Degree-Constrained Minimum Spanning Tree Problem Using Genetic Algorithm

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1 Degree-Constraned Mnmum Spannng Tree Problem Usng Genetc Algorthm Keke Lu, Zhenxang Chen, Ath Abraham *, Wene Cao and Shan Jng Shandong Provncal Key Laboratory of Network Based Intellgent Computng Unversty of Jnan, Jnan, P. R. Chna * Machne Intellgence Research Labs (MIR Labs), WA, USA * IT For Innovatons, VSB - Techncal Unversty of Ostrava, Czech Republc E-mal: dp_lukk@un.edu.cn, czx@un.edu.cn, ath.abraham@eee.org, @qq.com, ngshan@un.edu.cn Abstract Computer network technology has been growng explosvely and the multcast technology has become a hot Internet research topc. The man goal of multcast routng algorthm s seekng a mnmum cost multcast tree n a gven network, also known as the Stener tree problem, whch s a classcal NP-Complete problem. We measure the multcast capablty of each node through the degree-constrant for each node and dscuss the problem of multcast n the case of degree-constrant, whch has an mportant sgnfcance n the communcaton network. Lmtng the capacty of each node durng the replcaton process of nformaton transmsson can mprove the speed of the network, whch has an mportant sgnfcance n real-tme servce. In ths paper, we solve constraned multcast routng algorthm based on genetc algorthm. The dea s to smulate the Darwnan theory of bologcal evoluton. At the same tme, we mprove the generatng random tree and replace the varaton by the combnaton of the two varatons. On one hand, we mprove the effcency of generatng random tree and on the other hand, we can control the mutaton of dfferent varatons n a more flexble manner. I. INTRODUCTION Multcast s an effectve mechansm supportng for mult-pont communcaton that a communcaton form a sender to multple recpents. In multcast, you can reduce network traffc whch multple recevers at the same tme receved through send one sngle flow of data to multple recevers. Multcast routng algorthm s to dscuss how to fnd a multcast tree an n a relatvely short perod of tme and spendng a relatvely small resources and bandwdth whch connected to the source node and destnaton node. Data flows along the tree path and coped ust at the bfurcaton of the tree and the bandwdth can be shared n the common part of the path, thus savng network resources. Tradtonal multcast routng problem assumes that each node has a multcast functon, then to seek a mnmal cost multcast tree that contans the multcast nodes. Whle n the actual network, the capablty of multcast node s not the case. Many nodes do not support multcast and the ablty of nodes to copy the nformaton s lmted. Some nodes only store and forward functons wthout copy functon. Consderng generalty, we defne the multcast capablty of the network node as degree-constrant. We measure the multcast capablty of each node through the degree-constrant for each node and dscuss the problem of multcast n the case of degree-constrant, whch has an mportant sgnfcance n the communcaton network. Lmtng the capacty of each node durng the replcaton process of nformaton transmsson can mprove the speed of the network, whch has an mportant sgnfcance n real-tme servce. Degree-constraned [] multcast problem can be descrbed as a degree-constrant Stener tree problem, whch s also a NP-Complete problem. When compared wth the unconstraned Stener tree problem, lttle research has done on the degree-constraned Stener tree problem. Several heurstc algorthms were proposed for ths problem: the degree-constraned shortest path heurstc SPH [2] and the degree-constraned kruskal shortest path heurstc algorthm K.SPH [3]. The frst, we study genetc algorthm [4-6] n-depth and grasp the basc dea of genetc algorthms and problem-solvng steps. Fnally, we desgn and realze degree-constraned multcast routng algorthm based on genetc algorthm. The algorthm s based on the genetc evoluton of thnkng, smulatng Darwn's bologcal theory of evoluton. The algorthm uses the node matrx encodng a multcast routng tree. The encodng s smple and easy to mplement usng two-dmensonal array. In ths paper, we ntalze the populaton of ndvduals through random depth-frst algorthm. Usng crossover and mutaton, parent ndvduals