrum n SVNE s to augment a VN wth adequate backup capacty and to embed the VN on an SN accordngly []. More recently, [7] has studed a weaker verson of the SVNE problem that augments the VN topology to ensure connectvty under multple substrate lnk falures. However, [7] does not guarantee the affected vrtual lnks bandwdth, and only allows the VN to operate wth degraded QoS. In ths paper, we focus on the problem of augmentng a VN wth suffcent backup capacty and embeddng the VN on an SN n a way that ensures survvablty wth guaranteed bandwdth under sngle substrate lnk falure. The motvaton for provdng protecton at the VN level comes from the shft of control to SPs, as antcpated n future Transport Software Defned Networks (T-SDNs) []. T-SDNs are the next generaton of transport networks that leverage SDN technologes and promse to provde full fledged VN nstead of tradtonal end-to-end connectvty to SPs [], [6]. SPs can thus have more control over ther vrtual slce and deploy ther own routng, traffc engneerng, and falure recovery solutons. InPs can offload some of the falure recovery tasks to SPs by augmentng the VNs wth suffcent spare capacty for backup and embeddng the VNs usng necessary dsjont paths n the SN. A recent study emprcally evaluated the mpact of provdng survvablty at the SN level, compared to that at the VN level, n a real testbed []. The study shows that: () provdng survvablty at the VN level has smlar swtchng response tme durng a falure as compared to dong the same at the SN level, and () VN level survvablty can accommodate more VNs compared to dong the same at the SN, hence, s more proftable for InPs. The ntuton behnd the latter result s that InPs need to provson more dsjont resources when ensurng survvablty at the SN level []. Another motvatng applcaton for provdng survvablty at the VN level s to ensure strong IP layer survvablty n IP over Wavelength Dvson Multplexed (WDM) networks. Strong survvablty n ths context means ensurng both connectvty and bandwdth at the IP layer durng falures n ether the IP or the WDM layer [7]. By equppng the IP layer wth necessary backup resources needed to recover from a falure, part of the falure recovery tasks can be off-loaded to the IP

2 layer. However, a major dfference between NV and IP-over- WDM s that IP routers are provsoned n fxed locatons, whereas the vrtual nodes are not assumed to have been already placed and the embeddng algorthm determnes ther placement. Therefore, solutons for IP-over-WDM networks such as [7], [8] cannot be drectly appled n NV context. As a matter of fact, the problem n NV context s more general than ts nstance n IP-over-WDM networks. In ths paper, we study the problem of jontly optmzng spare backup capacty allocaton wthn a VN and embeddng the VN on an SN to ensure survvablty under sngle substrate lnk falure, whle mnmzng resource provsonng cost n the SN. We focus on the sngle lnk falure scenaro snce ths s the most common case [9], [0]. A major challenge n solvng the problem s to jontly optmze spare backup capacty allocaton and survvable embeddng. Spare capacty allocaton and VN embeddng, when performed ndependently of one another, may lead to suboptmal or nfeasble solutons. Hence, we propose a jont optmzaton model to solve spare capacty allocaton and VN embeddng smultaneously. Specfcally, we make the followng contrbutons: We formulate the optmal soluton to the jont optmzaton problem as a Quadratc Integer Program (QIP). We also present a transformaton of the QIP nto an Integer Lnear Program (ILP). We propose a heurstc soluton to tackle the computatonal complexty of the ILP-based optmal soluton. We perform extensve smulatons to evaluate our solutons. Smulaton results show that our proposed heurstc allocates % extra resources compared to the optmal. The rest of ths paper s organzed as follows. We present the related lterature n II and contrast our work wth the state-of-the-art. In III, we present the system model and problem statement followed by a dscusson on how spare backup capacty can be allocated along a vrtual lnk. Then, we present our QIP formulaton for the jont optmzaton problem and the ILP transformaton n IV. We present the desgn of our heurstc n V. The evaluaton of our solutons are presented n VI. Fnally, we conclude wth some future research drectons n VII. II. RELATED WORK Frst, we dscuss the state-of-the-art n SVNE n II-A. Then, we present the works focused on provdng protecton at the VN level n II-B. Fnally, we contrast our approach wth those from IP-over-WDM network survvablty n II-C. A. SVNE wth Protecton at SN Rahman et al., were the frst to address the SVNE problem by formulatng the problem as a mxed