Light-trail based Hierarchy

The Optimal Multicast Route in WDM Networks Without Splitters and Converters

Dinh Danh Le and Massinissa Merabet

LIRMM, University Montpellier 2, 161 rue Ada, 34095 Montpellier Cedex 5, France

Keywords:

WDM Network, Multicast Routing, TaC Cross-connect, Light-path, Light-trail, Light-hierarchy, Light-trail

based hierarchy, Wavelength Minimization, Integer Linear Programming (ILP).

Abstract:

Multicasting in WDM core networks is known as an efﬁcient way of communications in high-speed multime-

dia applications. However, costly and complicated fabrication prevents multicast capable switches (splitters)

from deploying in the proposed architectures. Besides, in practical routing cases, the state of the network

is given by a directed graph. Accordingly, this paper investigates the multicast routing without splitters in

directed asymmetric topologies. The objective is to minimize the number of wavelengths used and then ﬁnd

the best cost solution among those requiring the same number of wavelengths. In the case of no splitters, a

set of light-paths starting from the multicast source covering all the destinations is known as the traditional

solution. In this paper, we introduce two new concepts namely light-trail based hierarchy and light-path based

hierarchy, and develop two ILP formulations for them. Theoretical analysis and simulation results show that

the optimal solution is a set of light-trail based hierarchies. Particularly, our light-trail based solution achieves

fewer wavelengths required (up to 21.95% saved) while keeping slightly lower cost (up to 3.79% saved) com-

pared to light-path based solution.

1 INTRODUCTION

Being capable of supporting heavy load communica-

tions, all-optical networks are promising to be seri-

ous candidates for high-speed backbone networks. In

pure optical routing, the messages are transmitted us-

ing Wavelength Division Multiplexing (WDM) tech-

nology without electronic processing in the condition

that the computed routes should satisfy optical con-

straints (Zhang et al., 2000).

1.1 Multicast Routing Problems under

Optical Constraints

Multicast is known as the efﬁcient way of communi-

cations to perform data transmission from a source

to several destinations. In traditional IP electronic

networks, the solutions are known as spanning trees

computed in the topology graph. In optical networks,

however, the multicast routes do not necessarily cor-

respond to trees but some structures complying sev-

eral (optical) constraints. Among the constraints, the

availability of light splitters in the switches are of-

ten the most hard ones. Light splitters (or multicast

capable switches) are special nodes capable of split-

ting an incoming signal from a predecessor to sev-

eral successors. Nevertheless, splitters are expensive

and complicated in fabrication. Besides, splitting in-

duces considerable power loss (inverselyproportional

with the number of outgoing ports (Ali and Deogun,

2000b)). Therefore, we assume to study the multi-

cast routing on the WDM networks without splitters.

In addition, since wavelength converters (the devices

that can shift a passing signal from one wavelength to

the other (Mukherjee, 2006)) are costly and immature

enough, we also exclude them from this study.

Regarding the objectives of multicast problems,

the requirements of economizing the networks re-

sources (e.g., the wavelengths) are ﬁrst thing to con-

cern. Besides, among the possible routes, the least-

cost one is preferred. The total cost of the routes is

deﬁned as the summation of the costs of the individ-

ual links of the routes and the cost of each link could

be any types of metrics including distance, monetary

cost, etc., depending on the network entity that we are

trying to minimize. However, it is usually hard to si-

multaneously minimize both metrics. Therefore the

trade-off solution is more interesting.

Another important aspect to consider is the kind

of examined networks. Most of the studies in optical

Le D. and Merabet M..

Light-trail based Hierarchy - The Optimal Multicast Route in WDM Networks Without Splitters and Converters.

DOI: 10.5220/0004705300660073

In Proceedings of 2nd International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS-2014), pages 66-73

ISBN: 978-989-758-008-6

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

multicast routing problems are carried out on symmet-

ric networks under the assumption that there are two

opposite directed ﬁbers between every pair of con-

nected switches. In this case the networks are well

modeled by undirected (Zhang et al., 2000), (Din,

2009) or bidirected graphs (Ali and Deogun, 2000a),

(Zhou et al., 2010). However, it is more practical

and general to investigate the routing problems on

asymmetric networks that can be modeled by an ar-

bitrary directed graphs (or digraphs) where each arc

represents a directed ﬁber between a pair of nodes.

