August 2011, 18(4): 76–81 www.sciencedirect.com/science/journal/10058885 http://jcupt.xsw.bupt.cn

The Journal of China Universities of Posts and Telecommunications

Tree-based differential evolution algorithm for QoS multicast routing

KONG Sun ( ), CHEN Zeng-qiang

Department of Automation, Nankai University, Tianjin 300071, China

Abstract

Differential evolution (DE) algorithm has attracted more and more attention due to its fast optimization performance and good stability. When DE algorithm is applied into multi-constrained multicast routing optimization problem, a common solution to suchproblem is to merge the paths into a tree after finding paths from the source node to each destination node. This method maybe obtains the better result, but it can consume a lot of computational time. To solve the problem, a tree-based DE algorithm is introduced in this paper. The central operations of the algorithm are realized with tree structure. This method saves the time of finding paths andintegrating them to construct a multicast tree. The experiments show that the proposed algorithm can achieve higher success rate than several common algorithms with much smaller running time for different networks.

Keywords quality of service (QoS), multicast routing, DE, tree structure

1 Introduction

During the past years, more and more multimedia applications require strict QoS guarantee during the communication between a single source and multiple destinations. This gives rise to the need for an efficient QoS multicast routing (QMR) strategy. Hence, it is very important to research the development of the network theory and applications for QMR problem. Searching for such QoS-based optimal multicast routes basically leads to a multi-objective optimization problem. Unfortunately, this problem is computationally intractable in polynomial time due to the uncertainty of resources in networks [1]. Therefore, many researchers have presented some efficient algorithms to solve this problem. Intelligent optimization technology represented by intelligent optimization algorithms has developed rapidly in recent years. Some of these algorithms, which have special advantage in solving combinatorial optimization problem, have been applied to QMR problem.

Many QMR algorithms based on intelligent optimization have been derived in Refs. [2–15]. Genetic algorithm (GA) is Received date: 07-12-2010 Corresponding author: KONG Sun, E-mail: ksuser@mail.nankai.edu.cnDOI: 10.1016/S1005-8885(10)60087-8

one of the most popular intelligent optimization algorithms applied to QMR problem. Xiang et al. [8] have presented a GA-based algorithm for QoS routing. This algorithm adopts an N N one-dimensional binary code, where N represents

the number of nodes in the network. But in this encoding scheme, the coding/decoding operation is very complicated and the coding space increases rapidly with the increasing of the network size, especially for large networks. Hwang et al. [9] have proposed a GA-based algorithm for multicast routing problem. In this algorithm, a multicast tree is produced by the means of finding the paths from the source node to each destination node and merging the paths into a tree. The optimization of a multicast tree is finished through a serial path selection. This method can make the crossover and mutation operation easy to realize. However, when the network is large, the time of searching paths is so long that the algorithm is inefficient. A heuristic genetic algorithm for bandwidth-delay-constrained least-cost multicast routing problem has been given in Ref. [10]. They have used a tree structure for genotype representation, but the crossover scheme lead to discontinuity among sub-trees in encoding scheme and it is difficult to connect the sub-trees. In addition, some researchers use ant colony algorithm (ACO) to solve

Issue 4 KONG Sun, et al. / Tree-based differential evolution algorithm for QoS multicast routing 77

