HUANG Lin， LAI Jun－feng， HOU Jian， DU Xue－wu

（1.School of Mathematical Science，Dalian University of Technology，Dalian 116024，China；2.Department of Mathematics，China Jiliang University，Hangzhou 310018，China；3.College of Science，Inner Mongolia University of Technology，Huhhot 010021，China）

Multicast routing algorithm based on tabu search

HUANG Lin1，2， LAI Jun－feng＊3， HOU Jian1， DU Xue－wu1

（1.School of Mathematical Science，Dalian University of Technology，Dalian 116024，China；2.Department of Mathematics，China Jiliang University，Hangzhou 310018，China；3.College of Science，Inner Mongolia University of Technology，Huhhot 010021，China）

The delay and delay variation－bounded Steiner tree problem is an important multicast routing issue in real－time multimedia networks.Such a constrained Steiner tree problem is known to be NP－complete.A multicast routing algorithm is presented，which is based on tabu search to produce routing trees having a minimal network cost under delay and delay variation constraints.The simulation shows that the algorithm is efficient for actual networks.This approach makes IP multicast utilize resources efficiently in delivering data to a group of members simultaneously.

multicast；tabu search；delay constraint；delayvariation constraint

0 Introduction

Multicast is a mechanism to efficiently support multi－point communications［1］.In order to support real－time applications，network protocols must be able to provide QoS guarantees.Several delay－constrained heuristics have been proposed［2］.However，some of these heuristics may fail to provide a low cost tree as they assume that network links are symmetric.

There are several situations in which the need for bounded variation among the path delays takes place.During a teleconference，it is important that the speaker is to be heard by all participants at the same time，otherwise，the communication may lack the feeling of an interactive face－to－face discussion.Rouskas，etal.［3］presented a heuristic algorithm that was used to construct multicast trees，which guaranteed certain bounds on the end－to－end delays and delay variations.Sheu，etal.［4］presented a multicast routing scheme on core based tree（CBT）.But they do not attempt to optimize the multicast tree in terms of cost.

Artificial intelligence technique is used for solving this problem because of its success on similar difficult combinatorial problems.For the delay and delay variation－bounded minimum cost multicast routing problem，several heuristics have been proposed［5－10］.In this paper，an efficient algorithm is presented，which is based on tabu search［11］to produce a low－cost multicast tree with delay and delay variation constraints.This algorithm is called tabu search（TS）for delay and delay variation constraints low－cost multicast routing algorithm（TSDDVMA）.TSDDVMA belongs to source－based routing algorithm，because it is assumed that sufficient global information is available to the source.The algorithm starts with an initial shortest path tree constructed by using Sheu′s algorithm［4］.Then the algorithm constructs a backup－pathsset for each destination usingK－th shortest pathalgorithm，and generates neighborhood structure.

1 Problem definition

Mathematically，the delay and delay variation－bounded minimum cost multicast routing problem can be formulated as follows.Given a graphG＝（V，E），with node setVand edge setE，define two objective functions，c（u，v）andd（u，v），on each edge（u，v）∈E.Let

（u，v）be the cost of edge（u，v）andd（u，v）be ts delay.Assume thatc（u，v）＝c（v，u）andd（u，v）＝d（v，u）.On this graph，there are a source nodes，and a set of destination nodesM，called the multicast group.The set of vertices from the setV－M－｛s｝is called Steiner vertices.It tries to construct a delay and delay variation－bounded Steiner treeTrooted ats，that spans the destination nodes inMsuch that for each nodevinM，the delay on the path from

tovis bounded by a delay constraintΔ，and at the same time，the inter－destination delay variation is also bounded by a constraintδ.Formally，for eachv∈M，ifp（s，v）is the path nTfromstov，then

Delay and delay variation－bounded minimal cost multicast tree is a delay and delay variation constraints Steiner treeTsuch that

s minimized and satisfies the Inequalities（1），（2）.

2 Multicast routing algorithm with delay and delay variation constraints

The number of possible multicast trees in a computer network of a moderate size is extremely large.Furthermore，because of the multiobjective nature of the problem and the various cost parameters，it is not clear what constitutes the best tree.Modern iterative heuristics such as TS have been found effective n dealing with this category of problems which have an exponential and noisy search space with numerous local optima.These iterative algorithms are heuristic search methods，which perform a nondeterministic but intelligent walk through the search space.

