Traffic grooming in wdm ring networks to minimize the maximum electronic port cost,” Optic

Traf?c Grooming in WDM Ring Networks to Minimize the Maximum Electronic Port Cost

Bensong Chen,George N.Rouskas,Rudra Dutta

Department of Computer Science,North Carolina State University,Raleigh,NC27695-7534

E-mail:bchen,rouskas,dutta@https://www.360docs.net/doc/f218526039.html,

Abstract—We consider the problem of traf?c grooming in WDM ring networks.Traf?c grooming is a variant of the well-known logical topology design problem,and is concerned with the development of techniques for combining low speed traf?c components onto high speed channels in order to minimize network cost.Previous studies have focused on aggregate rep-resentations of the network cost.In this work,we consider a Min-Max objective,in which it is desirable to minimize the cost at the node where this cost is maximum.Such an objective is of high practical value when dimensioning a network for unknown future traf?c demands and/or for dynamic traf?c scenarios.We present new theoretical results which demonstrate that traf?c grooming with the Min-Max objective is NP-Complete even when wavelength assignment is not an issue.We also present a new polynomial-time traf?c grooming algorithm for minimizing the maximum electronic port cost in both uni-and bi-directional rings.We evaluate our algorithm through experiments with a wide range of problem instances,by varying the network size, number of wavelengths,traf?c load,and traf?c pattern.Our results indicate that our algorithm produces solutions which are always close to the optimal and/or the lower bound,and which scale well to large network sizes,large number of wavelengths, and high loads.We also demonstrate that,despite its focus on minimizing the maximum cost,our algorithm also performs well in terms of the aggregate electronic port cost over all ring nodes.

I.I NTRODUCTION

Wavelength division multiplexing(WDM)technology has the potential to satisfy the ever-increasing bandwidth needs of network users on a sustained basis.In WDM networks, nodes are equipped with optical cross-connects(OXCs)or optical add-drop multiplexers(OADMs),devices which can optically switch wavelengths,thus making it possible to es-tablish lightpath connections between pairs of network nodes. The set of lightpaths de?nes a logical topology,which can be designed to optimize some performance measure for a given set of traf?c demands.The logical topology design problem has been studied extensively in the literature.Typically,the traf?c demands have been expressed in terms of whole light-paths,while the metric of interest has been the number of wavelengths,the congestion(maximum traf?c?owing over any link),or a combination of the two.The reader is referred to[8]for a survey and classi?cation of relevant work.

With the deployment of commercial WDM systems,it has become apparent that the cost of network components, especially line terminating equipment(LTE),is one of the dominant costs in building optical networks,and is a more meaningful metric to optimize than,say,the number of wavelengths.Furthermore,with currently available optical technology,the data rate of each wavelength is on the order of 2.5-10Gbps,while40Gbps rates are becoming commercially available.In order to utilize bandwidth more effectively,new models of optical networks allow several independent traf?c streams to share the capacity of a lightpath.These observations give rise to the concept of traf?c grooming[5],[10],[19], [29],a variant of logical topology design,which is concerned with the development of techniques for combining lower speed components onto wavelengths in order to minimize network cost.

Given the wide deployment of SONET/SDH technology and the immediate practical interest of upgrading this infrastructure to WDM,early research in traf?c grooming focused almost entirely on ring topologies[2],[9],[13],[14],[18],[21],[25]. Since the development of MPLS and GMPLS standards makes it possible to aggregate a set of MPLS streams for transport over a single lightpath,more recently several studies have considered the problem of traf?c grooming in general network topologies[7],[17],[27],[28].A comprehensive survey and classi?cation of traf?c grooming strategies in rings as well as general network topologies can be found in[10]. Typically,the goal of existing traf?c grooming studies is to minimize the LTE cost,given a set of traf?c demands. To this end,the problem is formulated so that its objective re?ects the amount of LTE either directly(e.g.,by counting the number of lightpaths established)or indirectly(e.g.,by counting the amount of electronic–as opposed to optical –routing performed).However,most studies concentrate on some aggregate representation of the LTE cost.That is,the objective to be minimized is usually expressed as the sum,over all network nodes,of the LTE cost at each individual node. While a metric that accounts for the network-wide LTE cost is important,minimizing the total cost in the network without imposing any bound on the cost of individual nodes may result in a solution in which some nodes end up with a(very) large amount of LTE while some others with only a small amount of LTE.Such a solution has a number of undesirable properties.First,a node that requires a large amount of LTE may be too expensive or even impractical to deploy(e.g., due to high interconnection costs,high power consumption, or space requirements).Second,the resulting network can be highly heterogeneous in terms of the capabilities of individual nodes,making it dif?cult to operate and manage.Third,and more important,a solution minimizing the total LTE cost

can be extremely sensitive to the assumptions regarding the traf?c pattern,as previous studies[9]have demonstrated. Speci?cally,a solution that is optimal for a given set of traf?c demands may be far away from optimal for a different such set.Since LTE involve expensive hardware devices that are dif?cult to move from one node to another on demand, an approach that attempts to minimize total LTE cost may not be appropriate for dimensioning a network unless the network operator has a clear picture of traf?c demands far into the future and these traf?c demands are unlikely to change substantially over the life of the network.

In this work,we consider a variant of the traf?c grooming problem in which the objective is to minimize the LTE cost at the node where this cost is maximum.We believe that a Min-Max objective,such as the one we consider in this paper,is of high practical value to network designers and network operators.In particular,an approach that minimizes the maximum LTE cost at any network node is likely to be attractive because,from practical design considerations,all the network nodes are likely to be provisioned with identical equipment.Effectively,all nodes will have a cost that is dictated by the node with the maximum LTE.Such a homo-geneous network is easier to operate,manage and maintain, and is likely to be less expensive than a heterogeneous one due to the economies of scale that can be achieved when all nodes are subject to identical speci?cations.Furthermore, such an optimization approach can be of great importance to dimensioning the network for unknown and/or dynamic future demands.Speci?cally,the network designer may solve the optimization problem for a wide range of traf?c scenarios,and equip each node with an amount of LTE equal to the highest solution obtained(plus a certain fudge factor for making the solution future-proof).We also note that a similar approach was taken in[4]in a different context,namely for routing and wavelength assignment in the presence of converters. Speci?cally,an algorithm was developed for distributing a number of converters uniformly across the ring nodes rather than placing them at a single hub node;we make use of this algorithm in the heuristics we develop in this paper.

