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Multicast Routing and Wavelength Assignment in
WDM Networks with Limited Drop-offs
X.-D. Hu and T.-P. Shuai
Inst. of Applied Math.
Chinese Academy of Sciences
Beijing, China
{xdhu, shuaitp}@mail.amss.ac.cn
Xiaohua Jia
Dept. of Computer Sci.
City Univ. of Hong Kong
Kowloon, Hong Kong
jia@cs.cityu.edu.hk
Mu-Hong Zhang
Dept. of IE and OR
Univ. of California
Berkeley, USA
mhzhang@ieor.berkeley.edu
Abstract—In WDM networks with limited drop-offs, the route
of a multicast connection consists of a set of light-trees. Each of
the light-tree is rooted at the source node and contains no more
than a limited number, say k, destination nodes due to the power
loss of dropping optical signals off at destination nodes. We call
such a light-tree k-drop light-tree. In this paper we study the
multicast routing problem of constructing a set of k-drop light-
trees that have the minimal network cost. The network cost of a
set of light-trees is defined as the summation of the link cost of all
the light-trees. We first prove that this problem is polynomial-
time solvable for k = 2 and NP-hard for k ≥ 3. We then
propose a 4-approximation algorithm for the problem for k ≥ 3.
A wavelength assignment algorithm is also proposed to assign
wavelengths to the light-trees of a multicast connection. In the
end we give simulation results showing that k-drop multi-tree
routing can significantly save not only the network cost but also
wavelengths used. Moreover, when k ≥ 5 its performance is very
close to the case where k is infinite (i.e., the case of using a single
tree for a multicast connection).
I. INTRODUCTION
Multicast is a point-to-multipoint communication that a
node sends data to multiple destinations [4,13]. There are
many multicast applications, such as news feeds, video distri-
bution, multimedia conferencing, and so on. It is a challenging
job to implement multicast in Wide Area Networks (WANs)
due to high complexity of multicast routing [10,13]. Multicast
routing is to find a tree rooted from the source and connecting
all the destinations. The multicast data will be transmitted from
the source and propagated to all destinations along the tree.
An important objective of multicast routing is to minimize the
network cost of the routing tree, which is defined as the sum
of costs of all the links in the tree. The problem of finding a
tree in a general topology network, such that it connects a set
of nodes and has the minimal cost, is known as the Steiner
tree problem [2].
To facilitate multicast in wavelength-routed optical net-
works, the concept of a light-tree and the cross-connect
architecture of splitter-and-delivery were proposed in [1,14].
A light-tree is a point-to-multipoint all optical channel which
is rooted from a source and connects multiple destinations.
In the absence of wavelength converters, a light-tree would
occupy the same wavelength on all links of the tree. Each
intermediate node that has more than one child in a light-
tree must have a splitter which splits the incoming optical
signal into multiple copies outgoing to the child nodes. Each
time an optical signal is split, a splitting loss is incurred. If a
destination is an intermediate node in the light-tree, a splitting
loss is also incurred to drop-off a copy of the signal at the
destination. In practice, if a signal is split into n copies, the
signal power of at least one copy will be less than or equal
to
an optical signal must have a minimum amount of power in
order to be dropped-off at a destination or passed to the next
down-stream node. Therefore, due to this split loss, it is not
possible to drop off data at an arbitrary number of destinations
in a single light-tree [9]. A light-tree always has a limited
number of destinations that are allowed to drop-off data, and
a multicast connection with many destinations needs to use
multiple light-trees to transmit data to all destinations.
In this paper, we study how to establish a multicast con-
nection in an optical network under the k-drop multi-tree
model [9]. Under this model, there is a pre-specified integer k,
which limits the maximum number of destinations that optical
signals are allowed to drop-off at a light-tree. Multicast routing
becomes a problem of finding a set of light-trees such that at
most k destination nodes are designated to receive data in each
light-tree and every destination node is designated in one of
the light-trees in the set. The objective is that the total cost
of light-trees in the set is minimal. After the routing, we need
to assign a wavelength to each of the produced light-trees in
such a way that distinct wavelengths are used for two light-
trees if they share a common link. We assume no wavelength
converter is used in a light-tree.
