Sub-Graph p-cycle formation for span failures in all-Optical Networks
SSub-Graph p-cycle formation for span failures inall-Optical Networks
Varsha Lohani
Electrical EngineeringIndian Institute of Technology Kanpur
Kanpur, [email protected]
Anjali Sharma
Electrical EngineeringIndian Institute of Technology Kanpur
Kanpur, [email protected]
Yatindra Nath Singh
Electrical EngineeringIndian Institute of Technology Kanpur
Kanpur, [email protected]
Abstract —p-Cycles offer ring-like switching speed and mesh-like spare capacity efficiency for protecting network againstlink failures. This makes them extremely efficient and effectiveprotection technique. p-Cycles can also protect all the links ina network against simultaneous failures of multiple links. But ithas been mostly studied for single link failure scenarios in thenetworks with the objective to minimize spare capacity underthe condition of restorability. For large networks, useof p-cycles is difficult because their optimization requires anexcessive amount of time as the number of variables in thecorresponding Integer Linear Program (ILP) increase with theincrease in the network size. In a real-time network situation,setting up a highly efficient protection in a short time is essential.Thus, we introduce a network sub-graphing approach, in which anetwork is segmented into smaller parts based on certain networkattributes. Then, an optimal solution is found for each sub-graph. Finally, the solutions for all the sub-graphs is combinedto get a sub-optimal solution for the whole network. We achievedbetter computational efficiency at the expense of marginal sparecapacity increases with this approach.
I. I
NTRODUCTION
In the last few years, due to the proliferation of variety ofservices, including multimedia and cloud computing applica-tions, there is enormous growth in Internet traffic. Accordingto the Cisco Annual Internet Survey, nearly two-thirds of theworld’s population will have Internet connectivity by 2023.The number of nodes connected to the IP networks would bemore than three times the number of people worldwide. In2023, average of fixed global broadband speeds will hit 110.4Mbps, up from 45.9 Mbps in 2018 [1]. Such massive trafficvolume has been feasible due to optical networks, and hasbeen the driving force behind their further evolution.Optical networks need to carry an enormous amount oftraffic while maintaining service continuity even in the pres-ence of faults. Failure of even a single link will results in lossof a substantial amount of data if not protected automaticallyand restored in a very short time after the failure. Therefore,survivability against link or path failures is an essential designrequirement for the high-speed optical networks. The goal ofa survivability scheme is to offer reliable services for large span, link and edges are used interchangeably in this paper volume of traffic even on the occurrence of failures as wellas abnormal operating conditions [2].p-Cycles are one of the best methods to provide protectionand achieve much faster restoration speeds in case of failures.But as the network size becomes large, finding an optimumsolution for p-cycle based protection becomes a time consum-ing task. We are trying to resolve this problem by breakinga bigger problem into smaller sub problems which can besolved in parallel on different machines. But we need to findout the best strategy of partitioning the network graph intosub-graphs with optimum size, to minimize the computationtime without much increase in the spare capacity requirement(i.e., minimizing sub-optimality).In this paper, section II discusses related research doneon p-cycles and graph partitioning. In section III, fundamen-tal concepts of graph theory are explained. The essentialunderstanding of graph clustering and its significance insurvivability schemes has also been discussed in section III.In section IV, we present our graph partitioning methods, andin section V, we present resulting p-cycles and spare capacityrequirement after partitioning. These are compared with theresults for the formation of p-cycles in the whole networkwithout any partitioning. In section VI, results for differenttopologies are compared to understand if the partitioning ap-proach is a better and efficient strategy to solve the problem offinding p-cycles in a large network. Finally, the observations,future work, and conclusions are presented in the last section.II. P -C YCLES