The paper is organized as follows.In Section II,we review the general traf?c grooming problem,and we introduce the Min-Max objective we consider.We also provide a formulation of the traf?c grooming problem as an integer linear program (ILP).In Section III,we present new theoretical results which demonstrate that traf?c grooming with the Min-Max objective is an inherently more dif?cult problem than other subprob-lems of logical topology design e.g.,wavelength assignment. In Section IV,we present a polynomial-time algorithm for unidirectional ring networks to minimize the maximum LTE cost at any node;the algorithm consists of two distinct parts, one for traf?c grooming and one for wavelength assignment. In Section V we extend the algorithm for bidirectional rings in which the routing of traf?c is not predetermined.We present numerical results in Section VI,and we conclude the paper in Section VII.

II.P ROBLEM D EFINITION

We now review the known results on the complexity of the routing and wavelength assignment(RW A)problem,one of the subproblems of logical topology design,and we introduce the traf?c grooming problem;the review of known results can be found in our recent survey[10],but they are repeated here for completeness.

A.The RWA Problem

From a graph theory perspective,an optical network is viewed as a digraph where each(directed) edge represents an optical?ber link between its endpoints.

A request,for a connection from node to node ,is satis?ed by(i)assigning to a path of links in from to,and(ii)assigning to a wavelength to carry the information over the links of(the path and associated wavelength constitute a lightpath).Consider a multiset of connection requests.Then,the RW A problem is to satisfy all requests in in such a way that if two requests and have paths and,respectively,that share a common edge,then they are assigned different wavelengths.The goal is to satisfy all requests in using the minimum number of wavelengths. In its pure form,the RW A problem assumes no wavelength conversion,i.e.,the same wavelength is assigned on all links along the path of request.

First note that if the network is a tree,then every pair of nodes is joined by a unique path;therefore,part(i)of the RW A problem is trivial.It is known then that wavelength assignment to minimize the number of wavelengths can be solved in polynomial time in paths and stars,but that it is NP-hard in general trees.If is a path,part(ii)is equivalent to the interval graph coloring problem,which can be solved in linear time by a greedy algorithm[15].If is a star,part (ii)is equivalent to?nding a minimum edge coloring in a bipartite graph,which is solvable in polynomial time[22].If is a general tree,the problem of minimizing the number of wavelengths is known to be NP-hard[6].In ring networks,a request can be routed in two different ways,so both parts (i)and(ii)must be addressed;this problem is known to be NP-hard[23].Finally,the RW A problem is NP-hard in general topologies[16].

B.The Traf?c Grooming Problem

New models of optical networks allow several independent traf?c streams to share the bandwidth of a lightpath.If the multiplexing and demultiplexing of lower-rate traf?c compo-nents is performed at the boundaries of the network only (i.e.,at edge routers),and the aggregate traf?c transparently traverses the optical network,this problem is equivalent to RW A.However,it is in general impossible to set up lightpaths between every pair of edge routers,e.g.,due to wavelength constraints or constraints on the number of optical transceivers at each router.Therefore,it is natural to consider optical networks in which nodes have both optical and electronic switching capabilities.Such nodes let some lightpaths pass through transparently,while they may terminate others.Traf?c

on terminating lightpaths may then be electronically switched (groomed)onto new lightpaths towards the destination node. Introducing some amount of electronic switching within the optical network has several advantages:it signi?cantly en-hances the degree of virtual connectivity among the edge routers,which otherwise is limited by the number of optical interfaces at each router;it increases bandwidth utilization; and it may also drastically reduce the wavelength requirements within the optical network for a given traf?c demand.The trade-off is an increase in network cost due to the introduction of expensive active components(i.e.,optical transceivers and electronic switches).These observations motivate us to de?ne the following traf?c grooming problem.

Let be the capacity of each wavelength,expressed in units of some arbitrary rate(e.g.,OC3);we will refer to parameter as the grooming factor.Let be the number of wavelengths that each?ber link in the network can support. We represent a traf?c pattern by a demand matrix, where integer denotes the number of traf?c streams(each of unit demand)from node to node.(We allow the traf?c demands to be greater than the capacity of a lightpath,i.e.,it is possible that for some.)Given matrix,the traf?c grooming problem involves the following conceptual subproblems(SPs):

1)logical topology SP:?nd a set of lightpaths,

2)lightpath routing and wavelength assignment SP:solve

the RW A problem on,and

3)traf?c routing SP:route each traf?c stream through the

lightpaths in.

We note that the?rst and third subproblems together constitute the grooming aspect of the problem.Also,in this context, the number of wavelengths per?ber link is taken into consideration as a constraint rather than as a parameter to be minimized.

The goal we consider in this paper is to minimize the max-imum number of lightpaths originating from or terminating at any node.In our cost model,one unit of cost is incurred for each lightpath that terminates at,or originates from,a network node.Thus,this cost metric accurately re?ects the amount of LTE needed at each network node,and therefore our objective is to minimize the LTE cost at the node where it is maximum. Note also that this objective is equivalent to minimizing the maximum nodal degree in the logical topology.

In this work,we study the traf?c grooming problem in a unidirectional ring with nodes that carries traf?c in the clockwise direction;in other words,data?ows from a node to the next node on the ring,where denotes addition modulo-.The links of are numbered from0to, such that the link from node to node is numbered. We let denote the aggregate traf?c load on the physical link(from node to node)of the ring.The value of can be easily computed from the traf?c matrix.The component of the traf?c load due to the traf?c from source node to destination node is denoted by.If one or more lightpaths exist from node to node in the virtual topology,the traf?c carried by those lightpaths is denoted by

.The component of this load due to traf?c from source node to destination node is denoted by.In our formulation we allow for multiple lightpaths with the same source and destination nodes.We denote the lightpath count from node to node by,taking its value from.We also de?ne the potential lightpath set for a link to be the set of lightpaths that would pass through a given link,and denote it by=lightpath,if it existed,would pass through link.Finally,we let be the lightpath wavelength indicator,i.e.,is1if a lightpath from node to node uses wavelength,0otherwise.

Using the above de?nitions,we can now formulate the problem of designing a virtual topology for a unidirectional ring network with the stated Min-Max objective.The following formulation as an integer linear problem(ILP)consists of

constraints and variables, where is the number of nodes in the ring,and is the number of wavelengths.

Given:

The physical topology,a unidirectional ring of nodes. The traf?c matrix

.

The wavelength limit,which is the number of distinct wavelengths each link can carry.

Find:

Virtual topology,in terms of lightpath indicators,light-path wavelength indicators,and traf?c routing variables .

