İstabul Tcaret Üverstes Fe Blmler Dergs Yıl: 5 Sayı:0 Güz 006/ s. - OPTIMAL KNOWLEDGE FLOW ON THE INTERNET Bura ORDİN *, Urfat NURİYEV ** ABSTRACT The flow roblem ad the mmum sag tree roblem are both fudametal oeratoal research ad comuter scece. We are cocered wth a ew roblem whch s a combato of mamum flow ad mmum sag tree roblems. The aled terretato of the eressed roblem s to corresod a otmal owledge flow o the teret. Although there are olyomal algorthms for the mamum flow roblem ad the mmum sag tree roblem, the defed roblem s NP-Comlete. It s show that the otmal soluto of the roblem corresods a equlbrum state subroblem whch s a aulary roblem of Cuttg Agle Method solvg of the Global Otmzato Problems ad the develoed algorthms for solvg of the subroblem could be used to solve the eressg roblem. Keywords: Otmal Kowledge Flow, Mamum Flow Problem, Mmum Sag Tree Problem, Cuttg Agle Method, Global Otmzato İNTERNET ÜZERİNDE OPTİMAL BİLGİ AKIŞI ÖZET Aış ve Mmum Kasaya Ağaç roblemler Yöeylem Araştırması da ve Blgsayar Blmler de arşılaşıla temel roblemlerdedr. Yaıla çalışmada, masmum aış roblem ve mmum asaya ağaç roblem bleşm şelde ele alıablece ye br roblem celemştr. İfade edle roblemle, blg aışıı olduğu teret ortamıda arşılaşılmatadır. Masmum Aış Problem ve Mmum Kasaya Ağaç roblem ç olom zamada çözüm vere algortmalar bulumasıa rağme taımlaa roblem NP-Tam sııftadır. Problem otmal çözümü, Global Otmzasyo roblemler geş br sııfıı çözümüde arşılaşıla Yardımcı Altroblem çözümüde dege durumua arşı gelmetedr. Gösterlmştr, Yardımcı Alt roblem çözümü ç gelştrle algortmalar, bu çalışmada celee robleme de uyarlaablr. Aahtar Kelmeler: Otmal Blg Aışı, Masmum Aış Problem, Mmum Kasaya Ağaç Problem, Kese Açılar Yötem, Global Otmzasyo. * Ege ÜverstesFe Faültes, Matemat Bölümü,Blgsayar Blmler AB, Borova, İzmr, bura.ord@ege.edu.tr * * Ege ÜverstesFe Faültes, Matemat Bölümü,Blgsayar Blmler AB, Borova, İzmr, urfat.uryev@ege.edu.tr
Bura ORDİN, Urfat NURİYEV. INTRODUCTION A lot of roblems comuter scece ad ecoomy are terreted as usg of the mamum flow roblem ad the mmum sag tree roblem. Mamum flow roblem s to determe a least cost shmet of a commodty through a etwor G = (V, E) whch s a grah wth V vertces ad E edges order to satsfy demads at certa odes from avalable sules at other odes. As for the mmum sag tree roblem that s defed as follows: fd a acyclc subset T of E that coects all of the vertces the grah ad whose total weght s mmzed, where the total weght s gve by w(t) = sum of w(u,v) over all (u,v) T, where w(u,v) s the weght o the edge (u,v).t s called the sag tree (Paadmtrou, 98). I ths aer, t s studed a combato of above roblems. That s, we have a source ot, total flow ad we wat to fd ss m ots ad flow (caacty) for each le as dfferet from mamum flow roblem where s gve source ots, ss, the caactes of the edges for a grah G=(V, E) ad t s wated to be determed mamum flow. Also, the roblem s thought as a mmum sag roblem where the urose s to cover all ots by a chose tree havg a mmal total weght, whe we determe flows each le. The aled terretato