TSDLMRA: an efficient multicast routing algorithm based on Tabu search

TSDLMRA: an efficient multicast routing algorithm

based on Tabu search

Heng Wang*, Jie Fang, Hua Wang, Ya-Min Sun

Department of Computer Science and Technology, Nanjing University of Science and Technology,

608 Staff Room, Nanjing 210094, China

Received 17 June 2003; received in revised form 24 September 2003; accepted 2 October 2003

Abstract

As a NP-Complete problem, multicast routing with delay constraint is a research difficulty in

routing problem. Tabu Search is artificial intelligence algorithm, which is an extension of local

search algorithm and has simple realization and well properties. In this paper, an efficient algorithm

based on Tabu Search for Delay-Constrained Low-Cost Multicast Routing is proposed to solve

delay-constrained multicast routing problem. This problem is known to be NP-Complete. The

proposed heuristic algorithm makes use of the characteristics of flexible memory function and tabu

rule in TS algorithm, generates neighborhood structure base on ‘paths-switching’ operations, and

finds multicast tree satisfying constraint. A large number of simulations demonstrate that the

algorithm performs excellent performance of cost, rapid convergence and stable performance of

delay.

q 2003 Elsevier Ltd. All rights reserved.

Keywords: Multicast routing; Tabu search; Delay constrained; Constrained Steiner tree; NP-Complete

1. Introduction

Distributed real-time applications, such as audio- and video-conferencing, collabora-

tive environments and distributed interactive simulation, by and large, involve a source

sending messages to a selected set of destinations with varying Quality-of-Service (QoS)

delivery constraints. This requires the underlying network to provide multicasting and

QoS capabilities to efficiently support these applications. At the routing level, these

1084-8045/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jnca.2003.10.001

Journal of Network and

Computer Applications 27 (2004) 77–90

www.elsevier.com/locate/jnca

* Corresponding author. Tel.: þ86-25-4315660; fax: þ86-25-4315636.

E-mail addresses: [email protected] (H. Wang), [email protected] (J. Fang),

[email protected] (H. Wang), [email protected] (Y.M. Sun).

http://www.elsevier.com/locate/jnca

requirements are translated into the problem of determining a multicast tree, usually

rooted at the source node and spanning the set of receiver (destination) nodes. The QoS

constraints typically impose a restriction on the acceptable multicast trees.

With the advent of the real-time interactive applications, minimizing delay of the

multicast tree is also an important objective along with minimizing cost. Therefore, both

cost optimization and delay optimization goals are important for the multicast routing tree

construction. The problem of minim izing tree cost under the constrained that all path

delays are within a user-specified delay bound is referred to as the delay constraint

multicast routing problem. This problem is formalized into constrained Steiner tree

problem, which is known to be NP-Complete.

We briefly discuss some recently proposed delay-constrained Steiner tree heuristics.

KPP (Kompella et al., 1993) is the first heuristic algorithm for delay-constrained Steiner

tree problem. KPP extends KMB (Kou et al., 1981), an unconstrained Steiner tree

heuristic, to compute delay-bounded paths assuming the delay bound and link delays are

all integers. Then KPP converts the original network graph into a closure graph connecting

multicast group members. KPP uses Prim’s algorithm (Cormen et al., 2001) to obtain a

minimum spanning tree of the closure graph. Finally, KPP replaces the edges in the

minimum spanning tree with paths in the original graph. The time complexity of KPP is

OðDlV l3Þ: Zhu et al. (1995) proposed another delay constrained Steiner tree heuristic

algorithm, called BSMA. The algorithm starts by computing a least delay tree (LDT)

rooted at a given source and spanning all the group members. It then iteratively replaces

superedges in the tree with cheaper superedges not in the tree while not violating the delay

constraint, until the total cost of the tree cannot be further reduced. Cheaper superedges are

located by using a Kth shortest path algorithm. The time complexity for BSMA is

OðKlV l3log lV lÞ; where K may be very large in case of large, densely connected networks,

and it may be difficult to achieve acceptable running times (Salama et al., 1997).

