Research Article Minimum Cost Multicast Routing...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 432686 13 pageshttpdxdoiorg1011552013432686

Research ArticleMinimum Cost Multicast Routing Using Ant ColonyOptimization Algorithm

Xiao-Min Hu and Jun Zhang

Department of Computer Science Sun Yat-sen University Key Laboratory of Intelligent Sensor Networks Ministry of Education KeyLaboratory of Software Technology Education Department of Guangdong Province Guangzhou 510006 Guangdong Province China

Correspondence should be addressed to Xiao-Min Hu xmhuieeeorg

Received 7 February 2013 Accepted 18 April 2013

Academic Editor Yuping Wang

Copyright copy 2013 X-M Hu and J Zhang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Multicast routing (MR) is a technology for delivering network data from some source node(s) to a group of destination nodesThe objective of the minimum cost MR (MCMR) problem is to find an optimal multicast tree with the minimum cost for MRThis problem is NP complete In order to tackle the problem this paper proposes a novel algorithm termed the minimum costmulticast routing ant colony optimization (MCMRACO) Based on the ant colony optimization (ACO) framework the artificialants in the proposed algorithm use a probabilistic greedy realization of Primrsquos algorithm to construct multicast trees Movingin a cost complete graph (CCG) of the network topology the ants build solutions according to the heuristic and pheromoneinformationThe heuristic information represents problem-specific knowledge for the ants to construct solutions The pheromoneupdatemechanisms coordinate the antsrsquo activities bymodulating the pheromonesThe algorithmcan quickly respond to the changesof multicast nodes in a dynamic MR environmentThe performance of the proposed algorithm has been compared with publishedresults available in the literature Results show that the proposed algorithm performs well in both static and dynamic MCMRproblems

1 Introduction

Multicast routing (MR) is one of the most important com-munication network routing technologies It first appearedin ARPANET as selective broadcasting from a single sourcenode to a subset of the other nodes in the network [1]Different from unicast routing MR sends only one datacopy from the source node(s) to multiple destination nodesSince MR is resource efficient it has been implemented inmost of the modern communication networks includingoverlay multicast protocols [2ndash4] wireless networks [5ndash7] and satellite networks [8 9] These networks supportmulticast applications such as distributed data processinginternet telephone interactive multimedia conferencing andreal-time video broadcasting [4]

Different multicast applications have different definitionsof the cost such as minimum bandwidth [3 5 7] andminimum energy [6] The objective of the minimum costMR (MCMR) problem is to find an optimal multicast tree

with minimum cost for MR Such a tree is termed a Steinerminimal tree (SMT) in graphs [10] Finding the SMThas beenproven to be NP complete [11]

Various algorithms have been tried for solving theMCMRproblem Traditional heuristic algorithms [12ndash15] usegreedy strategies to construct a feasible multicast tree butthey lack effective guidance to improve the results Geo-graphic [7] and distributed algorithms [6] are only based onthe information from neighbors or nodes within a constanthop distance to compute multicast paths However it may bedifficult to build an efficient multicast tree without completeinformation [2] Computational intelligence (CI) methodssuch as neural networks (NNs) and genetic algorithms (GAs)have been applied to MCMR Gelenbe et al [16] proposed arandom neural network (RNN) to optimize a multicast treeThe RNN essentially enumerated the results by iterativelyadding the most potential node to the tree Leung et al[17] proposed a GA to train the population of individualsby simulating natural evolution Each individual represented

2 Mathematical Problems in Engineering

a multicast tree indirectly through an encoding schemeExcept for the evaluation of the fitness of individuals no treeconstruction information was utilized in the evolutionaryprocess of the GA

In recent years ant colony optimization (ACO) [18ndash20]has become an important CI method ACO is inspired by theforaging behavior of natural ants ACO dispatches a colonyof artificial ants to cooperatively search for the optimum ofthe optimization problem represented on a graph Each antin ACO incrementally builds a solution according to the con-struction information in a stochastic way [18] ACO has beenapplied successfully to various combinatorial optimizationproblems such as the traveling salesman [19 20] constraintsatisfaction [21] allocation problems [22] scheduling [23ndash27] data mining [28 29] water distribution systems [30]power electronic circuit design [31] and networks [32ndash37]UsingACO forMRoptimization is a promising research fieldwhich is still under development Singh et al [38] proposedan ant algorithm forMRoptimization In their algorithm oneant was initially placed at every node in the multicast groupand started to move to the other node via an edge If an antstepped on a node that had been visited by another ant itmerged into the latter ant When only one ant was left theedges passed by the ants forming amulticast treeThe authorshad tested three different sequences for moving the ants fromone node to another and found that the randomapproachwasthe best However their algorithm still could not always findthe optimal solutions of some of their test cases Shen andJaikaeo [39] and Shen et al [40] applied swarm intelligenceto a multicast protocol by connecting nodes in a multicastgroup through a designated node Each node in the multicastgroup periodically sent a packet that behaved like an ant toexplore different paths to the designated nodeThedesignatednode was not statically assigned and its location influencedthe optimality of the multicast tree

In order to make better use of ACO this paper pro-poses a novel minimum cost multicast routing ant colonyoptimization (MCMRACO) algorithm for solving MCMRproblemsThe algorithm has the following features (1) Basedon the ACO framework the proposed algorithm adoptsa probabilistic greedy realization of Primrsquos algorithm [41]for the ants to construct multicast trees (2) Moving ina cost complete graph (CCG) of the network topologythe ants build solutions according to the heuristic andpheromone information (3) The heuristic information isdesigned to represent problem-specific knowledge for theants to construct solutions and to bias the selection ofnodes in the multicast group (4) Representing the antsrsquoconstruction experience pheromones are modulated by thelocal and global pheromone update mechanisms The localpheromone update is applied after each ant has made aconstruction step whereas the global pheromone updatereinforces the pheromones in the bestmulticast tree after eachiteration Hence the algorithm can quickly respond to thechanges of multicast nodes in a dynamic MR environmentUtilizing heuristic and pheromone information effectivelythe proposed MCMRACO is more suitable for solvingMCMR problems than the other heuristic and CI algorithms

The comparison results show that the algorithm workssuccessfully in both static and dynamic MCMR problems

The rest of the paper is organized as follows In Section 2background information for the proposed algorithm is pre-sented Section 3 describes the implementation and maintechniques of the proposed MCMRACO algorithm InSection 4 the proposed algorithm is tested on static anddynamicMCMRproblems Its performance is comparedwiththe published results available in the literature Section 5concludes this paper and discusses some issues for futureresearch

2 Background

This section is composed of three parts In the first partthe MCMR optimization problem is formally defined Thena method for transforming a network graph into a CCG isillustrated The third part describes the ACO framework

21 Definition of the MCMR Optimization Problem In anMCMR optimization problem a network graph is denotedas 119866 = (119881

119866 119864119866) where 119881

119866is the set of nodes in the network

and 119864119866 is the set of edges which connect the nodes in119881119866 Theset 119881119866 is divided into three subsets 119881

119878 119881119863 and 119881

119868 where 119881

119878

is the set of source nodes 119881119863 is the set of destination nodesand 119881119868 is the set of intermediate nodes 119881119872 = 119881119878 cup 119881119863 is theset of nodes in a multicast group Intermediate nodes can beused for relaying multicast traffic but they do not belong tothe multicast group In backbone IP networks the nodes in119881119866 generally are routers Each edge 119890 isin 119864119866 has a positive costvalue 119888(119890) that measures the quality of the edge If there isno edge between two nodes in 119881

119866 the corresponding cost is

denoted as infin The optimization objective of the problem isto find a tree 119879 = (119881

119872+ 119881120579 119864120579) that minimizes the cost for

connecting the nodes in 119881119872

and that satisfies the multicastconstraintsΩ Formally the MCMR problem is defined as

minimize sum119890isin119864120579

119888 (119890) 119879 satisfies the constraints in Ω (1)

where 119881120579sube 119881119868is a subset of the intermediate nodes 119864

120579sube 119864119866

is a subset of the edges in the network graphFigure 1 shows an example ofMR in a networkwith twelve

nodes and some edges Node 1 is the source node The blacksolid nodes 2 4 6 12 are the destination nodes A multicasttree connecting the multicast nodes via the intermediatenodes 5 and 7 is shown in the figure In this paper a multicastgroup has only one source node and multiple destinationnodes

22 Cost Complete Graph (CCG) The network graph isusually a noncomplete but connected graph The edges in anetwork graph are termed physical edges In a CCG howevereach pair of nodes is connected by a logical edge with theminimum cost between the two nodes One of the classicalalgorithms for transforming a network graph into a CCG isFloydrsquos algorithm [42] The pseudocode of the algorithm isshown in Algorithm 1 The input cost matrix of the networkgraph is denoted by C

|119881G |times|119881G | where its element 119888

119894119895is the cost

Mathematical Problems in Engineering 3

Floydrsquos Algorithm

Input C|119881119866|times|119881119866|

[119888119894119895] the cost matrix of the network graph

Output C|119881119866|times|119881119866|

[119888119894119895] the cost matrix of the cost complete graph (CCG)