Subject to:

Traf?c Constraints

(1)

(2)

(3)

(4)

(5)

(6) Wavelength Constraints

(7)

(8)

(9) To minimize:

(10)

The traf?c constraint(1)ensures that a lightpath can carry traf?c for a source-destination node pair only if it is in the

physical route of the traf?c component.Constraint(3)states

that the physical traf?c on a link due to a source-destination node pair must be equal to the sum of the traf?c on all

lightpaths passing through that link due to that node pair. Constraints(4)and(5)de?ne the total traf?c on a lightpath

and relate it to the lightpath count,respectively.Because

of the de?nition of the quantities,constraints(1) and(3)together ensure that no traf?c component can be

routed completely around the ring before being delivered at the

destination node.Constraint(6)is an expression of traf?c?ow conservation at lightpath endpoints.Among the wavelength constraints,constraint(7)expresses the bound imposed by the number of wavelengths available,(8)relates the wavelength indicators to the lightpath counts,and(9)ensures that no wavelength clash can occur.Finally,the objective function(10) is the maximum nodal degree in the ring network.

The ILP formulation in(1)-(10)is for a unidirectional ring network.The formulation can be modi?ed to account for bidirectional rings in which the routing of traf?c is not predetermined,i.e.,traf?c routing is part of the problem(if routing is predetermined,the problem can be decomposed in two subproblems,one for each direction of the ring). Speci?cally,we make the following modi?cations to the above formulation:each variable is replaced by two variables, and,representing the clockwise and counterclockwise routing,respectively,of lightpath;similarly,each variable is replaced by two variables and;the potential lightpath set is calculated according to the respective routing direction;and,?nally,the wavelength constraints apply separately for each direction.

III.C OMPLEXITY R ESULTS

The traf?c grooming problem in ring networks is NP-

Complete since the RW A subproblem in rings is NP-Complete. In this section,we prove that the traf?c grooming prob-lem in path networks is also NP-Complete.Since the RW A problem can be solved in linear time in path networks,our results demonstrate that traf?c grooming with the Min-Max objective(10)is itself an inherently dif?cult problem.They also prove that the traf?c grooming problem with the Min-Max objective remains NP-Complete in rings or other general topologies even when full wavelength conversion is available at the network nodes

Let us consider a network in the form of a unidirectional path with nodes.There is a single directed?ber link from node to node,for each. An instance of the traf?c grooming problem is provided by specifying a number of nodes in the path,a traf?c matrix

,a grooming factor,a number of wavelengths,and a goal.The problem asks whether a valid logical topology may be formed on the path and all traf?c in routed over the lightpaths of the logical topology so that the number of incoming or outgoing lightpaths at any node in the path is less than or equal to.

We?rst consider the case where bifurcated routing of traf?c is not allowed.Speci?cally,for any source-destination pair such that,we require that all traf?c units be carried on the same sequence of lightpaths from source to

destination.On the other hand,if,it is not possible to carry all the traf?c on the same lightpath.In this case,we allow the traf?c demand to be split into

matrix.Speci?cally,to obtain an instance of a bidirectional

problem from the corresponding instance of a unidirectional problem,we add traf?c demands between adjacent nodes in the opposite direction,each demand equaling the full capacity

of the link in the opposite direction,and we add to the min-max objective.Also,we obtain an instance of a ring problem

from the corresponding path problem instance by merging the two nodes at the ends of the path into a single node,and modifying the traf?c matrix appropriately.

Corollary3.3:The decision version of the traf?c grooming

problem in bidirectional path networks with the Min-Max objective(bifurcated routing of traf?c allowed or not allowed) is NP-Complete.

Corollary3.4:The decision version of the traf?c grooming

problem in unidirectional ring networks with the Min-Max objective(bifurcated routing of traf?c allowed or not allowed) is NP-Complete,even when every node has full wavelength conversion capability.

Corollary3.5:The decision version of the traf?c grooming problem in bidirectional ring networks with the Min-Max objective(bifurcated routing of traf?c allowed or not allowed) is NP-Complete,even when every node has full wavelength conversion capability.

Finally,the following theorem shows that,when bifurcation of traf?c is allowed,the grooming problem with the Min-Max objective is not approximable in P,unless P=NP. Theorem3.3:If P NP,then no polynomial time approxi-mation algorithm for the traf?c grooming problem in uni-directional path networks with the Min-Max objective(bifur-cated routing of traf?c allowed)can guarantee

for a?xed constant integer

Proof:The proof is provided in Appendix C.

Since general network topologies,including most interesting topology families such as spiders,rings,grids,tori etc.,contain the path network as a sub-family,the above result shows that it is not practical to attempt optimal or constant ratio approximate solutions to the grooming problem with the Min-Max objective in these cases.The only family which does not include the path as a special case is the star topology.We have considered star networks elsewhere[3],and showed that the problem is again NP-Complete.Whether approximations are possible for star networks is a question that remains open at this time.

IV.M IN-M AX T RAFFIC G ROOMING A LGORITHM FOR

U NIDIRECTIONAL R INGS

We now present a polynomial-time algorithm for traf?c grooming in WDM ring networks with the goal of minimizing the maximum nodal degree(indegree or outdegree)in the logical topology,consistent with the objective function(10). As we mentioned earlier,the degree of a node is a re?ection of the LTE cost needed at that node.Rather than solving all three subproblems of the traf?c grooming problem simultaneously (refer to Section II-B),we decouple the logical topology and traf?c routing subproblems from the RW A subproblem and tackle them independently.Speci?cally,our algorithm consists of the following steps:

Step 1.Solve the logical topology and traf?c routing

subproblems on the ring network using the algorithm

in Section IV-A.The result of this step is a set of lightpaths(logical topology)and a routing of the traf?c

demands over the lightpaths in that minimize the maximum amount of LTE at any node.

https://www.360docs.net/doc/f218526039.html,e the algorithm in Section IV-B to color the

lightpaths of set.The result of this step is a wavelength assignment that does not use more than wavelengths.

However,at the end of this step,the amount of LTE at

one node,say,node,of the ring may increase beyond the corresponding value after Step1,by an amount equal to some value.

https://www.360docs.net/doc/f218526039.html,e the algorithm presented in[4]to distribute the additional LTE at node to other nodes in the ring network.

The following subsections explain the steps of our Min-Max

traf?c grooming algorithm in more detail.