produce offsprng, and then the excellent ndvduals are selected to form offsprng s accordng to the survval of the fttest (for the next generaton). In order to ensure convergence of the algorthm, we not only adopted proportonal selecton operator but also retan the best ndvdual strategy. At the same tme, we have mproved the random tree generaton algorthm and mutaton operaton. When we execute the random tree generaton algorthm, we often encounter nvald node, for whch the degree s nvald or consttute a loop. We record the node to avod search nvald node agan when executng random tree generaton next tme. In the mutaton operaton, we have abandoned the orgnal varaton, nstead of usng a combnaton of local varatons and global varatons. In ths way, we can adust the varaton of parameters to obtan the desred varaton effects easly. The codng of the algorthm s smple and solved the dffculty of encodng and decodng of multcast tree problem based on genetc algorthm effectve and can obtan suboptmal solutons quckly. The network model can be regarded as a connected undrected graph G(V, E). Where V s the set of network nodes, E s the set of network lnks. Let sze= V be the number of network nodes, E the number of the network lnks. Edge between node u and v s defned as Edge (u, v) c 202 IEEE 8

2 We defne the cost parameters: C (e), refers to all the costs of the edges n the tree (here, the cost s a generalzed defnton, t can be measured by dstance, channel bandwdth, average traffc, communcaton the overhead, the average queue length, delay and other factors). We defne the node degree constrant: Degcon (v). We defne Deg (v) as the degree of a node n the current multcast tree. Multcast tree T(s,M) s a generated sub-tree of the graph G(V, E). Ths subtree covers the source node s V and all the destnaton nodes M. We defne M as the set of all destnaton nodes: M = d,d 2,,d num, num = M s the number of multcast group. In addton, T (s,m ) nclude the ntermedate nodes n the multcast tree whch not belong to the set M and are not source node. P T (s,d ) s a path from the source node s to destnaton node d M n the multcast tree T. The RanPath T (u,v ) means that a random path from the tree T to the node v n the network topology, node u T. Crossover probablty: P C mutaton probablty: P M The total cost of the tree T (s,m ) s defned as the sum of the costs of all lnks n that tree and can be gven by: C (T (s,m )) C (e) e T (s,m ) The cost of the degree-constrant multcast routng problem may be defned as: MnC (T (s,m )) Deg (v ) Degcon(v ), v T (s,m ) II. GENETIC ALGORITHM The genetc algorthm uses the overall search strategy and optmzaton of the search [7-9], so the calculaton does not depend on the gradent or other auxlary knowledge and only need affect the search drecton of the obectve functon and the correspondng ftness functon. The genetc algorthm provdes a generc framework for solvng complex problems and t does not depend on the specfc areas of the problem and t has a strong robustness. So, genetc algorthm s wdely used n many subects. A. Codng Schemes The ndvdual n the multcast routng [0] problem can be as a spannng tree generated by the graphg (V,E ). In fact, we can dentfy a tree or ndvdual unquely f we determne each edge of the multcast tree. So, we encode ndvdual by node matrx and realze t by a two-dmensonal array. In Table, we defne the connecton matrx: Y n n E (v,v ),E (v,v ) 0,,v V,v V as an ndvdual and the Chromosome codng s llustrated. TABLE : CHROMOSOME CODING v v 2 v 3 v 4 v5 v 6 v v 2 v 3 v 4 v5 v B. Generaton of ntal populaton Generated ntalzaton of the populaton, we use the random depth-frst algorthm to get the degree-constraned spannng tree T (M) and the number of the ntal trees or ndvduals s pop_sze. We get each tree accordng to the followng steps. Frst, we ntalze a multcast tree contans only solated the source node. Then we select a destnaton node d from the set M. Then we generated a random path from the source node s to the destnaton node d. Then we merge the path nto the multcast tree T untl the entre destnaton node has been merged nto the multcast tree T. Among them, we use the random depth-frst search algorthm to generate a random path. Moreover, we mproved the random