O|119881119866|times|119881119866|

[119900119894119895] records the next node on the minimum cost route

from node 119894 to node 119895For 119894 = 1 to |119881

119866|

For 119895 = 1 to |119881119866|

119888119894119895= 119888119894119895

If 119888119894119895= infin

119900119894119895= minus1

Else119900119894119895= 119895

End IfEnd For

End ForFor 119896 = 1 to |119881

119866|

For 119894 = 1 to |119881119866|

For 119895 = 1 to |119881119866|

If 119894 = 119895 and 119888119894119896+ 119888119896119895lt 119888119894119895

119888119894119895= 119888119894119896+ 119888119896119895

119900119894119895= 119900119894119896

End IfEnd For

End ForEnd For

Algorithm 1 Pseudocode of Floydrsquos algorithm

1

2

34

5

67

8

910

11

Source node

Destination nodesIntermediate nodesData packets

Directions

12

Figure 1 An example of multicast routingThe source node and thedestination nodes belong to amulticast groupThe heavy black linesindicate the multicast tree

of the edge connecting nodes 119894 and 119895(119894 119895 = 1 2 |119881119866|)

The output of the Floydrsquos algorithm consists of two matrixesOne is the cost matrix C

|119881G |times|119881G |of the CCG where the value

of its elements 119888119894119895denotes the minimum cost from node 119894 to

node 119895 The other isO|119881G |times|119881G |

where each element 119900119894119895records

the next node of node 119894 on the minimum cost route from

node 119894 to node 119895 in the network graph Figure 2(a) presentsan undirected network graphThevalues in the figure indicatethe cost of the corresponding physical edges SupposeΩ =

and take the edges (1198871 1198872) and (1198871 1198874) as examples Nodes 1198871and 1198872are not adjacent but they are connected via node 119887

3

or 1198874or nodes 119887

5 1198876 1198874 In the CCG (Figure 2(b)) nodes 119887

1

and 1198872are logically adjacent with 119888

11988711198872= 11988811988721198871

= 25 and11990011988711198872

= 11990011988721198871

= 1198873 For nodes 119887

1and 1198874 although they are

already adjacent there is a shorter path via nodes 1198875and 1198876

So 11988811988711198874

= 11988811988741198871

= 15 whereas 11990011988711198874

= 1198875 11990011988741198871

= 1198876

The CCG of the network is obtained before the proposedMCMRACO algorithm starts Generally the CCG is main-tained by the network routers or a central network controllingapparatus For example in [2] master multicast routers areused to deal with all the multicast-related tasks They gatherrouting information make MR decisions and manage theother routers to perform MR The proposed MCMRACOalgorithm can be implemented by themastermulticast routerto find a high-quality multicast tree for MR

23 Ant Colony Optimization Once the CCG is available theants can be dispatched to construct multicast trees The antsin ACO canmove in parallel or sequentially to construct theirsolutions [18] Before introducing the proposed algorithmthe ACO framework is briefly presented

The ACO algorithm used in this paper is ant colonysystem (ACS) [20] which is characterized by its state tran-sition rule and the pheromone update mechanisms At eachiteration a colony of ants is dispatched to search for solutions


05

05

05

2

2

151

1198873

1198871

1198875

11988761198874

1198872

(a)

05

05

05

2

15

151

25

1198873

1198871

1198876

1198875

1198874

1198872

(b)

Figure 2 An example of transforming edge (1198871 1198872) and edge (119887

1 1198874) in the CCG (a)The original undirected graph showing the physical edges

and the edge cost (b) The graph showing the transformed edges (1198871 1198872) and (119887

1 1198874) and the corresponding edge cost in the CCG

Choosing an edge for the completion of the solution is termeda construction step Each ant makes every construction stepaccording to the state transition rule After an ant makes aconstruction step the local pheromone update is applied tothe selected edge The local pheromone update reduces thechances for the other ants to choose the same edge repetitivelyand thus ensures diversity in the search After every anthas constructed a complete solution the global pheromoneupdate mechanism is applied to the edges of the best-so-farsolution in order to intensify the attraction of the edges

Figure 3 illustrates the antsrsquo behavior in one iteration byshowing the local and global pheromone update operationson the edges of a graph When the ants are moving thepheromones on the visited edges are changed by the localpheromoneupdate resulting in the reduction of pheromonesAfter all the ants have completed their solutions the globalpheromone update reinforces the pheromones on the edgesof the best-so-far solution In a complex search environmentwith multiple ramifications the previous mechanisms wereshown to be able to find high-quality solutions [18]

3 ACO for Minimum Cost Multicast Routing

In this section the minimum cost multicast routing antcolony optimization (MCMRACO) algorithm is proposedfor solving the unconstrained MCMR problems A completeflowchart of the algorithm is shown in Figure 4(a)

31 Primrsquos Algorithm Pheromone and Heuristic MechanismsThe antrsquos construction behavior in MCMRACO is similarto the generation of a minimum spanning tree (MST) byPrimrsquos algorithm [41] Suppose that119881 is the set of nodes in anundirected connected graph and 119882 is a set containing onlyone node The classical Primrsquos algorithm continuously movesa node 119889 from 119881 minus 119882 to 119882 provided that 119904 isin 119882 and thecost of edge (s d) is minimum In the proposed algorithmthe selection criterion for the next node is not simply basedon the cost of the edges but is based on the product of thepheromone and heuristic values

The pheromone and heuristic values of an edge (119904 119889)

are denoted by 120591(119904 119889) and 120578(119904 119889) respectively The initial

pheromone value is 1205910 = 1119888(119879KMB) where c (119879KMB) is thecost of the initial tree generated by the Kou-Markowsky-Berman (KMB) method [12]

The heuristic value 120578(119904 119889) of the edge connecting nodes 119904and 119889 is a function of the cost of the edge (s d) and the typeof the node d that is

120578 (119904 119889) =

120583

119888119904119889

if 119889 isin 119881119872

1

119888119904119889

otherwise(2)

where 120583 (120583 gt 1) is a reinforcement rate to the nodes in themulticast group Heuristic values represent the quality of thecandidate edges Lower cost edges in the CCG are preferredMoreover multicast nodes have higher probabilities to beselected than intermediate nodes If 120583 rarr infin the antsperform similarly to KMB which only considers multicastnodes during the construction If 120583 = 1 the ants cannotdifferentiate multicast nodes from the intermediate nodes Sothe value of 120583 influences the antsrsquo sensitivity to the multicastnodes in the network

32TheAntsrsquo Search Behavior In this subsection we describean antrsquos search behavior step by step The correspondingflowchart is shown in Figure 4(b)

Step 1 (initialization) Initially an ant 119896 is placed on arandomly chosen multicast node 119904 The visited node set ofant 119896 is denoted by 119882

(119896) = 119904 The unvisited node set is119880(119896) = 119881Gminus119882

(119896)The product of the pheromone and heuristicvalues for every unvisited node 119894 isin 119880(119896) is computed as

120596119894= max119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) (3)

That is 120596119894= 120591(119904 119894) sdot 120578(119904 119894) as119882(119896) = 119904 By using 120594

119894to denote

the corresponding visited node that connects to node 119894 isin 119880(119896)with the maximum product we have

120594119894 = arg max

119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) = 119904 (4)


Source

Food

(a)

Source

Food

(b)

Source

Food

An ants route

The pheromone on the edge

(c)

Figure 3 An illustration of the antsrsquo behavior in one iterationThe thickness of the edges denotes the pheromone density (a)The distributionof pheromones from the last iteration (b) When the three ants are moving the pheromones on the passed edges are changed by the localpheromone update resulting in the reduction of pheromones (c) The global pheromone update reinforces the pheromones on the edges ofthe best-so-far solution

Step 2 (exploitation or exploration) Based on the statetransition rule [20] the ant probabilistically chooses to doexploitation or exploration which is controlled by

119889 =

arg max119903isin119880(119896)

120596119903 if 119902 le 119902

0(exploitation)

119863 otherwise (biased exploration) (5)

where 1199020 is a predefined parameter in [0 1] controlling the

proportion of exploitation to exploration 119902 is a uniformrandom number in [0 1) and119863 is a random number selectedaccording to a probability distribution as (6) If 119902 le 1199020 the antwill choose the next unvisited node 119889 that has the maximumproduct of the pheromone and heuristic values This is calledthe exploitation step Otherwise the next node 119889 is chosenaccording to (6) which is called the exploration step

119901119896 (119889) =

120596119889

sum119903isin119880(119896) 120596119903

if 119889 isin 119880(119896)

0 otherwise(6)

Step 3 (edge extension and local pheromone update) Whenan ant is building a solution the pheromone values on thevisited edges are reduced After choosing the next node d ant119896moves from node 119904 = 120594

