MOLECULAR biologists need to carry out reconciliation

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1 Proeeings of the Interntionl MultiConferene of Engineers n Computer Sientists 3 Vol I,, Mrh 3-5, 3, Hong Kong Distriute Moifie Extreml Optimiztion for Reuing Crossovers in Reonilition Grph Keiihi Tmur, Hjime Kitkmi, n Akihiro Nk Astrt To etermine the mehnism of moleulr evolution, reonilition grph is onstrute from two heterogeneous trees, whih re referre to s orere trees. In the reonilition grph, the lef noes of the two orere trees fe eh other. Furthermore, lef noes with the sme lel nme re onnete to eh other y n ege. To rry out reonilition work effiiently, it is neessry to fin the stte with the minimum numer of rossovers of eges etween lef noes. Reuing rossovers in reonilition grph is omintoril optimiztion prolem. In this pper, we propose novel io-inspire lgorithm lle istriute moifie extreml optimiztion (). This lgorithm is hyri of popultion-se moifie extreml optimiztion () n the istriute geneti lgorithm moel tht is use for reuing rossovers in reonilition grph. We hve evlute using tul t sets. shows etter performne ompre with. Inex Terms extreml optimiztion, istriute geneti lgorithm, evolutionry omputtion, reonilition grph I. INTRODUCTION MOLECULAR iologists nee to rry out reonilition work [], [2], [3], [4] in orer to etermine the mehnism of moleulr evolution. In reonilition work, the reltion etween two heterogeneous phylogeneti trees n the reltion etween phylogeneti tree n txonomi tree re ompre. To ompre two trees, we onstrut grph lle reonilition grph tht onsists of two phylogeneti trees or phylogeneti tree n txonomi tree. Phylogeneti trees n txonomi trees in reonilition grph re referre to s orere trees. The lef noes of these orere trees fe eh other. Moreover, lef noes with the sme lel nme re onnete to eh other y n ege. To rry out reonilition work effiiently, it is neessry to fin the stte with the minimum numer of rossovers of eges etween lef noes in the reonilition grph. In Fig., phylogeneti tree n phylogeneti tree 2 re inferre from ifferent moleulr sequenes with four ientil speies,,, n. The lef noes of phylogeneti tree n those of phylogeneti tree 2 fe eh other. Lef noes representing the sme speies re onnete to eh other. The reonilition grph shown in Fig. () hs two rossovers. If we reue rossovers in the reonilition grph, we n otin the reonilition grph shown in Fig. (), whih hs no rossovers. Reuing rossovers in reonilition grph is omintoril optimiztion prolem. The numer of omintions inreses exponentilly s the numer of lef noes inreses. There re some heuristis [5], [6] tht n e use for reuing rossovers in reonilition grph, n K.Tmur, H.Kitkmi, n A.Nk re with Grute Shool of Informtion Sienes, Hiroshim City University, 3-4-, Ozuk- Higshi, As-Minmi-Ku, Hiroshim Jpn, orresponing e-mil: (ktmur@hiroshim-u..jp). 3 2 phylogeneti tree phylogeneti tree 2 () Reonilition grph with two rossovers. 3 2 phylogeneti tree phylogeneti tree 2 () Reonilition grph with no rossovers. Fig.. Exmples of reonilition grphs (() shows reonilition grph tht hs two rossovers, n () shows reonilition grph tht hs no rossovers). they use geneti lgorithm(ga)[7], extreml optimiztion (EO) [8], [9], [], n moifie EO (MEO)[]. In our previous stuy[2], we propose popultion-se moifie extreml optimiztion (), whih is omintion of popultion-se pproh n MEO. shows etter performne ompre with MEO. However, it is iffiult to mintin iversity t the en of lterntion of genertions. To overome this