It is more practical because even if the network is

designed to be bidirected, when some demands hold

some of the network resources, the resultant topology

graph is then no longer bidirected but (arbitrarily) di-

rected, therefore the routing for subsequent demands

will be calculated on a digraph. It is also more gen-

eral because bidirected graphs are special cases of di-

graphs where every arc has its reverse one.

1.2 Related Works

Due to its interest, WDM multicast routing has been

investigated intensively in the literature and several

propositions exist to adapt multicast routing algo-

rithms to the optical constraints (cf. (Zhang et al.,

2000) for some basic algorithms and (Zhou and Poo,

2005) for a survey).

The problem of minimizing the number of used

wavelengths was investigated at ﬁrst in (Li et al.,

2000). The considered network is assumed to be

equipped with splitters and wavelength converters,

and it is considered as a set of wavelength graphs

where the arcs representing the wavelengths avail-

able in the corresponding ﬁbers. The objective is

to construct a light-tree satisfying optical constraints

such that the number of required wavelengths is min-

imized. The NP-hardness of the problem is proved,

and an approximation algorithm has been proposed.

The case of switching without splitters in sym-

metric networks has been discussed in (Ali and De-

ogun, 2000a). The problem is to ﬁnd a Multiple-

Destination Minimum Cost Trail that starts from a

source and spans all the destinations with minimizing

the total cost of the links traversed. To ensure a fea-

sible solution, a low-cost cross-connect architecture

called Tap-and-Continue (TaC) has been proposed to

replace splitters. TaC cross-connects can tap a sig-

nal with small power at the local station and forward

it to one of its output ports. The problem is proved

to be NP-hard and then a heuristic (namely MDT) is

proposed. The advantage of MDT heuristic is that

only one wavelength (and one transmitter) is sufﬁ-

cient for each multicast request (i.e, the wavelength

is minimized). However, due to multitude of round-

trip traversing, a large number of links is required in

both directions, and the total cost of the light-trail is

always very high.

The multicast routing problems without splitters

in asymmetric networks has been studied in (Le et al.,

2013). The problem is proved to be NP-hard, and

two heuristics namely Farthest First and Nearest First

are proposed. These heuristics based on light-trails.

Thanks to the interesting properties of light-trails, the

number of wavelengths can be considerably saved. In

comparison to the heuristics proposed in (Din, 2009),

they provide better solutions with fewer wavelengths

required and lower total cost. However there are no

exact solutions given to calculate their approximation

ratios.

In this paper, we study the multicast routing in

asymmetric WDM networks without splitters and

converters. Our objective is to minimize the number

of used wavelengths and then try to minimize the total

cost. To solve the problem we introduce a new con-

cept called light-trail based hierarchies and develop

two ILP formulations to search for the exact solutions.

We theoretically and experimentally show that the op-

timal solution is a set of light-trail based hierarchies.

The structure of the paper is the following. Section 2

presents the problem modeling and performance met-

rics. The concept of light-trail based hierarchy and

its beneﬁts are given in Section 3. Then the ILP for-

mulation for it is presented in Section 4, followed by

the experimental results on their performancesin Sec-

tion 5. We conclude our paper in Section 6.

2 PROBLEM MODELING AND

METRICS

The considered network is modeled by the topology

graph G = (V, A), a simple digraph in which each

arc represents the availability of a directed ﬁber be-

tween a pair of nodes (we suppose that there are at

most two opposite directed ﬁbers between any pair of

nodes). As mentioned in Section 1, we deal the multi-

cast problem in the networks which are not equipped

with any splitters but TaC cross-connects. We sup-

pose that each ﬁber has the same set W of available

wavelengths and each arc a ∈ A is associated with a

positive value cost(a). Given the multicast request

r = (s, D), in which s ∈ V is the source node and

D ⊆ V \ {s} is the set of destinations, the routing

problem is to compute the light-structures (e.g., light-

trees) from s covering all the destinations simultane-

ously. These light-structures must comply the follow-

ing constraints (Zhou and Poo, 2005):

Light-trailbasedHierarchy-TheOptimalMulticastRouteinWDMNetworksWithoutSplittersandConverters

• Wavelength Continuity Constraint: In the absence

of wavelength converters, the same wavelength

should be used continuously on all the links along

a light-structure.

• Distinct Wavelength Constraint: Two light-

structures should be assigned with different wave-

lengths unless there are edges (or arcs) disjoint.