The k-drop multi-tree model is a general model for multicast
routing. When there is no limit on the number of destinations
that optical signals are allowed to drop-off at (i.e., k can
be an arbitrary number), it becomes the single-tree model,
because all the destinations of a multicast can be reached in
a single tree. When no splitting nor drop-off is allowed at
switches (i.e., k = 1), it becomes the simple lightpath model,
where a separate lightpath is required from the source to each
of the multicast destinations. Recently some research work
has been done on how to establish a multicast connection
under the k-drop multi-tree model in optical networks. In
[9] the wavelength requirement for a multicast connection
in some special topology networks was analyzed, such as in
1
nof the original signal power [1]. On the other hand,
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rings, torus, and hypercubes. More analysis and the routing
algorithms for minimizing the number of wavelengths in the
special topology networks were given in [7]. Finding multi-
trees for a multicast connection was also discussed in [16,17].
A greedy algorithm was proposed in [16] to find multi-trees,
such that each tree uses the same wavelength (wavelength
continuity rule) and the total cost of multi-trees is minimized.
The size of a tree is constrained by wavelength continuity
rule and it did not consider the limited drop-offs in a tree.
Zhang et al. proposed in [17] a set of algorithms to construct
a source-based multicast light-forest consisting of one or
more multicast trees. The objectives of the algorithms include
minimizing the number of wavelengths and the number of hops
from the source to each destination. There is no guaranteed
performance bound for the algorithms proposed in [16,17] and
the performance analysis was done through simulations.
The rest of the paper is organized as the follows. In Section
2 we formulate the minimal multicast routing problem in k-
drop multi-tree model. In Section 3 we propose a polynomial-
time optimal algorithm for the case of k = 2. In Section 4,
we show the problem is NP-hard for k ≥ 3 and propose a
4-approximation algorithm to solve the problem. In Section 5,
we propose a wavelength assignment algorithm for the multi-
trees for a multicast connection. In Section 6, we conduct an
extensive simulations to study the performance of the proposed
algorithms. Finally, Section 7 concludes the paper.
II. PRELIMINARIES
A. Problem Specification
In this paper, we assume bidirectional transmission (but the
approach and analysis are both applicable for unidirectional
case). That is, for two nodes (switches/routers) A and B in
the network, there are two links between them, one carries
transmission from A to B while the other from B to A.
A multicast connection is represented by < s,D >, where
s is the source node from which data is sent to a set of
destination nodes D. Under the multi-tree model, at most k
destination nodes are allowed to receive the data in a light-
tree, where parameter k is dependent on the power budget of
light transmission.
B. Problem Formulation
According to the above specification and assumption of
our problem, we model the network under consideration as
an edge-weighted graph G(V,E), where vertex-set V is the
set of nodes in the network representing switches/routers and
edge-set E is the set of links between nodes. Weight function
c : E → ?+represents the network cost of using a particular
edge. We assume that the weight function c is additive over
the links in a path p(u,v) between u and v, i.e.,
c(p(u,v)) ≡
?
a∈p(u,v)
c(a).
For the simplicity of presentation, we denote by pG?(u,v) the
shortest path from u to v in subgraph G?of G.
We define a k-drop tree as a tree in G such that in the tree
at most k nodes in D are designated to receive the data. In
addition, we define a k-drop multi-tree routing of < s,D >,
denoted by R(s,D;k) = {Ti|i}, as a set of k-drop trees
Ti’s such that every destination in D must be designated to
receive the data in a Ti in R(s,D;k) for some i. Let m ≡
?|D|/k?, then the number of k-drop trees in R(s,D;k) is
|R(s,D;k)| ≥ m. Two k-drop trees in R(s,D;k) may share
a common edge, which will not cause any problem during
data transmission under the wavelength division multiplexing
(WDM) technology [3,15].