iffiulty, this pper proposes novel extreml optimiztion moel lle istriute moifie extreml optimiztion () for reuing rossovers in reonilition grph. is hyri of n the istriute geneti lgorithm (DGA) moel [3], [4]. In the DGA moel, we ivie popultion into two or more su-popultions n eh su-popultion evolves iniviully. Therefore, n mintin iversity t the en of lterntion of genertions. The min ontriutions of this stuy re s follows: Distriute moifie extreml optimiztion () is propose. is hyri io-inspire lgorithm tht omine n DGA. Mny stuies [8], [9], [], [5], [6], [7], [8] hve pplie EO to omintoril optimiztion prolems suh s the trveling slesmn prolem, grph prtitioning prolem, n imge rsteriztion. Reently, some stuies [9], [] hve fouse on integrting popultion-se pproh in EO. To the est of our knowlege, there is no stuy on popultion-se EO involving the use of the istriute geneti lgorithm moel. To evlute the propose, we implemente u u v v w w

2 Proeeings of the Interntionl MultiConferene of Engineers n Computer Sientists 3 Vol I,, Mrh 3-5, 3, Hong Kong Fig. 2. v 4 v 24 v v 2 3 v 2 v 5 v v w 2 v 3 v 23 v v 6 v 26 u v 7 v 27 phylogeneti tree phylogeneti tree 2 Prolem efinition. CM = MDEO for reuing rossovers in reonilition grph. Moreover, we evlute using two tul t sets for experiments. Experimentl results shows tht outperforms. The rest of the pper is orgnize s follows. Setion II presents the prolem efinition. Setion III explins. Setion IV proposes. Setion V presents experimentl results, n Setion VI onlues the pper. II. PROBLEM DEFINITION A reonilition grph (RG) onsists of two orere trees, OT = (V,E ) n OT 2 = (V 2,E 2 ), where V n V 2 re finite sets of noes n E n E 2 re finite sets of eges. A noe tht hs no hil noes is lef noe. The lef noe sets of OT n OT 2 re enote y L V n L 2 V 2, respetively. If the numer of speies is n, the numer of lef noes is n. A lef noe hs lel nme, whih is speies nme. The lel nme set is enote y L lef. In the reonilition grph, OT n OT 2 re lote fe to fe. If lef noe of OT hs the sme lel nme s tht of OT 2, then the two lef noes re onnete to eh other. In Fig.2, phylogeneti tree is OT n phylogeneti tree 2 is OT 2. The lef noe set L hs four noes, v 4, v 5, v 6, n v 7. Similrly, L 2 hs four noes, v 24, v 25, v 26, n v 27. There re four lel nmes in L lef,,,, n. Two lef noes v 4 n v 24 re onnete euse they hve the sme lel nme. Let OL n OL 2 e the orer lists of lef noes: OL = [ol,,ol,2,,ol,n ](ol,i L,L(ol,i ) L lef ), OL 2 = [ol 2,,ol 2,2,,ol 2,n ](ol 2,i L 2,L(ol 2,i ) L lef ), where funtion L returns the lel nme of n input noe. The funtion C(M) returns the numer of rossovers: C(M)= m j,β m k,α [ j < k n, α < β n], () where m i,j is (i,j)th-element of the onnetion mtrix M tht is efine s { if L(ol,i ) = L(ol m i,j = 2,j ), (2) otherwise. In Fig. 2, OL is given y OL = [v 4,v 5,v 6,v 7 ]. Similrly, there re four lef noes in phylogeneti tree 2, ol 2, = v 24, ol 2,2 = v 25, ol 2,3 = v 26, n ol 2,4 = v 27. Therefore OL 2 is given y OL 2 = [v 24,v 25,v 26,v 27 ]. Fig. 2 lso shows M. For exmple, the (,)th-element m, is eusel(v 4 ) equlsl(v 24 ). Similrly, the(,)th-element m, is euse L(v 5 ) oes not equl L(v 25 ). The tsk of reuing rossovers in the reonilition grph is efine s follows: min : C(M), sujet to : () M is the onnetion mtrix of the RG, (2) There re no rossovers on eges etween non-lef noes in the RG. There shoul e no rossovers on eges etween non-lef noes in the reonilition grph. For this onstrint, we nee to hnge orer of lef noes y hnging the orer