this problem. Wang et al. [11] have adopted an improved ACO and combined it with a heuristic algorithm for the computation of the multicast tree. The simulation results demonstrated that the algorithm can obtain better performance than standard ACO. However, the method is hard to apply in reality because they haven’t considered the case with constraints. A heuristic ant algorithm for the QoS constrained multicast routing problem has been proposed in Ref. [12], but the algorithm has been tested in only 20-node network. In this way, the simulation results can not fully explain the effectiveness of the algorithm for random networks. In order to overcome the shortage of the conventional ACO, a tree-based ACO has been developed in Ref. [13], which makes one ant directly find a multicast tree rather than path. The algorithm has higher convergence speed than basic ant colony algorithm. Nevertheless, the tree-based ACO easily leads to premature convergence. In recent years, several algorithms based on particle swarm optimization (PSO) have been proposed to solve multicast routing problem. Liu et al. [14] have obtained a path-based PSO, which performs better than standard GA in converging speed and searching capability for a 20-node network. The authors have presented a novel probability convergence based PSO for the optimization of QoS multicast routing in Ref. [15], and introduced a probability based updating process. The simulation results show that the algorithm has faster convergence and better global optimization performance than some other algorithms. Unfortunately, both the above-mentioned papers use the same method as Ref. [9] to generate a multicast tree, so that their algorithms also have the problem of longer operation time spent on searching paths. Although many intelligent optimization algorithms are adopted for multicast routing problem, most of them can’t solve this problem very well.

DE [16] is a heuristic stochastic search method based on individual differences. DE algorithm has been widely studied because of its simple principle, less parameter and good robustness [17–18]. The principle and process of DE algorithm are similar to that of GA [19]. The DE algorithm process contains three parts: mutation, crossover and selection. But, unlike GA, DE algorithm adopts differential strategy to realize mutation operation. The differential strategy makes use of colony distribution effectively and enhances search capability so that it can overcome the drawbacks of the mutation method in GA [20].

In this paper, we focus on the problem of tree-based differential evolution algorithm for QoS multicast routing. In

order to solve QMR problem better, we propose a multicast tree-based routing algorithm based on DE algorithm. In this algorithm, some novel methods based on tree structure are designed for mutation and crossover. We evaluate the performance and efficiency of the presented algorithm in comparison with other similar algorithms. Finally, experiments are provided to demonstrate that the proposed algorithm can solve QMR problem more effectively.

The remainder of this paper is organized as follows. The network model and problem formulation are given in Sect. 2. The original DE algorithm is introduced in Sect. 3. Sect. 4 describes the proposed DE algorithm in detail. The new algorithm is compared with several other common algorithms in Sect. 5. Sect. 6 concludes the research result of this paper.

2 Network model and problem formulation

A network is usually modeled as a weighted graph ( , )G V E , where V denotes the set of nodes and E denotes

the set of communication links connecting the nodes. Let s V be the source node and { { }}M V s be the set of

destination nodes. We refer to M as the multicast group. Generally, supposed there is only one link between a pair of nodes in network. Let be non-negative real numbers set. For the link ( )e e E between any two nodes, four QoS

eigenvalue can be defined: cost function ( ) :C e E ,

delay function ( ) :D e E , delay jitter function

( ) :J e E , bandwidth function ( ) :B e E .

For a given source node s and a set of destination nodes M,a multicast tree ( , )T s M which is a sub-graph of G spanning

the source node s and the set of destination nodes M has the following relations:

( , )

( , )

( , )

( ( , )) ( )

( ( , )) ( )

( ( , )) ( )

( ( , )) min{ ( ), ( , )}

T

T

e T s

Te P s t

Te P s t

T T

C T s M C e

D P s t D e

J P s t J e

B P s t B e e P s t

M

(1)

where ( , )TP s t is the path from source node s to each destination node ( )t t M in the multicast tree ( , )T s M .

The constraints of the multicast tree is defined by 1) Delay constraint: ( ( , ))T tD P s t D2) Delay jitter constraint: ( ( , ))T tJ P s t J3) Bandwidth constraint: ( ( , ))T tB P s t B

where tD is the delay constraint, tJ is the delay jitter

78 The Journal of China Universities of Posts and Telecommunications 2011

constraint, tB is the bandwidth constraint.

The multi-constrained least-cost multicast problem is minimizing the cost of the multicast tree ( , )T s M which

satisfies all of the constraint conditions above 1)–3). The expression of minimizing the cost of the multicast tree is of the form: min ( ( , ))C T s M (2)

In this paper, we assume every destination node has the same delay constraint, delay jitter constraint and bandwidth constraint.