depth-frst search algorthm. Due to the degree-constrant condtons n the path, we often meet nvald node, whch does not satsfy the degree-constraned condton. Ths moment, we wll records the nvald node down n order to avod search nvald node agan n the next path generaton. In a way, we mproved the effcency of the random path generaton. Random tree generaton algorthm s descrbed as follows: Procedure RandomTreeGenerater() Begn T (s,m ) s//ntal a multcast tree whch only contans a source node s For each d n M do /*Generate a random path whch from the tree T to the destnaton node d by callng the random path generaton functon RandomPathGenerater (d,t (s,m )) */ Generate a random path whch from the tree T to the destnaton node d as RanPath T (u,d )// u T 202 Fourth World Congress on Nature and Bologcally Inspred Computng (NaBIC) 9

3 Merge the path RanPath T (u,d ) to the tree T (s,m ) Random path generaton algorthm s descrbed as follows: Procedure: RandomPathGenerater (d,t (s,m )) Began: Vsted [sze] //Record the node n RanPath T (u,d )0not n the treen the tree. Current d; Vsted [d] //add the destnaton node nto path. Whle Current not nt (s,m ) do Get a node from V-Vsted as Current random; Path [current][d] //merge the Edge(Current,d ) nto the path; // Degrees constrant udgment to fork node If Deg (Current ) Degcon(current ) then Add current nto Vsted //recode the nvald node; Calls RandomPathGenerater(T (s,m )) recursve Compute the degree of node n the path P T (s,d )that s the degree of destnaton node and fork node and, the degree of ntermedate node and 2 Return RanPath T (current,d ) C. Ftness Functon Ftness functon s the standard for genetc algorthm to determne the ndvdual good or bad and smaller the cost s, hgher ts ftness be. It s a mnmum optmzaton problem. Ths problem can be converted nto the problem that seekng the maxmum value of the optmzaton of obectve functon by gets the recprocal of obectve functon. So for each multcast tree T (s,m ), ftness functon can be defned as the recprocal of the total cost of the treet (s,m ): F (T (s,m )) C (T (s,m )) C (e ) e T (s,m ) D. Selecton The genetc algorthm s based on Darwn's natural selecton process manly, so the selecton plays a key role n evoluton. The sample space of the algorthm s unform. At frst, we defne the number of offsprng equal parent group and then we select pop_sze ndvduals, whch have a hgh ftness as the next-generaton groups from the parent and offsprng usng roulette algorthm [6] and n accordance wth the prncple of survval of the fttest. In addton, takng nto account the prncple of the best ndvdual should be retaned, the best ndvdual of the parent coped to the next generaton drectly s not used for crossover wth other ndvdual. E. Crossover The crossover operaton s the core part of the genetc progress. The crossover operaton s the man way to produce ndvduals n the next generaton, t s necessary to passed the good character n the populaton to the next generaton, whle mantan the dversty of ndvduals n the populaton. Through the crossover operaton, the search capablty of genetc algorthm s able to leap to mprove. In ths algorthm, we draw on algorthm thnkng from the lterature [8] to complete the crossover operaton through mergng trees and we set the crossover probablty 0.5. Randomly selected two ndvduals: P F (s,m ), P M (s,m ), produce an offsprng ndvdual P G (s,m ). Frst, each tree T (s,m ) s decomposed nto a set P T (s,d ),d M whch contans path from source node to destnaton node. Then, we select the two cross-obect P F (s,m ) and P M (s,m ). We calculate P F (s,m ) and P M (s,m ) from the source node s to destnaton node d path ftness value F F (s,d ) and the F M (s,d ), we defne: F T (s,d ) c(e ) e P T (s,d ) We select P F (s,m ) or P M (s,m ) randomly as the path of offsprng P G (s,m ) n order to avod premature local convergence of populatons. After determne the path set of the offsprng P G (s,m ), we fnally merge the path and delete rng, constrants udgment. F. Mutaton Mutaton operaton n the evolutonary process s an mportant part. On the one hand, mutaton accelerates the move to the optmal soluton by small local varatons. On the other hand, fresh blood s nected nto the populatons non-stop to mantan the freshness of ndvduals n the populaton and avod fallng nto a local optmum. In ths paper, we use a combnaton of the two varants. The frst mutaton, small-scale varaton occurred n parent ndvduals. Frst, we select the outstandng ndvdual as Fourth World Congress on Nature and Bologcally Inspred Computng (NaBIC)