of the defed roblem s to corresod that: a owledge s to be carred from oe ma comuter to servers ad from the servers to m clets ( m ). The rocessg must be esured the followg codtos: The sedg tme s mmum. All servers are satsfed. Although there are olyomal algorthms for the mamum flow roblem ad the mmum sag tree roblem, the eressg roblem s NP-Comlete. It s show that the otmal soluto of the roblem corresods a equlbrum state subroblem whch s a aulary roblem of Cuttg Agle Method solvg of the Global Otmzato Problems. Also, a combatoral otmzato roblem that equals to the subroblem s used to solve the eressed roblem.. A GLOBAL OPTIMIZATION TECHNIQUE The Cuttg Agle Method (CAM) has bee develoed as global otmzato techque sce 997. I ths method the orgal global otmzato roblem s reduced to a sequece of subroblems where the obectve fucto s the mamum of secal m-tye fuctos (Adramoov et.al., 999). Ths method s a teratve oe requrg the soluto of a subroblem (mmzg fuctos f(), defed o the set S={ : =, 0, =,, }, where = (,, ) ), =
İstabul Tcaret Üverstes Fe Blmler Dergs Güz 006/ whch s, geerally, a global otmzato roblem. Comutatoal effcecy of the CAM method s sgfcatly affected by the effcecy of solvg the subroblem, whch s solved at each terato. The formulato of the Subroblem could be gve as follow. Let ( l ) be a (m ) matr, m, wth m rows l, =,, m, ad colums, =,...,. All elemets l are oegatve ad the frst rows of ( l ) matr form a dagoal matr,.e., l >0, oly for =, =,,. Itroduce the fucto h()= ma m I ( l l, where I( l )={: l >0}. ) Ad the subroblem Mmze h() (.) subect to S={: = =, 0, =,,.} (.) 3. FLOW BALANCING PROBLEM WITH MINIMUM SPANNING TREE Let us tae a grah G=(V, E) whch s drected ad weghted. t corresods to a l as follow for =,,,; =,,, +,, m. matr ( ) Accordg to the colums, the elemets of the matr ( ) (amely, last rows, =m-) are sorted le (3.). l for =+, +,...,m l l... l, =,, (3.) Here, the set of the vertces, V s defed as V = { 0,,...,, +, +,..., m}, V = m +, Verte 0 whch has degree s the root of the buldg a sag tree. I the other words, each of vertces,, coects verte 0 wth oe edge. The frst vertces (,,, ) corresod to the frst rows of the matr ad remag vertces (+, +,,m) corresod to the last rows of the matr. 3
Bura ORDİN, Urfat NURİYEV The set of the edges E s defed as E = { ( 0,), ( 0, ),...,( 0, ) ; (, ), (, ),..., (, ) ; (, ), (, ),..., (, );... ( ), (, ),..., (, ) ; ( ), (, ),...,(, ) },, I E, the edges the frst row coect verte 0 wth vertces,,,. The edges secod row o for = verte wth, wth ad fally wth accordg to the le (3.). The edges thrd row o for = verte wth, wth ad at last wth, accordg to the le (3.). At the ed, the edges the last row coect for = verte wth, wth ad fally wth. Clearly, there are edges row ad there are edges each other rows. The total umber of edges s +=(+)=[(m-)+]=m- +. 0, (=,...,) has t, t are equal to ( ) t l l t. Here, we assume that 0 =, (=,...,; t=,...,). The verte 0 that s the tal ot s the source (ma comuter) ad at the source ot, the flow whch esures codtos (.)