Tree Growth Based ACO Algorithm for Solving the Bandwidth-Delay-Constrained Least-Cost Multicast Routing Problem

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Abstract
Quality of service (QoS) multicast routing is an NP multi-objective optimization problem. This paper presents a treegrowth based ant colony optimization (ACO) algorithm (TGACO) for solving the least-cost multicast routing problem with three QoS constraints, namely: bandwidth, delay and delay jitter. In the proposed algorithm, each ant generates a multicast tree using tree growth, such that an edge is added to the tree if it satisfies only the bandwidth constraint. Then, the fitness of the constructed multicast tree is evaluated by using a cost function that includes the delay and delay jitter constraints. Depending on the fitness of the constructed multicast trees, the local and global best multicast trees can be determined. In the TGACO algorithm, the ants perform local and global pheromone updates. In the local pheromone update, pheromone evaporation is performed by all ants after each construction step, while the global pheromone update is performed at the end of each iteration by the local best and the global best ants. The paper also presents the results of the experiments that have been conducted to evaluate the performance of the proposed algorithm.
International Journal of Computer and Information Technology (ISSN: 2279 0764)
Volume 05 Issue 06, November 2016
www.ijcit.com 516
Tree Growth Based ACO Algorithm for Solving the
Bandwidth-Delay-Constrained Least-Cost Multicast
Routing Problem
Moheb R. Girgis*, Tarek M. Mahmoud, Ghada W. Hanna
Department of Computer Science, Faculty of Science, Minia University, El-Minia, Egypt
*Email: moheb.girgis [AT] mu.edu.eg
AbstractQuality of service (QoS) multicast routing is an NP
multi-objective optimization problem. This paper presents a tree-
growth based ant colony optimization (ACO) algorithm
(TGACO) for solving the least-cost multicast routing problem
with three QoS constraints, namely: bandwidth, delay and delay
jitter. In the proposed algorithm, each ant generates a multicast
tree using tree growth, such that an edge is added to the tree if it
satisfies only the bandwidth constraint. Then, the fitness of the
constructed multicast tree is evaluated by using a cost function
that includes the delay and delay jitter constraints. Depending on
the fitness of the constructed multicast trees, the local and global
best multicast trees can be determined. In the TGACO
algorithm, the ants perform local and global pheromone updates.
In the local pheromone update, pheromone evaporation is
performed by all ants after each construction step, while the
global pheromone update is performed at the end of each
iteration by the local best and the global best ants. The paper also
presents the results of the experiments that have been conducted
to evaluate the performance of the proposed algorithm.
Keywords-QoS multicast routing; Least-cost multicast tree; Ant
colony algorithm; Tree growth
I. INTRODUCTION
Due to rapid advances in the communication technologies
and the increased demand for various kinds of communication
services, many nowadays network applications require support
of multicast communication. Therefore, the issue of multicast
routing has become more and more important, especially with
the emergence of distributed real-time multimedia applications,
such as video conferencing, distance learning, and video on
demand. These applications involve multiple users, with their
own different quality of service (QoS) requirements in terms of
throughput, reliability, and bounds on end-to-end delay, delay
jitter, and packet loss ratio.
The main problem of QoS routing is to set up a least-cost
multicast tree, i.e. a tree covering a group of destinations with
the minimum total cost over all the links, which satisfies
certain QoS parameters. However, the problem of constructing
a multicast tree under multiple constraints is NP Complete [1].
Hence, the problem is usually solved by methods based on
computational intelligence such as meta-heuristic algorithms.
In recent years, many meta-heuristic algorithms have been
proposed for solving the QMR problem, such as ant colony
algorithm [2-7], genetic algorithm (GA) [8-11], simulated
annealing (SA) [12, 13], genetic simulated annealing [14], tabu
search algorithm [15, 16], particle swarm optimization (PSO)
[17, 18], and hybrid GA-PSO based algorithm [19].
However GA and SA algorithms have practical limitations
in real-time multicast routing, since the GA climbing capacity
is weak and premature easily, and both the efficiency and the
quality of the solution for the SA algorithm depend on
procedures that are sensitive to the influence of random
annealing sequence.