When multicasting, data is transmitted over the multiple k-
drop trees from the source to reach all the destinations. Data
is transmitted through each edge in a k-drop tree exactly once,
the network cost of multicasting data can then be defined as
the total network cost of k-drop trees in R(s,D;k), i.e.,
c?R(s,D;k)?≡
In this paper, we study how to, given a multicast connection
< s,D > and a positive integer k, find a k-drop multi-tree
routing R(s,D;k) of minimum network cost. We refer this
problem as k-MTR (k-drop Multi-Tree Routing) problem. Al-
though the optimization of wavelength usage is not expressed
explicitly as an objective, it is achieved implicitly. This is
because a R(s,D;k) with less network cost tends to have less
number of trees and less number of links in a tree, and thus
have less chances for trees to share a common links (which
results in less number of wavelengths required).
?
Ti∈R(s,D;k)
c(Ti).
III. OPTIMAL SOLUTION TO k-MTR FOR k ≤ 2
When k = 1, the optimal solution to the k-MTR problem
consists of |D| shortest paths from source s to each of |D|
destinations (which is the case of the lightpath model). It can
be found in polynomial-time. In the following, we will show
how to find an optimal solution to k-MTR for k = 2.
When k=2, each k-drop tree in the multicast routing can
contain no more than two destinations. The basic idea is to
introduce an auxiliary graph G?so that the k-MTR problem
in the original graph G can be reduced to the minimum
weighted matching problem [12] in G?, which can be solved
in polynomial-time. A matching of a graph is a set of edges
such that no vertex in the graph is incident to two edges in
the set. It is further called complete if every graph is incident
to an edge in the matching. The minimum weighted matching
problem is to, given an edge-weight graph that has complete
matchings, find a complete matching whose weight is minimal,
where the weight of a matching is the total weight of edges
in the matching.
Given a multicast connection < s,D > on network G,
where D = {d1,···,d|D|}. For each pair of destinations
(di,dj), compute the minimum Steiner tree from s to destina-
tions diand dj, denoted by T(s,di,dj). Note that T(s,di,dj)
consists of at most three paths, pG(di,u), pG(dj,u), and
pG(s,u), jointed at some node u ∈ V . (When u is di
or dj, T(s,di,dj) consists of two paths.) Hence computing
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T(s,di,dj) is to find a node u ∈ V such that
c(pG(di,u)) + c(pG(dj,u)) + c(pG(s,u)) =
minv∈V{c(pG(di,v)) + c(pG(dj,v)) + c(pG(s,v))}.
After computing the Steiner tree for each pair of destina-
tions, we introduce a new graph G?(D ∪ {s1,···,s|D|},E?).
Each siis a pseudo node of s associated with destination di.
There is an edge between diand djfor i ?= j and the weight of
the edge w(di,dj) = c(T(s,di,dj)), i.e., the cost of minimum
Steiner tree from source s to destinations diand dj. There is
an edge between siand sj for i ?= j and its weight is zero.
There is an edge between siand difor each i and its weight
w(si,di) = c(pG(s,di)), i.e, the cost of shortest path from
source s to destination di. There is no edge between diand
sjfor i ?= j.
Considering the new graph G?, edge (di,dj) ∈ E?corre-
sponds to the minimum Steiner tree from s to destinations di
and dj, and edge (si,di) ∈ E?corresponds to the shortest
path from source s to destination di. That is, each edge in G?,
except edges (si,sj), corresponds to a k-drop tree for k ≤ 2.
Thus a k-drop multi-tree routing R(s,D;k) corresponds to a
set of edges in G?such that each destination in D is incident
to exactly one edge in the set. Therefore, using pseudo nodes
si (so that G?has complete matchings) and setting weight
zero to edges between them, the problem of finding a set of
2-drop trees that include all destinations and have the minimal
total cost becomes the problem of finding a minimum weighted
complete matching in G?whose cost is minimal. The algorithm
is formally presented below.
Algorithm A Minimum Matching Based Algorithm
(A1) Compute the shortest path pG(u,v)
for each node pair u and v in V .
(A2) Compute the minimum Steiner tree T(s,di,dj)
for each destination pair diand djin D.
(A3) Construct new graph G?(D ∪ {s1,···,s|D|},E?).
(A4) Find the minimum weighted matching M in G?.
(A5) Output the set of 2-drop routing trees from M.