• Degree Constraint: In the absence of light-

splitters, all the nodes (except the source) in every

light-structure should have the degree that do not

exceed two.

Without loss of generality, let LS be the set of

light-structures LS

, i = 1, .. . , k computed for the

given request r. Since each light-structure consumes

a distinct wavelength, the number of wavelengths

needed to performthe multicast request r is equal to k:

No. Wavelengths(r) = k. The total cost of the light-

structures is the summation of cost all the arcs in all

light-structures LS

TotalCost(r) =

∑

i=1

cost(LS

) =

∑

i=1

∑

a∈LS

cost(a).

In our study, we ﬁrst minimize the number of used

wavelengths, then try to minimize the total cost

among the solutions with the same minimal wave-

lengths.

Traditionally, the solutions correspond to light-

trees in general cases or light-paths in the case of

no splitters and no converters (as it is considering in

this paper). However, the nodes can be traversed sev-

eral times with the same wavelength as long as there

are different incoming and outgoing ports for each

passing (Zhou et al., 2010). Consequently, the solu-

tions are not necessarily sets of light-paths but sets of

light-trails. In Section 3, we introduce a new light-

structure based on light-trails call light-path based hi-

erarchies. We will prove that the problem with light-

path based solutions is NP-hard. We then compare it

with the light-path based solution to ﬁnd a better solu-

tion for the considering problem. Its ILP formulation

is given in Section 4.

3 LIGHT-TRAIL BASED

HIERARCHIES

Before deﬁning the new concept light-trail based

hierarchy, let us ﬁrst introduce the concept light-

hierarchy proposed in (Molnar, 2011).

Based on the fact that the multicast routes are not

necessarily sub-graphs but any types of structures that

retain the connectivity and spanning properties, a hi-

erarchy is proposed to replace the traditional solutions

(e.g., paths, trees, etc.). It is a graph related structure

obtained by a homomorphism of a tree in a graph. Re-

call that in graphs, a homomorphism can be deﬁned

as follows. Let Q = (W, F) and G = (V, E) be two

(both undirected or directed) graphs. Q is called the

base graph, and G is the target graph. An applica-

tion h : W → V maps a vertex in W to each vertex

in V is a homomorphism if the mapping preserves the

adjacency: (u, v) ∈ F ⇒ (h(u), h(v)) ∈ E. If Q is a

connected graph without cycle (a tree) then the triple

(Q, h, G) deﬁnes a hierarchy in G. If both graphs Q

and F are directed, the triple (Q, h, G) deﬁnes a di-

rected hierarchy

in G (Molnar, 2011). In term of

optical routing, light-hierarchy is deﬁned as a hierar-

chy using a single wavelength. Equivalently, a light-

hierarchy is a hierarchy that has no duplicated arc but

is free of repetition of nodes (Zhou et al., 2010).

According to the deﬁnition of light-hierarchy,

when the base graph Q is a rootedtree without branch-

ing vertices (except the root correspondingto the mul-

ticast source

), i.e, Q is a star, the triple (Q, h, G) de-

ﬁnes a special light-hierarchy. It corresponds to a set

of rooted arc-disjoint trails in the target graph G, so a

single wavelength is needed to serve it. For this rea-

son, we call it light-trail based hierarchy (LTH).

Especially, if the mapping h is injective (i.e., each

vertexinW associates with only one vertex inV), then

the hierarchy has no duplicated vertices (and so no

duplicated arcs well as), and it corresponds to a set of

rooted elementary trails (trails without repetition of

vertices) or paths, in G. This has been considered as

the traditional solution for the problem we are exam-

ining (Din, 2009). We will call it a light-path based

hierarchy (LPH) in order to distinguish with a general

light-trail based hierarchy.

Figure 1 shows an example of a light-trail based

hierarchy. Each vertex of the star Q is associated with

an unique vertex of the graph G. In the reverse di-

rection, some vertices of G are mapped from several

vertices in Q (nodes a and f). A vertex in Q can be

labelled by the vertex in G which it is associated. To

distinguish the occurrences related to the same vertex

v in G, we will use the labels v

, v

, ..., v

in Q (and in

the hierarchy H as well). Notice that the degree of a

vertex occurrence v

in the hierarchy H is deﬁned as

the degree of the corresponding vertex occurrence v

in the base graph Q (Molnar, 2011). It is important to

verify the degree constraint stated in Section 2.