nteger lnear program []. A number of subsequent research works have addressed dfferent aspects of SVNE such as substrate node falure [], [6], leveragng mult-path embeddng [8], [9], shared backup protecton [0], [], and dedcated VN topology protecton [], [3]. However, these approaches address the SVNE problem from an InP s perspectve,.e., the InP provsons backup resources n the SN, dsjont from the prmary embeddng. They do not explore the soluton space where the VN can be augmented wth suffcent resources to survve substrate falures and embed the VN accordngly. B. SVNE wth Protecton at VN Recently, an emprcal study by Wang et al., [] compared dfferent protecton schemes for NV n T-SDNs. The results from [] show that provdng protecton at the VN level can ncrease VN acceptance rato. However, [] performed a two step backup capacty allocaton and embeddng rather than jontly optmzng them. We studed a weaker verson of the SVNE problem wth VN level protecton n [7]. Ths work proposed to embed a VN on an SN n such way that VN connectvty s ensured aganst multple substrate lnk falures. However, t does not guarantee any bandwdth n case of a falure and only allow the VN to operate n a best effort manner. Barla et al., proposed separate desgn models for cloud servces that provde reslency ether at the VN or at the SN layer []. Ther VN-based reslency model employs dedcated backup consstng of addtonal vrtual lnks to reach the recovery data center. In contrast, our model elmnates the need for changng the VN topology and uses spare capacty allocated to exstng vrtual lnks to survve lnk falures. C. IP-over-WDM Network Survvablty A smlar problem to our jont optmzaton has been studed n IP-over-WDM lterature, namely, Strongly Survvable Routng (SSR). The objectve of the SSR s to ensure both connectvty and bandwdth guarantee at the IP layer durng a falure n ether IP or WDM layer. One approach for solvng the SSR s to provson spare bandwdth at the IP layer to survve IP or WDM lnk falure [7], [8], [] []. However, these solutons do not jontly optmze spare bandwdth allocaton and routng. Moreover, IP-over-WDM networks assume a fxed placement of IP routers n the network, whereas an SVNE algorthm needs to determne both VNode and VLnk mappngs. Therefore, solutons from IP-over-WDM cannot be drectly appled to our jont optmzaton problem. III. SYSTEM MODEL AND BACKGROUND We frst present basc notatons n III-A and a formal statement of the problem n III-B. We explan the concept of shared rsk groups n III-C. We then dscuss how embeddng affects spare capacty allocaton on vrtual lnks n III-D. A glossary of notatons used n the paper s provded n Table I. A. Basc Notatons ) Substrate Network: We represent the substrate network (SN) as an undrected graph, G = (V, E), where V and E denote the set of substrate nodes (SNodes) and lnks (SLnks), respectvely. The set of neghbors of an SNode u V s denoted by N (u). We assocate the followng attrbutes wth each SLnk (u, v) E: () b uv : bandwdth capacty of the SLnk (u, v), () C uv : cost of allocatng unt bandwdth on (u, v) for a VLnk. We assume that the SNodes are network

3 G = (V, E) b uv C uv Ĝ = ( ˆV, Ê) bûˆv L(û) lûu {0, } ˆPûˆv Pûˆv Sûˆv d D dûˆv Ĥûˆv Ê zûˆv {0, } zûˆv uv {0, } yûu {0, } gûˆv () {0, } gûˆv uv {0, Sûˆv } TABLE I NOTATION TABLE Substrate Network (SN) Bandwdth capacty of SLnk (u, v) E Cost of unt bandwdth on SLnk (u, v) E Vrtual Network (VN) Bandwdth demand of VLnk (û, ˆv) Ê Locaton constrant set for VNode û ˆV lûu = f u L(û), u V, û ˆV A VPath between û and ˆv, edge dsjont from (û, ˆv) An SPath representng the mappng of (û, ˆv) Ê Spare backup bandwdth allocated to (û, ˆv) Ê An SRG consstng of a set of VLnks from Ê dûˆv = f (û, ˆv) Ê belongs to SRG d D Set of VLnks that have (û, ˆv) n ther backup VPaths = f (û, ˆv) Ê s on the backup VPath of (ˆx, ŷ) Ê uv = f (u, v) E s on the embedded SPath for (û, ˆv) Ê yûu = f û ˆV s mapped to u V () = 0 f zûˆv = and d = uv = S ûˆv f uv = nodes wth suffcent capacty to swtch traffc at peak rate between any par of ports. Therefore, we do not consder any node mappng cost or node capacty constrant. ) Vrtual Network: We represent the vrtual network (VN) as an undrected graph Ĝ = ( ˆV, Ê), where ˆV and Ê represent the set of vrtual nodes (VNodes) and vrtual lnks (VLnks), respectvely. The set of neghbors of a VNode ˆv ˆV s denoted by N (ˆv). Each VLnk (û, ˆv) Ê has a bandwdth demand bûˆv. We also have a set of locaton constrants, L = {L(û) L(û) V, û ˆV }, such that a VNode û ˆV can only be provsoned on an SNode u L(û). We use a bnary varable lûu ( f û ˆV can be provsoned on u V, 0 otherwse), to represent ths locaton constrant. We denote the spare