Theorem 1 Algorithm A solves the 2-MTR in time O(|V |3+
|D|4).
Proof In Step A1, the shortest path between each node pair
in V can be found in time O(|V |3). In Step A2, the minimum
Steiner tree for each node pair in D can be produced in
time O(|D|2|V |). In Step A3, constructing the auxiliary graph
requires no extra time after Step A1-2 are finished. In Step
A4, the minimum weighted matching problem in G?can be
formulated as a linear programming problem and thus solved
by suing the primal-dual method in time O(|D|4) (refer to
[12]). In Step A5, having found the matching M, the 2-MTR
for < s,D > can be obtained by substituting each edge
(di,dj) ∈ M by the Steiner tree T(s,di,dj), and each edge
(si,di) ∈ M by the shortest path pG(s,di). It costs linear
time to |D|. The proof is then finished.
IV. APPROXIMATE SOLUTION TO k-MTR FOR k ≥ 3
In general the k-MTR problem is NP-hard, because the
Steiner tree problem can be reduced to it by setting k ≥ |D|.
In this section we will propose an approximation algorithm
with a guaranteed performance ratio. The basic idea is to first
produce a directed trail of low cost including all nodes in
D ∪ {s}, and then break it into m small trails on which at
most k nodes in D are designated to receive data, in the end
for each small trail make a k-drop tree constituting of s and
those designated nodes in D. The directed trail can be obtained
by constructing a Hamilton circuit of low cost in an auxiliary
graph whose vertex-set is D ∪ {s}.
Algorithm B Hamilton Circuit Based Algorithm
(B1) Construct an auxiliary complete edge-weighted
graph Gaof D ∪ {s}. For u,v ∈ D ∪ {s} the weight
of edge (u,v) is c(pG(u,v)).
(B2) Construct a Hamilton circuit Hcof Gaby using
Christonfides’ method [5,12].
(B3) Obtain a directed trail T of D ∪ {s} in G by substi-
tuting each edge in Hcby the shortest path between
its two endpoints of the edge in G,
T = (s → d1→ ··· → d|D|−1→ d|D|→ s).
(B4) Partition T into m subtrails Tifor i = 0,1,···m − 1
such that dik+1,dik+2,···,dik+kare designated in
Tito receive the data. For each i, find viin Ti
which is closest to source s.
(B5) Construct a k-drop tree Tidesignating k destina-
tions dik+1,dik+2,···,dik+kin subtrail Ti, i.e.,
Ti:= Ti∪ pG(s,vi).
(B6) Output RB(s,D;k) := {T0,T1,···,Tm−1}.
Because in Step B1 the auxiliary graph Gais a complete
graph and the weight function defined on its edges satisfies
triangular inequality, in Step B2 Christonfides’ method can be
employed to construct a Hamilton circuit of Ga. The following
lemma comes directly from the well-known result due to
Christonfides [5].
Lemma 1 For any given multicast connection < s,D > on
G, the Hamilton circuit Hcof Gaproduced at (B1-2) has cost
at most 3/2 times that of the minimum Hamilton circuit of Ga.
We now prove that Algorithm B has a constant guaranteed
performance ratio in the worst case analysis. To do this,
we need the following lemma. Given a multicast connection
< s,D >, let Ropt(s,D;k) be an optimal k-routing and
c(Ropt(s,D;k)) be its cost.
Lemma 2 Let di? be the destination node in trail Tithat is
the closest to s. Then
m−1
?
i=0
c(pG(s,di?)) ≤ c(Ropt(s,D;k)).
Proof
trees, T∗
N ≥ m. We construct an auxiliary weighted bipartite graph
B(X,Y ), where X = {Ti|i = 0,1,···,m − 1} and Y =
{T∗
if and only if Tiand T∗
common and the weight of the edge is w(Ti,T∗
Suppose that the optimal k-MTR Ropt(s,D;k) has
1,T∗
2,···,T∗
N, where each T∗
iis a k-drop tree and
i|i = 1,···,N}. There exists an edge (Ti,T∗
jdesignate α ≥ 1 destination nodes in
j) in B(X,Y )
j) = α.