With LTH solutions, the considering problem is

In this paper we just consider directed hierarchies, but

sometimes the word directed is omitted for the sake of sim-

plicity.

Because the source can be equipped with multiple

transmitters, so it can inject the same wavelength to differ-

ent successors.

PHOTOPTICS2014-InternationalConferenceonPhotonics,OpticsandLaserTechnology

Figure 1: Mapping of vertices from a star for a light-trail

based hierarchy.

NP-hard. It can be deduced from the following theo-

rem.

Theorem 3.1. Let k be any ﬁxed positive integer. If

P 6= NP, then there is no polynomial time algorithm

to check whether W

∗

≤ k, where W

∗

is the minimum

number of wavelengths needed for the given multicast

request.

Proof. We reduce the well-known directed Hamilto-

nian Path problem to our problem. It is known that to

decide whether a given graph is Hamiltonian is NP-

complete (Garey and Johnson, 1979).

Let G = (V, A) be a given directed graph. We build

a graph G

′

by replacing each vertex v ∈ V by two

new vertices v

and v

and linking v

to v

by the arc

, v

). Each predecessor of v becomes a predecessor

of v

and each successor of v becomes a successor of

We build a graph H by making k copies of G

′

, G

′

, A

′

), G

′

= (V

′

, A

′

), ..., G

′

= (V

′

, A

′

), and adding

two new vertices s, z connected by the arc (s, z) (s

is considered as the source). Then we make z adja-

cent (predecessor) to all v

-vertices of each copy of

′

(Figure 2). We suppose that D = V

′

∪V

′

∪ ... ∪V

′

It is easy to check that H admits a solution W

∗

≤ k for

the light-trail based problem if and only if G admits a

directed Hamiltonian path.

Lemma 3.1. For the problem of minimizing the num-

ber of wavelengths, the path-based solution is not op-

timal.

Proof. Let us consider the topology given in Figure 1.

We suppose that the source node is s and D = V −{s}

(a) Input graph G

(b) Modiﬁed graph H

Figure 2: Illustration of the proof (polynomial transforma-

tion of G). For simplicity we suppose that k = 1.

is the set of destinations. The two solutions: light-

trail based hierarchy (LTH) and the light-path based

hierarchy (LPH) are shown in Figure 3. As shown

in Figure 3a), only one LTH is sufﬁcient to span all

the destinations satisfying the aforementioned degree

constraint. On the other side, as shown in Figure 3b),

two LPHs (each using one wavelength) are required,

i.e., two different wavelengths needed to span all the

destinations. Hence, in this case, the light-path based

solution can not be optimal.

b) LPH solutions

LTH

a) LTH solution

LPH

Figure 3: LTH and LPH solutions for the same multicast

request.

Theorem3.2. The optimal solution for the problem of

minimizing the number of wavelengths in non-splitter

WDM networks is a set of light-trail based hierar-

chies. The number of required wavelengths is at least

Light-trailbasedHierarchy-TheOptimalMulticastRouteinWDMNetworksWithoutSplittersandConverters

one (in the best case) and limited by that needed for

the optimal light-path based hierarchies.

Proof. Assume that the problem always has feasible

solutions, i.e., there is at least one directed path from

the multicast source to each destination and there are

enough wavelengths to route.

Let us ﬁrst recall the deﬁnition of light-trail based

hierarchy. The base graph Q is a star, so it satisﬁes the

aforementioned degree constraint. According to the

deﬁnition of vertex degree in the corresponding hier-

archy, the hierarchy H also satisﬁes the degree con-

straint.

Besides, to guarantee the distinct wavelength con-

straint, the mapping h associates a vertex of Q to a

vertex of G such that no two arcs in Q are associ-

ated with the same arc in G, i.e., no duplicated arcs in

H. The worst case happen when the mapping func-

tion (h) is injective, i.e., there are no duplicated ver-

tices (hence no duplicated arcs) in the resultant hierar-

chies. These light-trail based hierarchies correspond

to sets of light-paths (as we call light-path based hier-

archies). Thus, even if the duplication of nodes is not

possible to diminish the number of wavelengths, this

number of wavelengths needed for light-trail based

solution is equal to the number of light-path based hi-

erarchies in the worst case.