4 the varaton obect by replacng a small number of the path from destnaton node to the source node to complete the frst mutaton and the probablty s set to 0.2 that s 20% of the parent ndvdual producng offsprng ndvdual. For the second varaton, we use mult-pont mutaton and the mutaton probablty s set to 0.3 that s 30% of the parent ndvdual producng offsprng ndvdual. Accordng to our encodng, we generate a new ndvdual by replace some edge n the chromosome by ts allele and the probablty of the varaton n edge s set to 0.2. After varaton, the varaton of ndvdual T (s,m ) wll be dvded the tree nto a seres of subtrees T S (s,m ),T S (s,m )T 2 S (s,m ) and then merge the n subtree nto the mutated offsprng. Frst of all, each of subtrees delete ts loop and subtree n set of subtrees merged wth other tree untl there s one subtree T S (s,m ) n the set of subtrees. Fnally, we need to last traverse every destnaton nod, determne whether the node have oned the multcast tree T S (s,m ). Because the last process of mergng trees may come across falng of subtree s mergng due to the degree-constrant condtons. For the falure destnaton node n the tree, we then generated path whch from the source node to the destnaton node by the random path algorthm and then merge the path to the multcast tree TS (s,m ). last The frst mutaton operaton s descrbed as follows: Procedure: MutatonProcess_a ( ) Began Intalze the destnaton node subset of M '; // the ndvdual of T (s,m )or mutaton. Select the outstandng ndvdual T (s,m ) usng the roulette wheel; Select a leaf node meanwhle a destnaton node d T (s,m ) randomly and record the path P T (s,d ); Node d ons destnaton node subset of M '. Travers node n the path P T (s,d ) from the node d n reverse order, untl the node v s fork node and record the path P T (v,d ). At the same tme, determne whether the node v s destnaton node, f so, and then add the node v nto M '. For each d n M' do /*Call the random path generaton RandomPathGenerater( T (s,m ) ) to generate the path from the source node s to destnaton node d. */ Generate a random path from the source node s to destnaton node d the Path(s,d )randomly; Travers node n the path P T (s,d )from destnaton node n reverse order, untl the node already belongs to the tree T (s,m ) and then the merge Path(s,d ) nto the treet (s,m ); The second mutaton operaton s descrbed as follows: Procedure: MutatonProcess_b () Began: T (s,m ); // ndvdual T (s,m ) for varaton. Mutate T (s,m ), and thus obtaned a seres of subtree T s (s,m ); Subtree whch contans source and destnaton nodes get rd of loop and then prune the subtree. At the same tme, we should mark these subtrees that s mark subtree whch contans the source node: Mark T s (s,m ) = and subtree whch contans the destnaton node subtree labeled as 2,3 n turn. Compute the degree of node n the entre subtree T s (s,m ); Flter out nvald subtree not contanng the source node or destnaton node and then get the set of subtree fnally: subtree T S (s,m ),T S (s,m )...T 2 S (s,m ); n // Start to merge subtree n the set of subtree and udge degree-constrant Choose Ts (, s M ) from subtree whch Mark T s (s,m ) = 2; Whle the number of the set of subtree s greater than do Select another subtree T s (s,m ) from the set of subtree randomly; If Mark T s (s,m ) Mark T s (s,m ) then Generate the random path Path(T s (s,m ),T s (s,m )) whch connects the tree of Mark T s (s,m ) to the tree Mark T s (s,m ). If all the node n Path(T s (s,m ),T s (s,m )) satsfy the degree-constraned condton then Merger tree Ts (, s M ) nto Ts (, s M ) ; Modfy, Mark T s (s,m ) = Mark T s (s,m ) ; Delete tree T (, s M ) from the set of subtree; s 202 Fourth World Congress on Nature and Bologcally Inspred Computng (NaBIC)

