A NOVEL ANT COLONY ALGORITHM FOR MULTICAST ROUTING IN WIRELESS AD HOC NETWORKS

David C. Wyld, et al. (Eds): CCSEA, SEA, CLOUD, DKMP, CS & IT 05, pp. 309–317, 2012.
© CS & IT-CSCP 2012 DOI : 10.5121/csit.2012.2231
A NOVEL ANT COLONY ALGORITHM FOR MULTICAST
ROUTING IN WIRELESS AD HOC NETWORKS
Sunita Prasad1
, Zaheeruddin2
and D. K. Lobiyal3
,
1
Center for Development of Advanced Computing, India
2
Department of Electrical Engineering, JMI, Delhi, India
3
School of Computer and System Sciences, JNU, Delhi, India
ABSTRACT
The Steiner tree is the underlying model for multicast communication. This paper presents a
novel ant colony algorithm guided by problem relaxation for unconstrained Steiner tree in static
wireless ad hoc networks. The framework of the proposed algorithm is based on ant colony
system (ACS). In the first step, the ants probabilistically construct the path from the source to
the terminal nodes. These paths are then merged together to generate a Steiner tree rooted at
the source. The problem is relaxed to incorporate the structural information into the heuristic
value for the selection of nodes. The effectiveness of the algorithm is tested on the benchmark
problems of the OR-library. Simulation results show that our algorithm can find optimal Steiner
tree with high success rate.
KEYWORDS
Multicast routing, Steiner Tree, Wireless Ad Hoc Networks, Ant Colony Optimization (ACO)
1. INTRODUCTION
The rapid developments in multimedia applications like video/audio conferencing and distance
education require multicast communication. In multicast communication, the message is sent
concurrently to all members of the multicast group. The multicast routing problem intends to find
a minimum cost routing tree which is rooted at the source and links all the destinations. The
underlying model for multicast routing is Steiner tree. Given a undirected graph ),,( cEVG =
consisting of V as the node set and E as the edge set and a positive edge cost +
→ REc : . A
subset of terminal nodes VT ⊆ is also defined, the objective is to find a Steiner tree S which is
a subnetwork of G such that (i) there is a path between every pair of terminals T and (ii) total
cost )(∑∈Se
i
i
ec is minimized where Eei ⊆ . The vertices TV in S are called Steiner nodes.
Steiner tree is a well known NP-hard combinatorial optimization problem [1][2]. For most NP-
hard problems, the performance of deterministic algorithm is not satisfactory due to high
computational time required for even small instances. Metaheuristics such as Genetic Algorithm
(GA) [11] and the Ant colony Optimization (ACO) [3][4][5] has been applied to solve the Steiner
tree problem. In this paper, we present a novel ant colony algorithm based on problem relaxation
for unconstrained multicast routing. The framework of the proposed algorithm is based on ant
colony system [6]. Each ant is initially placed at the terminal node. In the first phase of the
algorithm, each ant builds a complete path from the destination node to the source node. In the