However, in general cases, the mapping can gener-

ate several duplicated vertices in the resultant hierar-

chies. As shown in the above example, these vertices

can help to visit more destinations in one trail. As the

result, these duplicated vertices reduce the number of

wavelengths required to cover all the destinations. In

the best case, a set of trails which can be colored by

just one wavelength is sufﬁcient (Figure 3a)).

Lemma 3.2. The optimal solution for the problem of

minimizing the number of wavelengths does not nec-

essarily minimize the total cost of the solution in non-

splitter WDM networks.

Proof. Consider an example in Figure 4 where

there is a trail that spans all destinations nodes

(s, 0, 1, 0, 2, 0, 3, 0, ..., k − 2, 0, k − 1, 0, k).

Just only one wavelength is sufﬁcient for this

trail. Therefore, this light-trail based hierarchy is

the optimal solution in term of number of wave-

lengths. In contrast, the light-path based solu-

tion can be found corresponding to the set of paths

{(s, 0, 1), (s, 0, 2), (s, 0, 3), ..., (s, 0, k −

1), (s, 0, k)}. All these paths share the arc (s, 0).

So the number of wavelengths needed to perform the

multicast is equal to k.

Now we suppose that the cost of arc (s, 0) is equal

to 1, all the others have costs of 10. Accordingly,

the light-trail based solution consumes cost(LTH) =

Figure 4: Graph G = (V,A), a source node s and a set D =

{1 , 2 , 3 , ... , k − 1 , k }.

1+2∗(k− 1)∗10+10) = 20∗(k− 1)+11. Whereas,

the light-path based solution consumes cost(LPH) =

k ∗ 1 + k ∗ 10 = 11 ∗ k. Obviously, cost(LTH) >

cost(LPH), ∀k > 1. Hence, the lemma follows.

4 ILP FORMULATION FOR

LIGHT-TRAIL BASED

HIERARCHIES

In this section, we formulate the considering problem

with the solution corresponding to a set of light-trail

based hierarchies. Let us recall that each LTH can

be composed by a set of rooted arc-disjoint trails (and

thus, each requires a distinct wavelength). The fact

that one wavelength can may not sufﬁcient to cover

all the destinations, several LTHs (i.e, several wave-

lengths) may be needed.

Notations and Network Parameters:

• G = (V, A): The directed graph with a set V of

nodes and a set A of arcs

• W: The set of wavelengths available on each arc

• λ: A wavelength λ ∈ W

• ∆: An big enough integer such that ∆ >

∑

a∈A

cost(a)

• In(m): The set of nodes which have incoming arcs

to node m in G

• Out(m): The set of nodes which have outgoing

arcs from node m ∈ V

• (s, D): A multicast request

• Indeg(m): The in degree of node m

• Outdeg(m): The out degree of node m

• a(m, n): The arc from node m to node n

• c

m,n

: The cost of the arc a(m, n)

ILP Variables:

• L

m,n

: Binary variable. Equal to 1 if wavelength λ

is used on arc a(m, n) on wavelength λ; equal to 0

otherwise.

PHOTOPTICS2014-InternationalConferenceonPhotonics,OpticsandLaserTechnology

• F

m,n

: Commodity ﬂow, integer variable. De-

notes the number of destinations served by the arc

a(m, n) on wavelength λ.

• w(λ): Binary variable. Equal to 1 if wavelength λ

is used by the light-trails, equal to 0 otherwise.

ILP Formulation:

The primary objective is to minimize the number of

wavelengths required. Secondly, among the wave-

length optimal solutions, the one with the lowest cost

will be chosen. To achieve this, a big enough integer

∆ is introduced which is superior to the summation

of costs of all the arcs in the graph, i.e., ∆ >

∑

a∈A

m,n

Accordingly, the general objectivefunction can be ex-

pressed as follows.

Minimize : ∆·

∑

λ∈W

w(λ) +

∑

λ∈W

∑

n∈V

∑

m∈Out(n)

m,n

· L

m,n

(1)

This objective function is subject to a set of con-

straints which are listed below.

LTH Structure Constraints:

Source Constraint:

∑

λ∈W

∑

m∈In(s)

m,s

= 0 (2)

1 ≤

∑

λ∈W

∑

n∈Out(s)

s,n

≤ |D| (3)

Constraints (2) and (3) ensure that the source s

must not have any incoming arcs in a LTH, but must

have at least one outgoing arc on some wavelength

and the total number of outgoing arcs from s should

not exceed the number of destinations, i.e., |D|.