-(.) s dvded to eces (servers): + +... + = (a corresods to each le (0,) (=,,)). Each edge (,) l weght ad the weghts of the edges ( ) By choosg s, ss are determed amog,,...,m vertces. The there are (m-) clets. I cocluso, there s a flow roblem whch has a source verte ad ss. Whe, s determed the weghts of the edges are l. for edges ( 0, ) (=,,,) t t ad ( l l ). for edges ( t, t ) ( =,,..., ; t =,,..., ). Chose tree for the grah determed by ew edges s the Mmal Weghted Sag tree. I other words, we have a ma comuter, servers ad (m-) clets. Also, we ow arrval costs from servers to each clet. The the roblem s to determe the best servers for clets. Flow Balacg Problem wth Mmum Sag Tree equals to subroblem ad the develoed methods for subroblem could be used to solve Flow Balacg Problem wth Mmum Sag Tree. 4
İstabul Tcaret Üverstes Fe Blmler Dergs Güz 006/ 4. TRANSFORMATION OF THE SUBPROBLEM TO AN EQUIVALENT PROBLEM The followg otato s used for smlcty, (Nuryev ad Ord, 003; Nuryev ad Ord, 004). = m, u =, =,,, ; =,,,. l l + Clearly, u s the cremet of the deomator of the fracto that eresses the fucto h the substtuto l + l. Let us defe the followg fucto: ( ) Sg :, f 0, = ad cosder varables 0, f < 0 +, f the substtuto l l s accomlshed = 0, otherwse, =,,, ; =,,, So the Subroblem (.)-(.) s trasformed to the followg Boolea (0 ) rogrammg roblem : = = = = u m (4.), =,,,, (4.), =,,,, (4.3) = = =, (4.4) y, =,,,, (4.5) = 0, =,,,; =,,,, (4.6) ( ma{ } ) =, y = Sg u u, =,,, ; =,,,, (4.7) 5
Bura ORDİN, Urfat NURİYEV THEOREM 4.. The Subroblem (.)-(.) ad the roblem (4.)-(4.7) are equvalet, (Nuryev, 005). Let us call the roblem (4.)-(4.7) as Domatg Subset wth Mmal Weght (DSMW). The roblem ca be terreted as follows: Let ( u ) be a matr, wth rows, =,,, ad colums, =,,, ad oegatve u for all,. 6 The tas s to choose some elemets of the matr such that: ) The sum of the chose elemets s mmal, ) Each row cotas a chose elemet, or cotas some elemet whch s less tha some chose elemet located ts colum. It s gve the followg aled terretato of ths roblem: A tas cosstg of ( =,,, ) oeratos ca be accomlshed by ( =,,,) rocessors. Suose that the matr ( u ) gves the tme ecessary for accomlshmet of the tas as follows: If u... u u (4.8) for colum, the u s the tme (or cost) for the accomlshmet of oerato by rocessor ; u s the tme for the accomlshmet of oeratos ad by rocessor, ad so o. At last u s the tme for the accomlshmet of all oeratos (,,..., ) by rocessor. The roblem s to dstrbute oeratos amog the rocessors mmzg the total tme (or the total cost) requred for the accomlshmet of all tass. Clearly, the roblem s geeralzed of the Assgmet roblem. Although the assgmet roblem ca be solved by Hugara method at a comlety of O(r 3 ) (r = ma{, }), Paadmtrou(98). THEOREM 4.. DSMW roblem s NP-Comlete, Nuryev et al (003). There are good heurstcs for solvg of the DSMW roblem Nuryev et al (004). EXAMPLE 4.. Let us tae a subroblem matr that s dagoal. 0 0 4 ( l ) = 3 8 5 6 4 7 Here, =, m=5 ad =3. For the matr, codto (3.) s determed as the followg:
İstabul Tcaret Üverstes Fe Blmler Dergs Güz 006/ l < 3 5 4 < l l, l < 4 5 3 < l l (4.9) Namely, = 3, = 5, 4 ad = 4, = 5, 3. For the above 3 = 3 = codtos, the corresodg grah s tae as follows. 0 4 3-= 6-4= 3 4 4-3= 8-7= 5-4= 7-6= 5 Fgure 4.. The Grah Notato of the Problem 7
Bura ORDİN, Urfat NURİYEV For the soluto of the roblem, subroblem matr s trasformed to Domatg Subset wth the Mmal Weght (DSMW) roblem matr by the eressed way. ( u 0.6 ) = 0.30 0.5 0. 0.08 0.0 The, DSMW roblem for the matr s solved by the method roosed Nuryev et al (004) ad for ths we tae = ad = 0,,. The we have the followg result. 6 ( ) = = 0 + 8 h, the values of : 6 8 6 = : =, = : 8 =. 0 0 0 0 8 A matr s tae as follows for =, = 0 0 8 6 = 0 0 0 8 4 3 = 0 0 8 40 5 = 0 0 8 3 4 = 0 0 0 4 0 8 0 6 0 7 0 8 = 0 6 = 0 = 0 4 = 0 The corresodg grah for the above matr s as follows: 8
İstabul Tcaret Üverstes Fe Blmler Dergs Güz 006/ 0 6/0 8/0 4/0-6/0=8/0 /0-0=4/0 3 3/0-4/0=8/0 6/0-4/0=/0 4/0-/0=/0 5 4 40/0-3/0=8/0 Fgure 4.. The Eressg of the Problem as Subroblem That s, verte s chose for = ad verte 3 s chose for =. I other words, flow s from 0 to ad from 0 to, from to 4, from 4 to 5, from 5 to 3. For the grah, The Mmal weghted sag tree s tae as: 0 6/0 8/0 3 4/0 /0 5 /0 4 Fgure 4.3. The Notato of Solvg of the Problem as Tree Grah 9
Bura ORDİN, Urfat NURİYEV NOTE 4.. I ths eamle, small values of m ad are tae for smlcty. Besdes, for grahs, we use the followg descrto : Source Pot (Ma Comuter), : Ma Dstrbuto Pots (Servers) : Pots (Clets) 5. CONCLUSION I ths wor t s eressed a ew roblem whch s a combato of flow roblem ad the mmum sag tree roblem. The roblem s to corresod a otmal owledge flow o the teret. Although there are olyomal algorthms for the mamum flow roblem ad the mmum sag tree roblem, the roblem s NP-Comlete. The otmal soluto of the roblem corresods a equlbrum state subroblem whch s a aulary roblem of CAM solvg of the Global Otmzato Problems ad the algorthms that are develoed for solvg of the DSMW roblem could be used effectvely for solvg of the eressed roblem. 6. REFERENCES Adramoov, M.Y., Rubov, A.M. ad Glover, B.M., (999), Cuttg Agle Methods Global Otmzato, Aled Mathematcs Letters,, 95-00. Babayev, D.A., (000), A Eact Method for Solvg the Subroblem of the Cuttg Agle Method of Global Otmzato, I Boo "Otmzato ad Related Tocs", I Kluwer Academc Publshers, Ser. "Aled Otmzato", Dordrecht, 47, 5 6. Nuryev, U.G., (005), A Aroach to the Subroblem of the Cuttg Agle Method of Global Otmzato, Joural of Global Otmzato, 3, 005, 353-370. Nuryev U.G. ad Ord B., (004), Comutg Near-Otmal Solutos for the Domatg Subset wth Mmal Weght Problem, Iteratoal Joural of Comuter Mathematcs, 8,. 0
İstabul Tcaret Üverstes Fe Blmler Dergs Güz 006/ Nuryev U.G. ad Ord B., (003), Aalyss of the Comlety of the Domatg Subset wth the Mmal Weght Problem, EURO / INFORMS Cofereces, July, 003, Istabul, Turey. Ord B., (004), O the Subroblem of the Cuttg Agle Method of Global Otmzato, The Euro Summer Isttute-ESI XXII Otmzato ad Data Mg, July 9-5, 004, Aara, Turey. Paadmtrou, C. H., Stegltz, K., (98), Combatoral Otmzato: Algorthms ad Comlety, Pretce Hall, 496.