按照城市主體功能區(qū)與自然環(huán)境區(qū)分化的思路，大都市區(qū)進(jìn)行了兩個同級別不同層面的劃分，即都市化地區(qū)和生態(tài)特色發(fā)展區(qū)。規(guī)劃通過形成主、次級區(qū)域，實(shí)行差異化發(fā)展引導(dǎo)。其中，都市化地區(qū)主要以南昌、緊鄰城市與縣鎮(zhèn)為重點(diǎn)形成的大都市核心區(qū)，生態(tài)特色發(fā)展區(qū)則主要包括九嶺山生態(tài)人文特色發(fā)展區(qū)等三大發(fā)展區(qū)，處于次級區(qū)域。奉新縣地處九嶺山生態(tài)發(fā)展區(qū)，未來將有環(huán)城旅游帶環(huán)繞，同時大都市區(qū)在整體上選擇了生態(tài)保護(hù)與資源開發(fā)共進(jìn)的模式，在奉新縣所屬區(qū)域重點(diǎn)建設(shè)九嶺山公園，發(fā)展定位為國家級養(yǎng)生度假勝地。

2.1 Initial solution

For the delay and delay variation－bounded Steiner tree problem，Rouskas，etal.［3］constructed a delay and delay variation constraints Steiner tree，but the algorithm′s complexity is high.In this paper，the initial solutionT0is constructed by using DDVCA［4，5］.If DDVCA returns false，then the algorithm may fail to obtain a feasible solution.

2.2 Backup－paths－set

For each destination nodev∈M，firstly compute least cost paths fromstovby usingK－th shortest path algorithm to construct a backup－paths－set［7，12－14］.LetBvbe the backup－paths－set for destination nodev，then

If there are nokdifferent paths from the source to destination nodevsatisfing the delay constraint，it is shown that the delay constraint is too small，then negotiate with destinations node above the delay constraint［3］.

2.3 Neighborhood structure

A neighbor structure of the solutionTnowis defined as：

whereis thes－th backup path for destination nodev.

See the following definitions about the multicast tree.Given a networkG＝（V，E）and a multicast treeT，p（s，v）is a path fromstov，fors，v∈V.

Definition 1Adding path（∪）.Add a pathp（s，v）intoT，denoted byT∪p（s，v），T∪p（s，v）＝｛e｜e∈Tore∈p（s，v）｝（6）

Theorem 1Given a networkG＝（V，E），a source nodes，destination node setM.Δandδare the delay bound and the delay variation bound of multicast session，respectively.SupposeTis a subtree，andΔT＜Δ，δT≤δ.Sub（M）is the destinations covered inTso far，andSub（M）≤M.Usedmaxanddminto represent the maximal delay and minimal delay of the path among the paths fromsto each destination ofSub（M）inT，respectively.m∈M，m貢Sub（M），ifp（s，m）satisfies max｛0，dmax－δ｝≤d（p（s，m））≤min｛dmin＋δ，Δ｝andT′＝T∪p（s，m），then

Theorem 1shows that the procedure of constructing a feasible tree meets delay and delay variation bounds，if the delay of a path fromsto next uncovered destination satisfies max｛0，dmaxδ｝≤d（p（s，m））≤min｛dmin＋δ，Δ｝，and then the tree after adding this path is still a feasible tree［5］.

2.4 Tabu moves

A tabu list is maintained to prevent returning to previously visited solutions.This list contains information that forbids the search to some extent from returning to a previously visited solution.In approach of this paper，a multicast tree is considered as an element of tabu list.

2.5 Aspiration criterion

Aspiration criterion is a device used to override the tabu status of moves whenever it is appropriate.It temporarily overrides the tabu status if the move is sufficiently good.The aspiration criterion must make sure that the reversal of a recently－made move（that is，a move in the tabu list）leads the search to an unvisited solution，generally a better one.In this approach，if the cost of a tabu candidate solution is better than current solution，then it replaces current solution and is considered as new current solution.

2.6 Termination rule

A fixed number of iterations have been used as a stopping criterion，and experimented with different values of iterations.It is found that for all the test cases，the TSDDVMA converges within a maximum of 200iterations.The pseudo code for TSDDVMA is as follows：

Procedure TSDDVMA（G＝（V，E），s，M，Δ，δ，c，d）

2.7 Time complexity

Theorem 2The time complexity of TSDDVMA isO（kmn3），wheremis group size andnis network size，kis the parameter ink－SPA.

