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Constructing Multiple Light Multicast Trees in WDM Optical Networks
Multicast routing is a fundamental problem in anytelecommunication network. We address the multicast is-sue in a WDM optical network without wavelength con-version capability. To realize a multicast request in suchnetwork, it is required to establish multiple light multicasttrees (MLMT) sometimes due to the fact that the requestednetwork resources are being occupied by other routing traf-fic and therefore it is impossible to establish a single lightmulticast tree for the request. In this paper we first pro-pose a multiple light multicast tree model. We then show theMLMT is NP-hard, and devise an approximation algorithmfor it which takes
� � � � � � � � � � � � � � � � � � � � time and
delivers an approximation solution within� � � � � �
times ofthe optimal, where
�,
�, and
�are the numbers of nodes,
links, and wavelengths in the network,�
is the set of des-tination nodes and � is constant, � � � " . We finallyextend the problem further with the end-to-end path delayis bounded by an integer # , and we call this latter prob-lem as the multiple delay-constrained light multicast treeproblem (MDCLMT), for which we propose two approxi-mation algorithms with the performance ratios of
� � �. One
of the proposed algorithms takes� � � � � � � # � � � � � � #
time and the path delay is strictly met; and another takes� � � � � ) � � � � � � � time and the path delay is no more than� " � � # , where � is constant with � � � " .
1 Introduction
Wavelength-division-multiplexing (WDM) is emergingas a key technology for next-generation networks by provid-ing unprecedented bandwidth in a medium that is free frominductive and capacitive loadings, thus, relaxing the limita-tions imposed on the bandwidth-distance product. In WDMoptical networks the fibre bandwidth is partitioned into mul-tiple data channels which may be transmitted simultane-ously on different wavelengths. In this paper we consider
circuit-switched WDM networks, which can be further clas-sified as either single-hop or multi-hop networks [10, 11]. Insingle-hop networks each message is transmitted from thesource to the destination without any optical-to-electronicconversion within the network. Therefore, single-hop com-munication can be realized by using a single wavelength toestablish a connection, but such connections may in generalbe difficult or impossible to find in the presence of other net-work traffic. Alternatively, all-optical wavelength convert-ers may be used to convert from one wavelength to anotherwavelength within the network. However, such convertersare likely to prohibitively expensive for most applications inthe foreseeable future [15]. In this paper we therefore onlyconsider the single-hop WDM optical network.
Multicast is a fundamental problem in telecommunica-tion networks, which arises in a wide variety of applicationssuch as video conferencing, entertainment distribution, tele-classrooms, distributed data processing, etc [12]. In WDMnetworks the multicast problem is also referred to one-to-many routing and wavelength assignment problem (RWA),which aims at finding a set of links and wavelengths onthese links to establish the connection from the source tothe destination nodes.
Lots of effort for multicast in multi-hop networks hasbeen taken in past years. The cost of using the network re-sources in such networks is measured as follows. A wave-length traversing a link incurs a cost and the wavelengthconversion at a node also incurs a cost. Thus, the multi-cast problem is to find a multicast tree rooted at the sourceand spanning the destination nodes in
�such that the tree
cost is minimized [6], where�
is the set of destinationnodes. When the size
� � �of the set of destination nodes
is 1, the multicast problem becomes the optimal semilight-path problem [2], for which Chlamtac et al [2] presented an� � � � � � � � �
time algorithm for it, where�
is the num-ber of wavelengths and
�is the number of nodes in the
network. Liang and Shen [7] later provided an improvedalgorithm, which takes
� � � � � � � � � � � 8 : < � � � time
and can be implemented in the distributed environment effi-
is the number of links in the network. For� � �, Liang and Shen [6] proposed an approximation
algorithm for finding a multicast tree with minimizing thetree cost. Sahin and Azizolgu [14] considered the multicastproblem under various fanout polices and Malli et al [9]dealt with this problem under a sparse splitting model. Sa-hasrabuddhe and Mukherjee [13] formulated the problemas a mixed-integer linear programming problem. Znati etal [17] dealt with the problem by decoupling the delay fromthe cost model, and presented several heuristic algorithmsfor finding a multicast tree under the constraint of end-to-end delay from the source to the destination nodes. In addi-tion, there have been several other studies for constructingconstrained multicast trees in the WDM optical networks.For example, Libeskind-Hadas [8] proposed a multi-pathrouting model, in which the multicast problem is to find aset of paths from the source to the destination nodes suchthat each path contains a subset of destination nodes, thenodes in the set of destination nodes are included by thesepaths, and the cost sum of these paths is minimized. Zhanget al [16] considered the multicast problem by focusing onthe limited splitting power of optical switches and providedfour heuristic algorithms for the problem.