backup bandwdth allocated to a VLnk (û, ˆv) Ê that serves as a backup for other VLnks by Sûˆv. We assume the VNs are -edge connected,.e., at least two edge dsjont paths exst between any two VNodes. -edge connectvty s a necessary condton to ensure that an edge dsjont backup vrtual path always exsts for each VLnk (û, ˆv) Ê [7]. B. Problem Statement Gven an SN G = (V, E), a VN Ĝ = ( ˆV, Ê), and a set of locaton constrants L: For each VLnk (û, ˆv) Ê, optmally allocate spare backup bandwdth along a non-empty path ˆPûˆv n the VN (VPath), edge dsjont from (û, ˆv) such that bûˆv bandwdth s avalable between VNodes û and ˆv even after (û, ˆv) s affected by an SLnk falure. Map each VNode ˆv ˆV to exactly one SNode, u V. Multple VNodes from the same VN request should not be mapped to the same SNode. Map each VLnk (û, ˆv) Ê onto a non-empty substrate path (SPath) Pûˆv havng suffcent bandwdth to accommodate the prmary demand of (û, ˆv) and the spare backup bandwdth provsoned on (û, ˆv). A VLnk (û, ˆv) Ê and the VLnks on ts backup VPath ˆPûˆv are edge dsjontedly mapped to ensure a sngle SLnk falure does not affect them at the same tme. The total cost of allocatng bandwdth on the SN to embed the VN along wth the spare bandwdth s mnmum. C. Shared Rsk Group VLnks that share at least one SLnk on ther mapped SPaths share the rsk of falure snce all of them can fal f the shared SLnk fals. In a context where only SLnk falure s consdered, a set of VLnks belong to the same shared rsk group (SRG) ff they share at least one SLnk on ther mapped SPaths. On the other hand, VLnks that do not share any SLnk on ther mapped SPaths belong to dfferent SRGs. To represent the SRG membershps, we partton the VLnks nto a number of SRGs represented by the set D = {d, d, d 3,... d D }, where D Ê. A VLnk belongs to exactly one SRG d D and shares at least one SLnk on ts mapped SPath wth other VLnks n d. We use the followng decson varable to decde on a VLnk s membershp to an SRG: { ff (û, ˆv) Ê belongs to SRG d D, dûˆv = 0 otherwse. D. Spare Capacty Assgnment Model e a c f Fg.. VLnk (a, b) on backup VPaths of VLnks (c, d), (e, f), (g, h) Based on how the VLnks form dfferent SRGs durng VN embeddng, the requrement for spare backup capacty on the VLnks can be dfferent. We explan ths fact wth a smple example llustrated n Fg.. In ths fgure, VLnk (a, b) s on the backup VPaths of three other VLnks: (c, d), (e, f), and (g, h). We can assgn dfferent spare capacty on (a, b) to protect (c, d), (e, f), and (g, h), based on how these three VLnks are mapped. Consder the followng scenaros regardng ther mappngs: All three belong to the same SRG. If all three VLnks are n the same SRG, then they share at least one SLnk on ther mapped SPaths (Fg. (a)). A sngle substrate falure can affect all three VLnks. Therefore, spare backup capacty allocated on (a, b) should be suffcent to support the bandwdth requrement of all three VLnks,.e., b cd + b ef + b gh. g b d h

4 Shared Physcal Lnk (a) (c, d), (e, f), (g, h) n the same SRG (b) (c, d), (e, f), (g, h) n dfferent SRGs Shared Physcal Lnk (c) (c, d), (e, f) n same SRG, (g, h) n a dfferent SRG Fg.. Dfferent Physcal Embeddng of the VLnks n Fg. All three belong to dfferent SRG. If all three VLnks belong to dfferent SRGs, then they do not share any SLnk on ther mapped SPaths (Fg. (b)). At most one of the VLnks wll be affected by a sngle SLnk falure. Therefore, the spare backup capacty allocated on (a, b) should be suffcent to support the maxmum bandwdth requrement of these three VLnks,.e., max(b cd, b ef, b gh ). Two belong to the same SRG, the thrd n a dfferent SRG. The mapped SPaths can create multple SRGs out of these three VLnks. For example, n Fg. (c), VLnks (c, d) and (e, f) belong to the same SRG, whereas VLnk (g, h) belongs to a dfferent SRG. A sngle SLnk falure wll then affect only one group. Therefore, spare capacty allocated on (a, b) should be suffcent to support the group wth the maxmum requrement. For the group wth (c, d) and (e, f), the bandwdth requrement s b cd + b ef. For the other group, the requrement s b gh. Therefore, spare backup bandwdth on (a, b) should be max(b cd + b ef, b gh ). More formally, f a VLnk (û, ˆv) Ê s present on the backup VPaths of a set of VLnks Ĥûˆv Ê and VLnks n Ê form a set of D = {d, d, d 3,... d D } SRGs, we can generalze the spare backup bandwdth allocated to (û, ˆv) as: Sûˆv = max d D (ˆx,ŷ) Ĥûˆv d b IV. PROBLEM FORMULATION () We frst provde a Quadratc Integer Program (QIP) formulaton for the jont spare capacty allocaton and survvable embeddng problem n IV-A, followed by a dscusson on the complexty of the QIP n IV-B. We then descrbe the transformaton of