Ant colony algorithm has high demand for parameter
setting in large scale optimization. The most obvious weakness
of ant colony algorithm is that it converges slowly at the initial
step and takes more time to converge. Researchers have been
working to improve the ant colony algorithm. For example,
Dorigo and Caro [20] have proposed radically based self-
adaptive ant colony algorithm, Zhao et al [21] have proposed
ant colony algorithm that employs mutation and dynamic
pheromone updating strategies. However, due to the
complexity of network environment, these algorithms are not
applicable to the multi constrained QMR.
Patel et al. [22] have proposed a hybrid ACO/PSO
algorithm to optimize the multicast tree. The algorithm starts
with generating a large amount of mobile agents in the search
space. The ACO algorithm guides the agents’ movement by
pheromones in the shared environment locally, and the global
maximum of the attribute values are obtained through the
random interaction between the agents using PSO algorithm.
Wang et al. [6] have proposed an algorithm which generates a
multicast tree by using tree growth and optimizes ant colony
algorithm parameters through orthogonal experiments.
The algorithm proposed by Wang et al. [6] has some
drawbacks concerning the evaluation of the QoS constraints at
each step during the construction of a multicast tree, which
slows down the convergence of the algorithm, and the local
pheromone update, which does not take into account the
evaporation of pheromone on the edges.
This paper proposes a tree-growth based ACO (TGACO)
algorithm for solving the least-cost multicast routing problem
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with three QoS constraints, namely: bandwidth, delay and
delay jitter that overcomes these drawbacks.
The proposed algorithm generates each multicast tree using
tree growth, such that an edge is added to the tree if it satisfies
the bandwidth constraint. Then, the fitness of the constructed
multicast tree is evaluated by using a cost function that
includes the delay and delay jitter constraints. Depending on
the fitness of the constructed multicast trees, the local and
global best multicast trees can be determined. In TGACO
algorithm, the ants perform local and global pheromone
updates. In the local pheromone update, pheromone
evaporation is performed by all ants after each construction
step, while the global pheromone update is performed at the
end of each iteration by the local and global best ants. Also, the
paper presents the results of the experiments that have been
conducted to evaluate the performance of the proposed
algorithm.
The remainder of this paper is organized as follows:
Section 2 presents the description and formulation of the QMR
problem. Section 3 describes the ACO algorithm. Section 4
describes the drawbacks of the algorithm proposed by Wang et
al. [6], and presents the proposed tree-growth based ACO
algorithm for solving the QMR problem, which overcomes
these drawbacks. Section 5 describes the operations of the
proposed algorithm. Section 6 presents the steps of the
proposed algorithm. Section 7 presents the results of the
experiments. Finally, Section 8 presents the conclusions of this
work.
II. PROBLEM DESCRIPTION AND FORMULATION
A network is modeled as a directed, connected graph G=
(V, E), where V is a finite set of vertices (network nodes) and E
is the set of edges (network edges) representing connection of
these vertices. Each link e=(x,y) ϵ E has three weights (B(e),
D(e) and C(e)) which correspond to the available bandwidth,
the delay and the cost of the link, respectively.
A multicast tree T(s,M) is a sub-graph of G spanning the
source node s ϵ V and the set of destination nodes M V S.
Let m = |M| be the number of multicast destination nodes. We
refer to M as the destination group and {{s}M} as the
multicast group. In addition, T(s,M) may contain relay nodes
(Steiner nodes), the nodes in the multicast tree but not in the
multicast group. Let PT(s,d) be a unique path in the tree T from
the source node s to a destination node d ϵ M. We now present
the parameters that characterize the quality of the tree. The
total cost of the tree T(s,M) is defined as sum of the cost of all
links in that tree and can be given by:
(1)
The total delay of the path PT(s,d) is simply the sum of the
delay of all links along that path:
(2)
The total delay of the tree T(s,M) is defined as the
maximum value of the delay on the paths from the source node
to each destination node:
(3)
The bottleneck bandwidth of the path (s,d) is defined as
minimum available residual bandwidth at any link along the
path:
(4)
The delay jitter of the tree T(s,M) is defined as the average
difference of delay on the path from the source to the
destination node:
(5)
where delay_avg denotes the average value of delay on the
path from the source to the destination node.