5 Generate the random path Path(T s (s,m ),T s (s,m )), whch connects the tree of Mark T s (s,m ) to the tree Mark T s (s,m ). If all the node n Path(T s (s,m ),T s (s,m )) satsfy the degree-constraned condton then Merger tree T s (s,m ) ntot s (s,m ); Modfy, Mark T s (s,m ) = Mark T s (s,m ) ; Delete tree T s (s,m ) from the set of subtree; Remove the last tree of T slast (s,m ) from the set of subtree; Whle there are stll the purpose of node, d_, s not oned T_ (s_last) (s, M) do Generate Path(s,d ) from the source node s to the destnaton node d randomly; If all the node n the path Path(s,d ) satsfy the degree constrant then Merge the path Path(s,d ) to the tree oft slast (s,m ); Calls the functon whch generate random Path(s,d ) from the source node s to destnaton node d recursve; G. Condton Ths algorthm, we set the termnaton condton for the generc: the varance of the best ndvdual s ftness value n last N group less than ther mathematcal expectaton of 0.0 tmes. The value of N should be set dynamcally wth the constant changes of network sze. When ths condton s reached, the group has evolved to a statonary state; the optmal ndvdual s close to the global optmal soluton most. Accordng to the expermental results, we can get more desrable results when the algorthm reaches a convergence state. 2Deg (v ) two algorthms, we set the value of Degcon(v) 3 that s lmtng the degree-constrant of node s 2/3 tmes as lager as the value of the actual degree of the nodes n the random network topology. Meanwhle we set Degcon (v) greater than or equal to. Populaton sze of pop_sze and termnaton condtons of N s set wth the network sze dynamcally. Fgure shows the cost of best ndvdual n each generaton changes n lne chart. We set the number of nodes s 40 and the destnaton nodes s 8. The genetc manpulaton acheves convergence after 6 generaton. From the fgure, we can fnd that the cost of the best ndvdual n each generaton decreasng untl a stable value, namely, the state of convergence. Fgure. Cost of best ndvdual n each generaton n GA and SPH Fgure 2 depcts the cost of best ndvdual n the generaton under the comparson between the genetc algorthm and the SPH algorthm n dfferent network sze when the state s convergence. We use the average value of the 0 costs of best ndvdual n the generaton generated by 0 tmes of executon of algorthm. We set the number of the destnaton nodes s 20% of the number of the all nodes n dfferent network sze. We can see the cost of the optmal tree generated by the genetc algorthm s always less than the SPH algorthm under dfferent network sze. And the larger the random network sze s, the more obvous advantages of genetc algorthms are. III. SIMULATION RESULTS In ths paper, we smulate our genetc algorthm n MRSIM. We realzed the degree-constrant SPH algorthm n order to analyze the results by comparng genetc algorthm wth SPH algorthm. Conductng comparatve analyss, In order to unfed smulaton envronment of the Fourth World Congress on Nature and Bologcally Inspred Computng (NaBIC)

6 Fgure 2. Cost of best ndvdual n the generaton of genetc algorthm and the SPH algorthm n dfferent network sze Fgure 3 llustrates the convergence tme of comparson between the genetc algorthm and the SPH algorthm under dfferent number of nodes. We set the number of the nodes s 20% of the number of the all nodes n dfferent network sze. We smulate 0 tmes n each network sze and calculate the average tme. We can see as the network sze ncreases, the convergence tme also ncreases. Ths s because as the number of node becomes large, the search space becomes larger and we need to enlarge the number of teratons n order to obtan better multcast tree and for the convergence tme grows too. Whle, the convergence tme of SPH algorthm s much faster than the genetc algorthm. In terms of convergence rate, the SPH s better than the genetc algorthm. Fgure 3. Impact of genetc algorthm and the SPH algorthm n convergence tme Fgure 4 shows the number of generatons when the state acheves convergence n the dfferent network sze. We set the number of the nodes s 20% of the number of the all nodes n dfferent network sze. We smulate 0 tmes n each network sze and calculate the average tme. We can see, see as the network sze ncreases, the number of generatons when the state acheves convergence ncreases. Ths s because as the network sze ncreasng, result n the search space larger and the search