A.Min-Max Traf?c Grooming Algorithm

In this section,we present a polynomial-time algorithm for

the logical topology and traf?c routing subproblems of the traf-?c grooming problem.Unlike previous studies,our algorithm attempts to minimize the maximum amount of LTE at any ring node by creating long lightpaths that bypass intermediate nodes whenever possible.Note that,because of the results we presented in Section III,the traf?c grooming subproblem is itself NP-Complete,and hence our polynomial-time algorithm will terminate without necessarily?nding an optimal solution. However,numerical results to be presented later indicate that the solutions obtained using our algorithm are close to the optimal and/or the lower bound on the objective function(10). Before we proceed,we introduce the concept of reduction of a traf?c matrix.Speci?cally,we reduce the matrix so that all elements are less than the capacity of a single wavelength,by assigning a whole lightpath to traf?c between a given source-destination pair that can?ll it up completely. The available wavelengths on the links of the path segment from the source to the destination node are also decremented by the number of lightpaths thus assigned.Since breaking such lightpaths would increase the amount of LTE at some inter-mediate nodes of the path,this procedure does not preclude us from reaching an optimal solution,nor does it make the problem inherently easier or more dif?cult.We continue using the same notation for the traf?c matrix and traf?c components, but in what follows they stand for the same quantities after the reduction process.

After the reduction,we initialize the logical topology to one

in which a suf?cient number of single-hop lightpaths is formed on each link of the ring network to carry the traf?c using this link.We note that this initial solution is a feasible solution to the logical topology and traf?c routing subproblems,in that it does not use more than lightpaths on any link.However, this initial topology yields a large value for the objective in

expression(10),which is equal to the number of single-hop lightpaths in the most congested link.Our approach,then,is to improve on this initial solution by joining short lightpaths to form longer ones,thus lowering the degrees at intermediate nodes.In the following,we summarize our algorithm for joining short lightpaths.

Let us de?ne the relationship between ring nodes to denote that node“precedes”node in the direction of traf?c ?ow;similarly,we will use the notation to denote that node precedes,or may be the same as,node.The main idea of our algorithm is to consider the node with the maximum degree,and to attempt to decrease its degree by one at each iteration;this process repeats until no more improvement is possible.Let be the node with the maximum degree.The algorithm searches for a pair of nodes such that there exist lightpaths and.The objective is to shift all the traf?c from lightpaths and/or to either an existing or a new lightpath in order to decrease the maximum of the indegree and outdegree of node by one.If the traf?c can be shifted entirely to an existing lightpath, then this procedure is always possible,since no wavelength limit constraints are violated,and also the degree of nodes and/or may also decrease in the process.However,if a new lightpath must be created,the above procedure is carried out only if the wavelength limit constraint(7)is not violated and the degrees of nodes and do not increase above the current maximum degree minus one(the minus one is necessary to ensure that the algorithm will not get into an in?nite loop).A more detailed description of the algorithm is provided in Figure1.

We now argue that at the end of each iteration of the repeat loop,the algorithm produces a solution to the logical topology and traf?c routing subproblems that is feasible(i.e., no link carries more than lightpaths–recall that we are not concerned with wavelength assignment at this point),and the maximum nodal degree is no larger(but possibly smaller)than that of the logical topology at the beginning of the iteration.At each iteration of the repeat loop,the algorithm tries to replace the lightpaths and for some nodes

with a(possibly)new and longer lightpath so as to decrease the nodal degree of node.However,no action is taken if replacing the two lightpaths with the longer one would violate any wavelength constraints or would increase the degrees of or to more than the maximum at the start of the iteration,as we can see at Step15.Since the initial topology at Step2of the algorithm is feasible,we conclude that the topology at the end of each iteration will be feasible and will not increase the maximum nodal degree.

The running time complexity of our algorithm is determined by the main iteration between Steps6and21.In turn,the complexity of the iteration is determined by the two for loops, the inner for loop from Step11to14,and the outer loop from Step8to20.Each of these loops takes time in the worst case,where is the number of nodes in the ring. Therefore,each iteration through the repeat loop from Step6 to21takes time in the worst case.The main iteration

Fig.1.Algorithm for logical topology and traf?c routing

of the algorithm(i.e.,the repeat loop)will be executed at most times,where is the maximum decrease in the degree of any node.Since the value of is always less than the number of wavelengths,the worst-case complexity of the algorithm in Figure1is.However,as we discuss in the next section,in practice,our algorithm runs much faster than the above worst-case analysis indicates;in fact it has never taken more than a few tens of milliseconds for any problem instance with nodes and wavelengths.

B.An Algorithm for Wavelength Assignment

The output of the algorithm we presented in the previous subsection is a set of lightpaths between pairs of ring nodes(i.e.,a logical topology),and a routing of the traf?c

elements over these lightpaths.While the algorithm guarantees that the resulting logical topology is such that no link in the ring network carries more than wavelengths, it may not be possible to color the lightpaths in using no more than wavelengths.In fact,the problem of deciding whether there exists a coloring of the set of lightpaths that uses no more than wavelengths is NP-Complete[20]. In this subsection,we present a polynomial-time algorithm to perform wavelength assignment with at most colors; the tradeoff in ensuring that the number of wavelengths does not exceed is a modi?cation of the logical topology(i.e., the set)which may result in an increase in the degree of some node in the ring.Consequently,the objective of our optimization problem may increase.Therefore,we then re?ne the new logical topology to decrease the objective.

Let us start by describing how to assign wavelengths to the lightpaths of set.Our approach is based on the observation that,while the wavelength assignment problem is hard for ring networks,it is solvable in linear time in paths[15].Consider some node of the ring.Let denote the lightpaths in which optically bypass node,and let

be the set of remaining lightpaths.The lightpaths in set can be viewed as the logical topology on a path network, and thus,can be colored using no more than wavelengths. Now consider all the lightpaths in set.It may be possible to color some of them without violating any wavelength continuity constraints;in general,however,there may be some lightpaths in this set that cannot be colored without the need for additional wavelengths.In this case,we break such a lightpath into two lightpaths and .The new lightpaths do not bypass node,and thus, can be colored along with the lightpaths in set using no more than wavelengths.While breaking such a lightpath will increase the indegree and outdegree of node by one, this approach guarantees a coloring of the new set of lightpaths that satis?es the wavelength constraints.The following steps describe our algorithm in more detail.

1)Let be the set of lightpaths that optically bypass

node.Let be the subset of remaining lightpaths.

2)Sort the lightpaths in in increasing order of their

length.