Let the delay, the delay jitter and bandwidth constraints are
Dmax, Dj and Bmin, respectively. The multi-constraint least-cost
multicast problem is defined as:
Min Cost(T(s,M)) subject to:
(6)
The QoS requirements described above can be classified
into link constraint (e.g., bandwidth), path constraint (e.g., end
to end delay) and tree constraint (e.g., delay-jitter). In our work
the QoS multicast evolution is driven by the fitness function
defined by (7), in which QoS constraints are considered except
the bandwidth constraint, because the link that does not meet
the bandwidth constraint is not chosen.
(7)
where 1 and 2 are punishment coefficients, their values
determine the punishment extent.
III. ACO ALGORITHM
The basic ideas of ACO are from the social search behavior
of biological ant colonies. In nature, ants move around in their
environment in a rather random way, but they have certain
tendency to follow the walk of other ants. They can recognize
these walks because, while moving, each ant leaves a chemical
substance called pheromone on the ground. Sensing pheromone
on a path increases the probability of an ant to follow it, which
further reinforces this path. This mechanism has the effect that
short paths between a starting point and a goal point are
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favored, leading to a kind of heuristic optimization behavior.
[7]
The described principle is exploited in ACO algorithms for
optimizing arbitrary objective functions of combinatorial
problems by simulating the walks of conceptual ants and by
doing the re-enforcement of good walks based on an evaluation
of the objective. Such ACO algorithms are based on the
following ideas. First, each path followed by an ant is
associated with a candidate solution for the given problem.
Second, when an ant follows a path, the amount of pheromone
deposited on that path is proportional to the quality of the
corresponding candidate solution for the given problem. Third,
when an ant has to choose between two or more paths, the
path(s) with a larger amount of pheromone are more attractive
to the ant. After some iteration, eventually, the ants will
converge to the path, which is expected to be the optimum or a
near-optimum solution for the target problem. [22]
IV. THE PROPOSED TREE-GROWTH BASED ACO
ALGORITHM FOR SOLVING THE QMR PROBLEM
Wang et al. [6] have proposed a tree growth based ACO
algorithm (TGBACA). It generates a multicast tree in the way
of tree growth and optimizes the ant colony parameters through
their most efficient combinations.
The basic idea of this algorithm is as follows: initially, the
multicast tree has only the multicast source node. Then, the ant
selects one link and adds it to the current tree according to an
edge selection probability. After adding the selected edge, the
path is checked to see whether it satisfies the specified QoS
constraints. If not, another edge is selected. When the tree
covers all the multicast members it stops growing. The tree
obtained is then pruned and rendezvous links are removed to
get the real multicast tree, then the delay jitter of the multicast
tree is calculated. If it is greater than the delay jitter bound,
then the multicast tree is rejected and another one is
regenerated. Then, pheromones on the links that have been
visited by the obtained local and global best multicast tree are
updated. The above mentioned steps are repeated until the
algorithm converges.
The drawbacks of the algorithm proposed by Wang et al.
[6] are as follows:
The rejection of the multicast tree after its construction,
if it does not satisfy the delay jitter constraint, and re-
generation of another one, slows down the
convergence of the algorithm.
During the construction of a multicast tree, each time
an edge e(i, j) is to be added, the values of delay,
packet loss ratio, and bandwidth, on the path from
multicast source to node j, are modified. If the new
values do not satisfy the related constraints, another
edge e(i, j) is selected, and the calculations are
repeated. Here, two odd situations may occur, which
slow down the convergence of the algorithm:
o If node j was a destination node, then the path to
that destination has to be reconstructed.
o If node j was the last destination node, i.e., at the
end of a multicast tree construction, then the whole
multicast tree has to be reconstructed.
The local pheromone update is performed by
computing the total pheromone on the candidate edge
each time that edge is traversed. It does not take into
account the evaporation of pheromone on the edges,
which is very important to increase the exploration of
edges that have not been visited yet and to prevent ants
from producing identical solutions during one iteration
[23].