119889to node 119889 Since the ant moves in

the CCG the logical edge (s d) can be extended to a physicalroute ⟨119887

0 1198871 1198872 119887

120595⟩ where 119887

0= 119904 119887

120595= 119889 119887

119897= 119900119887119897minus1 119889

and 119897 = 1 120595 minus 1 The edges (119887119894119887119895) (119894 = 0 120595 119895 =

0 120595 119894 = 119895) have the pheromones updated by

120591 (119887119894 119887119895) larr997888 (1 minus 120588) sdot 120591 (119887119894 119887119895) + 120588 sdot 120591min (7)

where 120588 isin (0 1) is the pheromone evaporation rate Thelarger the value of 120588 the more pheromones are evaporatedon the edges that the ant has just passedThe lower boundaryof the pheromone value is set as 120591min = 120591

0so that the updated

value will not be smaller than the initial pheromone valueThe unvisited nodes in 119887

1 1198872 119887

120595are marked as visited by

moving them from 119880(119896) to119882(119896)

Step 4 (has the ant finished themission) If an ant has visitedall the multicast nodes (119881

119872 sube 119882(119896)) the ant has finished

constructing a multicast tree Otherwise for all 119894 isin 119880(119896)update the values of 120596119894 and 120594119894 as

120596119894larr997888 max119895isin120594119894 1198871 1198872119887120595

(120591 (119895 119894) sdot 120578 (119895 119894))

120594119894 larr997888 arg max119895isin120594119894 1198871 1198872 119887120595

(120591 (119895 119894) sdot 120578 (119895 119894)) (8)

Then return to Step 2 for a further search

33 Redundancy Trimming After an ant has finishedbuilding a multicast tree connecting all multicast nodes


tree based on a probabilistic

Perform redundancy

Apply global pheromone updateto the best tree that has ever been found

No

No

No

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Return

Step

1St

ep 3

St

ep 2

St

ep 4

Start

Start

Ant 119896 randomly choosesa multicast node 119904

Initialize theparameters

119905 = 1

119896 = 1End

Primrsquo s method

Ant 119896 constructs a multicast

119896 = 119896 + 1

trimming to the antrsquo s tree

119905 = 119905 + 1

119905 le 119899

119896 le 119898

119894 = 1

119894 = 1

119894 = 119894 + 1

119894 = 119894 + 1

Select a next node 119889 by Select a next node 119889 byexploitation exploration

119902 = rand(0 1)

120596119894 = 120591(119904 119894) middot 120578(119904 119894)

120596119894 =

119902 le 1199020

Perform local pheromone update

Ant 119896 has found allmulticast nodes

and mark the nodes as visited by ant 119896

Node 119894 has not beenvisited by ant 119896

Any node 119895

120591(119895 119894) middot 120578(119895 119894)

120591(119895 119894) middot 120578(119895 119894)

gt 120596119894

119894 le |119881119866|

119894 le |119881119866|

119894 ne 119904

in 1198871 1198872 middot middot middot 119887Ψ satisfies

(a) (b)

Extend the edge (119904 119889) as ⟨1198870 1198871 middot middot middot 119887120595⟩

119883119894 = 119895

119904 = 119883119889

119883119894 = 119904

Figure 4 Flowchart of the proposed minimum cost multicast routing ant colony optimization (MCMRACO)

the tree must be checked for redundancy The flowchart ofthis process is given in Figure 5

Firstly apply the classical Primrsquos algorithm to the visitednodes in the network graph If the cost of the generated MST

is smaller than that of the tree built by the ant the antrsquossolution is replaced by the MST

Secondly check for useless intermediate nodes Delete theone-degree intermediate nodes and their connected edges


No

No

End

Start

119879119896 a tree built by an ant 119896

Yes

Yes

119879trim delete the redundantnodes and edges

119879119896larr119879trim

119879119896larr119879MST

119888(119879119896) gt 119888(119879MST )

to generate an MST

Redundant nodes exist

119879MST use the nodes in 119879119896

Figure 5 Flowchart of the redundancy trimming

34 Global Pheromone Update After all the119898 ants have fin-ished the tree construction the antsrsquo solutions are evaluatedand the best-so-far tree is updated The global pheromoneupdate is applied only to the best-so-far treeThe pheromonevalues on the edges (119894 119895) of the best-so-far (bsf) tree 119879bsf areupdated as

120591 (119894 119895) larr997888 (1 minus 120588) sdot 120591 (119894 119895) + 120572 sdot Δ120591 (9)

where Δ120591 = 1(119888(119879bsf))c (119879bsf) is the cost of 119879bsf and 120572 is thepheromone reinforcement rate with 120572 ge 120588

Moreover the pheromone values on some logical edgesin the tree 119879bsf are also updated If the extended physicalroute between a pair of nodes 119894 and 119895(119894 119895 isin 119879bsf) is⟨1198870 1198871 1198872 119887

120595⟩ which satisfies 119887

0= 119894 119887120595= 119895 119887

119903= 119900119887119903minus1 119895

119903 = 1 120595minus1 (119887

119897 119887119897+1) isin 119879bsf 119897 = 0 1 120595minus1 and120595 gt 1

is the number of edges in the route then the new pheromonevalue of edge (119894 119895) becomes

120591 (119894 119895) larr997888sum120595minus1

119897=0120591 (119887119897 119887119897+1)

120595 (10)

The updated pheromone value of edge (i j) is in accordancewith the average pheromone value of the correspondingphysical edges in the tree

35 The Complexity and Convergence of MCMRACO Thetime complexity of the proposed MCMRACO can be esti-mated by counting the number of multicast trees that aregenerated during the optimization process In each iterationof MCMRACO a colony of119898 ants construct119898 trees and theclassical Primrsquos algorithm is performed 119898 times for redun-dancy trimming As the time complexity for constructing amulticast tree is 119874(|119881

119866|2) and 2m trees are constructed in

each iteration the time complexity ofMCMRACO is approx-imately 119874(2119899119898|119881

119866|2) where 119899 is the predefined maximum

iteration numberAccording to [43] the convergence condition for the

proposed algorithm is to satisfy 0 lt 120591min lt 120591max lt +infinwhere 120591min and 120591max are the lower and upper boundaries ofthe pheromone value respectively Although the heuristicand pheromone mechanisms have been redesigned in theproposed algorithm the convergence of the algorithm is stillmaintained The lower boundary of the pheromone value is120591min = 1205910 gt 0 The upper boundary of the pheromone valueis (120572120588) sdot (1119888(119879opt)) where 119888(119879opt) is the cost of the optimaltree Furthermore every ant in MCMRACO builds a feasiblemulticast solutionTherefore the proposed MCMRACO canconverge to an optimal solution

4 Experiments and Discussions

Theexperiments in this paper are composed of two partsThefirst part is the experiment on the static MR cases whereasthe second part is the one on the dynamic MR cases All theresults in the experiments are computed by a computer withCPU Pentium IV 28GHZ memory 256MB

41 Static Multicast Routing Cases The static MR cases arethe Steiner problems in group B from the OR-Library [44]The eighteen problems are tabulated in Table 1 with the graphsize from 50 to 100 In the table |119881

119866| stands for the number

of nodes |119881119872| is the number of multicast nodes |119864

119866| is the

number of physical edges in the graph and 119888(119879opt) is thecost of the optimal tree The performance of the proposedalgorithm is compared with the KMB heuristic algorithm in[12] the randomneural network (RNN) algorithm in [16] thegenetic algorithm (GA) in [17] and the ant algorithm in [38]

Firstly the parameter settings of the proposed MCM-RACO algorithm are analyzed The proposed algorithmhas five parameters which are the number of ants 119898 thereinforcement rate to destination nodes 120583 the proportionof exploitation 1199020 the pheromone evaporation rate 120588 andthe pheromone reinforcement rate 120572 The number of ants 119898depends on the number of nodes in the network More antsmay perform better in a large network but it will take longertime in one iteration If the number of ants is not enoughthe algorithm may be trapped easily in suboptimal resultsThe value 119898 = 50 is suitable for most of the networks in ourempirical study

The other four parameters 120583 1199020 120588 and 120572 are analyzed by

testing120583 = 5 6 19 1199020= 02 03 04 05 06 07 08 085

09 095 120588 = 01 02 05 and 120572 = 01 02 11 with120572 ge 120588 Each combination of parameter values (120583 119902

0 120588 120572)


5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)

Figure 6 Average performance of MCMRACO with different parameter values in solving the static MR cases (a) Average value of Θ with120583 = 5 6 19 (b) Average value ofΘ with 119902

0= 02 03 04 05 06 07 08 085 09 095 (c) Average value ofΘ with 120588 = 01 02 05

(d) Average value of Θ with 120572 = 01 02 11

is termed a parameter configuration Each configuration istested in ten independent runs by MCMRACO on each ofthe eighteen instances Each run of MCMRACO terminateswhen it cannot find a better result in three consecutiveiterationsThe performance of MCMRACO by configuration120579 on instance 119894 is measured as