Destination Constraint:

1 ≤

∑

λ∈W

∑

m∈In(d)

m,d

≤ |D| − 1, ∀d ∈ D (4)

Constraint (4) guarantees that each destination

should be spanned in at least one LTH but at most

|D| − 1 LTHs.

Non-source Node Constraint:

∑

n∈Out(m)

m,n

≤

∑

n∈In(m)

n,m

, ∀λ ∈ W, ∀m ∈ V \ {s}

(5)

Since all the nodes are MI nodes that are equipped

with TaC option, they can be transited several times.

However, constraint (5) ensures that the number of

outgoingarcs should not exceed the number of incom-

ing ones for every LTH.

Non-member Nodes Constraint:

∑

n∈Out(m)

m,n

∑

n∈In(m)

n,m

, ∀λ ∈ W, ∀m ∈ V \ (s∪ D)

(6)

Constraint (6) makes sure that non-member nodes

can be either not used or served only as the intermedi-

ate nodes. In this case, the number of outgoing arcs is

equal to the number of incoming ones in every LTH.

Constraints (5) and (6) also imply that only destina-

tions can be leaf nodes.

Relationship between L

m,n

and w(λ):

w(λ) ≥ L

m,n

, ∀m, n ∈ V, ∀λ ∈ W (7)

w(λ) ≤

∑

m∈V

∑

n∈V

m,n

, ∀λ ∈ W (8)

Constraints (7) and (8) indicate that wavelength λ

is used in a LTH if and only if at least one arc uses it.

However, the above set of constraints is not

enough to guarantee the connectivity of the LTHs as

shown in (Zhou et al., 2010). To solve this problem,

we use the community method that is proposed

in (Yu and Cao, 2005). We introduce an other

variable, commodity ﬂow F

m,n

, as the support of the

variable L

m,n

in order to make sure the continuity and

connectivity of the resultant LTHs.

Connectivity Constraints:

Source Constraint:

∑

λ∈W

∑

n∈Out(s)

s,n

= |D| (9)

Constraint (9) indicates that the sum of ﬂows emit-

ted by the source is equal to the number of destina-

tions in the given multicast session.

Destinations Constraints:

∑

λ∈W

∑

n∈In(d)

n,d

∑

λ∈W

∑

n∈Out(d)

d,n

+ 1, ∀d ∈ D (10)

∑

n∈In(d)

n,d

−1 ≤

∑

n∈Out(d)

d,n

≤

∑

n∈In(d)

n,d

, ∀λ ∈W, ∀d ∈ D

(11)

Constraints (10) and (11) ensure that each destina-

tion must be consumed totally one and only one ﬂow

in all the LTHs. These constraints also guarantee that

each destination is reachable from the source.

Non-member Nodes:

∑

n∈In(m)

n,m

∑

n∈Out(m)

m,n

, ∀λ ∈ W, ∀m ∈ V \ (s∪ D)

(12)

Equation (12) ensures that non-member nodes are

only served as intermediate nodes without consuming

any ﬂows.

Light-trailbasedHierarchy-TheOptimalMulticastRouteinWDMNetworksWithoutSplittersandConverters

Relationship between L

m,n

and F

m,n

≥ L

m,n

, ∀m, n ∈ V, ∀λ ∈ W (13)

m,n

≤ |D| · L

m,n

, ∀m, n ∈ V, ∀λ ∈ W (14)

Equations (13) and (14) indicate that an arc should

carry a positive number of ﬂows if it is used in a LTH,

and this number should not exceed the total ﬂows

emitted by the source.

It is worth noting that with the supplementarycon-

nectivity constraints, the constraints (3) and (4) are

now relaxed.

5 EXPERIMENTAL RESULTS

In this section we present the experimental results of

the LTH solution for the concerned problem in com-

parison to the traditional LPH solution. In order to

make the comparison, we also develop an ILP formu-

lation for the LPH solution.

5.1 LPH Structure Constraint

Like the difference between light-path and light-trail

structures, LTH allows cycles whereas LPH does not.