3)Use the?rst-?t policy to color the lightpaths in.Note

that this step is always possible since it corresponds to a?rst-?t wavelength assignment for a path network. 4)Sort the lightpaths in in decreasing order of their

length.

5)Use the?rst-?t policy to color the lightpaths in.

If lightpath,cannot be colored, then:break into two lightpaths,and

which do not bypass node;increment the indegree and outdegree of node to accommodate the new lightpaths;and repeat from Step1with

and.

Note that we arbitrarily select the node in the above algorithm.One possible improvement would be to consider all possible nodes,run the algorithm for each of them, and then select the solution with the least maximum nodal degree.However,in our experiments we have found that the incremental improvement from such an approach does not justify the increase in running time,and thus we run the above algorithm just once.

Let denote the increase in the nodal degree of node after the termination of the above wavelength assignment algorithm.This increase is due to the fact that lightpaths which optically bypassed node under the initial logical topology de?ned by the set,have now been broken into two lightpaths each.Of the new lightpaths,terminate at node and originate from it.Therefore,node needs an additional pairs of LTE,one for each of the original lightpaths that used to bypass the node.Consequently,this increase of the objective by at node increases the LTE cost in the ring network.

We now show how we can re?ne the new logical topology at the end of the wavelength assignment algorithm to improve on the objective.Our approach is based on the observation that each additional pair of LTE at node,one for an incoming and one for an outgoing lightpath,can be thought of as a wavelength converter.Indeed,consider,one of the lightpaths that initially bypassed the node.This lightpath was broken by the algorithm into two shorter lightpaths that terminate at and originate from the node,respectively.This action was taken in Step6of the algorithm because it was not possible to assign the two shorter lightpaths on the two links in either side of node the same color.Therefore,the additional pair of LTE at node acts as a converter,changing the wavelength of the new incoming short lightpath to the wavelength of the new outgoing lightpath.

Consider a logical topology and corresponding feasible wavelength assignment on a ring network that requires a number of converters at some node.The recent study in[4]showed that it is possible to modify the wavelength assignment such that converters are uniformly distributed across all ring nodes(i.e.,each node has at most converters).Therefore,we use this algorithm to distribute the pairs of LTE(i.e.,“converters”)at node to the other ring nodes.As a result,the maximum degree at all the ring nodes will increase,but the maximum nodal degree of the network(i.e.,of node)will decrease,resulting in a new logical topology with a smaller value for the objective.For the details of this algorithm,the reader is referred to[4].

V.A LGORITHM FOR B IDIRECTIONAL R INGS

In bidirectional rings,we assume that between every two adjacent nodes there are now two links,each carrying wavelengths,in opposite directions.The objective is to mini-mize the maximum in/out nodal degrees in the logical topol-ogy,regardless of the lightpath direction.Also,make the following two assumptions;these assumptions are re?ected in the ILP formulation for the bidirectional ring problem we presented in Section II.First,we do not allow a single lightpath

+

=

Fig.2.Decomposition of a bidirectional ring using shortest path routing

to occupy both directions of a link.Such a lightpath uses wavelengths in both directions of the ring without contributing to a reduction in the Min-Max objective.Furthermore,by using ?bers in both direction,it unnecessarily complicates network protection;therefore,such a lightpath is clearly undesirable. Second,we do allow a traf?c component to be carried from its source node to its destination node on a sequence of lightpaths some of which are in one direction and some in the reverse direction;thus a traf?c component may traverse the same link multiple times in either direction.Such routing of traf?c components may have offer advantages in terms of the Min-Max objective we consider here,if it can make use of the remaining capacities in these lightpaths;in other words,not allowing a traf?c component to be routed in such a manner might require the setup of additional lightpaths,causing an increase in the nodal degree of some node.

We now present a traf?c grooming heuristic for bidirectional ring networks.The heuristic?rst applies shortest path routing to decompose the bidirectional problem instance into two uni-directional problem instances,as shown in Figure2.Shortest path routing forces traf?c demands to travel over the minimum number of links,and makes ef?cient use of the available network capacity(wavelengths).The heuristic then solves each unidirectional subproblem using the algorithm we presented in Figure1,and combines the individual solutions into a solution for the original problem by adding the in/out nodal degrees determined by each solution for each node.Note that,if the subproblems are solved independently of each other,it may happen that some node be the maximum degree node in both solutions.In this case,the?nal combined solution may not be a good one,since node may end up with a large overall degree. Therefore,we modify the algorithm to take into into account the overall objective(for the original bidirectional problem).In this approach,the solution to each unidirectional subproblem takes into account the solution to the other subproblem,and vice versa.Results to be presented in the next section indicate that this shortest path decomposition produces results close to the optimal solution in which the routing of traf?c components is determined by the solution to the ILP.

Speci?cally,our bidirectional traf?c grooming algorithm consists of the following steps.

https://www.360docs.net/doc/f218526039.html,e shortest path routing to determine the di-rection in which each traf?c component will be routed.

Decompose the original traf?c matrix into two traf?c matrices and containing the components routed in the clockwise and counterclockwise direction,respec-tively,and such that.This decomposition

creates two unidirectional ring subproblems with respec-tive traf?c matrices and.

https://www.360docs.net/doc/f218526039.html,e the heuristic we developed in the previous section(refer to Figure1)to solve the clockwise ring,and record the resulting in/out degrees at each node.Then, solve the counterclockwise ring,with the following minor change in the heuristic:when selecting the node with the maximum degree(refer to Step7in Figure1),we add the current degrees for both directions,so that is the node with the maximum aggregate degree.At the end of this step,we obtain a solution.

Step 3.Repeat Step2,but this time solve the coun-terclockwise ring?rst,then solve the clockwise ring accordingly,to obtain a solution.Return the best solution among and.

It is clear that the asymptotic complexity of the above algo-rithm is the same as that of the unidirectional ring algorithm.

VI.N UMERICAL R ESULTS

In this section,we present experiments to demonstrate the performance of the traf?c grooming algorithm we described in the previous section.The experiments are characterized by the following parameters:the traf?c pattern,the number of nodes in the ring,the number of wavelengths per link,the capacity of each wavelength,and the load on the link carrying the most traf?c.The maximum amount of traf?c that can?ow through a link is,hence we express the load as a percentage of.For each experiment(i.e.,each set of values for the above parameters),we generate50problem instances.