Θ (120579 119894) = 100120590 minus 119905 (11)

where 120590 is the successful percentage and 119905 denotes the averagetime in seconds of the ten independent runs by configuration120579 on instance 119894 for obtaining the optimal solution

Figure 6 shows the average performance of MCMRACOwith different parameter values in solving some static MRinstances Each point in the figure is drawn as follows For

example the total measurements of the performance Θ for120583 = 5 of B9 are 312354 When 120583 equals to 5 there are10 choices for 1199020 and 45 choices for the combinations of(120588 120572) So the total configuration number of (120583 1199020 120588 120572) with120583 = 5 is 450 The average measurement of performanceΘ for 120583 = 5 of B9 is thus computed as 312354450 =

69412 which has been plotted as a point in Figure 6(a)Only the instances that cannot be solved successfully by theKMB algorithm are considered in the figureThe curves showthat the parameter values have similar influences on differentinstances (1) The successful percentage increases when thevalue of 120583 becomes bigger It means that the ants should havea strong bias towards the multicast nodes (2) The desirablevalue of 119902

0is in the range of [0609] The results indicate

that the probability for performing exploitation is better to


Table 1 Eighteen Steiner B test cases

No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218

be higher than that of the exploration (3) With 120572 ge 120588 theinfluences for choosing different values of 120572 and 120588 are quitesmall for the same instance

After analyzing the influence of different parameter val-ues the values of the four parameters 120583 119902

0 120588 and 120572 are

selected based on a statistically sound approach that is aracing algorithm termed F-Race [45] According to the resultsby F-Race the parameter values of the proposedMCMRACOare set as119898 = 50 120583 = 19 119902

0= 08 120588 = 03 and 120572 = 05

The results for solving the Steiner B instances are tab-ulated in Table 2 As there is no stochastic factor in KMBand RNN the results are unique in different runs Theant algorithm in [38] the GA in [17] and the proposedMCMRACO are probabilistic algorithms so they are testedten times independently for each instance If the results areequal to the optima they are bold in the table

The results show that KMB only solves seven instancessuccessfully whereas RNN obtains eleven successful resultsout of the eighteen instances The ant algorithm in [37] canfind the best multicast trees of the instances at least onceexcept for B16 and B18 However it can only achieve 100success in eight instances GA terminates when it cannotfind a better result in twenty consecutive generations with apopulation size of 50 Note that the termination conditionof GA is more stringent than the one used in [17] butis looser than that of MCMRACO The results show thatGA can find the optima of all the eighteen instances butit can only solve eleven instances with 100 success Theproposed MCMRACO has the best performance among thealgorithms for it can solve all the static instances with 100success To further study the performance of the algorithmsa dynamic environment is designed in the next subsection

42 Dynamic Multicast Routing Cases Nodes in the networkmay join or leave the multicast group Suppose the networktopology is stable without failure We design the followingdynamic scenario to test the stability of the algorithm Whenthe multicast nodes are changed the algorithm is invoked tofind a new multicast tree

Suppose that V multicast nodes in the multicast groupleave and V new nodes join the group alternatively forminga dynamic situation Note that the nodes in the multicastgroup remain unchanged when the algorithm is still runningInitially there is a network 119866 = (119881119866 119864119866) including amulticast group 119881119872(0) sube 119881119866 At time 1 Vmulticast nodes arechosen randomly to become intermediate nodes indicatingthat V nodes leave the multicast group Then the multicastgroup becomes 119881

119872(1) = 119881

119872(0) minus Δ119881

119872(1) and a multicast

tree T(1) is computed At time 2 V intermediate nodes arechosen randomly to become multicast nodes indicating thatV nodes join the multicast group Then the multicast groupbecomes 119881

119872(2) = 119881

119872(1) + Δ119881

119868(2) and a new multicast tree

T(2) is computed Given the value of V the nodes in themulticast group are changed 2K times and the objective ofthe experiment is to minimize the total cost of the multicasttrees The formal definition is

ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881

119872 (119894 minus 1) minus Δ119881119872 (119894) if 119894 = 1 3 5 2119870 minus 1

119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of

the V randomly chosen multicast nodes to leave the multicastgroup when 119894 = 1 3 5 2119870 minus 1 Δ119881

119868(119894) is the set of the

V randomly chosen intermediate nodes to join the multicastgroup when 119894 = 2 4 6 2119870

There are eighteen dynamic multicast routing instanceswhich use the static Steiner B instances as their initialMR networks respectively For each instance the valuesV = 1 2 |119881119872| minus 3 are tested When the value of V growsthe multicast group becomes more and more unstable

The proposed MCMRACO takes advantage ofpheromones to encode a memory about the antsrsquo searchprocess After the search for a multicast tree is finished thepheromones are still maintained in the network reflectingthe previous routing information Once the multicastnodes are changed the antsrsquo construction is restarted Thepheromones still bias the ants to select the previous edges Asthe ants reduce pheromones on edges by the local pheromoneupdate the influences of the obsolete routing informationdecrease

The results of the summation cost of the 2K multi-cast trees which are computed by MCMRACO RNN andGA are denoted by Φ

(119872)

V Φ(119877)V and Φ(119866)V respectivelyThe comparison result of the tree cost between RNN andMCMRACO is denoted by 120575(119877) whereas the one between GA


Table 2 Optimization results for the static experiment

No 119888(119879opt) KMB RNN Ant GA MCMRACOBest Mean Best Mean Best Mean

1 82 82 82 82 82 82 82 82 822 83 88 83 83 83 83 83 83 833 138 138 138 138 138 138 138 138 1384 59 64 59 59 59 59 59 59 595 61 61 61 61 61 61 61 61 616 122 127 122 mdash mdash 122 122 122 1227 111 111 111 mdash mdash 111 111 111 1118 104 104 104 104 104 104 104 104 1049 220 221 220 220 220 220 220 220 22010 86 91 87 86 874 86 86 86 8611 88 92 90 88 89 88 882 88 8812 174 174 174 mdash mdash 174 174 174 17413 165 175 175 165 1673 165 1658 165 16514 235 237 237 235 2353 235 2373 235 23515 318 318 318 318 318 318 3186 318 31816 127 135 135 132 133 127 1273 127 12717 131 134 132 mdash mdash 131 1311 131 13118 218 221 219 224 2255 218 2184 218 218

and MCMRACO is denoted by 120575(119866) The definitions of 120575(119877)

and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)

Table 3 lists the tree cost of MCMRACO and the compar-ison results of the eighteen initial MR networks when V = 1lfloor(|119881M| minus 3)2 + 1rfloor |119881119872| minus 3 and119870 = 100 As the comparisonresults are all positive the proposed MCMRACO can obtainmulticast trees with less cost than RNN and GA Figure 7shows the comparison results of the tree cost with B18 as theinitial MR network when the values of V are changed from1 to 47 The differences between RNN and MCMRACO arealways larger than those between GA and MCMRACO Thefigure presents that the proposed MCMRACO can steadilyachieve better results than RNN and GA

When the multicast group is changed the time used forfinding the solution is the response delayThe time needed byRNN is determined by its enumeration subset The smallerthe multicast group the more nodes that may be used forenumeration resulting in longer computation timeHoweverfor GA and MCMRACO the time used for a small group isgenerally shorter than a large group in the same terminationcondition Figure 8 illustrates the average time used by RNNGA and MCMRACO with different values of V when theinitial multicast network is B18 The curves show that theaverage time needed by GA and MCMRACO reduces as thevalue of V becomes bigger but the time used by RNN isgrowing upwards

Table 4 tabulates the average time used for finding amulticast solution in the 18 instances when = 1 lfloor(|119881

119872| minus

3)2 + 1rfloor |119881119872| minus 3 Although the average time used by

002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

GA-MCMRACORNN-MCMRACO

Figure 7 Comparison results of the tree cost between RNN andMCMRACO and between GA and MCMRACO with differentvalues of V when the initial MR network is B18

Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA

Figure 8 Average time delay of RNN GA and MCMRACO withdifferent values of V when the initial MR network is B18


Table 3 Comparison results of the tree cost for the dynamic ex-periment

No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015

Table 4 Computation time delay for the dynamic experiment

No v Time (microsecond)RNN GA MCMRACO

11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961


RNN is shorter than that of MCMRACO in some cases theresults obtained by MCMRACO are always better than thoseof RNN (as Table 3) Moreover MCMRACO performs fasterthan RNN in instances Nos 7 10 13 14 16 and 17 For GA andMCMRACO the results show that even if GA takes longertime than MCMRACO the results obtained by GA are stillworse than those of the proposed algorithm For examplethe proposed MCMRACO uses only 742 microseconds (ms)on average to find the result of instance No 1 when V = 1whereas the GA needs 1742ms For the instance No 18 whenV = 47 the proposed MCMRACO uses only 2961ms toobtain the result whereas GA needs 7000ms and the resultis still 015 worse than the proposed algorithm Overall theproposedMCMRACO performs better than RNN and GA insolving the MCMR problems