Link protection schemes are used to restore the trafficthrough the faulty link via another path joining the twoendpoints of the link [3]. The other links on that primary pathremain as it is. In this strategy, every link is provisioned witha backup path. The link protection schemes are fast becauseof faster fault localization. Two of the link protection schemesare Ring cover [4] and p-cycles. In Ring Cover, cyclic pathsare identified in the network such that each edge is traversedby at least one cycle. In this scheme, some of the linksmay be covered with more than one cycle, which results in Fiber cut, human-made errors or natural disasters such as earthquakes,hurricanes, etc. Here, clustering and partitioning are used interchangeably a r X i v : . [ c s . N I] F e b ig. 1: (A) A p-cycle, p ( a-b-c-e-d-g-f-h-a ), (B) A failed on-cycle link i ( a-h ), p-cycle p provides single unit of backuppath ( a-b-c-e-d-g-f-h-a ), and (C) A failed straddling link i ( a-d ), p-cycle p provides two units of backup path ( a-h-f-g-d )and ( a-b-c-e-d )additional redundancy/ required spare capacity. Therefore tominimize spare capacity, the double-cycle ring cover scheme[5] was proposed. Each edge is equipped with protection bytwo unidirectional cycles. Where each cycle in the oppositedirection on the shared link. The spare capacity reduces to N in this configuration. There are no cycles formed with theworking capacity as done in Ring Cover protection. One ofthe drawbacks of double cycle ring covers is different signaldelays in the forward and backward directions. Further, therestoration times are also different in the two directions.In ring cover and double-cycle ring cover, only on-cyclefailures are restored. If the straddling link failures can also berestored by the cycle, we can further reduce the spare capacityrequirement. This kind of configuration is called p-Cyclebased protection. The concept of p-cycles was first introducedby Grover et al. [6]. p-Cycles in an optical mesh networkprovide the same switching speed as the ring based protection ( < ms ) and capacity efficiency as in mesh networks [6],[7]. Whenever there is a link failure, each p-cycles based onwhether the failed link is on-cycle or straddling to the p-cycle,will provide a single unit (on-cycle) or two units (straddling )of protection to working capacity as shown in fig. 1.A good amount of work has been done in the past thathighlights the advantages of p-cycles for the protection andrestoration of traffic [6], [7]. The p-cycles can also be used forprotecting nodes [8], paths [9], [10] and path-segments [11] inaddition to links in an optical mesh network. For singlelink protection, optimum required p-cycles can be computedusing an ILP (integer linear program). The main objective ofthe ILP is to choose the number of p-cycles to provide protection while minimizing the spare capacity to be used.The concept of the Hamiltonian cycle was introducedin the literature for a single link failure scenario while allthe links are carrying some traffic to be protected. It is anefficient solution when all the links are almost equally loaded,as compared to the multiple simple arbitrary p-cycles [12]based protection. It also mitigates the problem of loop-back only end-nodes are part of the cycle, and not the link Bi-directional Line Switched Ring A straddling link must have its edge nodes on the p-cycle, but it’s not partof the cycle. cycle that traverses each node once [13].The p-cycle protection in large networks have also beenexplored using Multi-Domain approach [15], [16], [17], [18],[19]. The idea is to partition the whole network into multipledomains using graph clustering algorithms. p-Cycles can becomputed in each domain independently, forming intra-domainp-cycles. For protecting the links connecting different do-mains, separate inter-domain p-cycles are computed. Normally,each domain is an independently administered entity, andhence they are clearly defined by using some parameterslike hop-length. The time taken by individual (small sized)domains to calculate the number of p-cycles (or spare capacity)is also expected to be less.We would like to take clue from multi-domain protectionand use partitioning of large sized optical networks to findthe near optimal solutions using p-cycles in each partitionindependently. We can allow the node and links to be sharedacross two partitions to avoid inter-domain p-cycles. We willnow explore how the partitions should be created and whatshould be their optimal sizes. One should note that thiswill normally be feasible only when the whole network isadministered by single entity.III. O PTICAL N ETWORK G RAPH AND C LUSTERING
The knowledge of graph theory is imminent to appreciatethe work reported in this paper. A network graph G is definedas a set of vertices V , indexed by v and set of edges E , indexedby e . Each edge is connected to a pair of vertices e.g. ( i, j ) ∈ E , where i ∈ V , j ∈ V .In this paper, we represent an optical network as a graph G(V, E) . The optical add-drop multiplexers (OADMs) in thenetwork are the vertices, and optical fiber cables connectingthese optical multiplexers are the edges. The optical edgescontain the wavelength slots for WDM optical links for WDMoptical networks, or frequency slots for flexi-grid (or elastic)optical networks [21]. In this paper, we consider the undirectedgraphs representation of optical networks, where each span hastwo directed links (representing a fiber) in each direction.