310 Computer Science & Information Technology (CS & IT)
second phase, the algorithm iteratively adds an entire path to the partially constructed tree rather
than edges to form the tree. The probability of node selection is influenced by both pheromone
and heuristic information. We utilize the structural information provided by problem relaxation to
guide the decision of ants for node transition.
The ant algorithm described Singh et. al. is based on ant system (AS). Each ant starts its journey
from the terminal node. The ants merge when one ant collides with other ant or it steps into the
route of the other ant. In [5], collision detection mechanism is incorporated as there is a
possibility of the collision of the ants. But our algorithm does not require any such mechanism.
In DCACS [3], is based on Prims algorithm in the framework of ant colony system (ACS). The
algorithm is applied on a distance complete graph (DCG). The ant starts with a randomly chosen
terminal node. The ant probabilistically builds the solution after which the both the actual and the
virtual edges are subjected to pheromone updation.
The rest of the paper is organized as follows. Section 2 discusses the Ant Colony Optimization
(ACO) metaheuristics. Section 3 describes the proposed algorithm in detail along with the
formulation. Section 4 presents the results, followed by conclusion in section 5.
2. ANT COLONY OPTIMIZATION (ACO) METAHEURISTICS
ACO was proposed by [6] and is a population based stochastic optimization technique. It is
inspired by the foraging behaviour of ants and is based on stigmergic learning. In this, a
population of artificial agents (ants) work collectively to generate the shortest path from the
source to the destination. The solution is built step by step going through several probabilistic
decisions which depends on (i) long term joint population memory (pheromone) and (ii) some
additional information about the problem (heuristic information). After the solution has been
constructed by the ants, some pheromone is deposited on the edges of the path which is biased
towards better solution i.e. more pheromone is deposited on the edges of good solutions.
Gradually, the concentration of pheromone on the edges corresponding to good solutions builds
up evolving a global optimum solution. The exploitation of the pheromone value on the edges of
the good solution may lead to premature convergence. To facilitate the exploration of the entire
search space, pheromone trail evaporation is also incorporated in ACO. [12] gives an overview of
the recent developments in ACO. Convergence proofs for ACO can be found in [7]. For better
results and faster convergence, ACO are usually combined with local search algorithms. In this
paper, we use problem relaxation to gain insights into the structural information of the problem.
2.1 Problem Relaxation
The minimum spanning tree (MST) with edge cost is essentially a Steiner tree without Steiner
nodes. The Steiner tree heuristics are based on MST heuristics [8][9]. The approximate Steiner
tree is obtained in two steps (i) generate the MST of the network and (ii) prune the MST. The
distributed versions of the classical MST algorithm – Prim and Kruskal are used to obtain the
Steiner tree using this method. There are two disadvantages of this technique (i) the
computational cost is high since all the nodes are involved in the execution of MST algorithm and
(ii) the result obtained is suboptimal.
The approach proposed in this paper uses problem relaxation to gain insights into the structure of
the Steiner tree. The edges contained in MST are very likely to be part of the Steiner tree. The
paper incorporates this information into the heuristic value of the ants. Thus, the transition
probability of the ants is guided by the edge information provided by MST.

Computer Science & Information Technology (CS & IT) 311
3. PROPOSED ALGORITHM
The proposed algorithm applies ant colony optimization to obtain Steiner tree for multicast
routing. The algorithm is initialized by placing each ant on the terminal node. The algorithm
consists of two phases (i) Forward set initialization and (ii) Merge path.
Forward Set Initialization: In this phase, each ant starts from the terminal node and builds a
shortest path from the terminal node to the source node. The node transition probability depends
on pheromone and the heuristic information. Since the input graph is not complete, it is possible
that the set N of the entire available alternative that go out from the node v lead to already
visited nodes. In this case, the ant is relocated to a node within its own tabu list such that it is
nearest to a node in the tabu list of any other ant.
Merge Path: In this phase, we merge the path to obtain the minimum cost Steiner tree. Given a
terminal it we first find all the nodes in the path iP from the source to the destination it that are
already in the existing tree. The path iP can be joined to the tree at any of these points. The node
that joins the subpath at the minimum cost is selected as the point of attachment.
2.1. Ant Colony Based Tree Construction
In this section, we describe the search behaviour of ant to build a tree. The algorithm is as
follows:
Step 1: Initialization
The multicast group consist of a source node s, and a set of terminal nodes { }mtttT ,......,, 21= .
Let || Tn = be the number of group members. The number of ants antnum is equal to n. The
pheromone value on the link is initialized to a constant 0τ . The iteration is set to a constant MAX.
Each ant maintains its own tabu list to record the list of nodes already visited. This avoids the ant
revisiting the same node again and forming a cycle. The ants are placed at each destination node
it where 1≤ i ≤ n that needs to be connected and the tabu list of the ant is initialized with it.
Step 2 : State transition probability
The ant m at node i, probabilistically determines the next node j based on the state transition rule
given below:



 ≤
= ∉
on)(explorati
ion)(exploitat,][][maxarg 0,,)(
otherwiseJ
qqif
j kikimtabuk
βα
ητ
(1)
where
• ki,τ is a positive real quantity of the pheromone value associated with the edge connecting
node i and k where k is a set of feasible nodes in the neighbourhood of node i. The pheromone
value ki,τ represents the accumulated knowledge about the goodness of the edge and indicates
how useful it is to move to a feasible node j from the current node i.