of hil noes in intermeite noes. We nnot hnge the orer etween v 5 n v 7 (Fig. 2) euse it will le to the presene of rossovers on eges etween non-lef noes. If we wnt to hnge the orer etween v 5 n v 7, it is neessry to reple v 5 n v 3, whih re hil noes of v 2. If we reple v 5 n v 3, the numer of rossovers in the reonilition grph eomes zero, n OL is hnge to OL = [v 4,v 6,v 7,v 5 ]. III. POPULATION-BASED MODIFIED EXTREMAL OPTIMIZATION EO[8], [9], [] follows the spirit of the Bk-Sneppen moel, upting vriles tht hve one of the worst vlues in solution n repling them y rnom vlues without ever expliitly improving them. EO ivies n iniviul I into n omponentso i ( i n). Letλ i e the fitness vlue ofo i. First, EO selets O worst, whih hs the worst fitness vlue. Seon, the stte of omponento worst is hnge t rnom. Heneforth, seletion n hnge stte of omponent re repete. The omponent with the worst fitness vlue hs high possiility tht the fitness vlue of it will eome etter y hnging stte. Consequently, the fitness vlue of the iniviul lso gets etter euse the fitness vlue of the omponent with worst fitness vlue gets etter. Moifie EO (MEO) [] genertes two or more neighor iniviuls s nites for the next genertion iniviul. The est neighor iniviul mong the nites is selete s the next genertion iniviul. Moreover, MEO uses roulette seletion to selet omponent. First, MEO selets O selete with roulette seletion. The seletion rtes of roulette seletion re reiprols of fitness vlues with omponents. Seon, MEO genertes new iniviuli from I y hnging the stte of O selete. Thir, the generte I is store into Cnites. Finlly, MEO selets the est iniviul from Cnites. Popultion-se MEO () [2] involves popultion-se pproh. There re two or more iniviuls in popultion. Alterntion of genertion is repetely performe for every iniviul y using MEO. To improve the serh effiieny, iniviuls opy sustruture of n iniviul tht hs goo su-strutures t eh lterntion of genertions. This opertion resemles the rossover opertion in geneti progrming (GP). However, one sie only opies su-struture of nother sie. Copying of goo su-strutures les to high proility of genertion of goo iniviul. As result, effiient serh n e performe y mintining iversity. IV. DISTRIBUTED MODIFIED EXTREMAL OPTIMIZATION This setion explins the min onept of n the lgorithm of for reuing rossovers in reonilition grph.

3 Proeeings of the Interntionl MultiConferene of Engineers n Computer Sientists 3 Vol I,, Mrh 3-5, 3, Hong Kong A. Min Conept is hyri of n DGA moel. ivies the entire popultion into two or more su-popultions. A su-popultion evolves iniviully y. Moreover, from eh su-popultion, some iniviuls re selete n trnsferre to nother su-popultion. In return, the sme numer of migrnts re reeive from nother supopultion. repets the following two steps: () Su-popultions shoul e me to evolve through one or more genertions y using. (2) Some iniviuls of su-popultion re migrte to nother su-popultion. Eh su-popultion evolves iniviully. Eh supopultion onverges to the seprte est solution. Therefore, n mintin iversity t the en of lterntion of genertions. B. Definition of Iniviul n Component A reonilition grph is efine s n iniviul. A omponent of n iniviul is efine s pir of lef noes with the sme lel nme: O i = {ol,i,ol 2,δ(i) } (L(ol,i ) = L(ol 2,δ(i) )). (3) Let ol,i e lef noe of OL n ol 2,δ(i) e lef noe of OL 2. The funtion δ(i) returns the susript numer of n element of OL 2 whose lel nme is the sme s the lel nme of ol,i. To hnge the stte of O i, it is neessry to hnge the