3 Differential evolution algorithm

DE, originally proposed by R. Storn and K. Price [16], is one of the latest evolutionary optimization methods. Like other evolutionary-type algorithms, DE is a population-based, stochastic global optimizer. In original DE algorithm, candidate solutions are represented as chromosomes based on real numbers. In the mutation process of the algorithm, a mutated solution vector is generated by adding the weighted difference between two population vectors to a third vector. Then the elements of the mutated vector are mixed with the elements of another predetermined vector to obtain a trial solution vector. After completion of the previous crossover operation, the trial vector is compared with the target vector by evaluating the fitness function value. According to the comparison result, the following procedure will select one vector for the next generation.

All population direct search methods use a population set S.The initial set 1 2{ , ,..., }S mx x x consists of m random

points in domain n . ( {1,2,..., })ix i m is a

D-dimensional vector. The procedure of the DE algorithm can be divided into the following steps:

1) Mutation The weighted difference of any two points is then added to

the third point which can be mathematically described by

1 2 3( )Ft + 1 t t t

i a a ay x x x (3)

where 1a , 2a and 3 {1,2,..., }a m are random indexes in Sand mutually different; 1a , 2a and 3a are chosen to be different from the index i; [0,2]F is a real factor which

controls the amplification of the differential variation

2 3( )t t

a ax x .

2) Crossover The trial vector t+1

iz is generated from its parents t+1iy

and tix using the following crossover rule:

1rnbr1

rnbr

; if or ( )

; if and ( )

t jji Rt

ji t jji R

y R C j N iz

x R C j N i (4)

where 1,2,...,j D represents the jth element of respective

vector; (0,1)jR is the jth evaluation of a uniform random number; rnbr ( ) 1,2,...,N i D is a randomly chosen index;

[0,1]RC is the crossover constant.

3) Selection 1 1

1 ; if ( ) ( )

; otherwise

f ft+ t+ ti i it+

i ti

z z xx

x ( 5 )

At this phase, the trial point 1t+iz is compared with the

target vector tix using the greedy criterion. If vector 1t+

iz

corresponding to the target tix , satisfies the criterion

1( ) ( )f ft+ ti iz x then the vector 1t+

iz replaces tix .

Otherwise, the old value tix is retained.

4 Tree-based DE algorithm (TBDE) for QoS multicast routing

4.1 Coding

We choose N N binary matrix Q as the coding scheme

of the algorithm, where N is the number of network nodes and the matrix represents a multicast tree. The element ( , )Q i j in

the matrix is defined as follows: 1; if the link from node to is selected

( , )0; otherwise

ijl i ji jQ (6)

where ijl represents the link from node i to node j in

network.

4.2 Initialization of population

Before constructing initial population, we first remove the links which don’t satisfy the bandwidth constraint. In the process, we must ensure that the path from the source node to any of the destination nodes always exists in the refined network. Then the network graph is simplified. A randomized depth-first search algorithm is employed to form a random multicast tree. The searching process terminates until all the destination nodes are added to the tree. We use the same method to generate m initial multicast trees 1 2{ , ,...,T T T

}mT . Perhaps some leaf nodes don’t belong to the multicast

group. In such a case, the leaf node and its edge should be deleted. Perform a similar operation until the parent node is either the destination node or its out-degree is greater than 0.