ProofThe time complexity of generating initial solution by using DDVCA isO（mn2）［4］，and the complexity of constructing backup－paths－set by usingk－SPA isO（kmn3）［3，7－10，13，14］.Because one iteration costsO（k），thus，forQiterations，the cost becomesO（Qk）.So the worst time complexity of the algorithm isO（mn2＋kmn3＋Qk）.The termQkis usually much smaller thankmn3，so the time complexity of TSDDVMA isO（kmn3）.

3 Simulation results and discussion

The TSDDVMA algorithm described in this paper has been tested on several randomly generated networks based on the Waxman′s algorithm［15］.

In the first set of experiments，TSDDVMA is compared with DDVCA［4］and SADDVMA［5，6］for cost performance，where DDVCA is a delay and delay variation－bounded Steiner tree without considering the tree cost.Fig.1shows the tree costctfor varying network sizenwith the group sizem＝4and 6respectively with an average node degree of 3，Δ＝0.40andδ＝0.20.The source nodes and destination nodes vary in each time experiments.It can be seen from Fig.1that TSDDVMA has a better cost performance than DDVCA，and is close to SADDVMA，and could construct low－cost trees which satisfy the given delay and delay variation bound and manage the network resources efficiently.

In the second set of experiments，F(xiàn)ig.2shows the tree cost for varying network sizenwith the group sizem＝5，an average node degree of 3.5（around）and 4.0respectively，Δ＝0.40andδ＝0.20.In general，TSDDVMA has good cost performance and is feasible and effective.

Finally，consider the iteration times of the algorithm.Fig.3shows the tree cost for varying iteration times with the number of network nodesn＝40and 50respectively，group sizem＝4，5and 6.The algorithm converges quickly，and has desirable characteristics of approximation iterative heuristics，which satisfies the real－time requirements of multimedia network.

Fig.1 Tree cost ctversus network size nforΔ＝0.40，δ＝0.20and average node degree 3

Fig.2 Tree cost ctversus network size nforΔ＝0.40，δ＝0.20and m＝5

Fig.3 Tree cost ctversus iteration times i for example network

4 Conclusions

Simulations demonstrate that the algorithm of this paper performs excellent performance of cost，rapid convergence and better real－time property.

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1000－8608（2010）05－0801－05

基于禁忌搜索的組播路由算法

黃 林1，2， 賴俊峰＊3， 侯 劍1， 杜學(xué)武1

（1.大連理工大學(xué)數(shù)學(xué)科學(xué)學(xué)院，遼寧大連 116024；2.中國計(jì)量學(xué)院數(shù)學(xué)系，浙江杭州 310018；3.內(nèi)蒙古工業(yè)大學(xué)理學(xué)院，內(nèi)蒙古呼和浩特 010021）

實(shí)時多媒體網(wǎng)絡(luò)中，帶延遲與延遲抖動約束的斯坦利樹問題是一個研究熱點(diǎn).這種帶約束的斯坦利樹被證明是NP－完全問題.提出了一種基于禁忌搜索的帶延遲與延遲抖動約束最小代價組播路由算法.實(shí)驗(yàn)結(jié)果表明，該算法對于實(shí)際網(wǎng)絡(luò)是有效的.這種方法使得IP組播把數(shù)據(jù)同時發(fā)送到組成員時有效地利用了網(wǎng)絡(luò)資源.

組播；禁忌搜索；延遲約束；延遲抖動約束

TP393

A

2008－05－04；

2010－06－05.

內(nèi)蒙古工業(yè)大學(xué)科研基金資助項(xiàng)目（X200829）.

黃 林（1979－），男，博士，E－mail：goodluckyoume＠yahoo.com.cn；賴俊峰＊（1978－），男，講師，E－mail：ljfimpu＠sina.com.

by：2008－05－04； Revised by：2010－06－05.

Supported by：Fund of Inner Mongolia University of Technology（X200829）

s：HUANG Lin（1979－），Male，Doc.，E－mail：goodluckyoume＠yahoo.com.cn；LAI Jun－fen＊（1978－），Male，Lecturer，E－mail：ljfimpu＠sina.com.