As an extension of local neighborhood search, Tabu Search (TS) has been successful in

a variety of problem settings like scheduling, transportation, layout and circuit design, and

graphs (Wang, 2001). Procedures based on TS explores the search space by moving from a

solution to its best neighbor (the one with the best objective value among all examined

candidates), even if this results in a deterioration of the objective function value, in order

to increase the likelihood of escaping from a poor local optimum.

In this paper, we propose an efficient algorithm based on TS for generating a low-cost

multicast tree subject to delay constraints. We call this algorithm Tabu Search for Delay-

Constrained Low-Cost Multicast Routing (TSDLMRA). The algorithm starts with an

initial shortest path tree constructed by using Zhang and Mouftah’s DDSP algorithm

(Zhang and Mouftah, 2002). Then algorithm constructs a backup-paths-set for each

destination using Kth shortest path algorithm, and generates neighborhood structure by

‘paths-switching’ operation. For those trees that violate delay constraint, we assign an

extra penalty by increasing their cost, so that final solutions are feasible. Simulation results

show our algorithm has features of well performance of cost, fast convergence and stable

delay. TSDLMRA belongs to source-based routing algorithm, because it assumes that

sufficient global information is available to the source.

The rest of the paper is organized as follows. In Section 2, delay-constrained low-cost

multicast routing problem is formulated. Section 3 introduces briefly TS method.

H. Wang et al. / Journal of Network and Computer Applications 27 (2004) 77–9078

The proposed algorithm is described in Section 4. The complexity analysis of TSDLMRA

is introduced in Section 5. Simulation results and comparison with other reported

heuristics are presented in Section 6. Section 7 concludes the paper.

2. The delay-constrained multicast routing problem

The communication network is modeled as a undirected graph G ¼ ðV ;EÞ; where V

denotes the set of nodes and E is the set of edges (links) representing physical connectivity

between nodes. Any link e [ E has a cost cðeÞ : E ! Rþ and a delay dðeÞ : E ! Rþ

associated with it. The function cð·Þ represents the utilization of the link and the function

dð·Þ represents the delay that the packet experiences on link including queuing,

transmission, and propagation delay. Let s [ V be a source node and M # V 2 {s} be

the set of destination nodes, called the multicast group. A multicast tree TðT # GÞ is a tree

rooted at s and spanning all members of M: Assume that pðu; vÞ represents the path from u

to v, then there exist following relationships in multicast tree T :

delayðpðs;mÞÞ ¼X

e[pðs;mÞ

dðeÞ; m [ M ð1Þ

costðTÞ ¼X

e[T

cðeÞ ð2Þ

delayðTÞ ¼ maxm[M

ðdelayðpðs;mÞÞÞ ð3Þ

Given these definitions, we can formally present the delay-constrained multicast

routing problem as follows:

Definition 1. Delay-constrained low-cost multicast routing problem. Given a network

G ¼ ðV ;EÞ; a source node s; destination node set M; a link delay function dð·Þ; a link cost

function cð·Þ; and a positive delay constraint D; the objective of the delay-constrained low-

cost multicast routing problem is to construct a multicast tree TðVT ;ET Þ such that the delay

constraint is satisfied, i.e.

delayðTÞ # D ð4Þ

and that the tree cost costðTÞ is minimized.

3. Tabu search method

TS was introduced by Glover as a general iterative metaheuristic for solving

combinatorial optimization problems (Wang, 2001). TS is conceptually simple and

elegant. It is a form of local neighborhood search. Each solution x [ V has an associated

set of neighbors NðxÞ; where V is the set of feasible solutions. A solution x0 [ NðxÞ can be

reached from x by an operation called a move to x0: TS moves from a solution to its best

H. Wang et al. / Journal of Network and Computer Applications 27 (2004) 77–90 79

admissible neighbor, even if this causes the objective function to deteriorate. To avoid

cycling in the course of the search, the reverses of the last certain number of moves,

formed as a tabu list, are forbidden or declared as tabu restricted for certain number of

iterations. To prevent a too strict setting the tabu restriction, aspiration criteria are usually

introduced to override the tabu restriction and thereby to lead the search to a promising

region of the solution space. Intensification and diversification strategies are used to

improve the search. The general TS algorithm is given as follows.