orer of hil noes of nestor noes of ol,i or ol 2,δ(i). Here, AS(T,lnme) is set of nestor noes of lef noe in T tht hs the lel nme lnme. The numer of rossovers etween ol,i n ol 2,δ(i) is enote y C(M, i). The following re the efinitions of C(M,i) n the fitness vlue λ i of O i : λ i = C(M) C(M,i), (4) C(M) δ(i) n m l,m i n m l,m C(M,i) = (5) l=i+ m= l= m=δ(i)+ In Fig. 2, there re four omponents, O = {ol,,ol 2, }(= {v 4,v 24 }), O 2 = {ol,2,ol 2,4 }(= {v 5,v 27 }), O 3 = {ol,3,ol 2,2 }(= {v 6,v 25 }), n O 4 = {ol,4,ol 2,3 }(= {v 7,v 26 }), with δ() =, δ(2) = 4, δ(3) = 2, n δ(4) = 3. The fitness vlues of the omponents re λ =, λ 2 = /2, λ 3 = 3/4, n λ 4 = 3/4. C. Algorithm The lgorithm of for reuing rossovers in reonilition grph onsists of two steps: () Evolution Step n (2) Migrtion Step (Algorithm ). First, n initil popultion ivie to p su-popultions (p is the numer of su-popultions). In the Evolution Step (step 5), ll su-popultions re me to evolve through m genertions y using the funtion (SuP i,m) (m is migrtion intervl). In Migrtion Step (step 6), some iniviuls of su-popultion re migrte to nother su-popultion. Finlly, the est iniviul from ll su-popultions (step 7 n step 8). Algorithm : Generte initil popultion P init t rnom. 2: I est BEST(P init ) 3: Divie P init into p su-popultions SuP i. 4: for i = to mx genertions/m o 5: (Evolution Step) For ll su-popultions, supopultion SuP i shoul e me to evolve through m genertions y using the funtion (SuP i,m). 6: (Migrtion Step) For ll su-popultions, migrte some iniviuls of su-popultion to nother supopultion. 7: if F(BEST(SuP SuP p )) > F(I est ) then 8: I est BEST(SuP SuP p ) 9: en if : en for Algorithm 2 (P, m) : for i = to m o 2: for ll I P o 3: Evlute fitness vlue λ i of eh omponent O i of I. 4: C φ 5: n 6: while n < num of nites o 7: Selet O selete y roulette seletion (seletion rtes re the reiprol of fitness vlues with omponents). 8: C C GNI(I,O selete ) 9: n n+ : en while : I BEST(C) 2: en for 3: CSS(P) 4: en for D. Evolution Step In the Evolution Step, ll su-popultions re me to evolve throughmgenertions y using the funtion (Algorithm 2). First, for eh iniviul, the stte of the iniviuls in P is hnge y using MEO. Seon, the funtion CSS opies goo su-struture of n iniviul to nother iniviul. In the MEO steps, for eh iniviul, the following steps re exeute. Initilly, the funtion evlutes the fitness vlue λ i (step 3). Next, the following three steps re repete while n is less thn num of nites. First, omponent O selete in I is selete y using the roulette seletion (step 7). Seon, the funtion genertes n neighor iniviul from I with the funtion GNI. The funtion GNI genertes neighor iniviul y hnging the stte of omponent O selete. Thir, the neighor iniviul is store in C (step 8). Finlly, the est iniviul in C is selete n I is reple y it (step ). The stte of O selete is hnge y hnging the orer of hil noes in n intermeite noe tht is n nestor noe of O selete. The proessing steps of GNI re s follows. First, the outputs of AS(T,L(O selete )) or AS(T 2,L(O selete )) re store in the set Anestors. Then,