• ki,η is the heuristic function which represents the desirability of choosing a feasible node j
from current node i. The heuristic value for ant m is defined as:
m
i
ijm
ji
jic ψγ
η
.),(
1
,
+
Ω+
= (2)
where γ is a constant and m
iψ is the minimum cost path from node i to all the vertices in the
tabu list of other ants. This causes the current ant m merge into the path of other ants as
quickly as possible to form the tree. ijΩ is 1 if the edge is included in MST, 0 otherwise.
• Parameter βα and weigh the relative importance of pheromone value and the heuristic
function.
• q is a random number chosen with a uniform probability in [0,1] and 0q is a parameter such
that 10 0 ≤≤ q . If q is smaller than 0q , the ant will choose the next unvisited node with the
maximum product of pheromone and heuristic value (exploitation step). Otherwise, the next
node j is chosen as given by (3) with a probability distribution (exploration step)






∉
= ∑∉
otherwise0
if
][][
][][
),( ,,
,,
m
tabuk
kiki
jiji
m
tabuj
jip
m
βα
βα
ητ
ητ
(3)
The next node j is determined stochastically but the process favours the minimum cost edges
having high amount of trail.
Step 3: Pheromone updation rule
The updating of the trail intensity on the edges is defined as follows.
ji,,, )1( τρτρτ ∆+−= jiji (4)
where ρ is a constant, called the trail evaporation rate. The increment in updating is given by the
following formula.





∈
=∆
otherwise0
),(if
)(,
t
tji
Eji
Sc
Q
τ
where )( tSc is the cost of the current tree tS , tE is the edge set of the current tree and Q is a
constant that matches the tree cost.
The high level description of the proposed algorithm is shown in Fig. 1. The notation used in the
algorithm are given as follows
1. JoinPath ( )uSP ii ,, 1− : joins the path iP to the existing tree 1−iS at point u to return the
current Steiner tree iS .

2. FindCommonNode ( )1, −ii SP : Given an existing tree 1−iS and iP be the path from the source
to the destination it . The function returns a sequence of nodes in path iP that are already a part
of the existing tree 1−iS .
3. Subpath_Cost ( )itu, : the function returns the cost of the subpath from the common node u to
the selected destination it .
4. Shortest_Subpath ( )itu, : The function returns the subpath ip that joins the destination it to
the tree iS at a tree node u .
Fig. 1. Ant Colony Based Algorithm for Steiner Tree
Main procedure
Input : A connected graph ),,( cEVG = , terminal set T and a source s
Output : A minimal cost Steiner tree S
1. /* Initialization phase */
Place the ant on each node in the terminal set T and put the node into its
tabu-list
Compute the MST of G
2. /* Main Algorithm */
while loop < MAX do
ConstructSteinerTree (G, T, s)
Update the trail intensity on every edge (i,j) by (4)
Update the current best solution
loop++
Return the current best solution

Fig. 2. ConstructSteinerTree subprocedure
Procedure ConstructSteinerTree (G, T, s)
Input : A connected graph ),( EVG = and a terminal set T and a source s
Output : A Steiner tree S
1. /* Phase 1 : Construction of the initial forwarding path from the destination it to the source */
for m=1 to antnum
currentnode=Tm
while currentnode!= s do
determine the nextnode j based on (1)
if nextnode !=φ
currentnode=nextnode
else
relocate(m)
end-if
Add the edge ( )ji, into the path mP of ant m
end-while
end-for
2. /* Merge Path */
11 PS =
for i=2 to n
mincost=inf;
Z = FindCommonPoint ( )1, −ii SP
if Z >0
for each u in Z do
cost = Subpath_Cost( )itu,
if mincost > cost
mincost=cost
ip = Shortest_Subpath ( )itu,
end-if
end-for
iS =JoinPath ( )uSP ii ,, 1−
else
iS =JoinPath ( )sSP ii ,, 1−
end-if
end-for
3. Prune (T) /* Prune the tree to obtain the minimal Steiner tree */