Step 1. Start with an initial solution xnow; and initialize tabu list H ¼ B;

Step 2. If satisfy termination rule, then procedure stop; otherwise, generate candidate

solution set Can_NðxnowÞ satisfied tabu restriction in xnow’s neighbors NðxnowÞ; find a

best solution xnext in Can_NðxnowÞ; xnow U xnext; update tabu list H; repeat Step 2.

4. Delay-constrained low-cost multicast routing algorithm based on tabu search

4.1. Encoding and initial solution

In TSDLMRA, a solution is encoded as an array of lMl elements, where each element is

a path from source s to destination node m [ M in multicast tree T ; i.e. x ¼ ðp1; p2;…; pkÞ;

where k ¼ lMl; pi ¼ pðs;miÞ; mi [ M; 1 # i # k:

The initial solution T0 is a shortest path tree, which is constructed by using DDSP

algorithm (Zhang and Mouftah, 2002). DDSP algorithm aims to construct a low-cost SPT

by considering link sharing between different destinations, and can be seen as an extension

of Dijkstra’s shortest path algorithm (Cormen et al., 2001). DDSP algorithm integrates the

destination-driven characteristics of Shaikh and Shin’s DDMC algorithm (Shaikh and

Shin, 1997) and the shortest path characteristics of SPT, so it outperforms SPT in terms of

cost performance.

A ‘best soulution so far’ Tbest is memorized during the whole procedure. If T0 satisfies

the delay constraint, then initial Tbest is T0; otherwise Tbest is a LDT rooted at source by

using Dijkstra algorithm. If a LDT’s delay violates the delay constraint, then algorithm

returns.

For ease of presentation, we consider an example network as shown in Fig. 1(a), where

s ¼ {A} and M ¼ {B;C;D;E}; the numbers in the parentheses along each edge represent

the cost and delay for that edge. Given delay constraint D ¼ 7: The initial solution T0 and

the initial best solution Tbest are shown, respectively, in Fig. 1(b) and (c). Tbest is a LDT

because of its violating delay constraint. The encoding for T0 and Tbest are given in Table 1.

4.2. Evaluation function

The evaluation function is used for evaluating state of search, in our algorithm, we take

objective function as evaluation function, i.e. f ðxÞ ¼ costðxÞ:


Fig

.1

.E

xam

ple

net

work

,in

itia

lso

luti

on

T0

and

init

ial

bes

tso

luti

on

Tb

est:


4.3. Backup-paths-set

For each destination node m [ M; we compute least-cost paths from s to m by using

Kth shortest path algorithm to construct a backup-paths-set for generating neighbors. Let

Pi be paths set for destination node i; then

Pi ¼ {p1i ;…; p

ji;…; pK

i } ð5Þ

where pji is the jth path for destination node i: Table 2 shows backup-paths-set from source

A to destination nodes in example network.

4.4. Neighborhood structure

For generating neighbors, we choose a neighborhood structure based on ‘paths-

switching’ operations. We randomly select a destination node m [ M; and delete the path

from source s to m in current solution Tnow; then generate different neighbor solutions by

selecting other paths from s to m in backup-paths-set. Each iteration begins by generating

neighbor solutions NðTnowÞ corresponding to Tnow; and the size of neighbor set is dynamic

in our algorithm. For Tbest in Fig. 1(c), we assume that destination node B is selected and

path from A to B is deleted, then we select other paths from Table 2 to construct new

neighbors. The two neighbors of Tbest are shown in Fig. 2 and are encoded in Table 3.

Among the neighbors, the one with the best cost is selected, and considered as new

current solution for the next iteration. The ‘paths-switching’ fails to consider delay

constraint, so it might happen the some of the trees violate delay constraint. In that case,

we assign an extra penalty by increasing its cost, so that it is less likely to be accepted in

the candidate list as shown in the tree of Fig. 2(b). Obviously, the new multicast tree of

Fig. 2(a) is selected for the next iteration and considered as new current solution.