The proposed tree-growth based ACO (TGACO) algorithm
for solving the QMR problem overcomes these drawbacks.
In our TGACO algorithm, during the construction of a
multicast tree, we check only the bandwidth of the edge that
has been chosen from the candidate edge set to be sure that it
satisfies the bandwidth constraint. The other QoS constraints,
i.e. delay and delay jitter, are included with the cost in the
fitness function (7), which is used to evaluate the quality of the
constructed multicast tree. Depending on the fitness of the
constructed multicast trees, the local and global best multicast
trees can be determined. This way we avoid the overhead of
calculating the QoS parameters for each path when a new edge
is to be added to it. Accordingly, no path or tree rejection
occurs. This speeds up the convergence of the algorithm.
In addition, our algorithm performs pheromone evaporation
on each traversed edge in the local pheromone update step, as
described below, to improve its performance.
V. THE OPERATIONS OF THE PROPOSED
ALGORITHM
The proposed TGCAO algorithm for solving the QMR
problem has the same three operations: tree growth, tree
pruning and pheromone update, as TGBACA [6], but here they
are carried out differently, as described below.
A. Tree Growth
This operation creates a tree T(ET, VT), where ET and VT
are the sets of edges and nodes of the tree, respectively, such
that each edge in ET satisfies the bandwidth constraint. Initially,
ET and VT are set as follows: ET = NULL, VT ={s}, where s is
the multicast source. Then, a link set E' is created. Initially, E' =
{e(s, i)}, where each link e(s, i) belongs to the given network
and satisfies the bandwidth constraint: B(e(s, i)) Bmin. Then,
the following steps are performed:
Step 1: Select an edge from the set E' according to the
following selection probability equation:
(8)
where ei is the ith link of E', i is the pheromone intensity of ei,
and i is the heuristic function of ei. We have selected i =
1/costi, where costi is the cost value of ei, α and β are
parameters used to adjust the effect of the pheromone intensity
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and the heuristic function. Suppose the edge (i, j) is selected,
then it is added to the set ET of tree T, and node j is added to
the node set VT of T.
Step 2: Update the links set E' to be E' = E' E1 + E2, where E1
= {e(k, j)| e(k, j) E'}denotes the link set which takes node j as
the destination and belongs to E', and E2 = {e(j, k)| k VT}
denotes the link set which takes node j as starting node and
satisfy the bandwidth constraint. It can be seen that one link at
a time is added to tree T.
Step 3: Repeat the previous two steps of tree growth process
until tree T covers all the destination group nodes.
B. Tree Pruning
Although the tree found covers all the destination group
nodes, it may not be a multicast tree, because it contains some
leaf nodes which are not destination nodes. Therefore, a
pruning process is performed to remove those leaf nodes. Then,
the delay, cost and delay jitter of the pruned multicast tree are
modified accordingly, and its fitness is calculated using (7).
C. Pheromone Update
The most interesting contribution of ant colony system
(ACS), [24], is the introduction of a local pheromone update in
addition to the global pheromone update performed at the end
of the construction process for all ants, which is called offline
pheromone update.
The local pheromone update is performed by all ants after
each construction step. Each ant applies it only to the last edge
traversed. The main goal of the local update is to diversify the
search performed by subsequent ants during one iteration. In
fact, decreasing the pheromone concentration on the edges as
they are traversed during one iteration encourages subsequent
ants to choose other edges and hence to produce different
solutions. This makes less likely that several ants produce
identical solutions during one iteration. [25]
In our algorithm, after obtaining a multicast tree, the ant
reduces the pheromone trial ij of each traversed edge e(i, j), by
using the following local pheromone update equation:
(9)
where [0,1] is the pheromone evaporation parameter,
which is used to control the evaporating speed of pheromone.
Then, the offline pheromone update is performed at the end
of each iteration by two ants, the iteration-best (local-best) ant
and the best-so-far (global-best) ant using the following
equation:
(10)
where ij = Q / L denotes the pheromone increase for each
edge e(i, j) that belongs to the local best and global best tree
obtained, Q is the pheromone strength coefficient, and L can be
either the cost of the local or global best tree.