the QIP to an ILP n IV-C. A. Quadratc Integer Program Formulaton We frst present our decson varables ( IV-A). Then we ntroduce the constrants ( IV-A) followed by the objectve functon of our formulaton (( IV-A3). ) Decson Varables: For each VLnk (û, ˆv) Ê, there s a backup VPath ˆPûˆv that provdes protecton to that VLnk from a sngle SLnk falure. When any SLnk on the VLnk s mapped SPath fals, ˆPûˆv provdes the same bandwdth bûˆv between the VNodes û and ˆv. The followng decson varable defnes whether a VLnk (û, ˆv) Ê belongs to the VPath protectng a VLnk (ˆx, ŷ) { Ê: zûˆv f (û, ˆv) Ê s on the backup VPath of (ˆx, ŷ) = Ê, 0 otherwse. Note that, zûˆv ûˆv = 0, snce a VLnk s backup VPath has to be edge dsjont from tself. The followng decson varable ndcates the mappng between a VLnk (û, ˆv) Ê and an SLnk (u, v) E: { f (û, ˆv) Ê s mapped to (u, v) E, uv = 0 otherwse. The VNode to SNode mappng s denoted usng the followng decson varable: { f û ˆV s mapped to u V, yûu = 0 otherwse. VLnks that share at least one SLnk on ther mapped SPaths belong to the same SRG ( III). SRG membershp s defned usng the decson varable dûˆv, defned n III-C. ) Constrants: a) VNode Mappng Constrants: () and (3) ensure that each VNode of a VN s provsoned on an SNode satsfyng the provded locaton constrants. Moreover, () constrants an SNode to host at most one VNode from the same VN. Note that VNode mappng follows from the VLnk mappng, snce there s no cost assocated wth the VNode mappng. û ˆV, u V : yûu lûu () û ˆV : u V yûu = (3) u V : yûu () û ˆV b) Backup VPath Contnuty Constrants: A VLnk n a VN s protected by a VPath n the VN to survve a sngle SLnk falure. () ensures contnuty of a backup VPath protectng a VLnk (ˆx, ŷ) Ê: (ˆx, ŷ) Ê : f û = ˆx (zûˆv zˆvû ) = f û = ŷ ˆv N (û)\{ŷ} 0 otherwse () c) VLnk Mappng Constrants: Frst, we ensure that every VLnk s mapped to a non-empty set of SLnks usng (6). Then, we ensure that the n-flow and out-flow of each SNode s equal, except for the SNodes where the endponts of a VLnk are mapped usng (7). (7) ensures that the non-empty set of SLnks correspondng to a VLnk s mappng form a sngle contnuous SPath. (û, ˆv) Ê : û, ˆv ˆV, u V : (u,v) E v N (u) uv (6) ( uv vu) = yûu yˆvu (7)

5 The bnary nature of the VLnk mappng decson varable and the flow constrant prevents any VLnk from beng mapped onto more than one SPaths, thus, restrctng the VLnk mappng to the Mult-commodty Unsplttable Flow Problem [6]. We also need to ensure that we do not over-commt the bandwdth resources we have on the SLnks. To do so, we frst compute the spare backup bandwdth allocated to a VLnk (û, ˆv) Ê usng () as follows: Sûˆv = max d (ˆx,ŷ) Ê\{(û,ˆv)} zûˆv d (û,ˆv) Ê b (8) Then, the followng constrant prevents any over-commt of the bandwdth resource n the SLnks: (u, v) E : uv (bûˆv + Sûˆv ) b uv (9) Note that (9) s a cubc constrant, snce Sûˆv s quadratc accordng to (8). Therefore, we take the followng steps to lnearze Sûˆv n order to ensure that (9) remans quadratc. Frst, we ntroduce a new varable gûˆv (), defned as follows: { gûˆv 0 f z ûˆv () = = and d =, otherwse. Essentally, for a gven VLnk (û, ˆv) Ê, the zero values of gûˆv () nduce a set of VLnks such that they belong to the same SRG and have (û, ˆv) on ther backup VPaths. The value () s set usng the followng constrant: of gûˆv (û, ˆv) Ê, (ˆx, ŷ) Ê, d D : zûˆv + d + gûˆv (0) We can use gûˆv () to rewrte (8) n a lnear form as follows: Sûˆv = max ( gûˆv ()) b () d (ˆx,ŷ) Ê\{(û,ˆv)} Snce our objectve functon wll be a mnmzaton functon, we defne gûˆv () so that settng t to mnmzes the value of Sûˆv, unless t s constraned to be 0 accordng to (0). Ths constraned case wll only occur when both zûˆv and d are, as defned by (0). d) Dsjontedness Constrants: The mapped SPaths of the VLnks from an SRG d, must be edge dsjont from the mapped SPaths of the VLnks from a dfferent SRG d j ( j ). Ths s ensured by (). (3) ensures that two VLnks from the same SRG share at least one SLnk on ther mapped SPaths. Note that a VLnk (û, ˆv) Ê cannot be present n more than one SRGs, whch we ensure by (). (u, v) E, (û, ˆv) d, (ˆx, ŷ) d j s.t. j : dûˆv + d j + uv + vu + x uv + x () vu 3 d D, ((û, ˆv), (ˆx, ŷ)) d d s.t. (û, ˆv) (ˆx, ŷ), (u, v) E : dûˆv + d (ˆx, ŷ) Ê : + uv + vu + x uv + x vu = (3) d = () d D To ensure survvablty of the VN under sngle SLnk falure, the mapped SPath of a VLnk cannot share any SLnk wth the mapped SPaths of the VLnks present on ts backup VPath. The followng constrant ensures ths dsjontedness: (u, v) E, ((û, ˆv), (ˆx, ŷ)) Ê Ê s.t. (û, ˆv) (ˆx, ŷ) : z ûˆv + uv + vu + x uv + x vu () 3) Objectve Functon: As per the problem statement presented n III-B, we do not consder