Table 1

The encoding for T0 and Tbest

0 1 2 3 Delay Cost

T0 A;F;D;B A;F;C A;F;D A;F;D;B;E 8 5

Tbest A;H;E;B A;F;D;C A;F;D A;H;E 5 10

Table 2

The backup-paths-set of example network, K ¼ 5

Destination 1 2 3 4 5

B A;F;D;B A;F;C;B A;H;G;B A;F;C;E;B A;F;C;D;B

C A;F;C A;F;D;C A;F;D;B;C A;F;D;B;E;C A;H;E;C

D A;F;D A;F;C;D A;F;C;B;D A;F;C;E;B;D A;H;G;B;D

E A;F;C;E A;F;D;B;E A;F;C;B;E A;H;E A;F;D;C;E


4.5. Tabu list and tabu length

To avoid local minimum, a tabu list is constructed where forbidden moves are listed.

This list contains information that to some extent forbids the search from returning to a

previously visited solution. In our algorithm, a multicast tree is considered as an element

of tabu list, and the tabu length is set to 9.

4.6. Aspiration criterion

While central to TS, tabus are sometimes too powerful: they may prohibit attractive

moves, even when there is no danger of cycling, or they may lead to an overall stagnation

of the searching process. It is thus necessary to use algorithmic devices that will allow one

to revoke (cancel) tabus. These are called aspiration criteria. The simplest and most

commonly used aspiration criterion consists in allowing a move, even if it is tabu, if it

results in a solution with an objective value better than that of the current best-known

solution (since the new solution has obviously not been previously visited). So, if the cost

of a tabu candidate solution is better than Tbest’s, then it is considered as new current

solution, and Tbest is updated at once.

4.7. Termination rule

We use a fixed iteration number as stopping criterion, and maximum iteration

number is 100.

Fig. 2. Two neighbors from Tbest:

Table 3

The encoding for neighbors of Tbest

0 1 2 3 Delay Cost

(a) A;F;D;B A;F;D;C A;F;D A;H;E 6 10

(b) A;H;G;B A;F;D;C A;F;D A;H;E 8 12 þ penalty


The pseudo code of our TSDLMRA algorithm is given in Fig. 3. Steps 1–11 show the

initialization phase. Steps 12–26 show the iteration phase.

5. The complexity analysis

Theorem 1. The time complexity of TSDLMRA is OðKmn3Þ; where m is group size and n is

network size.

Fig. 3. The pseudo code for TSDLMRA.


Proof. In the initialization phase, the time complexity of generating LDT by using Dijkstra

and initial solution by using DDSP are both Oðn2Þ: The time complexity of constructing

backup-paths-set by using Kth shortest path algorithm is OðKmn3Þ: One iteration costs

OðKÞ; thus, for Q iterations, the cost becomes OðQKÞ: So the worst time complexity of our

algorithm is Oðn2 þ n2 þ Kmn3 þ QKÞ: The term QK is usually much smaller than Kmn3;

so the time complexity of TSDLMRA is OðKmn3Þ: A

6. Simulation results

To evaluate the efficiency of TSDLMRA algorithm, we use the random link

generator developed by Salama (1996), which yields networks with an average node

degree of 4–6. The positions of the nodes are fixed in a rectangle of size

4000 £ 2400 km2, and the capacity of each link is 155 Mbps. The Euclidean metric

is then used to determine the distance between each pair of nodes. Edges are introduced

between pairs of nodes u; v with a probability that depends on the distance between

them. The edge probability is given by Pðu; vÞ ¼ b expð2dðu; vÞ=aLÞ; where dðu; vÞ is

the distance from node u to v; L is the maximum distance between two nodes. a and b

are parameters, and are set to 0.15 and 2.2, respectively. Larger values of b result in

graphs with higher edge densities, while small values of a increase the density of short

edges relative to longer ones.

The link delay function dðeÞ is defined as the propagation delay of the link, and queuing

and transmission delays are negligible. The propagation speed through the links is taken to

be two third the speed of light. The link cost function cðeÞ is defined as the current total

bandwidth reserved on the link, which is random variable uniformly distributed between

10 and 120 Mbps.