A. Graph Clustering
Graph clustering is a method of assorting the nodes ofa graph into clusters such that there should be more intra-cluster links and fewer inter-cluster links [20]. Let a clusterrepresented by graph G s be composed of set of vertices V s ( V s ⊆ V ) and set of edges E s ( E s ⊆ E ). Therefore, the G s is sub-graph of G . Various methods of clustering a graphhave been used in data science, biological and sociologicalnetworks, data transformations, database systems, etc. [20].In this paper, we are using clustering of the graph (wealso call it graph partitioning) for resolving the problem ofsurvivability in the optical networks with lesser computation.For large networks, graph partitioning can be very helpfulin finding the near-optimal solution. Faster convergence (i.e.,smaller run-time) of Integer Linear Programs (ILP) can beachieved by running the algorithm in parallel to find a solutionfor all the sub-graphs.n the literature for p-cycle based protection, the spectralclustering algorithm has been used to partition a connectedgraph into multi-domain sub-graphs [17]. In the next section,we will compare the partitioning methods of [15], [16], [17],[18], [19] with our proposed partitioning technique.IV. P ROPOSED M ETHOD :S UB -G RAPHING USING MINIMUMCUT - SET ALGORITHM
The method proposed in this section is for both WDMbased Optical Networks and Elastic Optical Networks. Here,we consider that Routing and Resource Allocation is done apriori. In WDM based optical network, single fixed grid slotsare allocated to the connection request, which is a workingdemand, whereas, in EON, the set of slots can be allocated tothe connection request; in this case, each request is a workingdemand [22]. The set of these working demands is the workingcapacity as shown in table I. The method is only dependenton the working capacity traversing through the network.In this method, the entire graph is divided into two partsbased on minimum cut set algorithm. The pairs of nodes whichare endpoints of links in the cut-set, are used to form sub-graphs, after performing the partitioning on the super-graphusing the minimum cut-set algorithm as shown in fig.2. Oneof the sub-graph contains both the nodes in all such pairs.The other sub-graph contains only those nodes from the pairs,which already belong to it. Spectral partitioning method [23] isused for finding minimum cut-set as explained in Algorithm 1.The Algorithm 2, is used for finding the sub-graphs by furthersubdivisions.Every time out of all sub-graphs, the one with mostnumber of p-cycles is taken up for further partitioning therebyincreasing the number of sub-graphs by one. The processiterates until we obtain fundamental cycles .In algorithms 2 and 3, PCYCLE() function is used tocalculate the number of p-cycles within a graph. Algorithm2 is a recursive function used for the sub-graphing of agraph based on the p-cycles count. In each recursion, the sub-graphs with the maximum number of p-cycles is separatedfrom the list and further partitioned into two using SpectralPartitioning Function(SPF). The partitioning continues until allthe sub-graphs are simply fundamental p-cycles. Algorithm3 gives the list of p-cycles for every partition. This list isused in subsequent section for spare capacity optimization andcompute time analysis.Intuitively, as the number of partitions increases, the sizeof the sub-graphs reduces. The run time for optimizing thep-cycle is reduced, considering that the number of candidatep-cycles and the number of all other variables decrease. Theoptimizations for all the partitions are assumed to be donein parallel. However, the spare capacity requirement willincrease. Nevertheless, for real-time scenarios, a much fastercomputation of protection and hence provisioning will bepossible. Resource can be either Wavelength or Spectrum Slots Fundamental cycles are those cycles which does not have straddling links
Algorithm 1
Spectral Partitioning Function (SPF) function SPF(
G(V,E) ) (cid:46) v ∈ V , e ∈ E A = ADJ ( G ( V, E ) ); (cid:46) ADJ is used to find AdjacencyMatrix(A) of the G(V,E) D = DIAG ( G ( V, E ) ); (cid:46) DIAG is used to find DiagonalMatrix(D) of the G(V,E) L = D-A ; (cid:46) Laplacian Matrix ( L ) Compute eigenvector( ei v ) of the vertices using L ma-trix; for each vertices v do if ei v < then place vertix v to V (1) else place vertix v to V (2) end if end for Using V (1) and V (2) find Minimum cut-set; Mark Minimum cut-set as