Due to currently it is prohibitively expensive to employthe wavelength conversion switches in optical networks, inthis paper we consider a circuit-switched single-hop WDMoptical network. We extend the multi-path routing con-cept [8] further by proposing a multiple light multicasttree model, on which we deal with the multicast problem.Specifically, given a source � and a set of destination nodes�
, if the wavelength conversion at nodes is not allowed,then it is impossible to realize a multicast request throughthe construction of a single light multicast tree rooted at thesource including all the nodes in
�due to lack of the avail-
able resources that are being used by other routing traffics.Instead, to realize a multicast request, it necessitates to con-struct several light multicast trees that cover the nodes in�
, here a light multicast tree is referred to a tree in whichevery link has the same wavelength. To minimize the net-work resource consumption for a multicast session, the ob-jective is to minimize the cost summation of all the lightmulticast trees. We thus refer to the multicast problem onthis cost model as the multiple light multicast tree problem(MLMT for short). When the set of destination nodes in-cludes all other nodes except the source � , the problem be-comes a broadcast problem which is called the light broad-cast tree problem. If the end-to-end delay of a routing pathin each of the trees from the root (source) to a destinationnode is bounded by an integer � , then the problem is re-ferred to the multiple delay-constrained light multicast treeproblem (MDCLMT for short). Under our model we firstshow the mulicast problem (MLMT) is NP-Complete evenwhen the set
�of destination nodes is
� � � � � . We
then provide an approximation algorithm for it which takes� � � � � � � � � � � � � � � � � � time and delivers a solution
within� � � � � �
times of the optimal, where � is constant,� � � � � . We finally deal with the MDCLMT, for whichtwo approximation algorithms with performance ratios of� � �
are proposed, which trade-offs between the runningtime and the accuracy of the path delay. One of the proposedalgorithms takes
� � � � � � � � � � � � � � � time and the path
delay is strictly met; and another takes� � � � � $ � � � � � � �
time which is independent of � , and the path delay is nomore than
� � � � � , where � is constant with � � � � � .The rest of the paper is organized as follows. In Sec-
tion 2 notations are introduced and the problems are de-fined. In Section 3 the MLMT is shown to be NP-Completeeven when the destination set is
� � � � � , an approx-imation algorithm for it is proposed, and the running timeas well the performance ratio of the algorithm is analyzed.In Section 4 the MDCLMT is considered, and two approxi-mation algorithms for it with performance ratios of
� � �are
presented.
2 Preliminaries
The network model: The optical network is modeledby a directed graph � � � � � � � �
, where�
is a set ofnodes (vertices) representing switches,
�is a set of directed
links (edges) representing the optical fibers, and�
is a setof wavelengths in � ,
� � � � �,
� � � � �, and
� � � � �. Let� � � � � � � � � " " " � � & . Associated with each link ' ) �
, awavelength set
� � ' ( * �
) is given, and for each� ) � � '
,a non-negative weight + � ' � �
is assigned which is the costof traversing ' using wavelength
�. This cost reflects the
communication bandwidth consumption on ' . Sometimes,the routing congestion factor on the link is also incorporatedin the cost by assigning different weights to different wave-lengths. In some cases it is also assumed that an integraldelay , � '
is associated with each link ' ) �.
Problem formulation: A multicast request is an orderedpair
� � / � where � ) �
is the source of the multicast ses-sion and
�( * � � � ) is the set of destination nodes. As-
sume that multicast requests are made and released dynam-ically. To realize a multicast communication request, theideal case is to establish a single light multicast tree. How-ever, due to the fact that some specific network resource,e.g., a specific wavelength, is occupying by other commu-nication traffic at this moment, and it is impossible to estab-lish such a tree for the current multicast request. Instead, itis required to establish a collection of light multicast trees torealize the request. Specifically, we deal with the followingtwo multicast problems.