In the first set of experiments, TSDLMRA is compared with KPP, BSMA and LD

for cost performance. Fig. 4 shows the tree cost for varying network size with the

group size ¼ 5; average node degree of network ¼ 4; and D ¼ 40 and 60 ms,

respectively. It can be seen from Fig. 4 that our algorithm has a better cost

performance than the compared algorithms. Fig. 5 shows the cost performance

measures versus group size for a 50-node network and node degree of 4, delay

constraint of 40 and 60 ms. As the group size increases, the tree cost of all algorithms

grow. Fig. 6 shows the performance of different heuristics for varying delay bound, for

a 50-node network and a group size of 5. In general, TSDLMRA has best cost

performance among all algorithms.

In the second set of experiments, we observe the delay of tree that comes from applying

for TSDLMRA and LD algorithm versus varying network size for a group size of 5 nodes,

with node degree of 4. We can see from Fig. 7 that the delay of tree generated using by our

algorithm always maintains at 24 ms as network size increases whether delay constraint is

so stringent or not.

In the third set of experiments, TSDLMRA is compared with other two algorithms

(Wang and Wang, 2002; Shi et al., 2000) based on genetic algorithms for convergence

performance. We call the algorithm in (Wang and Wang, 2002) WGA, and the

algorithm in (Shi et al., 2000) SGA. Fig. 8 shows the best solutions for example


network in Fig. 1(a) when delay constraint is 6 and 7, respectively, and their cost is 7

and 6, respectively. Fig. 9 shows the tree cost for varying iteration number. In Fig. 9(a),

WGA and SGA’s iteration number is 89 and 107, respectively, for getting best solution

when D ¼ 6; and our algorithm converges just within 10 steps. Fig. 9(b) shows that

WGA and SGA’s iteration number is 220 and 405, respectively, for D ¼ 7; and

TSDLMRA converges still within 10 steps.

Fig. 4. Tree cost versus network size for group size ¼ 5; node degree ¼ 4:

Fig. 5. Tree cost versus group size for network size ¼ 50; node degree ¼ 4:


Finally, we track the cost of the best solution over time. Fig. 10 shows cost of best

solution found by TSDLMRA versus iteration number for 90-, 60- and 30-node

networks with group size ¼ 5; D ¼ 60 ms and node degree ¼ 4: As is clear, the

algorithm converges quickly, which satisfies the real-time requirement of multimedia

network.

Fig. 6. Tree cost versus delay constraint for network size ¼ 50; group size ¼ 5:

Fig. 7. Tree delay versus network size for group size ¼ 5; node degree ¼ 4:


Fig. 9. Tree cost versus iteration number for example network.

Fig. 8. The best solution for example network when delay constraint is 6 and 7, respectively.

Fig. 10. Cost of best solution found by TSDLMRA versus iteration number for three networks. Group size ¼ 5;

delay constraint D ¼ 60 ms; node degree ¼ 4:


7. Conclusion

In this paper, we presented an efficient algorithm based on TS for obtaining delay-

constrained low-cost multicast trees. The proposed TSDLMRA algorithm has following

features: (1) the time complexity is low, i.e. OðKmn3Þ; (2) it was always able to find a

multicast tree if one exists; (3) simulation results show that TSDLMRA performs

excellent performance of cost, rapid convergence and stability of delay.

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Heng Wang is currently a PhD candidate at Nanjing University of Science and

Technology. He received his BE from Department of Computer Science and

Engineering, Nanjing University of Science and Technology, China in 1999. His

research interests include QoS routing algorithm and multicast routing with QoS

constraint.


Jie Fang is currently a Master student at Nanjing University of Science and

Technology, her research interests include QoS routing algorithm.

Hua Wang is currently a PhD candidate at Nanjing University of Science and

Technology, his research interests include MPLS traffic engineering and QoS

routing.

Ya-Min Sun is professor and PhD supervisor. His current research interests are computer network and

communications.


TSDLMRA: an efficient multicast routing algorithm based on Tabu search

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