4 Proeeings of the Interntionl MultiConferene of Engineers n Computer Sientists 3 Vol I,, Mrh 3-5, 3, Hong Kong TABLE I DATA SETS Txonomi tree Phylogeneti tree Numer of noes Numer of lef noes Numer of noes Numer of lef noes Housekeeping Moss Algorithm 3 CSS(P) : Selet iniviul SI P y roulette seletion (seletion rtes re the fitness vlues of omponents). 2: for ll I P,I SI o 3: for i = to n o 4: Clulte the ifferene iff i etween the fitness vlue of O i in SI n the fitness vlue of O j in I, where O i n O j hve the sme lel nme. 5: en for 6: Selet O selete y roulette seletion (seletion rtes re iff i ). 7: A AS(T,L(O selete )) or AS(T 2,L(O selete )) 8: C φ 9: for ll A o : Generte new iniviul I from I y hnging the orer of hil noes in. : C C I 2: en for 3: I BEST(C) 4: en for noe is selete t rnom from A. Finlly, the orer of the hil noes in is hnge. Suppose tht the selete omponent is O 2 in Fig. 2. The funtion AS(T,L(O 2 )) returns {v 2,v } n AS(T 2,L(L 2 )) returns {v 22,v 2 }. If Anestors = {v 2,v } n v 2 is selete s, the orer of hil noes in v 2 is hnge. In this se, orer of noe v 5 n v 3 re hnge. As result, new iniviul I is otine y the hnge of stte. Algorithm 3 shows the funtion CSS. At the eginning, n iniviul SI in P is selete y roulette seletion (step ). Eh iniviul of P opies su-struture of SI y the following steps. First, the funtion lultes the ifferene iff i etween the fitness vlue of O i of SI n the fitness vlue of O j of I, where O i n O j hve the sme lel nme (steps 3, 4, n 5). Seon, O selete is selete y roulette seletion (step 6). Next, AS(T,L(O selete )) or AS(T 2,L(O selete )) is store in A (step 7). Then, for ll A, new iniviul I is generte from I y hnging the orer of hil noes in, n I is store in C (steps 9,,, n 2). Finlly, the funtion selets the est iniviul from C (step 3). E. Migrtion Step In the Migrtion Step, some iniviuls of supopultion re migrte to nother su-popultion. The istriute geneti lgorithm moel requires numer of su popultions, migrtion rte, migrtion intervl, n migrtion moel. The first three items re usergiven prmeters. The lst item onsists of two things: seletion metho n topology. The metho use for the seletion of iniviuls for migrtion is referre s seletion metho. The struture of the migrtion of iniviuls etween su-popultions is referre s topology. In this stuy, we use uniform rnom seletion s the seletion metho. Moreover, the propose lgorithm uses the rnom ring migrtion topology. The most si migrtion topology is the ring migrtion topology. In this topology, iniviuls re trnsferre etween iretionlly jent su-popultions. In the rnom ring migrtion topology, n rrivl su-popultion to whih iniviuls re to e migrte is eie t rnom. V. PERFORMANCE EVALUATION We performe four experiments for evluting the performne of. In the experiments, the two t sets liste in Tle I re use. The Housekeeping t set onsists of phylogeneti tree of the housekeeping gene n its txonomi tree. The Moss t set onsists of phylogeneti tree of the rps4 gene n its txonomi tree. The numer of speies in the Housekeeping t set is 4 n tht in the Moss t set is 7. Experiment mesure the numer of rossovers of the est iniviul t eh genertion to ompre n. Experiment 2 lso mesure the numer of rossovers of the est iniviul t eh elpse time to ompre n. Experiment 3 mesure frequeny of the numer of rossovers of est iniviuls in fixe genertions. Experiment 4 mesure the numer of rossovers of the est iniviul t eh genertion y hnging the numer of su-popultions. In n, the numer of iniviuls in the popultion ws set to. The user prmeter num of nites ws set to n m ws set to in. In, the user prmeter num of nites, migrtion intervl(m), numer of su popultions(p), n migrtion rte were set to e,, 5 n.5, respetively. The numer of iniviuls in su-popultion is. The numer of rossovers ws the verge of three trils. Experiment In Experiment, we mesure the numer of rossovers of the est iniviul in eh genertion. Figure3() n Figure3() show the numer of rossovers (vertil xis: the numer of rossovers, horizontl xis: genertions). Fig. 3() n Fig. 3() show tht the numer of rossovers of in eh genertion ws smller thn tht in the se of. showe etter performne ompre with. The numer of rossovers in is onverging into roun when we use M oss t set. On the other hn, in, the numer of rossovers is onverging into roun 2. The iversity of is smll, euse the numer of su-popultions is one. Therefore, the fitness