VI. THE OVERALL TGACO ALGORITHM
The steps of the proposed TGACO algorithm are as
follows:
Input: A network G= (V, E), s (multicast source), M
(destination group), QoS bounds (Dmax, Dj and Bmin)
maxIteration (maximum number of iterations), nAnts
(number of ants), constants (, Q, α, β, 1 and 2)
Output: A bandwidth-delay-constrained least-cost multicast
tree (Tbest)
Begin
1. For iter = 1 to maxIteration Do
2. For ant = 1 to nAnts Do
2.1 Tree Growth
a) Assign an initial value 1 to the pheromone trial
ij on each edge e(i, j) E.
b) Initialize set of nodes of tree Tant: VT = {s}, and
its set of edges: ET = NULL
c) Create a link set E'. Initially, set E' = {e(s, i)},
where each link e(s, i) satisfies the bandwidth
constraint
d) Initialize number of covered destination nodes:
mCount = 0
e) For each node n in the given network, set
visited[n] = false
f) Select one edge e(s, i) from set E', according to
the selection probability (8)
g) if i M then mCount ++
h) visited[i] = true
i) Add node vi to VT and e(s, i) to ET
j) While mCount < |M| do
- Update the set E' as described in Sec. V.A
- Select one edge e(i, j) from set E', according
to the selection probability (8)
- if j M then mCount ++
- visited[j] = true
- Add node vj to VT and e(i, j) to ET
End While
2.2 Prune tree Tant
2.3 Evaporate pheromone on the edges used by Tant
using (9)
2.4 Calculate cost, delay, delay jitter, and bandwidth
of the tree Tant using equations (1) to (5)
2.5 Calculate the fitness F(Tant) using (7)
2.6 Get local best tree Tlbest:
if ant = 1 or F(Tant) < F(Tlbest) then Tlbest = Tant
End For
3. Get global best tree Tbest :
if iter = 1 or F(Tlbest) < F(Tbest) then Tbest = Tlbest
4. Update pheromone on edges used by Tlbest and Tbest
using (10)
End For
End.
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Figure 1. Network Model 1
Figure 2. Network Model 2, showing the multicast tree obtained using our
algorithm
Figure 3. Network Model 3
VII. EXPERIMENTAL RESULTS
This section presents the results of the experiments that
have been conducted to evaluate the performance of the
proposed TGACO algorithm in solving the QMR problem
compared to the TGBACA proposed by Wang et al. [6], and to
study the effect of the number of ants used, the network size,
and the destination group size, on the convergence of our
algorithm. The algorithm has been implemented using C++.
In these experiments we have used three network models
shown in Fig. 1, 2 and 3, where each edge is labeled with
(delay, bandwidth, and cost). The first network model, shown
in Fig.1, has 23 nodes with node 0 being the source node, the
second network model, shown in Fig.2, has 14 nodes with node
5 being the source node, and the third network model, shown in
Fig.3, has 8 nodes with node 1 being the source node.
Figure 4. An example of multicast tree construction using the TGBACA [6],
which led to violation of the specified bounds when the last destination node
was to be added
The experiments configurations were as follows: Number
of ants is 15 and number of iterations is 20; the value of α and
β, which are used in (8), were set to 0.9 and 3.0, respectively;
and the value of which is used in (9) and (10) was set to 0.01.
The performance of the algorithm was measured from
perspective of the best multicast tree obtained, the fitness value
and run time.
Before describing the experiments, we show an example of
the case that may occur during the multicast tree construction
using the TGBACA algorithm [6], where the specified bounds
are violated when the last destination node is to be added,
which requires the reconstruction of the multicast tree, (see
Section IV). The network model used was the first one, with
the destination group M = {4, 9, 14, 19, 22}, and the delay
bound equals to 25. As shown in Fig. 4, the ant started at the
source node 0, and reached the destination node 4 along the
path (0-6-7-8-4), with path delay equals 18, then it reached the
destination node 9 along the path (0-6-7-8-9), with path delay
equals 18, then it reached the destination node 14 along the
path (0-6-7-8-13-14), with path delay equals 22, and it reached
destination node 19 along the path (0-6-7-8-13-18-19), with
path delay equals 25. Finally, when the ant tried to reach the
last destination node 22 along the path (0-6-7-8-13-18-22), the
delay of this path was 26, which violates the delay bound.