The Multiple Light Multicast Tree Problem (MLMT) isto construct a collection of multicast trees 1 � � 1 � � " " " 1 & 3such that (i) tree 1 5 is a light multicast tree rooted at �
.In some real-time QoS applications, not only the cost of
multicast trees is the main optimization objective, but alsothe response time (the path delay) from the source to eachdestination node in a multicast session is paramount. Forthis latter case, another extra metric - an integral delay � � � �is assigned, in addition to the cost metric for each link � ��
. Thus, given a source � , an integral delay � , and a set ofdestination nodes
�, the Multiple Delay-Constrained Light
Multicast Tree Problem (MDCLMT) is to find a collectionof light multicast trees � � � � � � � � � � � � �
such that (i) tree � �is a light multicast tree rooted at � covering a subset of
�,
and the wavelength� � �
is on each link of � � , � � � � � �,
� � � � � �; (ii) each node � � �
as a leaf node is includedin one of the
� �trees at most once but it may appear as a
relay node in other trees, and the path delay in the tree from
� to � is bounded by � ; (iii) �� �
� � � � � � �is minimized,
where � � � � � � � � � � � � � � � � � � � �.
3 Approximation Algorithm for the MLMT
3.1 The Light Broadcast Tree Problem is NP-Hard
When the destination set� � � � � � , the MLMT be-
comes the light broadcast tree problem. In contrast to thatthe broadcast tree problem (the minimum spanning tree) inconventional networks is polynomially solvable, the lightbroadcast tree problem in WDM optical networks on themultiple tree model is NP-Complete. We state it in the fol-lowing theorem without proof due to the space limit. Themajor technique is to reduce the 3-CNF SAT problem to it.
Theorem 1 The light broadcast tree problem in WDM net-works on the multiple light multicast tree model is NP-Complete.
3.2 Approximation Algorithm
We have shown that the light broadcast tree problem isNP hard, so is the MLMT. We thus focus on devising anapproximation algorithm for it. The basic idea is to trans-form the problem to a directed Steiner tree problem in anauxiliary directed graph � � . As a result, a feasible solu-tion for this latter problem will give a feasible solution forthe concerned problem.
For each distinct wavelength� � � �
, a subgraph � # %of � containing � is induced by the links that the wave-
length� �
is on them and every other node in the subgraphis reachable from � . Let � # % � � � � � � �
be the subgraph andlet node � be included in � # % , then � in � # % is relabeledas
� � � � � �. The weight of a corresponding link � �
represents the node � � �, the set of wavelengths on link) � � � � � � � � � - in � � is
� � � � , and the set of wavelengths onlink
) � � � � � � � � � - in � � is� � � � . The weight assignment of
links in � � is defined as follows. For each link � � � �,
its weight in � � is � � � � � �, for each link
) � � � � � � � � � - , itsweight is � ) � � � � � � � � � - � � � � � , and for each � � �
,the weight of link
) � � � � � � � � � - is � ) � � � � � � � � � - � � � � � .The auxiliary graph � � has the following properties. (i)For each node � � �
, there is at least a directed path in� � from � �
to � �, using the links in a specific � # % , links) � � � � � � � � � - and
) � � � � � � � � � - in � � . (ii) Each � # % is a sub-graph of � � . The nodes in � � between � # % and � # �
arenot reachable from each other, if
� 5 � . We then have thefollowing theorem.
Theorem 2 The multicast tree in � � rooted at � �includ-
ing the nodes in� � � � � � � corresponds to an optimal
solution for the MLMT in � .
Proof Let 6 8 : < be an optimal solution for the MLMTin � consisting of the collection of light multicast treesand � � 6 8 : < �
be the cost sum of the light multicast treesin 6 8 : < . Following the construction of � � , it is easy toshow that there is a corresponding multicast tree � � in
� � in which each tree � � in 6 8 : < corresponds to a sub-tree rooted at
� � � � � � �of � � , and � � is a multicast tree
rooted at � �including the nodes in
� � � � � � � . Let � � � �
be the weighted sum of the links in � � . Then, � � � � � � 6 8 : < �
.Assume that � @ B8 : < is a directed Steiner tree in � � rooted
at � �including the nodes in
� � � � � � � . Following the de-finition of directed Steiner trees, the weighted sum � � @ B8 : < �of the links in � @ B8 : < is � � @ B8 : < � � � � � � � � 6 8 : < �
. Thetheorem follows. D
It is well known that the directed Steiner tree problemis NP hard [3]. In the following we aim to find an approx-imate, directed Steiner tree in � � rooted at � �
includingthe nodes in
� � � � � � � , which in turn will give an ap-proximation solution for the MLMT. The detailed algorithm(Fig 1) is described as follows.