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Now we prove, by using Hall’s Theorem (refer to [12]), that
B(X,Y ) has a perfect matching such that each Tiis incident
to an edge in the matching. Suppose, by contradiction, that
there exists a subset X0 ⊆ X such that X0’s neighbor set
Y0⊆ Y , which consists of vertices adjacent with some vertices
in X0, satisfies |Y0| ≤ |X0| − 1. Since each Tidesignates at
most k destination nodes and each of them is designated in
exactly one optimal k-drop tree, then the total weight of edges
incident to Ti is at most k. For each T∗
result. Now for X?⊆ X and Y?⊆ Y , let w(X?) and w(Y?)
denote the total weights of edges incident to some Ti∈ X?
and T∗
w(X0) ≤ w(Y0) ≤ k|Y0|. In addition, we have
w(X \ X0) ≤ k|X \ X0| = k(m − |X0|).
Hence we obtain the following contradiction.
jwe have the same
j∈ Y?, respectively. Then w(Y?) ≤ k|Y?|, this implies
|D|
=
≤
≤
w(X0) + w(X \ X0)
k|Y0| + k(m − |X0|)
k(m − 1) < |D|.
Therefore, there exists a desired matching. Without loss of
generality, we denote this matching by M = {(Ti,T∗
This means that for each i there exists a destination node
designated in both Ti and T∗
less than the cost of the shortest path from s to that common
designated destination node, which, by the definition of di?, is
not less than the cost of the shortest path from s to di?, i.e.,
c(T∗
obtain the desired inequality. The proof is finished.
Theorem 2
Given a multicast connection < s,D > and
k ≥ 3, Algorithm B produces a k-MTR RB(s,D;k) in time
O(|D||V |2) whose cost is at most four times that of the optimal
k-MTR Ropt(s,D;k).
Proof Let Hoptbe the minimum Hamilton circuit of Ga. Then
we have
i)|i}.
i. Thus the cost of T∗
i is not
i) ≥ c(pG(s,di?)). To sum up this inequality over i, we
2c(Ropt(s,D;k)) ≥ c(Hopt),
since two Ropt(s,D;k)s correspond a Hamilton circuit of Ga.
In addition, by Lemma 1 and inequality (1), we have
c(Hc) ≤3
Thus by the rules of Algorithm (B5) and Lemma 2, we have
(1)
2c(Hopt) ≤ 3c(Ropt(s,D;k)).
(2)
c(RB(s,D;k))=
m−1
?
m−1
?
c(T ) + c(Ropt(s,D;k))
c(Hc) + c(Ropt(s,D;k))
4c(Ropt(s,D;k))
i=0
c(Ti)
=
i=0
c(Ti) +
m−1
?
i=0
c(pG(s,vi))
<
=
≤
by (2).
Now consider the running time of Algorithm B. First, notice
that the shortest path between a pair of vertices in G can be
found in time O(|V |2), thus the auxiliary graph Gaat (B1) can
be constructed in time O(|D||V |2). Secondly, the Hamilton
circuit Hc of Ga can be produced in time O((|D| + 1)3)
(refer to [12]). Thirdly, the directed trail T at (B3) and its
partition into m subtrails at (B4) can be obtained in time
O(|V |). In the end, every k-drop tree can be produced in time
O(|V |2). Therefore, Algorithm B outputs RB(s,D;k) in time
O(|D||V |2). The proof is finished.
When m = 1 the k-MTR problem becomes the Steiner tree
problem in networks, which has an approximation algorithm
with performance ratio less than 2 [4]. When m = 2,
Algorithm B can be modified slightly so that its approximation
ratio 4 could be reduced.
Corollary 1 For the case of m = 2, there is a polynomial-time
algorithm that produces a k-MTR whose cost is at most three
times that of the optimal k-MTR and the minimum Steiner tree
of D ∪ {s}.
Proof At (B4) of Algorithm B, after partitioning T into 2
subtrails T0and T1, we construct a k-MTR consisting of two
k-drop trees,
T1
T2
=
=
{s → d1→ d2→ ··· → dk−1→ dk},
{s → d|D|→ d|D|−1→ ··· → dk+2→ dk+1}.