Slider ; return g (1) , g (2) (cid:46) g ( i ) = g ( i ) ( V ( i ) , E ( i ) ) where i =1,2. end functionAlgorithm 2 Function for Sub-Graphing using p-cycle function PC YCLE S UB ( G ( V, E ) , P, P s ) Call [ g (1) , g (2) ] = SPF( G(V,E) ) p g (1) = PC YCLE ( g (1) ); p g (2) = PC YCLE ( g (2) ); if p g (1) > = 1 && p g (2) > = 1 then P s .A PPEND ([ g (1) , p g (1) ]); P s .A PPEND ([ g (2) , p g (2) ]); Sum p = null; Sum p = S UM ( P s ); (cid:46) SUM function is to find outthe sum of pcycles (not graph) P .A PPEND ( Sum p ); maxP s = M AX ( P s ); (cid:46) MAX function is tofind out the element([graph, pcycle]) with maximum p g () value from P s P s .R EMOVE ( max P s ); end if PC YCLE S UB ( maxP s [0] , P, P s ) (cid:46) input argumenttakes sub-graph with highest number of p-cycles end functionAlgorithm 3 Sub-Graphing Algorithm p org = PC YCLE ( G(V,E) ); (cid:46) original graph p-cycle count P = [ p org ]; (cid:46) p-cycle count for each partition P s = []; (cid:46) contains sub-graphs p-cycle count PC YCLE S UB ( G ( V, E ) , P, P s )ig. 2: Sub-Graphing using minimum cut-set algorithm. A. ILP Optimization for Sub-graphs
The Sub-Graphing Algorithm 3 returns the partitionedgraph with the p-cycle count for each partition. In this section,with the help of Integer Linear Programming (ILP), we arecalculating spare capacity requirement and analyzing computetime for each partition. The set of candidate p-cycles are usedas an input to the ILP.The ILP is used here for spare capacity optimization for protection against the single link failure in the networksub-graph. It may be noted that as optimization is done foreach sub-graph, and therefore failure happening in differentsub-graphs simultaneously, can be protected independently.The objective is to minimize the total spare capacity [6] foreach sub-graph, G s . In this approach, all candidate p-cyclesfor each sub-graph are pre-computed. SETS L s : Set of link in sub-graph s , indexed by l . P s : Set of p-cycles in sub-graph s , indexed by p . PARAMETERS C sl : Cost of link l in sub-graph s , W sl : Working capacity on link l in sub-graph s , N sp : Number of p-cycles p in sub-graph s , S sl : Spare capacity on link l in sub-graph s , δ pl = (cid:40) , if p-cycle p crosses link l , , otherwise, x pl = , if p-cycle p protects link l as on-cycle span, , if p-cycle p protects link l as straddling span, , otherwise, Single Link Failure - The objective is to minimize thespare capacity which is used to form the p-cycles in thesub-graph to protect all possible single link failures.
Objective:
Minimize (cid:18)(cid:80) l ∈ L C sl ∗ S sl (cid:19) Subject to: • All the working capacity of every link ( protection)is protected against any possible single link failure. W sl ≤ (cid:88) p ∈ P s x pl ∗ N sp , ∀ l ∈ L s • Enough spare capacity exists on each link to form thep-cycles. S sl ≥ (cid:88) p ∈ P δ pl ∗ N sp , ∀ l ∈ L s • N sp ≥ .The above ILP assumes that all nodes have full wavelengthconversion capability.V. R ESULTS FOR DIFFERENT TOPOLOGY
We consider the four network topologies as shown infigs.3a, 3b, 3c, and 3d. SCILAB 5.5.2, and CPLEX 12.9.0,are used for network parameters calculation and optimizationrespectively. Table I displays the amount of total workingcapacity assigned to all links in each network. For differentpartitions of network topologies, different attributes (Numberof p-Cycles, Spare Capacity, and Compute Time) are computedand plottted in figs. 4, 5, 6 and 7. We estimated two computetimes- the overall computation time in fig. 6 and the maximumcomputation time in fig. 7. The overall computation time is thesum of all sub-graphs’ computation time in the i th iteration ofpartitioning. If we run the optimization on all the sub-graphsin parallel, then the maximum computation time is selectedfrom the times taken by each of them.The working capacity distribution between the commonlinks of the two partitions is k or k : 0 (Type I) based onspare capacity minimization, where k is the working capacityon the common link. But the spare capacity for this sort ofdistribution turns out to be very high. So, we chose equaldistribution of working capacity on the common link to thetwo partitions, i.e. k/ k/ (Type II). As we can seefrom fig. 5 the value of spare capacity is high when lessnumber of partitions are made for both Type I and Type IIworking capacity distributions. But with the increasing numberof partitions, the spare capacity requirement reduces for TypeII. So, opting for k/ k/ working capacity distribution isbetter. The dynamic partitioning is also