In order to demonstrate the performance and generality of our algorithm,we consider three traf?c patterns in our study. For each traf?c pattern,the traf?c matrix of each of the50problem instances is generated by drawing random numbers(rounded to the nearest integer) from a Gaussian distribution with a given mean and standard deviation that depend on the traf?c pattern.The three traf?c patterns are:

1)Uniform pattern.To generate this traf?c pattern,we?rst

determine the mean value of the Gaussian distribution according to the desired load,and we let the standard deviation be10%of the mean.As a result,most elements of the traf?c matrix take values close to the mean.The traf?c is also balanced across the links of the network,with each link carrying a load close to.

2)Random pattern.The traf?c matrix for this pattern is

generated in a manner very similar to that of the uniform pattern.The main difference is that we let the standard deviation of the Gaussian distribution be150%of the mean.Consequently,the traf?c elements take values in a wide range around the mean,and the loads of individual links also vary widely.With such a high standard deviation,the random number generator may return a negative value for some traf?c element;in this case,we set the corresponding value to zero.Also,

if a traf?c matrix generated in this manner is infeasible

(i.e.,the load on some link exceeds the value),

then we discard it and we generate a new matrix for the corresponding problem instance.

3)Locality pattern.This traf?c pattern is designed to cap-

ture the traf?c locality property that has been observed in some networks.Speci?cally,the traf?c elements

are generated such that,on average,50%of the traf?c sourced by any node is destined to node,30%is for node,and10%is for node.The remaining 10%of the traf?c generated by node is distributed equally among the other nodes of the ring.

A.Results for Small Unidirectional Ring Networks

We were able to use to CPLEX to solve the ILP we presented in Section II-B only for ring networks with up to nodes;in fact,for problem instances with nodes,CPLEX run on a Sun Ultra workstation for several days without terminating.In this section we present experiments with8-node rings in order to compare the results of our algorithm to the optimal solution.

In the?gures shown in this section,we compare the following values for each instance:

The lower bound on the objective given by:

(11)

To obtain the lower bound,we note that each node must source and terminate a suf?cient number of lightpaths to carry the traf?c demands from and to this node, respectively.In the above expression,is the maximum of this value over all ring nodes,and can be determined directly from the traf?c matrix.

The optimal value of the objective,obtained by using CPLEX[1]to solve the ILP we presented in Section II-B.

The value of the objective returned by our algorithm.

The value of the objective for a network using all-electronic routing.This is the value of the objective for a logical topology that consists of only single-hop lightpaths,i.e.,one in which no optical switching of wavelengths takes place.The value of corresponds to the number of wavelengths needed to carry the traf?c on the link with the heaviest traf?c load,and can be determined from the traf?c matrix as:

There are several observations that we can make regarding the relative behavior of the four curves in the three?gures. We?rst observe that the value of the objective returned by our algorithm is close to the optimal,and that the optimal is close to the lower bound.On the other hand,the value of the all-electronic solution is signi?cantly higher than the other three values,anywhere from twice to four times larger.This relative behavior of the four curves is consistent across all three traf?c patterns,and has been observed for a wide range of values for the system parameters.These results demonstrate the effectiveness of our algorithm,which can?nd a solution close to the optimal in a tiny fraction of the time required for CPLEX to terminate;in fact,for the experiments presented in this section,the running time of our algorithm never exceeds a few milliseconds,whereas CPLEX,depending on the values of the parameters,may take anywhere from a few minutes to a few hours.

From the?gures,we can also see that the value for the all-electronic solution is always close to.This result is exactly what is predicted from expression(12):since the traf?c load on the most congested link is,the number of lightpaths needed to carry this traf?c is,and these lightpaths must terminate at the nodes at the two ends of this link,requiring an equal amount of LTE at each node. On the other hand,the solution returned by our algorithm, as well as the optimal one,are signi?cantly lower than the all-electronic value.This result indicates that our approach of minimizing the maximum LTE cost in the network can produce signi?cant cost savings.Another related observation is that,by using a Min-Max objective,we can ensure that the cost of any individual network node(as well as the overall network cost) is determined by the traf?c demands of the node,and will not scale with the number of wavelengths.Speci?cally,the values of the objective in Figures3-5are small compared to the number of wavelengths we used for the experiments (),and are close to the lower bound,which represents the minimum LTE cost to accommodate the traf?c demands for any ring node(see(11).This result demonstrates that,with an appropriate network design approach,a network with optical switching capabilities can scale to large numbers of wavelengths without a corresponding increase in cost;we reached a similar conclusion in a different context in another recent study[24].

Finally,from the three?gures we see that the optimal value of the objective depends on both the traf?c pattern and the maximum link load.This is more evident in Figure5,where the high degree of randomness in the traf?c matrices produces a variety of patterns and maximum link loads across the50 problem instances,resulting in more jagged curves.(Note, however,that even in this case,the curve produced by our algorithm tracks the optimal curve very closely,attesting to the quality of our approach.)This observation suggests that,for the purposes of dimensioning the network for unknown future demands,the network designer should run our algorithm for a variety of traf?c patterns and loads,and then equip each node with an mount of LTE determined by the highest value of.B.Results for Large Unidirectional Ring Networks

In this section,we present results for ring networks with nodes and wavelengths;such networks are of practical interest because the maximum size of a SONET ring is16nodes,and also because WDM links supporting128 wavelengths are becoming commercially available.Since we were not able to use CPLEX to obtain the optimal solution, the?gures in this section only contain three curves:one for the all-electronic value,one for the value of returned by our algorithm,and one for the lower bound.Our algorithm, on the other hand,needed less than a second to?nd a solution for each problem instance that we present in this section. Figures6,7,and8present the results of50different prob-lem instances for each of the uniform,locality,and random traf?c patterns,respectively.The relative behavior of the three curves in these?gures is very similar to that we observed in Figures3-5.In particular,the values of the objective returned by our algorithm are close to the lower bound,and track it well across the different traf?c patterns and problem instances within each pattern.The all-electronic solutions,on the other hand,are again quite high with values at around .Therefore,all the conclusions regarding the scalability and cost-effectiveness of our traf?c grooming approach are valid for large WDM ring networks as well.

C.The Effect of the Traf?c Load

Let us now investigate how our solutions scale with the load on the most congested link of the network.In Figures9 and10,we plot the values of,,and against the load for a uniform traf?c pattern;each point in the?gures is the average over50problem instances for the given value of (the optimal values are also plotted for the small network in Figure9).The experiments presented in the two?gures are identical in all respects except that Figure9is for6-node rings,while Figure10is for16-node rings.We have obtained similar results for the other two patterns as well.