any node mappng cost n our VN embeddng. Thus, our cost functon mnmzes the total cost of provsonng the workng and backup bandwdth for the VLnks of a VN on the SLnks of an SN. Ths gves us the followng objectve functon: mnmze uv C uv (bûˆv + Sûˆv ) (û,ˆv) Ê (u,v) E B. Complexty of the QIP (6) Our formulaton for the jont optmzaton problem has a quadratc constrant (9) and a quadratc objectve functon (6). Therefore, the QIP presented n IV-A s a Quadratcally Constraned Quadratc Program (QCQP) and falls nto the general category of the Quadratc Assgnment Problem (QAP) [7]. Solvng a QAP s computatonally expensve and s known to be N P-hard [8]. Sahn et al., proved that even fndng an ɛ approxmate soluton of QAP s NP-hard [9]. In the next secton, we present the steps to lnearze the QIP by usng a technque smlar to the one dscussed n [30]. C. ILP Transformaton For the purpose of lnearzaton, we frst put a bound on the spare backup bandwdth of a VLnk (û, ˆv) Ê,.e., S ûˆv: 0 Sûˆv λ, where λ s a very large nteger. We also ntroduce a new nteger varable uv, { defned n terms of uv as follows: uv = Sûˆv f uv =, 0 f uv = 0. The followng constrants enforce the relatonshp between uv and uv: (u, v) E, (û, ˆv) Ê : uv 0 (7) (u, v) E, (û, ˆv) Ê : S ûˆv λ ( uv) uv (8) To elaborate, when uv = 0, constrants (7) and (8) become uv 0 and Sûˆv λ uv, respectvely. Snce λ s a very large number by defnton, the constrants fnally reduce to uv 0. On the other hand, when uv =, constrant (7) and (8) become uv 0 and Sûˆv uv, respectvely. In ths later case, constrant (8),.e., Sûˆv uv domnates. Fnally, f we nclude uv n the mnmzaton objectve functon, the smallest possble value of uv wll be used to mnmze the value of the objectve functon, yeldng uv = Sûˆv. We now rewrte the capacty constrant (9) as the followng lnear constrant usng uv. (u, v) E : ( uv bûˆv + uv) b uv (9) (û,ˆv) Ê Smlarly, the quadratc objectve functon can be wrtten n a lnearzed form as follows: mnmze uv C uv bûˆv + C uv uv) (û,ˆv) Ê (u,v) E (0)

6 V. HEURISTIC DESIGN There are several challenges n desgnng a heurstc for the jont optmzaton of spare backup capacty allocaton and survvable embeddng problem. We frst dscuss these challenges n detal and brefly explan how we addressed them n V-A. Then, we present our heurstc algorthm n V-B. A. Challenges ) Selecton of Backup VPath: The frst challenge s the selecton of a backup VPath from an exponental number of VPaths. One of the most effcent shared backup path protecton schemes for sngle layer protecton s p-cycle based protecton [3]. A p-cycle s formed by connectng the spare capacty n a network n a rng lke structure. For nstance, a Hamltonan p-cycle that passes through all the nodes n a network once, can provde recovery paths for ether an on-cycle VLnk falure or a straddlng VLnk falure [3]. However, fndng a Hamltonan cycle s N P-Complete. Addtonally, a longer p-cycle requres all the VLnks on the p-cycle to be mapped on dsjont SPaths to ensure that the VLnks belong to separate SRGs. Such constrants can lead to longer mapped SPaths or even falure to embed the VN. We address ths ssue by keepng an overlap between the backup allocatons and mappng phases. We frst fnd a mappng usng estmated backup allocatons and then re-optmze the backup allocatons usng a p-cycle based technque. ) VNode Mappng: The next challenge comes from node mappng, a combnatoral optmzaton problem [33]. Although we are not consderng any cost for VNode mappng, the order of VNode mappng and VNode mappng tself have a profound mpact on subsequent VLnk mappng. Selecton of an SNode for mappng a VNode nfluences later VNodes mappngs and the possble SPaths that can be chosen for the VLnks. Such constrants subsequently mpact the cost of a soluton. In our approach, we map the VNodes from the most constraned to the least constraned, to mnmze chances of mappng falure at a later stage due to resource exhauston. We measure the constrant of a VNode û by ts nodal degree.e., N (û). To fnd the mappng of a VNode, we select the SNode ncurrng the least cost from the locaton constrant set of the VNode to mnmze total cost of the embeddng. 