5 Conclusion

This paper proposes a minimum cost multicast routing antcolony optimization (MCMRACO) algorithm for solvingthe minimum cost multicast routing (MCMR) problemsDifferent from the traditional algorithms for MCMR theproposed MCMRACO utilizes the ant colony optimization(ACO) technique to search for an optimal multicast tree inthe network graph The artificial ants in the algorithm arebased on a modified Primrsquos algorithm to build a tree Theheuristic information represents problem-specific knowledgefor the ants to construct solutions whereas the pheromoneson edges reserve the previous routing information Bydesigning effective heuristic and pheromone mechanismsthe proposed MCMRACO is very competitive for solvingMCMR problems The performance of MCMRACO hasbeen compared with the published results available in theliterature The comparison results show that the proposedMCMRACO works successfully in both static and dynamicMCMR cases The proposed algorithm is protocol indepen-dent so that it can be implemented conveniently in differentnetwork environments Extending the proposed algorithm toheterogeneous networks is a promising future research topic

Acknowledgments

This work was supported in part by the National High-Technology Research and Development Program (ldquo863rdquoProgram) of China under Grant no 2013AA01A212 by theNational Science Fund for Distinguished Young Scholarsunder Grant 61125205 by the National Natural ScienceFoundation of China under Grants 61070004 61202130 bythe NSFC Joint Fund with Guangdong under Key ProjectsU1201258 and U1135005 by the Guangdong Natural ScienceFoundation S2012040007948 by the Fundamental ResearchFunds for the Central Universities no 12lgpy47 and by theSpecialized Research Fund for the Doctoral Program ofHigher Education 20120171120027

References

[1] K Bharath-Kumar and J M Jaffe ldquoRouting to multiple destina-tions in computer networksrdquo IEEE Transactions on Communi-cations vol 31 no 3 pp 343ndash351 1983

[2] Y Yang J Wang and M Yang ldquoA service-centric multicastarchitecture and routing protocolrdquo IEEE Transactions on Par-allel and Distributed Systems vol 19 no 1 pp 35ndash51 2008

[3] D-N Yang andW J Liao ldquoOn bandwidth-efficient overlaymul-ticastrdquo IEEE Transactions on Parallel and Distributed Systemsvol 18 no 11 pp 1503ndash1515 2007

[4] A Ganjam and H Zhang ldquoInternet multicast video deliveryrdquoProceedings of the IEEE vol 93 no 1 pp 159ndash170 2005

[5] D-N Yang and M-S Chen ldquoEfficient resource allocation forwireless multicastrdquo IEEE Transactions on Mobile Computingvol 7 no 4 pp 387ndash400 2008

[6] S Guo and O Yang ldquoLocalized operations for distributed min-imum energy multicast algorithm in mobile ad hoc networksrdquoIEEE Transactions on Parallel and Distributed Systems vol 18no 2 pp 186ndash198 2007

[7] J A Sanchez PM Ruiz J Liu and I Stojmenovic ldquoBandwidth-efficient geographic multicast routing protocol for wirelesssensor networksrdquo IEEE Sensors Journal vol 7 no 5 pp 627ndash636 2007

[8] F Filali G Aniba and W Dabbous ldquoEfficient support of IPmulticast in the next generation of GEO satellitesrdquo IEEE Journalon Selected Areas in Communications vol 22 no 2 pp 413ndash4252004

[9] E Ekici I F Akyildiz and M D Bender ldquoA multicast routingalgorithm for LEO satellite IP networksrdquo IEEEACM Transac-tions on Networking vol 10 no 2 pp 183ndash192 2002

[10] E N Gilbert and H O Pollak ldquoSteiner minimal treesrdquo SIAMJournal on Applied Mathematics vol 16 pp 1ndash29 1968

[11] M R Garey andD S JohnsonComputers and Intractability WH Freeman and Co San Francisco Calif USA 1979

[12] L Kou G Markowsky and L Berman ldquoA fast algorithm forSteiner treesrdquo Acta Informatica vol 15 no 2 pp 141ndash145 1981

[13] V J Rayward-Smith and A Clare ldquoOn finding Steiner verticesrdquoNetworks vol 16 no 3 pp 283ndash294 1986

[14] V J Rayward-Smith ldquoThe computation of nearly minimalSteiner trees in graphsrdquo International Journal of MathematicalEducation in Science and Technology vol 14 no 1 pp 15ndash231983

[15] P Winter and J M Smith ldquoPath-distance heuristics for theSteiner problem in undirected networksrdquo Algorithmica vol 7no 2-3 pp 309ndash327 1992

[16] E Gelenbe A Ghanwani and V Srinivasan ldquoImproved neuralheuristics for multicast routingrdquo IEEE Journal on Selected Areasin Communications vol 15 no 2 pp 147ndash155 1997

[17] Y Leung G Li and Z-B Xu ldquoA genetic algorithm for themultiple destination routing problemsrdquo IEEE Transactions onEvolutionary Computation vol 2 no 4 pp 150ndash161 1998

[18] M Dorigo and T Stutzle Ant Colony Optimization MIT PressCambridge Mass USA 2004

[19] MDorigo VManiezzo andA Colorni ldquoAnt system optimiza-tion by a colony of cooperating agentsrdquo IEEE Transactions onSystems Man and Cybernetics B vol 26 no 1 pp 29ndash41 1996

[20] M Dorigo and L M Gambardella ldquoAnt colony system a coop-erative learning approach to the traveling salesman problemrdquoIEEETransactions on Evolutionary Computation vol 1 no 1 pp53ndash66 1997


[21] C Solnon ldquoAnts can solve constraint satisfaction problemsrdquoIEEE Transactions on Evolutionary Computation vol 6 no 4pp 347ndash357 2002

[22] Y-C Liang and A E Smith ldquoAn ant colony optimizationalgorithm for the redundancy allocation problem (RAP)rdquo IEEETransactions on Reliability vol 53 no 3 pp 417ndash423 2004

[23] W-N Chen and J Zhang ldquoAnt colony optimization for softwareproject scheduling and staffing with an event-based schedulerrdquoIEEE Transactions on Software Engineering vol 39 no 1 pp 1ndash17 2013

[24] W N Chen J Zhang H S H Chung R Z Huang and O LiuldquoOptimizing discounted cash flows in project scheduling-an antcolony optimization approachrdquo IEEE Transactions on SystemsMan and Cybernetics C vol 40 no 1 pp 64ndash77 2010

[25] S Guo H-Z Huang Z Wang and M Xie ldquoGrid servicereliability modeling and optimal task scheduling consideringfault recoveryrdquo IEEE Transactions on Reliability vol 60 no 1pp 263ndash274 2011

[26] W-N Chen and J Zhang ldquoAn ant colony optimizationapproach to a grid workflow scheduling problem with variousQoS requirementsrdquo IEEE Transactions on Systems Man andCybernetics C vol 39 no 1 pp 29ndash43 2009

[27] D Merkle M Middendorf and H Schmeck ldquoAnt colony opti-mization for resource-constrained project schedulingrdquo IEEETransactions on Evolutionary Computation vol 6 no 4 pp333ndash346 2002

[28] R S Parpinelli H S Lopes and A A Freitas ldquoData miningwith an ant colony optimization algorithmrdquo IEEE Transactionson Evolutionary Computation vol 6 no 4 pp 321ndash332 2002

[29] D Martens M de Backer R Haesen J Vanthienen M Snoeckand B Baesens ldquoClassification with ant colony optimizationrdquoIEEE Transactions on Evolutionary Computation vol 11 no 5pp 651ndash665 2007

[30] A C Zecchin A R Simpson H R Maier and J B NixonldquoParametric study for an ant algorithm applied to water distri-bution systemoptimizationrdquo IEEETransactions on EvolutionaryComputation vol 9 no 2 pp 175ndash191 2005

[31] J Zhang H S-H Chung AW-L Lo and THuang ldquoExtendedant colony optimization algorithm for power electronic circuitdesignrdquo IEEE Transactions on Power Electronics vol 24 no 1pp 147ndash162 2009

[32] A Konak and S Kulturel-Konak ldquoReliable server assignment innetworks using nature inspired metaheuristicsrdquo IEEE Transac-tions on Reliability vol 60 no 2 pp 381ndash393 2011

[33] K M Sim and W H Sun ldquoAnt colony optimization forrouting and load-balancing survey and new directionsrdquo IEEETransactions on Systems Man and Cybernetics A vol 33 no 5pp 560ndash572 2003

[34] R Schoonderwoerd O Holland and J Bruten ldquoAnt-likeagents for load balancing in telecommunications networksrdquo inProceedings of the 1st International Conference on AutonomousAgents (Agents rsquo97) pp 209ndash216 New York NY USA February1997

[35] G di Caro and M Dorigo ldquoAntNet distributed stigmergeticcontrol for communications networksrdquo Journal of ArtificialIntelligence Research vol 9 pp 317ndash365 1998