By applying the same argument used in the proof of Theorem
2, we deduce c(RB(s,D;k)) ≤ 3c(Ropt(s,D;k)). Moreover,
since the minimum Hamilton circuit of Gahas cost less than
two times that of the minimum Steiner tree Toptof D ∪ {s}
in G, we have c(RB(s,D;k)) ≤ 3c(Topt).
V. WAVELENGTH ASSIGNMENT FOR MULTI-TREES
In this section, we consider how to assign a wavelength
to each of k-drop trees such that two trees are assigned
with distinct wavelengths if they share a common link. This
problem can be reduced to the vertex-coloring problem of
graphs as follows: Construct a graph G?(V?,E?) such that V?
is a set of k-drop trees and there is an undirected edge in E?
between two nodes in V?in graph G?if the corresponding
k-drop trees share a common physical link in G(V,E).
Since the vertex-coloring problem is NP-hard which has
no polynomial-time algorithm with a constant approximation
performance ratio [2], we use a simple heuristic based on the
sequential coloring approach proposed in [11].
Algorithm C Wavelength Assignment Algorithm
(C1) Construct graph G?(V?,E?).
(C2) Choose a vertex in V?that has the least degree.
(C3) Find a maximal set of vertices in current V?that
are not adjacent to the selected vertex and there is
no edge between any pair of vertices in the set.
(C4) Assign a wavelength to the vertices in the set and
removes them from V?.
(C5) Repeat (C2-4) until every vertex V?is assigned
a wavelength.
Theorem 3
Given a k-MTR R(s,D;k) that has m k-
drop trees, Algorithm C assigns wavelengths to it in time
O(m2|V |3).
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Proof
O(m2|V |2), because Tihas at most (|V | − 1) edges. In the
loop of Step C2-5, all vertices in V?are assigned a color
one by one, and two vertices are assigned the same color if
and only if they are not adjacent. Choosing a vertex of the
least degree and finding a maximal set both can be done by
checking adjacency among vertices in V?. Thus Algorithm C
can finish wavelength assignment in time O(m2|V |3).
VI. SIMULATION STUDY
In the previous sections, we have proposed algorithms for
constructing a k-MTR and allocating wavelengths to them.
At the same time we have made theoretical analysis of their
performances, respectively. In this section, we will simulate
the proposed algorithms in various network environment.
At Step C1, G?(V?,E?) can be produced in time
A. Simulation Model
In the simulations we use two network topologies. One is the
backbone of NSFnet [3]. It consists of 14 nodes representing
14 states in the USA. The network cost of a link joining
two states is the driving distance between them. The other
is generated by using the approach introduced in [15] to
emulate wide-area sparse networks deliberately. 100 nodes are
distributed randomly over a rectangular coordinate grid. Each
node is placed at a location with integer coordinates. A link
between two nodes u and v is added by using the probability
function P(u,v) = λexp(−p(u,v)/γδ), where p(u,v) is the
distance between u and v, δ is the maximum distance between
any two nodes, and 0 < λ,γ ≤ 1. Larger values of λ produce
graphs with higher link densities, while small values of γ
increases the density of short links relative longer ones. In
our simulations, γ and λ both are set to 0.9. As a result, the
nodes in generated graphs have average degrees of 6.83. Cost
function c on an edge in the generated graphs is the distance
between its two end nodes on the rectangular coordinated grid.
The multicast connections are generated randomly. The
source node s and set D of destination nodes are randomly
picked up from the nodes in the networks.
There are three objectives to conduct simulation work:
(i) To determine how much k-MTR can save the network
cost. For this purpose, we use the cost of Steiner trees as a
performance benchmark. A Steiner tree is the optimal tree (in
terms of minimal network cost) for a multicast connection for
the case when k is infinite (i.e., a Steiner tree contains all
destinations). Since it is NP-hard to compute Steiner trees,
we use a simple 2-approximation algorithm proposed in [4].