Theorem 3 Given a directed network � � � � � � � �with
a given source � and a node set�
, there is an approxima-tion algorithm for the MLMT with a performance ratio of
orem 2, where � and � are constants and � � � � � . �
4 Approximation Algorithm for the MD-CLMT
In this section we start with a routine for finding a delay-constrained shortest path in a graph bounded by an end-to-end integral delay ' , which will be used in the proposedalgorithms. We then provide approximation algorithms forthe MDCLMT.
4.1 The delay-constrained shortest path problem
The delay-constrained shortest path problem is to find ashortest path in � from � to
�under the constraint that the
end-to-end delay of the path is bounded by ' . If the pathdelay ' is not an integer, the problem is NP-hard [3]; other-wise, it is polynomial solvable, and one such a dynamic pro-gramming algorithm due to Kompella et al [5] is describedas follows.
Let � � � � � � �be the cost of a shortest path from
�to �
with the delay exactly � and � � � � � �be the cost of the short-
est path in � with a bounded delay ' . If there are multipleshortest paths with the same cost, then the one with the leastdelay is chosen. Thus, � � � � � � �
� � � � � � and � � � � � � � � � � � � � � � � � � � . Then, the so-lution for the problem is to find the value of � � � � � �
. UseDijkstra’s algorithm, it takes
� � ' � �time to find a solution
for this dynamic programming problem.Let � � � � � � � �
be a delay-constrained shortest path ina subgraph � � of � from � to
�with the end-to-
end delay ' , the cost of � � � � � � � �is � � � � � � � � � � �
� � � � � � � � � � � � � � � �, where the subgraph � � is induced by
the links in � containing wavelength�
and � is a link in� � � � � � � �
. It is easy to see that the dynamic programmingalgorithm for the delay-constrained shortest path problemis a pseudo-polynomial algorithm, because its running timedepends on the delay ' . However, if the end-to-end delay isnot met strictly, there is a strongly polynomial approxima-tion algorithm for this problem, which is stated as follows.
Lemma 1 Given a WDM optical network � � � � � � � �with
a pair of nodes � and�, and an integral delay ' , assume
that each link � has been assigned a set� � � �
( � �) of
wavelengths, each wavelength� � � � �
traversing � incursa cost � � � � � �
, and the delay � � � �for traversing on � is an
integer and � � � � � ' . There is an� � � � + � �
approxima-tion algorithm for finding a delay-constrained shortest pathfrom � to
�with the relaxation of the bounded delay, which
finds a shortest path with the bounded delay� � � � � ' , where
� is constant, � � � � � .
Proof The approximation algorithm [4] for the delay-constrained shortest path problem uses a scaling technique,which delivers an optimal solution in terms of the path costbut the path delay is bounded by
� � � � � ' , where � is con-stant, � � � � � . �
4.2 Approximation algorithm with strictly delay
We here propose an approximation algorithm for theMDCLMT such that the end-to-end delay of the routingpaths must be met strictly. The idea behind the proposedalgorithm is as follows. Let the approximation solutionconsist of
times ofthe optimal. So, the running time for Steps 5 and 6is � � � � � � � � � � � � � � � � � � � � � � � � � �
. Thealgorithm therefore takes
� � � � � � � � � �time.
4.3 Approximation algorithm without strictly de-lay
We now provide a truly polynomial approximation algo-rithm for the MDCLMT if the end-to-end delay of a pathis not met strictly. The basic idea is similar to the one inthe preceding section except the following modifications.At Step 2 an approximation algorithm instead of an exactalgorithm for finding a delay-constrained shortest path willbe used. The set
�of destination nodes is then partitioned
into�
disjoint subsets� � �
, � � � � � and � � � � � �
.For each subset
� � �, at Step 6 a new approach called stick-
ing approach instead of the algorithm due to Kompella etal [5] will be employed to find an approximate, light multi-cast tree in � � � � rooted at � including all nodes in
� � �. In
the following we present the sticking approach.Let � � � � � � � � � �
be a shortest path in � � from � to�
with a bounded delay no more than� � � � , founded
by applying the Goel et al [4] algorithm on � � . Anode � � �
to each subset to construct a light multicast tree. As results,the
� �light multicast trees are constructed, which form a
solution for the MDCLMT. The time used for this step is� � �� �
� � � � � � � � � � � � � �, following Lemma 3. The
theorem then follows. �
5 Acknowledgment
It is acknowledged that the work by Weifa Liang wassupported by a research grant #DP0449431, funded by Aus-tralian Research Council under its Discovery Schemes.
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