3) Dsjont SPath Computaton: Fnally, VLnk mappng on unsplttable SPath wthout the dsjontedness constrants s at least as hard as solvng the N P-Hard Mult-commodty Unsplttable Flow Problem [6]. Furthermore, fndng the optmal set of dsjont SPaths s an N P-Complete problem [3]. In our soluton, we teratvely compute the dsjont SPaths usng a modfed verson of Djkstra s shortest path algorthm [3]. B. Heurstc Algorthm Our heurstc algorthm s presented as a pseudocode n Alg.. Alg. solves the jont optmzaton problem n two steps: () t estmates the spare backup bandwdth on the VLnks, determnes the dsjontedness requrements based on ths estmaton, and performs a VN Embeddng, () t dentfes the longest cycle consstng of VLnks belongng to separate Algorthm : Embed VN wth Protecton functon VNEmbeddng(G, Ĝ, C, σ) D {d, d,... d Ê } // Set of all SRGs 3 û ˆV : nmapû NIL (û, ˆv) Ê: Sest ûˆv 0, Λ ûˆv σ, SRGûˆv d, emapûˆv φ, backupûˆv φ ˆV Sort û ˆV n decreasng order of N (û) 6 foreach û ˆV do 7 foreach ˆv N (û) do 8 backupûˆv GetBackup(Ĝ, (û, ˆv), Λ, emap) 9 foreach (ˆx, ŷ) backupûˆv do max(sest, b ûˆv) 0 S est f SRGûˆv = SRG then Fnd d j D s.t. SRGûˆv d j, (û, ˆv) Ê 3 SRGûˆv d j Ĥ {(â, ˆb) Ê (ˆx, ŷ) backup âˆb } foreach (â, ˆb) Ĥ do f SRGûˆv = SRGâˆb then 6 Fnd d j D s.t. SRGûˆv d j, (û, ˆv) Ê 7 SRGûˆv d j 8 bestû NIL, Q best φ, c best û 9 foreach l L(û) do 0 foreach ˆv N (û) do W C (m, n) {(u, v) E SRG SRGûˆv (u, v) emap, (ˆx, ŷ) Ê}: W mn 3 f nmapˆv NIL then Qûˆv CWSP(G, l, nmapˆv, bûˆv, W ) else Qûˆv mn m L(ˆv) {CWSP(G, l, m, b ûˆv, W )} 6 f ˆv N (û) : Qûˆv = φ then c 7 else c ˆv N (û) Q ûˆv 8 f c < c best then 9 bestû l, Q best û Qû, c best c 30 f bestû = NIL then return {φ, φ, φ, φ} 3 nmapû bestû 3 foreach ˆv N (û) and nmapˆv NIL do 33 emapûˆv Q best ûˆv, Λ ûˆv Cost(Q best ûˆv ) 3 {backup, S} UpdateBackup(Ĝ, backup, SRG) 3 return {nmap, emap, backup, S} SRGs and re-optmzes backup bandwdth allocatons usng the cycle. Upon success, Alg. returns nmap, emap, backup, and S representng the VNode mappng, VLnk mappng, backup VPaths, and spare backup capactes, respectvely. Alg. starts by ntalzng the estmated spare backup bandwdth of each VLnk, Sûˆv est to 0 and by placng all the VLnks nto a sngle SRG d. It then proceeds to map the VNodes from the most constraned to the least constraned ones,.e., n decreasng order of ther degrees. If two VNodes have equal degrees then we arbtrarly select one of them. For a VNode û, Alg. frst fnds estmated backup VPaths for each VLnk ncdent to û by teratvely nvokng GetBackup procedure (Alg. ). Alg. nvokes Constraned Weghted Shortest Path (CWSP) procedure to compute a VPath wth at

7 Algorthm : Compute Backup VPath of a VLnk functon GetBackup(Ĝ, (û, ˆv), Σ, emap) foreach (ˆx, ŷ) Ê do 3 f S est b ûˆv then W eght est else f emap = φ or mn b resdual uv (u,v) Q then W eght est (b ûˆv S est) Σ 6 else W eght est 7 W eght est ûˆv 8 return CWSP(Ĝ, û, ˆv, b ûˆv, W eght est ) bûˆv least bûˆv bandwdth between û and ˆv n the VN Ĝ, accordng to a provded weght functon. Alg. frst computes the weght functon W eght est for all the VLnks and nvokes CWSP to obtan the backup VPath between û and ˆv. Alg. assgns lower weghts to VLnks wth already assgned backups to enhance the sharng of spare bandwdth by more VPaths (Lne 3). The weght functon also takes the mappng cost of an already mapped VLnk (ˆx, ŷ) nto account and assgns (ˆx, ŷ) a weght proportonal to the mappng cost Λ. In lne, a specal case occurs when a VLnk (ˆx, ŷ) s not yet mapped. For ths case, we set Λ to use the average SPath length (σ) as an ndcator of future cost. Fnally, an nfnte weght s set to the VLnks whose mapped SPaths do not have adequate resdual capacty to exclude them from the search space (Lne 6). After computng the estmated backup VPath backup est ûˆv, Alg. updates S est for all (ˆx, ŷ) backup est ûˆv wth the maxmum value of bûˆv (Lne 0). It then places (û, ˆv) and (ˆx, ŷ) nto dfferent SRGs (Lne 3). Fnally, t places (û, ˆv) and all other VLnks that use (ˆx, ŷ) n ther backup VPaths nto dfferent SRGs (Lne 7). After fndng the backup VPaths and SRGs of all the ncdent VLnks of a VNode û, Alg. fnds the mappng of û and VLnks ncdent to û. It terates over all canddate SNodes l L(û) and selects the one that results n the least cost mappng for all the VLnks ncdent to û (Lne 0 9). For a specfc l L(û) and ˆv N (û), f ˆv s already mapped to nmapˆv, Alg. computes CWSP from l to nmapˆv (Lne ), whle satsfyng capacty constrants and SRG constrants n the SN (usng the weghts n W ). To do so, Alg. dentfes the set of SLnks that the mappng of (û, ˆv) should be dsjont from and assgns as ther weghts (Lne ). On the other hand, f ˆv s not mapped yet, t computes CWSPs from l to the SNodes m L(ˆv) and selects the CWSP wth the mnmum cost (Lne ). After mappng a VNode û, Alg. maps the VLnks whose both endponts have already been mapped and updates Λ of the mapped VLnks (Lne 33). The last phase (Alg. 3) of our heurstc leverages the concept of p-cycle based protecton to optmze the spare backup bandwdth Sûˆv for each mapped VLnk (û, ˆv) Ê. Alg. 3 frst fnds the longest cycle ˆR n Ĝ such that any par of VLnks n ˆR do not share any SLnk n ther mappngs (Lne 3). Recall from III-D that each (ˆx, ŷ) ˆR belongs to dstnct