[36] S S Iyengar H-C Wu N Balakrishnan and S Y ChangldquoBiologically inspired cooperative routing for wireless mobilesensor networksrdquo IEEE Systems Journal vol 1 no 1 pp 29ndash372007

[37] OHHussein T N Saadawi andM J Lee ldquoProbability routingalgorithm for mobile ad hoc networksrsquo resources managementrdquoIEEE Journal on Selected Areas in Communications vol 23 no12 pp 2248ndash2259 2005

[38] G Singh S Das S V Gosavi and S Pujar ldquoAnt colonyalgorithms for Steiner trees an application to routing in sensornetworksrdquo in Recent Developments in Biologically Inspired Com-puting L N de Castro and F J von Zuben Eds pp 181ndash206Idea Group Publishing Hershey Pa USA 2004

[39] C-C Shen and C Jaikaeo ldquoAd hoc multicast routing algorithmwith swarm intelligencerdquoMobile Networks andApplications vol10 no 1 pp 47ndash59 2005

[40] C-C Shen K Li C Jaikaeo and V Sridhara ldquoAnt-baseddistributed constrained steiner tree algorithm for jointly con-serving energy and bounding delay in ad hocmulticast routingrdquoACMTransactions on Autonomous and Adaptive Systems vol 3no 1 article 3 2008

[41] R C Prim ldquoShortest connection networks and some general-izationsrdquo Bell System Technical Journal vol 36 pp 1389ndash14011957

[42] R W Floyd ldquoShortest pathrdquo Communications of the ACM vol5 no 6 p 345 1962

[43] T Stutzle andM Dorigo ldquoA short convergence proof for a classof ant colony optimization algorithmsrdquo IEEE Transactions onEvolutionary Computation vol 6 no 4 pp 358ndash365 2002

[44] J E Beasley ldquoOR-Library distributing test problems by elec-tronic mailrdquo Journal of Operational Research Society vol 41 no11 pp 1069ndash1072 1990

[45] M Birattari T Stutzle L Paquete andKVarrentrapp ldquoA racingalgorithm for configuring metaheuristicsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference (GECCOrsquo02) pp 11ndash18 2002

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


a multicast tree indirectly through an encoding schemeExcept for the evaluation of the fitness of individuals no treeconstruction information was utilized in the evolutionaryprocess of the GA

In recent years ant colony optimization (ACO) [18ndash20]has become an important CI method ACO is inspired by theforaging behavior of natural ants ACO dispatches a colonyof artificial ants to cooperatively search for the optimum ofthe optimization problem represented on a graph Each antin ACO incrementally builds a solution according to the con-struction information in a stochastic way [18] ACO has beenapplied successfully to various combinatorial optimizationproblems such as the traveling salesman [19 20] constraintsatisfaction [21] allocation problems [22] scheduling [23ndash27] data mining [28 29] water distribution systems [30]power electronic circuit design [31] and networks [32ndash37]UsingACO forMRoptimization is a promising research fieldwhich is still under development Singh et al [38] proposedan ant algorithm forMRoptimization In their algorithm oneant was initially placed at every node in the multicast groupand started to move to the other node via an edge If an antstepped on a node that had been visited by another ant itmerged into the latter ant When only one ant was left theedges passed by the ants forming amulticast treeThe authorshad tested three different sequences for moving the ants fromone node to another and found that the randomapproachwasthe best However their algorithm still could not always findthe optimal solutions of some of their test cases Shen andJaikaeo [39] and Shen et al [40] applied swarm intelligenceto a multicast protocol by connecting nodes in a multicastgroup through a designated node Each node in the multicastgroup periodically sent a packet that behaved like an ant toexplore different paths to the designated nodeThedesignatednode was not statically assigned and its location influencedthe optimality of the multicast tree

In order to make better use of ACO this paper pro-poses a novel minimum cost multicast routing ant colonyoptimization (MCMRACO) algorithm for solving MCMRproblemsThe algorithm has the following features (1) Basedon the ACO framework the proposed algorithm adoptsa probabilistic greedy realization of Primrsquos algorithm [41]for the ants to construct multicast trees (2) Moving ina cost complete graph (CCG) of the network topologythe ants build solutions according to the heuristic andpheromone information (3) The heuristic information isdesigned to represent problem-specific knowledge for theants to construct solutions and to bias the selection ofnodes in the multicast group (4) Representing the antsrsquoconstruction experience pheromones are modulated by thelocal and global pheromone update mechanisms The localpheromone update is applied after each ant has made aconstruction step whereas the global pheromone updatereinforces the pheromones in the bestmulticast tree after eachiteration Hence the algorithm can quickly respond to thechanges of multicast nodes in a dynamic MR environmentUtilizing heuristic and pheromone information effectivelythe proposed MCMRACO is more suitable for solvingMCMR problems than the other heuristic and CI algorithms

The comparison results show that the algorithm workssuccessfully in both static and dynamic MCMR problems

The rest of the paper is organized as follows In Section 2background information for the proposed algorithm is pre-sented Section 3 describes the implementation and maintechniques of the proposed MCMRACO algorithm InSection 4 the proposed algorithm is tested on static anddynamicMCMRproblems Its performance is comparedwiththe published results available in the literature Section 5concludes this paper and discusses some issues for futureresearch

2 Background

This section is composed of three parts In the first partthe MCMR optimization problem is formally defined Thena method for transforming a network graph into a CCG isillustrated The third part describes the ACO framework

21 Definition of the MCMR Optimization Problem In anMCMR optimization problem a network graph is denotedas 119866 = (119881

119866 119864119866) where 119881

119866is the set of nodes in the network

and 119864119866 is the set of edges which connect the nodes in119881119866 Theset 119881119866 is divided into three subsets 119881

119878 119881119863 and 119881

119868 where 119881

119878

is the set of source nodes 119881119863 is the set of destination nodesand 119881119868 is the set of intermediate nodes 119881119872 = 119881119878 cup 119881119863 is theset of nodes in a multicast group Intermediate nodes can beused for relaying multicast traffic but they do not belong tothe multicast group In backbone IP networks the nodes in119881119866 generally are routers Each edge 119890 isin 119864119866 has a positive costvalue 119888(119890) that measures the quality of the edge If there isno edge between two nodes in 119881

119866 the corresponding cost is

denoted as infin The optimization objective of the problem isto find a tree 119879 = (119881

119872+ 119881120579 119864120579) that minimizes the cost for

connecting the nodes in 119881119872

and that satisfies the multicastconstraintsΩ Formally the MCMR problem is defined as

minimize sum119890isin119864120579

119888 (119890) 119879 satisfies the constraints in Ω (1)

where 119881120579sube 119881119868is a subset of the intermediate nodes 119864

120579sube 119864119866

is a subset of the edges in the network graphFigure 1 shows an example ofMR in a networkwith twelve

nodes and some edges Node 1 is the source node The blacksolid nodes 2 4 6 12 are the destination nodes A multicasttree connecting the multicast nodes via the intermediatenodes 5 and 7 is shown in the figure In this paper a multicastgroup has only one source node and multiple destinationnodes

22 Cost Complete Graph (CCG) The network graph isusually a noncomplete but connected graph The edges in anetwork graph are termed physical edges In a CCG howevereach pair of nodes is connected by a logical edge with theminimum cost between the two nodes One of the classicalalgorithms for transforming a network graph into a CCG isFloydrsquos algorithm [42] The pseudocode of the algorithm isshown in Algorithm 1 The input cost matrix of the networkgraph is denoted by C

|119881G |times|119881G | where its element 119888

119894119895is the cost



Input C|119881119866|times|119881119866|


Output C|119881119866|times|119881119866|


O|119881119866|times|119881119866|



119866|

For 119895 = 1 to |119881119866|

119888119894119895= 119888119894119895

If 119888119894119895= infin

119900119894119895= minus1

Else119900119894119895= 119895

End IfEnd For

End ForFor 119896 = 1 to |119881

119866|

For 119894 = 1 to |119881119866|

For 119895 = 1 to |119881119866|

If 119894 = 119895 and 119888119894119896+ 119888119896119895lt 119888119894119895

119888119894119895= 119888119894119896+ 119888119896119895

119900119894119895= 119900119894119896

End IfEnd For

End ForEnd For


1

2

34

5

67

8

910

11

Source node


Directions

12











3

or 1198874or nodes 119887


1


11988711198872= 11988811988721198871

= 25 and11990011988711198872

= 11990011988721198871

= 1198873 For nodes 119887



So 11988811988711198874

= 11988811988741198871

= 15 whereas 11990011988711198874

= 1198875 11990011988741198871

= 1198876





05

05

05

2

2

151

1198873

1198871

1198875

11988761198874

1198872

(a)

05

05

05

2

15

151

25

1198873

1198871

1198876

1198875

1198874

1198872

(b)














120578 (119904 119889) =

120583

119888119904119889

if 119889 isin 119881119872

1

119888119904119889

otherwise(2)