possible k/i : k/j i and j are the whole numbers. a) Net1: N06L11 (b) Net2: N8L14(c) Net3: COST239 (d) Net4: NSFNET Fig. 3: Different Network Topologies used for Simulation andOptimization.but the results are almost same as k/ k/ for less partitions.Since the spare capacity values are much higher for morepartitions, we are ignoring them.Here, we are partitioning the networks to get faster con-figuration of protection, which is desirable for large networksas well as real-time scenarios. As we can see in figs.4, 6,and 7, that after fourth partitioning iteration the results arealmost constant for all the network topologies. But, here weseek to keep partitioning as minimal as possible so that thespare capacity used is not high and the remaining capacitycan be used for another failed link which is disjoint to theprotection of first failed link. We can see in fig.4, that thenumber of candidate p-cycles are significantly reduced for allthe network topologies. As the candidate p-cycles are reduced,the number of variables in the ILP and hence the computetime also decreases. So, we can go for partitions starting from2 until the results are almost constant (i.e., the near-optimalpartition, in our case four partitions).Sub-graphing method reduces the compute time for findingp-cycle based protection configuration in real-time scenario. p-Cycles have not found practical use only due to large computetime for finding optimal strategy for big networks. But now,the protection configuration can be determined quickly as aconsequence of sub-graphing. The partitioning methods used[15], [16], [17], [18], [19] considers inter-domain and intra-domain. Each intra-domain is provisioned with different setsof p-cycle whereas for inter-domain either FIPP p-cycles areused or different approach is used to provision protection. Thisincreases the compute time for ILP.Another alternative approach in real time protection is todo local information based decision using heuristics at allnodes. Our algorithm can be compared with such actual real-time scenarios where the protection is computed dynamicallyon need basis. In such dynamic algorithms, when a p-cycle Fig. 4: Number of Candidate p-cycles Vs Number of Partitionsfor different network topologies.Fig. 5: Spare Capacity Vs Number of Partitions for differentnetwork topologiesis not protecting any working path, it should be dismantledto recover free capacity from other p-cycles on need basis.The existing p-cycles should be used whenever possible. It ispossible that we can set up a working path but can’t protectall of its links. These p-cycles can be merged or expanded tokeep the protection capacity to a minimum in such a scheme.It may not always work with optimum capacity, but can bepushed towards optimality.By partitioning, we can now periodically compute theoptimal configuration and correct the operating p-cycles. Thus,the dynamic creation, dismantling, merger and expansion ofp-cycles is not needed which will always give sub-optimalperformance. But the ILP for the partition should consider thetotal fixed capacity in all the links. The objective will change toABLE I: Assigned capacities for various network topologies Networks Total Capacity Working Capacity Maximum allowable Spare Capacity
Net1 880 300 580Net2 1120 390 730Net3 1680 510 1170Net4 1760 550 1210
Fig. 6: Sum of Compute Time Vs Number of Partitions fordifferent network topologies.Fig. 7: Maximum Compute Time Vs Number of Partitions fordifferent network topologies. provide maximum protection, and consequently sometimes theoptimal solution will not provide single fault tolerance.For real-time scenario, we can write heuristics where wecan assign p-cycles for protection without interrupting existingtraffic on all other routes, though there is no guarantee of restorability. However, we can try to merge p-cycles ofdifferent partitions and get optimal partitioning until we getenough spare capacity to protect the failed links to the extentpossible. VI. C
ONCLUSION
We have formulated sub-graphing method that splits thenetwork graph into several partitions, so that optimization foreach partition can be made in parallel. But, with this, the keyissue is the increase in the amount of spare capacity needed.So, we have tried to find out near-optimal partitions with whichthe both the requirement of minimum compute time and sparecapacity can be met. The results indicate that this method holdpromise in making p-cycles practically feasible.R
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