Thus, the ant must search for different paths to reconstruct a
multicast tree, although it has reached 80% of the destination
nodes.
In our algorithm, during the construction of a multicast tree,
we only make sure that each selected edge satisfies the
bandwidth constraint. This way, the ant constructs a multicast
tree, then its fitness is evaluated using the fitness function, Eq.
(7), which includes the other QoS constraints, i.e. delay and
delay jitter, with the cost.
In the first experiment, we have applied the two algorithms
to the three network models. The destination groups for the
first, second and third network models were {4, 9, 12, 14, 19,
22}, {0, 2, 6, 13} and {3, 5, 7}, respectively. Table I shows the
best multicast trees obtained by using the two algorithms for
(3,2,2)
(2,3,3)
(3,2,4)
(2,2,2)
(2,2,2)
(7,1,7)
(1,2,4)
(2,3,1)
(7,3,2)
(1,3,3)
(5,1,3)
5
7
0
1
2
4
3
6
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the three network models, with the cost, delay, delay jitter, the
fitness value and running time in minutes. The table shows
that, for the first two network models, the proposed TGACO
algorithm produced multicast trees with less cost than the
TGBACA algorithm [6], and for the third network model both
algorithms produced multicast trees with same cost. But, in all
cases, our algorithm has taken less time.
Fig. 5 and 6 shows the multicast trees obtained for Network
Model 1 (Fig. 1), in the first experiment, by using the proposed
TGACO and TGBACA [6] algorithms, respectively.
In the second experiment, we have applied the two
algorithms to the first network model (Fig. 1) with four
different destination groups: {3, 7,19}, {4, 9, 14, 19, 22}, {4, 9,
14, 19, 22, 12, 6}and{4, 9, 14, 22, 12, 16, 3, 21, 17},
representing 13%, 21%, 30% and 43% of the network nodes,
respectively. Table II shows the best multicast trees obtained
using the two algorithms with their fitness, cost, delay and
delay jitter. It can be seen that our proposed algorithm
produced multicast trees with less cost in all cases.
In the third experiment, we have applied our TGACO
algorithm to the first network model (Fig. 1) with destination
group {3, 7, 9}, to study the relationship between the number
of ants used and the fitness of the best multicast tree obtained.
Fig. 7 shows the results of this experiment. It indicates that as
the number of ants increases, the fitness of the best multicast
trees decreases, until it reaches a certain number, 10 ants in this
case, where the fitness value starts to stabilize.
Figure 5. The multicast tree obtained by using the proposed TGACO
Figure 6. The multicast tree obtained by using TGBACA [6].
TABLE I. COMPARISON BETWEEN BEST MULTICAST TREES, WITH QOS CONSTRAINTS VALUES AND RUNNING TIME, OBTAINED USING THE TWO ALGORITHMS
FOR THE THREE NETWORK MODELS.
Network
model
Algorithm
used
Multicast tree
Cost
Delay
Delay
jitter
Fitness
function
Run time
(min)
Network Model 1
(Fig. 1)
Proposed
TGACO
0-1-6-7-8-4.
0-1-6-7-8-9.
0-1-6-7-8-13-12.
0-1-6-7-8-13-14.
0-1-6-7-8-13-18-19.
0-1-6-7-8-13-18-22.
120
28
49.8
120
1.64
TGBACA [6]
0-6-7-8-4.
0-6-7-8-13-14-9.
0-6-7-8-13-12.
0-6-7-8-13-18-22.
0-6-7-8-13-18-19.
124
27
47.37
124
4.56
Network Model 2
(Fig. 2)
Proposed
TGACO
5-4-2-0.
5-6-9-13.
32
23
20.3
32
0.187
TGBACA [6]
5-6-9-8-12-13.