As we can see,,,and all increase linearly with the load.However,the curve corresponding to the all-electronic solution has a slope much steeper than that of the curves corresponding to the lower bound and our solution.This behavior demonstrates that the cost bene?t of our optimization approach increases with the load of the network.This property is an important one,and it implies that network operators will be able to operate the network at high loads with only an incremental increase in cost.

Another important property of our Min-Max optimization approach becomes evident once we compare the curves of Figure9to those of Figure10.Speci?cally,we note that the curves corresponding to the all-electronic solutions have similar slopes in both?gures.On the other hand,the curves corresponding to the lower bound and our solution have a steeper slope in Figure9(which corresponds to a6-node network)than in Figure10(for a16-node network).Equally important,for a given load,the actual values of the lower bound and our solution are lower for the larger network than the respective values for the smaller network;this is true

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despite the fact that link loads of both networks are similar and that the 16-node network carries a much larger aggregate amount of traf?c than the 6-node network.Based on this observation,we conclude that our solution approach scales well not only with the load and number of wavelengths,as we explained above,but also with the size of the network.Finally,in Figure 11,we present results for problem in-stances with nodes,a uniform traf?c pattern,and a high load .Let us compare Figure 11to Figure 6which plots results for the same values of the system parame-ters,except that the load

in Figure 6.As we can see,the all-electronic solution is about 20%higher in Figure 11,as expected,and the lower bound is also somewhat higher.However,there is little difference in the values of returned by our algorithm when the load increases from 80%to 95%.We have observed similar behavior for the other two traf?c patterns,reinforcing our earlier conclusion that the bene?t of our Min-Max optimization approach increases with the traf?c load,enabling cost-effective network operation at very high loads.

D.Results for Bidirectional Ring Networks

We now present results for bidirectional rings with

nodes.We have found that,due to the fact that the ILP formulation for bidirectional rings contains far more variables and constraints than in the unidirectional case (refer also to Section II,for rings with more than ?ve nodes,CPLEX takes too long (several days)to ?nd an optimal solution for rings with more than ?ve nodes.We also note that,since the routing of traf?c is part of the problem,it is not possible to a priori determine the loading of each link from the traf?c matrix as in the unidirectional case.Therefore,we use the concept of grooming effectiveness to characterize the results.The grooming effectiveness is de?ned as the ratio of the maximum nodal degree of a solution to the one required by an all-electronic solution with shortest path routing.By de?nition,the smaller the grooming effectiveness,the better the solution in terms of the Min-Max objective.

For each experiment,we again generate 50different prob-lem instances,and we compare the traf?c grooming effective-ness values for the following four values:

The optimal solution returned by CPLEX.

The decompose optimal solution.In this case,we use the shortest-path decomposition we described in Section V and we solve the two unidirectional ring instances in-dependently using CPLEX on the corresponding unidi-rectional ILPs.We then combine the two solutions,and compute the overall Min-Max result (and corresponding grooming effectiveness value).

The decompose heuristic solution.This is similar to the decompose optimal method,except that we use the unidirectional heuristic in Figure 1instead of CPLEX to solve each unidirectional ring instance.

The bidirectional algorithm solution,which is obtained by running the algorithm we presented in Section V.Recall that this algorithm focuses on the overall objective

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for the bidirectional ring as a whole,and is such that the solution to each unidirectional ring instance takes into account the solution to the other.

Figures 12,13,and 14present the grooming effectiveness values of 50instances for each of the uniform,locality,and random traf?c patterns,respectively.It is not surprising to see that,overall,the grooming effectiveness is best for the locality pattern,since the pattern itself favors shortest path routing.For both the uniform and locality patterns,all four solution approaches give similar results.However,for the random pattern,we can see that the decompose optimal method performs the worst,whereas the decompose heuristic approach approach performs well.This behavior may be due to the fact that CPLEX stops after ?nding the ?rst optimal solution for each of the unidirectional ring instances,which may be such that maximum degree exists in the same node;our algorithm,on the other hand,returns solutions in which the nodal degrees are more balanced.Finally,the bidirectional algorithm is always close to the optimal,as expected,since it takes into consideration the overall objective.

From the three ?gures,we observe that our shortest-path de-composition always works well for the ?ve-node ring networks we have considered.To further evaluate the effectiveness of the decomposition,we conducted additional experiments on ?ve-node bidirectional rings with the random traf?c pattern.We randomly generated 100,000problem instances and obtained the optimal solutions using CPLEX.The resulting maximum nodal degrees range from 9to 74,with an average around 33.We then run the bidirectional heuristic and recorded the dif-ference in maximum nodal degree from the optimal solution.Of the 100,000instances,our algorithm found the optimal solution in 38,895cases (39%);in 55,519cases (55.5%)our algorithm found a solution whose maximum degree is one more than the optimal;in 5,550cases (5.5%)our solution are two more than the optimal;and in 36cases,our solution are three more than the optimal.These results indicate that using shortest path routing in bidirectional rings works well,at least for small rings and for a wide range of traf?c patterns.Although we cannot use CPLEX to solve larger problem in-stances,we believe that shortest path decomposition combined with the bidirectional algorithm we presented in Section V will work well for moderate size rings (i.e.,rings with up to 16nodes,the maximum size of SONET networks).To see why this is so,recall that the results in Figures 6-8indicate that the solution to each unidirectional problem is close to the lower bound.Also,the sum of the lower bounds for each unidirectional subproblem is a lower bound for the original bidirectional problem (since the lower bound 11only accounts for the total amount of traf?c originating or terminating at each node,which is independent of the routing of traf?c).Therefore,the maximum nodal degree resulting from combining (adding)the solutions to the individual unidirectional problems will also be close to the lower bound.

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E.The Aggregate LTE Cost

The algorithm we presented in Figure 1terminates once it is determined that no further improvement (reduction)of the maximum nodal degree is possible.The resulting solution has a low maximum nodal degree,but it may be such that the sum of the degrees over all network nodes is relatively high.Since the sum of nodal degrees represents the total amount of LTE that will need to be deployed in the network,it is desirable to obtain solutions with a low aggregate LTE cost.We now show how to extend the algorithm in Figure 1to reduce the network-wide sum of nodal degrees without increasing the maximum degree.