In other words, for LPH structures, there is at most

one incoming arc to every node (except the source)

for every given wavelength. Thus, to make ILP for-

mulation for LPH we just add one more constraint to

the constraints of the ILP formulation for LTH pre-

sented in Section 4. This constraint can be formulated

as follows:

∑

n∈In(m)

n,m

≤ 1, ∀λ ∈ W, ∀m ∈ V \ {s} (15)

5.2 Simulation Settings

The two ILP formulations are implemented in C++

using GLPK v4.45 package (Makhorin, 2010). We

have carried out series of simulations with ran-

dom graphs generated using LEDA v.6.3 library

(Mehlhorn and Naeher, 2010). All the considered

graphs are directed. Due to the fact that ILP pro-

grams do not scale well, we just test with relative

small graphs in which the number of nodes N =

{20, 30, 40, 50}. The density value (the ratio between

the number of arcs and the number of nodes) is ﬁxed

to 2. Graphs with this density are considered as sparse

graphs. We suppose that sparse graphs well reﬂect the

common core optical networks.

The costs of arcs are randomly selected from the

set of integer {1, 2, .., 20}, and the set of destina-

tions D are also randomly selected with different size

|D| = {10%, 20%, .., 50%} of the number of nodes.

To ensure that there is a feasible solution for all in-

stances, the selected graph must be connected and

have at least one directed path from the source to each

destination. Moreover, in order to guarantee an ac-

ceptable conﬁdence interval, for a certain graph and

for each size |D|, we run 100 simulations with differ-

ent sources and destination sets. For each simulation,

the number of wavelengths used and the total cost of

the routes (hierarchies) are computed as the resultant

performance metrics to evaluate the two ILP formula-

tions.

Besides, to accelerate the ILP computation speed,

we ﬁrst employ the Farthest First heuristic proposed

in (Le et al., 2013) to the light-trail based hierarchy

ILP and the Farthest Greedy heuristic proposed in

(Din, 2009) to the light-path based hierarchy ILP to

get the upper bound for the number of wavelengths

used. These heuristics are known to be good ones for

the same concerned problem applying the two consid-

ering approaches respectively. With these heuristics,

we beneﬁted much of the time saved to accomplish

the simulations.

5.3 Simulation Results

The overall simulation results are presented in Table

1. As it is expected, light-trail based hierarchy solu-

tion (marked as LTH in the table) outperforms light-

path based hierarchy counterpart (marked as LPH) in

both number of wavelengths used and the total cost.

For the number of wavelengths used, the LTH solu-

tion always consumes fewer wavelengths than LPH

one. In particular, the ratio of saved wavelengths of

LTH solution is up to 19.84% (N = 20), 21.47% (N =

30), 16.09% (N = 40) and 21.95% (N = 50). On aver-

age, this ratio is 11.43%, 10.75%, 9.82% and 12.71%

respectively. This is obvious, because in general, al-

lowing the repetition of nodes in a LTH, more des-

tinations can be covered by a LTH than by a LPH.

Therefore, in term of minimizing the number of wave-

lengths, LTHs requires fewer or at most equal to the

number of wavelengths needed by LPH solution. This

is compatible with the Theorem 3.2.

For the total cost, there are few instances in which

the total costs are better with LPH solution, shown

as some negative reduced ratios in the table. This is

compatible with the Lemma 3.2. However, in general

the LTH solution results in lower costs with the saved

ratio up to 3.79% (N=50), and on average this ratio

is 1.31% (N=20), 0.46% (N=30), 0.90% (N=40), and

2.49% (N=50). In short, even though the total cost is

the second optimized objective, it is also better with

LTH based solution.

PHOTOPTICS2014-InternationalConferenceonPhotonics,OpticsandLaserTechnology

Table 1: Performance comparison between light-trail based

hierarchy (LTH) and light-path based hierarchy (LPH) so-

lutions.