120596119894= max119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) (3)


119894to denote


120594119894 = arg max

119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) = 119904 (4)


Source

Food

(a)

Source

Food

(b)

Source

Food

An ants route


(c)



119889 =

arg max119903isin119880(119896)

120596119903 if 119902 le 119902

0(exploitation)




119901119896 (119889) =

120596119889

sum119903isin119880(119896) 120596119903

if 119889 isin 119880(119896)

0 otherwise(6)




0 1198871 1198872 119887

120595⟩ where 119887

0= 119904 119887

120595= 119889 119887

119897= 119900119887119897minus1 119889







1 1198872 119887






120596119894larr997888 max119895isin120594119894 1198871 1198872119887120595

(120591 (119895 119894) sdot 120578 (119895 119894))

120594119894 larr997888 arg max119895isin120594119894 1198871 1198872 119887120595

(120591 (119895 119894) sdot 120578 (119895 119894)) (8)





Perform redundancy


No

No

No

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Return

Step

1St

ep 3

St

ep 2

St

ep 4

Start

Start



119905 = 1

119896 = 1End

Primrsquo s method


119896 = 119896 + 1


119905 = 119905 + 1

119905 le 119899

119896 le 119898

119894 = 1

119894 = 1

119894 = 119894 + 1

119894 = 119894 + 1


119902 = rand(0 1)

120596119894 = 120591(119904 119894) middot 120578(119904 119894)

120596119894 =

119902 le 1199020





Any node 119895

120591(119895 119894) middot 120578(119895 119894)

120591(119895 119894) middot 120578(119895 119894)

gt 120596119894

119894 le |119881119866|

119894 le |119881119866|

119894 ne 119904


(a) (b)


119883119894 = 119895

119904 = 119883119889

119883119894 = 119904







No

No

End

Start


Yes

Yes


119879119896larr119879trim

119879119896larr119879MST

119888(119879119896) gt 119888(119879MST )

to generate an MST









0= 119894 119887120595= 119895 119887

119903= 119900119887119903minus1 119895

119903 = 1 120595minus1 (119887



120591 (119894 119895) larr997888sum120595minus1

119897=0120591 (119887119897 119887119897+1)

120595 (10)













119866| is the




testing120583 = 5 6 19 1199020= 02 03 04 05 06 07 08 085


0 120588 120572)


5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)





Θ (120579 119894) = 100120590 minus 119905 (11)









No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Input C|119881119866|times|119881119866|


Output C|119881119866|times|119881119866|


O|119881119866|times|119881119866|



119866|

For 119895 = 1 to |119881119866|

119888119894119895= 119888119894119895

If 119888119894119895= infin

119900119894119895= minus1

Else119900119894119895= 119895

End IfEnd For

End ForFor 119896 = 1 to |119881

119866|

For 119894 = 1 to |119881119866|

For 119895 = 1 to |119881119866|

If 119894 = 119895 and 119888119894119896+ 119888119896119895lt 119888119894119895

119888119894119895= 119888119894119896+ 119888119896119895

119900119894119895= 119900119894119896

End IfEnd For

End ForEnd For


1

2

34

5

67

8

910

11

Source node


Directions

12











3

or 1198874or nodes 119887


1


11988711198872= 11988811988721198871

= 25 and11990011988711198872

= 11990011988721198871

= 1198873 For nodes 119887



So 11988811988711198874

= 11988811988741198871

= 15 whereas 11990011988711198874

= 1198875 11990011988741198871

= 1198876





05

05

05

2

2

151

1198873

1198871

1198875

11988761198874

1198872

(a)

05

05

05

2

15

151

25

1198873

1198871

1198876

1198875

1198874

1198872

(b)














120578 (119904 119889) =

120583

119888119904119889

if 119889 isin 119881119872

1

119888119904119889

otherwise(2)






120596119894= max119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) (3)


119894to denote


120594119894 = arg max

119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) = 119904 (4)


Source

Food

(a)

Source

Food

(b)

Source

Food

An ants route


(c)



119889 =

arg max119903isin119880(119896)

120596119903 if 119902 le 119902

0(exploitation)




119901119896 (119889) =

120596119889

sum119903isin119880(119896) 120596119903

if 119889 isin 119880(119896)

0 otherwise(6)




0 1198871 1198872 119887

120595⟩ where 119887

0= 119904 119887

120595= 119889 119887

119897= 119900119887119897minus1 119889







1 1198872 119887






120596119894larr997888 max119895isin120594119894 1198871 1198872119887120595

(120591 (119895 119894) sdot 120578 (119895 119894))

120594119894 larr997888 arg max119895isin120594119894 1198871 1198872 119887120595

(120591 (119895 119894) sdot 120578 (119895 119894)) (8)





Perform redundancy


No

No

No

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Return

Step

1St

ep 3

St

ep 2

St

ep 4

Start

Start



119905 = 1

119896 = 1End

Primrsquo s method


119896 = 119896 + 1


119905 = 119905 + 1

119905 le 119899

119896 le 119898

119894 = 1

119894 = 1

119894 = 119894 + 1

119894 = 119894 + 1


119902 = rand(0 1)

120596119894 = 120591(119904 119894) middot 120578(119904 119894)

120596119894 =

119902 le 1199020





Any node 119895

120591(119895 119894) middot 120578(119895 119894)

120591(119895 119894) middot 120578(119895 119894)

gt 120596119894

119894 le |119881119866|

119894 le |119881119866|

119894 ne 119904


(a) (b)


119883119894 = 119895

119904 = 119883119889

119883119894 = 119904







No

No

End

Start


Yes

Yes


119879119896larr119879trim

119879119896larr119879MST

119888(119879119896) gt 119888(119879MST )

to generate an MST









0= 119894 119887120595= 119895 119887

119903= 119900119887119903minus1 119895

119903 = 1 120595minus1 (119887



120591 (119894 119895) larr997888sum120595minus1

119897=0120591 (119887119897 119887119897+1)

120595 (10)













119866| is the




testing120583 = 5 6 19 1199020= 02 03 04 05 06 07 08 085


0 120588 120572)


5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)





Θ (120579 119894) = 100120590 minus 119905 (11)









No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










05

05

05

2

2

151

1198873

1198871

1198875

11988761198874

1198872

(a)

05

05

05

2

15

151

25

1198873

1198871

1198876

1198875

1198874

1198872

(b)














120578 (119904 119889) =

120583

119888119904119889

if 119889 isin 119881119872

1

119888119904119889

otherwise(2)






120596119894= max119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) (3)


119894to denote


120594119894 = arg max

119895isin119882(119896)

(120591 (119895 119894) sdot 120578 (119895 119894)) = 119904 (4)


Source

Food

(a)

Source

Food

(b)

Source

Food

An ants route


(c)



119889 =

arg max119903isin119880(119896)

120596119903 if 119902 le 119902

0(exploitation)




119901119896 (119889) =

120596119889

sum119903isin119880(119896) 120596119903

if 119889 isin 119880(119896)

0 otherwise(6)




0 1198871 1198872 119887

120595⟩ where 119887

0= 119904 119887

120595= 119889 119887

119897= 119900119887119897minus1 119889







1 1198872 119887






120596119894larr997888 max119895isin120594119894 1198871 1198872119887120595

(120591 (119895 119894) sdot 120578 (119895 119894))

120594119894 larr997888 arg max119895isin120594119894 1198871 1198872 119887120595

(120591 (119895 119894) sdot 120578 (119895 119894)) (8)





Perform redundancy


No

No

No

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Return

Step

1St

ep 3

St

ep 2

St

ep 4

Start

Start



119905 = 1

119896 = 1End

Primrsquo s method


119896 = 119896 + 1


119905 = 119905 + 1

119905 le 119899

119896 le 119898

119894 = 1

119894 = 1

119894 = 119894 + 1

119894 = 119894 + 1


119902 = rand(0 1)

120596119894 = 120591(119904 119894) middot 120578(119904 119894)

120596119894 =

119902 le 1199020





Any node 119895

120591(119895 119894) middot 120578(119895 119894)

120591(119895 119894) middot 120578(119895 119894)

gt 120596119894

119894 le |119881119866|

119894 le |119881119866|

119894 ne 119904


(a) (b)


119883119894 = 119895

119904 = 119883119889

119883119894 = 119904







No

No

End

Start


Yes

Yes


119879119896larr119879trim

119879119896larr119879MST

119888(119879119896) gt 119888(119879MST )

to generate an MST









0= 119894 119887120595= 119895 119887

119903= 119900119887119903minus1 119895

119903 = 1 120595minus1 (119887



120591 (119894 119895) larr997888sum120595minus1

119897=0120591 (119887119897 119887119897+1)

120595 (10)













119866| is the




testing120583 = 5 6 19 1199020= 02 03 04 05 06 07 08 085


0 120588 120572)


5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)





Θ (120579 119894) = 100120590 minus 119905 (11)









No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Source

Food

(a)

Source

Food

(b)