3. RESULTS
The effectiveness of the proposed algorithm is tested using MATLAB simulations. The problem
set B from the OR-library is used as the data set [10]. The parameters of ant colony is set
empirically as 1=α , 4=β , 1.0=ρ , 9.00 =q , 100=Q . The trail on all edges is initialized
to a very small value 0τ at the beginning of the algorithm. The maximum iteration is set as 500.
The stopping criterion of our algorithm is either the maximum iteration or a fixed number of
generations without improvement in the solution. Such a number is fixed as 100. Initially the
movement of ants is primarily based on the heuristic information but subsequently the pheromone
information is also used to build the solution. The simulation scenario for B01 is shown in Fig. 3.
The nodes are randomly placed in an area of 50 x 50 m2
. The obtained results are tabulated in
Table 1. The results of the proposed algorithm are compared with the ant based algorithm
reported in [5] using a fixed sequence approach for selection. The results suggest that the
proposed algorithm is able to find the optimal results with high success rate.
Table 1: Results for B-Test Data
Graph Data Results
Test
Data
Set
V E T Ant
Algo
[5]
Proposed Algo
Best Value
Proposed Algo
Average Value
B01 50 63 9 82 82 82
B02 50 63 13 83 83 83
B03 50 63 25 138 138 140
B06 50 100 25 - 122 125
B08 75 94 19 110 104 104
B09 75 94 38 230 225 226
B11 75 150 19 103 88 88
B12 75 150 38 - 176 179
B14 100 125 25 242 235 236
B15 100 125 50 350 320 321
B16 100 200 17 145 127 132
[-] results not available in [5]

Fig 3. The simulation scenario for B01 test data set
4. CONCLUSION
The paper proposed a novel ant colony based algorithm for unconstrained Steiner tree in wireless
ad hoc networks. The proposed ant based algorithm uses problem relaxation to incorporate the
structural information into the heuristic value for node transition. The algorithm was tested on the
standard test data set of the OR-library. The results suggest that the proposed algorithm is able to
find the optimal results with high success rate. The future work is to further enhance the
algorithm for constrained Steiner tree in wireless ad hoc networks and also extend it for dynamic
multicast groups.
REFERENCES
[1] M. R. Garey and D.S. Johnson, “ Computers and Intractability : A Guide to the Theory of NP
Completeness”, W. M. Freeman, 1979.
[2] F. Hwang and D. Richards, “ Steiner Tree Problems”, Networks, vol 22, pp 55-89, 1992
[3] X. Hu, J. Zhang and L. Zhang, “Swarm Intelligence Inspired Multicast Routing : An Ant Colony
Optimization Approach”, LNCS, pp. 51-60, 2009.
[4] L. Luyet, S. Varone and N. Zufferey, “An Ant Algorithm for Steiner Tree Problem in Graphs”,
LNCS, pp. 42-51, 2007
[5] G. Singh, S. Das, S. Shekhar and S. Pujar, “Ant Colony Algorithms for Steiner Trees: An Application
to Routing in Sensor Networks”, Book Chapter in Recent Developments in Biologically Inspired
Computing, IGI global publishing, pp. 181-206, 2005
[6] M. Dorigo, G. Caro and L. Gambardella, “ Ant Algorithms for Discrete Optimization”, Artificial Life,
vol 5(2), pp. 137-192, 1999.
[7] T. Stuetzle and M. Dorigo, “ A Short Convergence Proof for a class of ACO Algorithms” IEEE
Transactions on Evolutionary Computation, vol 6(4), pp. 358-365, 2002
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Simulation Scenarion for B01 Test Set

[8] L. Kuo, G. Markowsky and L. Berman, “A fast algorithm for steiner trees”, Acta Informatica, vol 15,
pp. 141-145, 1981.
[9] H. Takahashi and Matsuyama, “An Approximate solution for Steiner Tree problem in Graphs”, Math
Japonica, vol 24(6), pp. 573-577, 1980
[10] J. Beasley: “OR-Library-Distributing Test Problem by electronic mail” Journal of Operational
Research Society, vol 41, pp. 1061-0172, 1990.
[11] A. Haghighat et. al., “GA Based heuristic Algorithms for Bandwidth Delay Constrained Least Cost
Multicast Routing”, Computer Communications, vol 27(1), pp. 111-127, 2004
[12] M. Dorigo and T. Stutzle, “Ant Colony Optimization”, MIT Press, 2004

A NOVEL ANT COLONY ALGORITHM FOR MULTICAST ROUTING IN WIRELESS AD HOC NETWORKS

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