5-4-2-0.
37
30
29.32
37
0.48
Network Model 3
(Fig. 3)
Proposed
TGACO
1-4-5
1-7-0-3
10
7
5.2
10
0.15
TGBACA [6]
1-4-5
1-7-0-3
10
7
5.2
10
0.209
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TABLE II. COMPARISON BETWEEN THE RESULTS OBTAINED BY USING
THE TWO ALGORITHMS FOR NETWORK MODEL 1 WITH DIFFERENT
DESTINATION GROUPS
Destination
Group
Algorithm
Cost
Delay
Delay
jitter
Fitness
function
{3, 7,19}
Proposed
TGACO
79
28
27.6
79
TGBACA [6]
88
27
31.2
88
{4, 9, 14, 19,
22}
Proposed
TGACO
104
33
46.9
104
TGBACA [6]
109
29
48.8
109
{4, 9, 14, 19,
22, 12, 6}
Proposed
TGACO
120
27
48.8
120
TGBACA [6]
124
27
47.37
124
{4, 9, 14, 22,
12, 16, 3, 21,
17}
Proposed
TGACO
163
24
49.7
163
TGBACA [6]
171
35
56.4
171
In the fourth experiment, we have applied the two
algorithms, our TGACO algorithm and the TGBACA
algorithm [6], to the three network models, to show the effect
of the variation of the network size on the cost of the obtained
multicast tree. Fig.8 shows this relationship, with network
sizes, 23, 14 and 8, and the multicast group with ratio 43%. It
can be seen that as the network size increases the cost
increases. It can be seen also that the costs of the multicast
trees obtained by our algorithm were less than those obtained
by TGBACA [6].
In the fifth experiment, we have used our TGACO
algorithm to show the effect of the size of distention group on
the multicast tree cost. Fig. 9 shows the relationship between
the percentage of destination group and cost for the three
network models. It can be seen that as the percentage of
destination group increases the cost of the multicast tree
increases.
Finally, we have evaluated the two algorithms in terms of
run time. The last column of Table I and Fig.10 show the run
time of the two algorithms with the three network models with
destination group size equal to 30% of the total nodes of each
network. It can be seen that our algorithm takes considerably
less time than TGBACA [6]. It can be seen also that as the size
of the network increases the time increases.
Figure 7. Multicast tree fitness values with different number of ants
Figure 8. The relation between number of nodes and cost of the multicast
tree obtained by using the two algorithms.
Figure 9. The relation between percentage of destination nodes and cost
Figure 10. A comparison between the time taken by our TGACO algorithm
and the TGBACA algorithm [6] for the three network models
VIII. CONCLUSION
This paper presented a proposed tree-growth based ACO
(TGACO) algorithm for solving the QMR problem. In this
algorithm, during the construction of a multicast tree, we only
make sure that each selected edge satisfies the bandwidth
constraint. This way, the ant constructs a multicast tree, then its
fitness is evaluated using a proposed fitness function, which
includes the other QoS constraints, i.e. delay and delay jitter,
with the cost. Also, in addition to the pheromone update
performed at the end of the construction process, the algorithm
performs a local pheromone update. The effect of the local
International Journal of Computer and Information Technology (ISSN: 2279 0764)
Volume 05 Issue 06, November 2016
www.ijcit.com 523
pheromone updating rule is that each time an ant uses a link its
pheromone trial is reduced, so that the link becomes less
desirable for the following ants, this allows an increase in the
exploration of arcs that have not been visited. So, the algorithm
does not show a stagnation behavior.
The performance of the algorithm has been evaluated
through experiments, which showed that it is efficient in
producing least cost multicast trees that satisfy the specified
QoS constraints.
The experiments showed that the costs of the multicast
trees obtained by our algorithm were less than those obtained
by TGBACA [6], and our algorithm takes considerably less
time.
It showed also that as the number of ants increases, the
fitness of the best multicast trees decreases, and stabilizes after
reaching a certain number of ants. It showed also that as the
network size and the percentage of destination group increase
the cost of the multicast tree increases, and as the size of the
network increases the time increases.
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