Issue 4 KONG Sun, et al. / Tree-based differential evolution algorithm for QoS multicast routing 79

4.3 Mutation

After initializing population, three trees 1aT ,

2aT ,3aT are

selected randomly from the initial population and they are mutually different. There are two steps for the mutation operation: the first is searching and recording the different edges between

2aT and 3aT ; the second is adding these

different edges to 1aT according to a certain probability. In

this paper, F is treated as a probability value and it ranges from 0 to 1. After adding some different edges from

2aT and

3aT to 1aT , the current tree iT obtained is probably not a

tree. If so, iT needs to be pruned and reconstructed later. Now we reconstruct the tree iT . Before this, the evaluation

of edge fitness function can be defined as ave ave ave( ) / ( ) / ( ) /

1 2 3( ) e e eC e C D e D J e Jf e m m m ( 7 ) where the parameters 1m , 2m , 3m ( 1 2 30 , , 1m m m ) are

the weight values of the cost, delay and delay jitter function of the link respectively. The variables aveC , aveD and aveJ

represent the average cost, delay and delay jitter value of the set of links in network topology.

The edges in iT are evaluated by Eq. (7). Starting with the source node, we select edges from iT to set up a new tree in

the way of roulette wheel scheme, once the probabilities of selection are given by edge fitness function. However, this approach may result in no feasible edge in iT to choose. If

so, we choose a feasible edge from the simplified network graph and add the edge to iT . While all the destination nodes

are added to the tree, those useless edges and possible circles in iT need to be eliminated. After the operation, we can make iT become a real tree.

4.4 Crossover scheme

For any multicast tree, a destination node has only one corresponding path from the source node to itself. So the total number of the corresponding paths is the same as the number of destination nodes. It can be said that all trees in optimization process contain the same number of paths from the source node to all the destination nodes. Unlike conventional crossover, path is used as crossover object in our crossover scheme. The rule is given in Eq. (4). The ultimate aim of this operation is to obtain a trial tree with paths coming from the paths of the target tree and the mutated tree iTaccording to the crossover probability RC . After crossover

operation, the multicast tree may contain circles. Once this

situation occurs, the circles have to be eliminated.

4.5 Selection

A new multicast tree is obtained through crossover operation. An objective function is adopted to evaluate the performance of the new tree and the target tree from the last population set. The objective function Tf is defined as

follows:

D J

DD

J

1 ( )[ ( , )]1; ( ( , )) 0

; ( ( , )) 0

1; ( ( , )) 0; ( ( , )) 0

T

T t

T t

T tJ

T t

fC T s M

D P s t Dr D P s t D

J P s t Jr J P s t J

( 8 )

where D and J are the penalty function of delay and delay-jitter; and are adjustable parameters, which

are used to control the penalty scale of the delay and delay jitter function respectively ; The values of Dr and Jr

determine the degree of penalty. In the real algorithm, different values can be set according to different network. In our experiment simulation, we select D J 0.5r r .

According to the computed objective values, some of the new trees are selected. If the objective value of a new tree is higher than the target one, it will replace the target tree for the next generation.

5 Experiment simulation

In order to testify the effectiveness of the proposed algorithm, we adopt a 20-node network topology structure as shown in Fig. 1. The ranges of the eigenvalues for each link and node are given in Table 1. The average degree of each node in this network is 4.2.

Fig. 1 A 20-node Network

Table 1 The ranges of the eigenvalues Range Edge Node Delay 2–12 1–11

Delay-jitter 0–3 0–3 Bandwidth 20–130 –

Cost 1–21 1–11

80 The Journal of China Universities of Posts and Telecommunications 2011

In this experiment, we assume the source node s to be node 1and the destination node set to be {2,6,10,18}M . The

weight values of the fitness function in Eq. (7) are set as follows: 1 0.8m , 2 0.15m , 3 0.05m . The adjustable parameters in Eq. (8) are set: 4 , 3 . The parameter

settings in DE algorithm are achieved: mutation factor 0.5F , crossover probability 0.5RC .

We compare the proposed algorithm (TBDE) with the tree-based GA (TBGA) in Ref. [10], the tree-based ACO (NACO) in Ref. [13] and the path-based PSO (PBPSO) in Ref. [14]. The four algorithms are tested in the same constraint conditions: 40, 46, 8t t tB D J .