SRGs for a sngle SLnk falure. Therefore, Alg. 3 allocates the maxmum of the demands of all the VLnks n ˆR to each Algorthm 3: Reconfgure Backup VPaths of all VLnks functon UpdateBackup(Ĝ, backup, SRG) (û, ˆv) Ê : S ûˆv 0 3 ˆR longest cycle n Ĝ such that no VLnk par n ˆR shares an SLnk on ther mapped SPaths (ˆx, ŷ) ˆR : S max (û,ˆv) ˆR{bûˆv } foreach (û, ˆv) Ê do 6 foreach (ˆx, ŷ) Ê \ {(û, ˆv)} do 7 f SRGûˆv = SRG then W eght 8 else f S bûˆv then W eght 9 else W eght (bûˆv S ) 0 W eghtûˆv backupûˆv CWSP(Ĝ, û, ˆv, b ûˆv, W eght) (ˆx, ŷ) backupûˆv : S max(s, bûˆv ) 3 return {backup, S} S ˆR (Lne ). It then recomputes backup VPath backupûˆv for each (û, ˆv) Ê usng a process smlar to Alg.. However, Alg. 3 utlzes the mappng nformaton to better compute the backup VPaths. It does so by settng as the weght of the VLnk (ˆx, ŷ) f (û, ˆv) and (ˆx, ŷ) are n the same SRG (Lne 7). Alg. 3 also enhances the spare capacty sharng by settng unt weghts to the VLnks havng already assgned spare capactes (Lne 8). Alg. 3 then nvokes the CWSP procedure wth the weght functon to compute backupûˆv. Fnally, S for each (ˆx, ŷ) backupûˆv s updated accordngly (Lne ). C. Runnng Tme Analyss The CWSP procedure s mplemented usng a modfed Djkstra s shortest path algorthm, takng nto account the constrants and weghts. Djkstra s algorthm usng a mnprorty queue on G runs n O( E + V log V ) tme. CWSP s nvoked O( ˆV Lδ ) tmes n Alg., where L and δ are the maxmum sze of a locaton constrant set and maxmum degree of a VNode, respectvely. Therefore, the overall runnng tme of the heurstc s O( ˆV Lδ ( E + V log V )). VI. EVALUATION We evaluate our proposed solutons for the jont optmzaton problem through extensve smulatons. We brefly dscuss the compared approaches n VI-A, smulaton setup n VI-B followed by the performance metrcs n VI-C. Fnally, we descrbe our evaluaton results focusng on the followng aspects: () mpact of SN ( VI-D), () mpact of VN ( VI-E), and () scalablty of the solutons ( VI-F). A. Compared Approaches We mplemented the ILP-based optmal soluton, Opt-ILP, presented n IV-C usng IBM ILOG CPLEX C++ lbrares and compare that wth a C++ mplementaton of the heurstc. However, Opt-ILP was unable to scale beyond very small problem nstances. Therefore, we mplemented a smpler varant of Opt-ILP, called Max-ILP, to use as a baselne. Desgn of Max-ILP s motvated by an observaton from the results that for a VLnk (û, ˆv), Opt-ILP places the VLnks n Ĥûˆv nto separate SRGs whenever possble, thus preferrng more

8 Cost (x0 3 ) Cost (x0 3 ) Executon Tme (s) Opt-ILP Max-ILP Heurstc SN Sze (Number of SNodes) Opt-ILP Max-ILP Heurstc VN Sze (Number of VNodes) Opt-ILP Max-ILP Heurstc VN Sze Cost (x0 3 ) Cost (x0 3 ) Executon Tme (s) Opt-ILP Max-ILP Heurstc SN LNR Fg. 3. Impact of SN Topology Opt-ILP Max-ILP Heurstc VN LNR Fg.. Impact of VN Topology Opt-ILP Max-ILP Heurstc VN LNR Mean SPath Length Mean SPath Length Opt-ILP-SPath Opt-ILP-VPath Max-ILP-SPath Max-ILP-VPath Opt-ILP-SPath Opt-ILP-VPath Fg.. Scalablty Analyss sharng of the spare capacty. Hence, we constraned any par D. Impact of SN Topology of VLnks n Ĥûˆv to be n separate SRGs durng embeddng. Dong so allowed us to exclude the decson varable d from (8) and reduce complexty. For Max-ILP, (8) was modfed to take the maxmum demand of the VLnks n Ĥûˆv as Sûˆv. B. Smulaton Setup We evaluate the compared approaches on both small and large scale settngs. Snce VNs are stll not wdely deployed, the topologcal propertes of VNs and SNs are not well understood yet. Hence, we vary number of nodes (sze) and lnk to node rato (LNR) of VNs and SNs. For each smulaton run, we generate an SN and random VNs wth the desred property. In small scale, SN sze s vared between 0 90 nodes, whle VN sze ranges from 3 nodes. For larger scale, VN sze ranges from 0 00 nodes on 00 and 000 node SNs. We vary the connectvty of both SNs and VNs by varyng the LNR from.0 to.60. We set VLnk demand 0% of the SLnk bandwdth. For each SN, the performance metrcs are measured by takng the mean over all VNs. Smulatons are performed on a machne wth 8-core.0 Ghz Intel Xeon E-60 processors and 6GB of RAM. C. Performance Metrcs ) Cost: The cost of embeddng a VN computed usng (0). We set a unt cost for allocatng bandwdth on an SLnk, therefore, (0) drectly represents resource consumpton. ) Executon Tme: The tme requred for an algorthm to fnd an embeddng of a VN. 3) Mean SPath Length: The mean length of the SPath used to map a VLnk of a VN. ) Mean VPath Length: The mean length of the VPath used as a backup path for a VLnk n a VN. Heurstc Executon Tme (s) SN LNR Max-ILP-SPath Max-ILP-VPath Heurstc-SPath Heurstc-VPath SN-00 SN-000 VN LNR