Size Wavelengths Total cost

N=20

|D| LPH LTH ց LPH LTH ց

2 100 100 0% 5758 5630 2.22%

4 138 113 18.12% 9762 9543 2.24%

6 123 121 1.63% 10591 10565 0.25%

8 126 101 19.84% 8997 8884 1.26%

10 199 164 17.59% 14004 13922 0.59%

AVG 11.43% 1.31%

N=30

|D| LPH LTH ց LPH LTH ց

3 102 100 1.96% 6416 6281 2.10%

6 114 109 4.39% 11474 11253 1.93%

9 163 128 21.47% 14701 14377 2.20%

12 161 151 6.21% 20598 20277 1.56%

15 203 163 19.70% 22069 23281 -5.49%

AVG 10.75% 0.46%

N=40

|D| LPH LTH ց LPH LTH ց

4 120 109 9.17% 9198 9132 0.72%

8 170 154 9.41% 18436 18349 0.47%

12 174 146 16.09% 22640 22158 2.13%

16 196 171 12.76% 26584 26255 1.24%

20 302 297 1.66% 33441 33467 -0.08%

AVG 9.82% 0.90%

N=50

|D| LPH LTH ց LPH LTH ց

5 114 105 7.89% 14417 13981 3.02%

10 151 126 16.56% 19859 19165 3.49%

15 246 192 21.95% 28187 28310 -0.44%

20 204 179 12.25% 33614 32750 2.57%

25 266 253 4.89% 54694 52620 3.79%

AVG 12.71% 2.49%

6 CONCLUSIONS

In this paper we address the multicasting problem in

all-optical networks without splitters and converters.

The problem is to ﬁnd the multicast routes which min-

imize the number of required wavelengths with a low

cost. The problem is proved to be NP-hard, and two

exact solutions are presented in the forms of ILP for-

mulations (one for light-path based hierarchy, and the

other for light-trail based hierarchy). The theoretical

analysis pointed out that the optimal solution for the

problem in term of wavelength minimization corre-

sponding to a set of light-trail based hierarchies. The

simulations are carried out to verify it. Once again,

the experimental results showed that not a set of light-

path based hierarchies but a set of light-trail based

hierarchies are the optimal solution. Moreover, al-

though it does not necessarily deduce the cost optimal

solution, the light-trail based solution also appears to

be a good solution when consuming a lower cost com-

pared to the traditional light-path based solution in

general.

REFERENCES

Ali, M. and Deogun, J. S. (2000a). Cost-effective imple-

mentation of multicasting in wavelength-routed net-

works. IEEE/OSA Journal of Lightwave Technology,

18:1628–1638.

Ali, M. and Deogun, J. S. (2000b). Power-efﬁcient design of

multicast wavelength routed networks. IEEE Journal

on Selected Areas in Communications, 18:1852–1862.

Din, D.-R. (2009). Heuristic Algorithms for Finding Light-

Forest of Multicast Routing on WDM Network. In-

formation Science and Engineering, 25:83–103.

Garey, M. R. and Johnson, D. S. (1979). Computers

and Intractability: A Guide to the Theory of NP-

Completeness. W. H. Freeman & Co, New York, NY,

USA.

Le, D. D., Molnar, M., and Palaysi, J. (2013). Multicast

Routing in WDM without Splitters. Infocommunica-

tions Journal, 5(2):1–10.

Li, D., Du, X., Hu, X., Ruan, L., and Jia, X. (2000). Min-

imizing number of wavelengths in multicast routing

trees in WDM networks. Networks, 35(4):260–265.

Makhorin, A. O. (2010). GNU Linear Programming Kit

(GLPK).

Mehlhorn, K. and Naeher, S. (2010). LEDA-a Library of

Efﬁcient Data Types and Algorithms.

Molnar, M. (2011). Hierarchies to Solve Constrained Con-

nected Spanning Problems. Technical Report 11029,

LIRMM.

Mukherjee, B. (2006). Optical WDM Networks. Springer.

Yu, O. and Cao, Y. (2005). Mathematical formula-

tion of optical multicast with loss-balanced light-

forest. In Global Telecommunications Conference,

2005. GLOBECOM ’05. IEEE, volume 4, pages 5 pp.–

1972.

Zhang, X., Wei, J., and Qiao, C. (2000). Constrained multi-

cast routing in WDM networks with sparse light split-

ting. IEEE/OSA Journal of Lightwave Technology,

18:1917–1927.

Zhou, F., Molnar, M., and Cousin, B. (2010). Light-

Hierarchy: The Optimal Structure for Multicast Rout-

ing in WDMMesh Networks. In Computers and Com-

munications (ISCC), 2010 IEEE Symposium on The

15th IEEE Symposium on Computers and Communi-

cations (ISCC2010), 2010, pages 611 – 616, Riccione

Italie.

Zhou, Y. and Poo, G.-S. (2005). Optical multicast over

wavelength-routed WDM networks: A survey. Op-

tical Switching and Networking, 2(3):176 – 197.

Light-trailbasedHierarchy-TheOptimalMulticastRouteinWDMNetworksWithoutSplittersandConverters