Table 2 shows the average objective function value generated by the four algorithms. Compared with other algorithms, TBDE can find the optimal multicast tree faster.

Table 2 Average fitness values of four algorithms Generation TBDE NACO PBPSO TBGA

20 0.316 7 0.310 1 0.311 0 0.306 440 0.328 7 0.325 6 0.322 1 0.317 460 0.333 3 0.332 5 0.327 1 0.326 680 0.333 3 0.333 3 0.333 3 0.333 3

The success rates of TBDE, TBGA, NACO and PBPSO are shown in Fig. 2. It can be seen from the figure that TBDE can obtain higher success rate and faster convergence speed than other algorithms in this 20-node network.

Fig. 2 Comparison of success rate

In the following simulation, we test the robustness of TBDE by varying the network size. The method in Ref. [21] is adopted to generate larger networks with 40 nodes, 80 nodes, 120 nodes, 160 nodes and 200 nodes. The size of multicast group is set to be 25% of the number of the network nodes. In this simulation, the proposed algorithm is still compared with the three algorithms mentioned before. The average success rate and the relative execution time are used as the measures of performance.

For each algorithm, the average success rate changes with the increasing number of network nodes. All of the algorithms

adopt the same number of iterative time. Among the solutions obtained by all the algorithms, the best one is recorded for each random network. The success rate is defined as

best reqtS S , where bestS is the number of the best solution,

reqtS is the total number of all solutions. Fig. 3 shows the

results.

Fig. 3 Comparison of average success ratio in different network scales

Fig. 4 shows the relative execution time ratio of TBGA, NACO and PBPSO algorithm. The simulation results are obtained when each algorithm converges to a steady solution. The relative execution time ratio is defined as

execution TBDE_executiont t , where executiont represents the

average execution time for each algorithm, TBDE_executiont is

the average execution time of the proposed algorithm in this paper.

Fig. 4 Comparison of average execution time ratio in different network scales

It can be seen from Figs. 3 and 4 that TBDE performs better than other algorithms. The path-based PSO (PBPSO) algorithm has higher success ratio but it takes longer execution time. Most of the time is spent on searching the possible paths from a source node to each destination node before optimization. The execution time of the tree-based ACO (NACO) algorithm is close to TBDE, but NACO algorithm easily gets into local optima with the increase of network size, which can result in decline of the success ratio. Compared with the three algorithms, the new algorithm presented by this paper has the highest success rate in networks of 40–200 nodes. Overall, TBDE algorithm can spend less execution time with guaranteeing higher success ratio. It can be said that the new algorithm shows the better convergence performance by comparison.

Issue 4 KONG Sun, et al. / Tree-based differential evolution algorithm for QoS multicast routing 81

There are three main reasons for this good performance. Firstly, TBDE can directly find a multicast tree on the network, overcoming drawbacks of the algorithms based on searching for paths. Therefore, TBDE can save a lot of operation time relatively. Secondly, the differential strategy makes TBDE have better global search ability than several comparable algorithms mentioned before. Finally, basic DE algorithm is simple and easy to realize. It is popular for its fast convergence. TBDE algorithm retains this good property so that it has faster convergence speed.

6 Conclusions

In this paper, we have developed a differential evolution algorithm based on tree structure scheme to solve the QoS multicast routing problem. By adopting connectivity matrix of edges for representation of a multicast tree, an idea of tree re-shaping has been used to realize mutation and crossover operation. To verify the effectiveness of the algorithm, we have compared it with several common multicast routing algorithms based on intelligent optimization. The simulation has been implemented in a given 20-node network and random networks with different sizes, respectively. Finally, the experiment results have been provided to demonstrate that the proposed algorithm in this paper has not only better capability of global optimum and stability but also faster convergence.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (60774088); the Hi-Tech Research and Development Program of China (2009AA04Z132); the Specialized Research Foundation for the Doctoral Program of National Education Ministry (20090031110029).

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