Heurstc-SPath Heurstc-VPath VN Sze Fg. 3(a) presents embeddng costs for dfferent SN szes, whle keepng the VN sze and SN LNR fxed at and.8, respectvely. We frst observe that Max-ILP very closely approxmates Opt-ILP. Compared to Max-ILP, the heurstc provsons % addtonal resources on average over all test cases. For a fxed LNR, wth ncreasng SN sze, embeddng costs ncrease for all approaches. As SN szes ncrease, canddate SNodes of the VNodes of a VN are placed far apart from one another, thus, contrbutng to larger costs. However, for a fxed SN sze, wth ncreasng SN LNR, costs decrease for all three approaches as observed n Fg. 3(b). We explan ths behavor wth the help of Fg. 3(c) n the followng. Fg. 3(c) presents mean SPath and VPath lengths by varyng SN LNR. At the lowest end of the LNR spectrum, Max-ILP and heurstc fal to fnd suffcent dsjont SPaths, mposed by the SRG constrants, resultng n nfeasble solutons. However, Opt-ILP s able to fnd a soluton wth a very hgh cost by reducng the number of SRGs. Reducng the number of SRGs s benefcal n SNs wth lower LNRs, snce t s hard to fnd suffcent dsjont SPaths n such SNs. In addton, a dsjont SPath becomes sgnfcantly longer than the correspondng non-dsjont SPath between the same par of SNodes n an SN wth lower LNR. As SN LNR ncreases, the number and length of the dsjont SPaths ncrease and decrease, respectvely. As a consequence, cost decreases for all three approaches as shown n Fg. 3(b). Fg. 3(c) shows that mean VPath lengths for all three approaches ncrease ntally wth ncreasng SN LNR. However, when VPath lengths get very close to the VN dameter, they reman almost constant wth ncreasng SN LNR. The ntal ncrease n mean VPath length s due to the Mean VPath Length Mean VPath Length

9 use of the same VLnks by more VPaths, leadng to more sharng of spare backup bandwdth. On the other hand, SNs wth lower LNRs cannot satsfy the dsjontedness constrants mposed by longer VPaths, hence, all the approaches select shorter VPaths resultng n more spare bandwdth and more cost (Fg. 3(b)). E. Impact of VN Topology Fg. (a) presents embeddng costs for dfferent VN szes, whle keepng the SN sze, SN LNR, and VN LNR fxed at 0,.8, and., respectvely. Fg. (b) and Fg. (c) compare embeddng cost, mean SPath and VPath lengths by varyng the VN LNR. The key takeaway from these fgures s that both cost and mean SPath length ncrease wth ncreasng VN sze and VN LNR. Ths s due to more dsjontedness constrants mposed by the hgher number of SRGs and shorter VPaths nduced by both larger and denser VNs (Fg. (c)). F. Scalablty Analyss Fg. shows the executon tmes for the compared approaches to demonstrate ther scalablty. As the problem sze ncreases n terms of VN sze, VN LNR, and SN sze, the executon tme grows for all the approaches except for the case of heurstc executon tmes aganst VN sze. The decrease n heurstc s executon tmes wth ncreasng VN sze s due to the reducton of the SN soluton space mposed by the hgher number of SRG constrants. In contrast, the executon tme of Opt-ILP and Max-ILP ncrease exponentally, lmtng ther applcablty to smaller problem nstances. As we can see from Fg. (a) and Fg. (b), wth our current hardware, both Opt- ILP and Max-ILP ht a celng n terms of VN sze or VN LNR. Whereas, the heurstc can solve much larger problem nstances. Even for the successful cases, Opt-ILP and Max-ILP requre several orders of magntude more tme to solve smlar problem nstances compared to the heurstc. Furthermore, the heurstc s able to fnd solutons wthn a reasonable tme lmt for much larger problem nstances (Fg. (c)),.e., VNs wth 0 00 nodes and 8 lnks on 00 and 000 node SNs. VII. CONCLUSION AND FUTURE WORK In ths paper, we have proposed a novel soluton to the SVNE problem. Instead of addressng the problem at the SN level, we have addressed the problem at the VN level,.e., the VN s augmented wth suffcent spare backup bandwdth and embedded on the SN accordngly to ensure survvablty aganst sngle substrate lnk falure. We have formulated the optmal soluton to ths jont optmzaton problem as a QIP and transformed t nto an ILP. We have also proposed a heurstc to tackle the computatonal complexty of the optmal soluton. Smulaton results show that our heurstc allocates % addtonal resources compared to the optmal soluton, whle executng several orders of magntude faster. In the future, we plan to extend ths work to vrtual fabrcs where the bandwdth requrement s expressed as parwse bandwdth rather than as per lnk bandwdth requrement. We ntend to study the resource allocaton challenges to ensure survvablty at the fabrc level compared to dong the same at the VN level. 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