It works as follows: (1) construct a complete weighted graph
Gawhose vertex-set is {s} ∪ D and each edge (u,v) in Ga
is the shortest path between u and v in original graph G;
(2) compute a minimum spanning tree Tminon Ga; and (3)
obtain a Steiner tree TSby substituting each edge in Tminby
the corresponding shortest path in G.
(ii) To see the advantages of k-MTR over other multi-drop
routing models, we use the k-drop Multi-Path Routing (k-MPR
for short) as a performance benchmark. k-MPR model [8,9]
is for the networks where switching nodes do not have splitter
to split signals but are still drop-off capable. A k-drop path is
a path that is from the source to some destinations in which
at most k destinations are designated to receive the data. In
the k-MPR model, multicast routing is to find a set of k-drop
paths such that every destination is designated to receive data
in a k-drop path in the set and the total cost is minimal. We
compare our method for k-MTR with an algorithm for the k-
MPR model, which is a 4-approximation algorithm proposed
in [8]. It is based on the Steiner tree algorithm described in
the above (i).
(iii) To see how much k-MTR can reduce the number of
wavelengths used. For this purpose, we use the lightpath model
(i.e., k = 1) as a performance benchmark. In the lightpath
model, a route of a multicast connection is a tree of lightpaths
(lightpath-tree for short), where there is a lightpath from the
source to each of the destinations. We simply use the shortest
path from the source to a destination as a lightpath.
We simulate the network cost and the number of wave-
lengths used against two parameters: |D| the number of
destination nodes and the drop number k in k-MTR and k-
MPR. The results presented in the figures are the mean values
of 50 simulation runs.
B. Analysis of Simulation Results
Fig. 1-4 are the results in the NSFnet. Fig. 1 and Fig. 3
show the network costs and the number of wavelengths used
by lightpath-tree, Steiner tree, k-MTR and k-MPR against the
drop number k, respectively. Two sets of results are displayed.
The solid line is for the case of 13 destinations and the
dashed line is for 7 destinations. Notice that the network cost
and the number of wavelengths used by lightpath-tree and
Steiner tree do not vary with the drop number k, i.e., they are
constant (shown as horizontal lines). Fig. 2 and Fig. 4 show the
network costs and the number of wavelengths used by these
four routing methods against the number of destinations when
k = 4, respectively. Notice that the network costs and the
number of wavelengths used by all routing methods increase
proportionately as the number of the destinations increases.
Fig. 5-8 show the parallel results of randomly generated
network.
In the simulation, when k = 2, k-MTR and k-MPR are
produced by using methods described in the proofs of Theorem
1 of [8] and Algorithm A. Both of them have minimal costs in
each case, respectively. When k ≥ 3, k-MTR and k-MPR are
produced by applying the Steiner tree based 4-approximation
algorithm of [8] and Algorithm B, respectively. In Fig. 1, Fig.
3, Fig. 5, and Fig. 7, accordingly we do not join the results
of k = 2 with the results of k = 3. From those eight figures
we can draw the following conclusions.
(1) The network cost of lightpath-tree is about two times
that of k-MTR. These ratios are independent of the size of
a multicast connection and very stable. This can be observed
from Fig. 1-2 and Fig. 5-6. In addition, the network cost of
Steiner tree routing is about two times that of k-MTR. In fact,
when k becomes large enough, the average performance of
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Page 6
our algorithm is much better than the guaranteed ratio 4. This
can be observed from Fig. 1 and Fig. 5.
5000
8000
11000
14000
17000
20000
23000
26000
29000
32000
45
6
78
9
1023
Steiner tree
Drop number
Network cost
Lightpath-tree
-MTR
Steiner tree
13 destinations
7 destinations
k
-MPR
k
-MTR
k
-MPR
k
Lightpath-tree
Fig. 1.Network costs vs. drop numbers in NSFnet.
(2) The network costs of k-MTR decrease as k increases.
However, increasing k is not very effective in decreasing the
network cost. This can be seen from Fig. 1 and Fig. 5. The
reason behind this interesting result is that when k becomes
larger, although k-MTR will consist of less number of trees,
each of them will become bigger so that it includes more
destinations. This make each tree more costly.