Speci?cally,rather than terminating after minimizing the maximum nodal degree,the algorithm continues to the main iteration between Steps 6and 21.The main difference is in the manner in which the node is selected in Step 7and in the criterion for termination in Step 21.Note that,although it may not be possible to reduce the degree at the node where it is maximum,it may be possible to reduce the degree at some other ring node;doing so will not affect the Min-Max objective,but will reduce the aggregate nodal degree.Therefore,the algorithm uses a greedy approach,and selects the ring node whose degree can be reduced by the maximum amount,and performs an additional iteration to reduce ’s degree,taking care not to increase the maximum degree.This process is repeated until no further progress can be made in reducing the degree of any node in the network.Hence,this version of the algorithm achieves two objectives:the primary objective is to minimize the maximum nodal degree,and the secondary objective is to minimize the total degree over all network nodes.

In order to characterize the performance of the new algo-rithm with respect to the two objectives,we have conducted experiments with 16-node unidirectional rings.For each set of parameters we generated 50problem instances,and we recorded the maximum nodal degree and the total degree over all network nodes for two algorithms:the algorithm in Figure 1after making the modi?cations we described above,and the algorithm by Zhang and Qiao [26].The latter algorithm was developed speci?cally for minimizing the total number of SONET ADMs (i.e.,the total degree)in the network.For each problem instance,we also computed the lower bound on the maximum and total degrees.

The three Figures 15-17plot the maximum nodal degree of each problem instance for the uniform,locality,and random traf?c patterns,respectively.As we can see,our algorithm clearly outperforms Zhang and Qiao’s algorithm for the lo-cality and random patterns,while the results for the uniform pattern are mixed,with each algorithm performing better than the other over some of the problem instances.Interestingly,the relative behavior of the two algorithms is similar in terms of the total degree (over all network nodes)for the three traf?c patterns,as shown in Figures 18-20.Again,our algorithm outperforms Zhang and Qiao’s algorithm for the locality and random patterns,while the latter performs better for the

uniform pattern.The fact that Zhang and Qiao’s algorithm performs well when traf?c is uniform is not surprising,due to the nature of the algorithm which attempts to form full circles of unit traf?c components which are then groomed into wavelengths;this operation is most successful when traf?c demands are symmetric.However,the important observation is that,for the asymmetric traf?c scenarios that are more likely to be encountered in practice,our algorithm performs better, not only in terms of the maximum nodal degree,but also in terms of the total degree over all network nodes.

Overall,the results we presented in this section demonstrate that our min-max optimization approach for traf?c grooming is successful in obtaining solutions that keep the maximum

nodal degree low.Moreover,our solutions also tend to keep the overall network cost(in terms of the total degree over all network nodes)low,and compare favorably to solutions obtained by algorithms whose main objective is network-wide cost minimization.

VII.C ONCLUDING R EMARKS

We have studied a new variant of the traf?c grooming problem in ring networks,where the objective is to minimize the maximum LTE cost at any node.We have developed new polynomial-time algorithms for traf?c grooming and wavelength assignment in both unidirectional and bidirectional rings.We have presented numerical results which demonstrate that our algorithms perform well under a wide range of traf?c patterns and system parameter values,not only in terms of the maximum LTE cost,but also in terms of the total LTE cost over all network nodes.Our algorithms have applications in dimensioning networks for unknown and/or dynamic traf?c demands.Currently,we are extending our work to networks with general topologies.

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A PPENDIX A

P ROOF OF T HEOREM 3.1

The reduction is from the Subset Sum problem [12].An instance of the Subset Sum problem consists of elements of size ,and a goal .The question is whether there exists a subset of elements whose sizes total .Let .Construct a path network using the following transformation:,,

,and let the objective be .The

nodes of the path are labeled ,and the traf?c matrix is:

Fig.21.Example of path construction for the proof of Theorem3.1, ,

or

or

or

otherwise

The traf?c matrix is set up so that we are forced to use the virtual topology shown in Figure21.Speci?cally,the?rst of the three wavelengths is used to form two lightpaths,one from node to node,and one from node to node;both these lightpaths are?lled to capacity carrying the corresponding traf?c demands of magnitude.Therefore, this?rst wavelength cannot be used to carry any other traf?c. The second wavelength is used to form single-hop lightpaths between adjacent pairs of nodes in the path;these lightpaths are denoted by the straight arrows in Figure21.However, note that the traf?c demands in the top row of the traf?c matrix above are greater than the capacity of a wavelength. Therefore,the third wavelength is used to form single-hop lightpaths between nodes

,in the path.The single-hop lightpaths on this third wavelength between any such pair of nodes carry the following traf?c components:(1)the one unit of traf?c from node to node(the other units of such traf?c are carried by the single-hop lightpath on the second wavelength), and(2)the units of traf?c from node,to node.Since,the single-hop lightpath on the third wavelength from node to has enough capacity to carry this traf?c,for all such.Finally,the third wavelength is also used to form a direct lightpath from node to node that bypasses node,and the result is the virtual topology in Figure21.Note that no more lightpaths can be added to this virtual topology,and no existing lightpaths can be split without violating the objective.

After grooming all the traf?c except for,which have to go through node with or without stopping,the capacity left in the lightpath from node to node is equal to ,and the capacities left in the lightpaths from node to node and from node to node are both equal to.Since bifurcation is not allowed,it is possible to use the virtual topology above to groom the traf?c, if and only if there is a subset of whose sum is exactly.Since deciding the satis?ability of the Subset Sum problem is NP-Complete,then the new grooming problem is also NP-Complete.

(a)

(b)

Fig.23.The logical topology for the proof of Theorem3.2:(a)the topology after forming lightpaths for each demand of size,and(b)the topology after forming additional lightpaths for each demand between node pairs in set

two lightpaths from node6to node15,each of them carrying

units of the traf?c demand.

Because of the lightpaths already set up,the only way to

satisfy the unit traf?c demands between node pairs in set in

the above traf?c matrix,is to form a direct lightpath for each

such traf?c demand.The resulting logical topology is shown

in Figure23(b).For example,consider the source-destination

pair(refer to Figure22(a)).As we can see,a

direct lightpath has been formed between nodes1and3in

Figure23(b);similarly for other node pairs in set.

We now see that the logical topology in Figure23(b)is

such that exactly two lightpaths originate from,and terminate

at,each node,of the path.Therefore,it is not

possible to add any new lightpath with such a node as the

origin or termination point without violating our stated goal

.Consequently,we have to use the remaining capacities

left on the lightpaths formed between node pairs in set in

order to carry(groom)the demands in the above traf?c matrix

due to node pairs in set.This can be done if and only if

the original constrained MCF problem has a solution.Since

the MCF problem is known to be NP-Complete,so is the path

grooming problem with bifurcation of traf?c allowed.