5 Proeeings of the Interntionl MultiConferene of Engineers n Computer Sientists 3 Vol I,, Mrh 3-5, 3, Hong Kong f rossovers Numer of f rossovers Numer of Genertions. Proessing time(se) () Housekeeping t set () Housekeeping t set f rossovers Numer of f rossovers Numer of Genertions Proessing time(se) () Moss t set () Moss t set Fig. 3. Experiment (() n () shows the numer of rossovers of the est iniviul). Fig. 4. Experiment 2 (() n () shows the numer of rossovers of the est iniviul). vlue of iniviul will not e improve in the en of lterntion of genertions. Experiment 2 In Experiment 2, we mesure the numer of rossovers of the est iniviul t ifferent time instnts. The omputtion time of ws longer thn tht in the se of euse the former inlue the Migrtion Step. Therefore, it ws neessry to ompre the numer of rossovers for the sme omputtion time. Fig. 4() n Fig. 4() show the numer of rossovers t ifferent time instnts (vertil xis: the numer of rossovers, horizontl xis: proessing time). The numer of rossovers of eomes smller thn tht of t the en of lterntion of genertions. Experiment 3 The numer of rossovers of the est iniviul ws mesure times for the,th lterntion genertion. Figure5() n Figure5() show frequeny of the numer of rossovers when Housekeeping t set is use. The numer of rossovers of the optiml solution of Housekeeping t set is 9. Both of them n otin the est solution y %. Fig. 6() n Fig. 6() show the frequeny of the numer of rossovers for the Moss t set. In, ll the numers of rossovers of optiml solutions were etween n 299. On the other hn, they were istriute etween n 4 for. Aove ll, lthough 9% of optiml solutions were etween n 249 in the se of, only few optiml solutions were otine etween n 249 in the se of MEO. Experiment 4 In Experiment 4, we hnge the numer of supopultions in. Fig. 7 shows the results of Experiment 4 using Moss t set. When the numer of su-popultins is four, it hs fllen into the lol optiml solutions. On the other hn, when the numer of su-poupltions is five or ten, onvergene is not erly. Therefore, they n otin etter solutions. VI. CONCLUSION This pper proposes istriute moifie extreml optimiztion () for reuing rossovers in reonilition grph. The propose lgorithm is io-inspire lgorithm of popultion-se moifie extreml optimiztion () n the istriute geneti lgorithm moel. We hve evlute y using tul t sets. Experimentl results show tht is etter performne ompre with. In the future work, we will evelop extene for mking it pplile to other omintion optimiztion prolems. ACKNOWLEDGMENT This work ws supporte in prt y Grnt-in-Ai for Young Reserh (B) (No.23724) from the Ministry of Eution, Culture, Sports, Siene n Tehnology in Jpn n Grnt-in-Ai for Sientifi Reserh (C) (2) (No.37) from the Jpnese Soiety for the Promotion of Siene, Jpn. REFERENCES [] M. Goomn, J. Czelusnik, G. Moore, A. Romero-Herrer, n G. Mtsu, Fitting the gene linege into its speies linege, prsimony strtegy illustrte ylogrms onstrute from gloin sequenes, Systemti Zoology, vol. 28, pp , 979.

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