(3) The number of wavelengths used by lightpath-tree is
about four times that of k-MTR. This ratio is independent of
the size of multicast connection and very stable. This can be
observed from Fig. 4 and Fig. 8.
(4) The number of wavelengths used by k-MTR decreases
as k increases. However, increasing k is not very effective in
reducing the network cost for k-MTR. This can be seen from
Fig. 3 and Fig. 7. The reason is that when k becomes larger,
a tree can contain more destinations, which makes a bigger
tree. Therefore, trees with larger k would have more chances
to share links among them, which prevent them from using
the same wavelength.
(5) k-MTR is more effective in saving the wavelengths than
the network cost, although it is designed for achieving minimal
cost. The reason is that k-MTR of less network cost consists
of less trees and less links for a tree (averagely), which results
less chance of wavelength conflict (i.e., two trees share a
common link, but assigned with the same wavelength).
(6) In general, k-MTR uses considerably less network cost
and wavelengths than k-MPR. The reason is that k-MPR is
4000
7000
10000
13000
16000
19000
22000
25000
28000
31000
456789 1011 12 13
Lightpath-tree
-MTR
Steiner tree
Number of destinations
Network cost
Drop number = 4.
k
-MPR
k
k
Fig. 2.Network cost vs. number of destinations in NSFnet.
45
6
78
9
1023
Drop number
Number of wavelengths used
1
2
3
4
5
Steiner tree
13 destinations
Lightpath-tree
Steiner tree
7 destinations
-MTR
k
-MTR
k
-MPR
k
-MPR
k
Lightpath-tree
Fig. 3. Number of wavelengths vs. drop numbers in NSFnet.
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Page 7
456789 1011 12 13
Lightpath-tree
Steiner tree
Number of destinations
Number of wavelengths used
1
2
3
4
5
Drop number = 4.
-MTR
k
-MPR
k
k
Fig. 4.Number of wavelengths vs. number of destinations in NSFnet.
200
400
600
800
1000
1200
1400
1600
2345678910
Drop number
Network cost
Steiner tree
20 destinations
Lightpath-tree
-MTR
Steiner tree
5 destinations
k
-MTR
k
-MPR
k
-MPR
k
Lightpath-tree
Fig. 5.Network costs vs. drop numbers in random networks.
200
400
600
800
1000
1200
1400
1600
23456789 10
Drop number
Network cost
Steiner tree
20 destinations
Lightpath-tree
-MTR
Steiner tree
5 destinations
k
-MTR
k
-MPR
k
-MPR
k
Lightpath-tree
Fig. 6. Network costs vs. number of destinations in random networks.
12
11
10
9
8
7
6
5
4
3
2
1
23456789 10
Drop number
Number of wavelengths used
Steiner tree
Lightpath-tree
-MTR
Steiner tree
20 destinations
5 destinations
k
-MTR
k
-MPR
k
-MPR
k
Lightpath-tree
Fig. 7. Number of wavelengths vs. drop number in random networks.
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1
4
7
10
13
16
19
22
25
28
31
51015 20 25
Number of destinations
30 3540 4550
Lightpath-tree
-MTR
Steiner tree
Number of wavelengths used
Drop number = 5.
k
-MPR
k
k
Fig. 8.
networks.
Number of wavelengths vs. number of destinations in random
a special case of k-MTR since every k-drop path can be
considered as a k-drop tree.
VII. CONCLUSIONS
In this paper, we have studied multicast routing problem
under the k-drop multi-tree model. We have proposed a
polynomial-time optimal algorithm for the cases when k ≤ 2
and a 4-approximation algorithm for the cases when k ≥
3. The extensive simulation studies show that our proposed
algorithm can save considerable amount of network cost and
wavelengths than the lightpath-tree method, and the perfor-
mance is close to the Steiner tree method when k becomes
large enough. Simulation results also show that the k-drop
multi-tree model (k-MTR) generally outperforms the k-drop
multi-path model (k-MPR).
ACKNOWLEDGMENT
This work was supported in part by the National
Natural Science Foundation of China under Grant No.
60273071,70221001, 60373012. The author Xiaohua Jia also
works with Computing School, Wuhan Univ, China.
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