Source

Food

An ants route


(c)



119889 =

arg max119903isin119880(119896)

120596119903 if 119902 le 119902

0(exploitation)




119901119896 (119889) =

120596119889

sum119903isin119880(119896) 120596119903

if 119889 isin 119880(119896)

0 otherwise(6)




0 1198871 1198872 119887

120595⟩ where 119887

0= 119904 119887

120595= 119889 119887

119897= 119900119887119897minus1 119889







1 1198872 119887






120596119894larr997888 max119895isin120594119894 1198871 1198872119887120595

(120591 (119895 119894) sdot 120578 (119895 119894))

120594119894 larr997888 arg max119895isin120594119894 1198871 1198872 119887120595

(120591 (119895 119894) sdot 120578 (119895 119894)) (8)





Perform redundancy


No

No

No

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Return

Step

1St

ep 3

St

ep 2

St

ep 4

Start

Start



119905 = 1

119896 = 1End

Primrsquo s method


119896 = 119896 + 1


119905 = 119905 + 1

119905 le 119899

119896 le 119898

119894 = 1

119894 = 1

119894 = 119894 + 1

119894 = 119894 + 1


119902 = rand(0 1)

120596119894 = 120591(119904 119894) middot 120578(119904 119894)

120596119894 =

119902 le 1199020





Any node 119895

120591(119895 119894) middot 120578(119895 119894)

120591(119895 119894) middot 120578(119895 119894)

gt 120596119894

119894 le |119881119866|

119894 le |119881119866|

119894 ne 119904


(a) (b)


119883119894 = 119895

119904 = 119883119889

119883119894 = 119904







No

No

End

Start


Yes

Yes


119879119896larr119879trim

119879119896larr119879MST

119888(119879119896) gt 119888(119879MST )

to generate an MST









0= 119894 119887120595= 119895 119887

119903= 119900119887119903minus1 119895

119903 = 1 120595minus1 (119887



120591 (119894 119895) larr997888sum120595minus1

119897=0120591 (119887119897 119887119897+1)

120595 (10)













119866| is the




testing120583 = 5 6 19 1199020= 02 03 04 05 06 07 08 085


0 120588 120572)


5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)





Θ (120579 119894) = 100120590 minus 119905 (11)









No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Perform redundancy


No

No

No

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Return

Step

1St

ep 3

St

ep 2

St

ep 4

Start

Start



119905 = 1

119896 = 1End

Primrsquo s method


119896 = 119896 + 1


119905 = 119905 + 1

119905 le 119899

119896 le 119898

119894 = 1

119894 = 1

119894 = 119894 + 1

119894 = 119894 + 1


119902 = rand(0 1)

120596119894 = 120591(119904 119894) middot 120578(119904 119894)

120596119894 =

119902 le 1199020





Any node 119895

120591(119895 119894) middot 120578(119895 119894)

120591(119895 119894) middot 120578(119895 119894)

gt 120596119894

119894 le |119881119866|

119894 le |119881119866|

119894 ne 119904


(a) (b)


119883119894 = 119895

119904 = 119883119889

119883119894 = 119904







No

No

End

Start


Yes

Yes


119879119896larr119879trim

119879119896larr119879MST

119888(119879119896) gt 119888(119879MST )

to generate an MST









0= 119894 119887120595= 119895 119887

119903= 119900119887119903minus1 119895

119903 = 1 120595minus1 (119887



120591 (119894 119895) larr997888sum120595minus1

119897=0120591 (119887119897 119887119897+1)

120595 (10)













119866| is the




testing120583 = 5 6 19 1199020= 02 03 04 05 06 07 08 085


0 120588 120572)


5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)





Θ (120579 119894) = 100120590 minus 119905 (11)









No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










No

No

End

Start


Yes

Yes


119879119896larr119879trim

119879119896larr119879MST

119888(119879119896) gt 119888(119879MST )

to generate an MST









0= 119894 119887120595= 119895 119887

119903= 119900119887119903minus1 119895

119903 = 1 120595minus1 (119887



120591 (119894 119895) larr997888sum120595minus1

119897=0120591 (119887119897 119887119897+1)

120595 (10)













119866| is the




testing120583 = 5 6 19 1199020= 02 03 04 05 06 07 08 085


0 120588 120572)


5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)





Θ (120579 119894) = 100120590 minus 119905 (11)









No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 97 11 13 15 17 19120583

100

80

60

40

20

Aver

age v

alue

ofΘ

(a)

02 03 04 05 06 07 08 09 1

100

80

60

40

20

Aver

age v

alue

ofΘ

1199020

(b)

0201 03 04 05

100

80

90

70

60

50

Aver

age v

alue

ofΘ

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

120588

(c)

0201 03 04 05 06 07 08 09 1 11

100

80

90

70

60

50

Aver

age v

alue

ofΘ

120572

B2

B4

B6

B16B11

B9B13B17

B10

B14B18

(d)





Θ (120579 119894) = 100120590 minus 119905 (11)









No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











No |119881119866| |119864

119866| |119881

119872| 119888(119879opt)

1 50 63 9 822 50 63 13 833 50 63 25 1384 50 100 9 595 50 100 13 616 50 100 25 1227 75 94 13 1118 75 94 19 1049 75 94 38 22010 75 150 13 8611 75 150 19 8812 75 150 38 17413 100 125 17 16514 100 125 25 23515 100 125 50 31816 100 200 17 12717 100 200 25 13118 100 200 50 218



0 120588 and 120572 are


0= 08 120588 = 03 and 120572 = 05





119872(1) = 119881

119872(0) minus Δ119881



119872(2) = 119881

119872(1) + Δ119881



ΦV = min2119870

sum119894=1

sum119890isin119864(119894)

119888 (119890)

119879 (119894) = (119881119872 (119894) + 119881120579 (119894) 119864 (119894)) 119894 = 1 2 2119870

119881119872 (119894) = 119881


119881119872 (119894) = 119881

119872 (119894 minus 1) + Δ119881119868 (119894) if 119894 = 2 4 6 2119870

(12)

where 119881120579(119894) sube 119881

119866minus 119881119872(119894) 119864(119894) sube 119864

119866 Δ119881119872(119894) is the set of







(119872)







and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














and 120575(119866) are

120575(119877)

=Φ(119877)V minus Φ(119872)V

Φ(119872)

V

120575(119866)

=Φ(119866)V minus Φ(119872)V

Φ(119872)

V

(13)




119872| minus


002040608

11214

120575(

)

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46



Tim

e (m

s)

RNN

MCMRACO

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46

20406080

100120

GA




No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











No v Φ(119872)V 120575(119877) 120575(119866)

11 16381 133 0004 12570 099 0006 10426 055 000

21 17345 132 0006 17553 188 00110 14399 151 003

31 25656 027 00012 23588 045 00122 17162 041 002

41 10756 178 0014 9243 136 0026 7736 131 000

51 12687 114 0006 10374 133 00310 8320 091 002

61 22206 043 00412 18168 062 00422 13142 069 002

71 22392 064 0026 18322 166 00410 14574 122 006

81 25664 148 0009 22945 116 00216 17236 068 002

91 44687 041 00118 38390 051 00235 26371 040 004

101 16525 111 0116 13361 263 00810 10803 202 019

111 16823 081 0089 15501 190 01916 11748 203 014

121 31642 063 00218 27443 089 00835 19069 079 006

131 30795 190 0258 25326 303 00814 21049 291 011

141 40368 097 01312 33929 145 01822 24475 103 024

151 62009 084 00924 52308 113 01047 35118 081 006

161 19557 171 0258 17001 206 03814 13228 181 026

171 24107 108 01712 18215 118 01522 13003 112 020

181 45115 068 02324 36000 067 02747 24589 069 015



11 469 1742 7424 500 1508 5396 508 1407 500

21 438 1992 8606 461 1860 73410 485 1680 602

31 305 3305 140612 352 2883 122722 422 2344 891

41 508 1688 6104 532 1500 4926 539 1430 430

51 469 2071 8446 500 1805 70310 516 1617 547

61 336 3578 139112 391 2977 118022 445 2391 852

71 1508 2742 13286 1555 2461 112510 1602 2203 891

81 1375 3625 17429 1453 3133 150816 1547 2680 1148

91 914 6805 301618 1094 5648 265635 1305 4391 1829

101 1563 2774 11886 1610 2454 100010 1664 2211 789

111 1438 3586 15869 1516 3180 142216 1586 2696 1071

121 1047 6844 289918 1211 5672 249235 1399 4438 1750

131 4016 4211 22748 3648 3625 196914 3820 3297 1649

141 3110 5985 336012 3305 4984 275022 3540 4094 2016

151 2195 11688 518024 2602 9196 453247 3031 6992 3039

161 3508 4250 21188 3633 3852 186814 3750 3336 1453

171 3164 5844 304712 3375 4774 256322 3578 3946 1836

181 2227 12109 534424 2711 9110 425047 3149 7000 2961



5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusion


Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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