harder. Fgure 4. Number of generatons when the state acheves convergence n the dfferent network sze. IV. CONCLUSIONS In ths paper, we proposed a degree-constraned multcast routng algorthm based on genetc algorthm. The algorthm s based on Darwn's genetc theory of evoluton, accordng to the natural law of survval of the fttest. Frst, ths paper mproved random path generaton algorthm, ncreasng tree generaton effcency. Second, the paper uses a combnaton of two varants to better smulate the varaton process n order to adust the varaton parameter expedently. Fnally, we realzed our algorthm n MRSIM and then evaluate the smulaton results. Not the same as wth ordnary heurstc search algorthm, genetc algorthm s a stochastc global search algorthm whch focusng on achevng global parallel search wth a large search space and adust the search drecton n order to fnd to the optmal soluton or quas-optmal soluton through the search process. General heurstc algorthms are poor search algorthms or local search algorthms mostly. Such as the SPH algorthm, t ust concentrated n the shortest edge whle not consder the overall stuaton. From the smulaton results, the results of multcast routng algorthm based on genetc algorthm obtaned usually better than common heurstc algorthms such as the SPH algorthm. Especally the larger network sze, our algorthm tends to fnd a multcast tree wth a lower cost than the average heurstc algorthm. However, the genetc algorthm also has ts own shortcomngs. In the case of convergence tme, genetc algorthm needs a longer tme. Because the genetc algorthm s global search, whle the SPH algorthm s only concentrated n the shortest edge. From the expermental results, ths convergence tme of the algorthm ncreases but the performance decreased wth the ncreasng n network sze and nodes. ACKNOWLEDGEMENTS Ths work was supported by the Natonal Natural Scence Foundaton of Chna No and No , the Program for New Century Excellent Talents n Unversty No.NCET , the Scence and 202 Fourth World Congress on Nature and Bologcally Inspred Computng (NaBIC) 3

7 Technology Development Program of Shandong Provnce No.20GGX06, the Natural Scence Foundaton of Shandong Provnce No.ZR200FQ028 and No.ZR20FL02, the Program for Youth and mld-lfe scentst s award fund n Shandong provnce under Grant No.BS2009DX037 and the Program for Youth scence and technology star fund of Jnan No.TNK08. REFERENCES [] Yng Lu, You-an Zhao,Jan-png Wu. Degree-Constraned Multcastng Algorthm [J]. Journal of Chnese Computer Systems, 2004, vol. 3: [2] K. Vk, P. Halvorsen, C. Grwodz.Evaluatng Stener-tree heurstcs and dameter varatons for applcaton layer multcast Orgnal Research Artcle.Computer Networks,2008, Vol52(5): [3] Jn Zhang, Lang Ma, Lantang Zhang. Algorthms for degree-constraned Eucldean Stener mnmal tree[j]. Journal of Systems Engneerng and Electroncs, 2008, Vol 9(4): [4] Mara Angelova, K. Atanassov, T. Pencheva. Purposeful model parameters geness n smple genetc algorthms[j]. Computers & Mathematcs wth Applcatons, 202, Vol 64(3): [5] A Sadrzadeh. A genetc algorthm wth the heurstc procedure to solve the mult-lne layout problem Orgnal Research Artcle[J].Computers & Industral Engneerng, 202, Vol 62(4): [6] W. Yang, Felx T.S. Chan, V. Kumar. Optmzng replenshment polces usng Genetc Algorthm for sngle-warehouse mult-retaler system[j]. Expert Systems wth Applcatons, 202, Vol 39(3): [7] B. Ramkumar, M. P. Schoen, Feng Ln. Hybrd enhanced contnuous tabu search and genetc algorthm for parameter estmaton n colored nose envronments[j]. Expert Systems wth Applcatons, 20, Vol 38(4): [8] Yun Wen, Hua Xu, Jadong Yang. A heurstc-based hybrd genetc-varable neghborhood search algorthm for task schedulng n heterogeneous multprocessor system[j]. Informaton Scences, 20, Vol 8(3): [9] M.J. Abedn, M. Nasser, D.H. Burn. The use of a genetc algorthm-based search strategy n geostatstcs: applcaton to a set of ansotropc pezometrc head data[j].computers & Geoscences, 202, Vol 4: [0] F. A. Samman, T. Hollsten, M. Glesner. Planar adaptve network-on-chp supportng deadlock-free and effcent tree-based multcast routng method[j]. Mcroprocessors and Mcrosystems, 202, Vol 36(6): Fourth World Congress on Nature and Bologcally Inspred Computng (NaBIC)