Task Assignment Problem Solved by Continuous Hopfield Network

Transcription

1 IJSI Ieraioal Joural of ompuer Siee Issues, Vol. 9, Issue, o, arh ISS (Olie: Task Assigme Problem Solved by oiuous Hopfield ework ETTAOUIL ohamed, LOQA hakir, HAI Youssef 3 ad HADDOUH Khalid 4,3,4 UFR: Sieifi ompuig ad ompuer siees, Egieerig siees, odelig ad Sieifi ompuig Laboraory, Fauly of Siee ad Tehology. Uiversiy Sidi ohamed Be Abdellah, Bo, Fez, oroo Deparme of ompuer Egieerig, High Shool of ehology, oulay Ismail Uiversiy, B. P. 33, 5, Toulal, ekes, oroo Absra The ask assigme problem wih o uiform ommuiaio oss (TAP osiss i fidig a assigme of he asks o he proessors suh ha he oal eeuio ad ommuiaio oss is miimized. This problem is aurally formulaed as - quadrai programmig subje o liear osrais (QP. I his paper, we propose a ew approah o solve he ask assigme problem wih o uiform ommuiaio oss usig he oiuous Hopfield ework (H. This approah is based o some eergy or Lyapuov fuio, whih dimiishes as he sysem develops uil a loal miimum value is obaied. We show ha his approah is able o deermie a good soluio for his problem. Fially, some ompuaioal eperimes solvig he ask assigme problem wih o-uiform ommuiaio oss are show. Keywords: ombiaorial opimizaio, oiuous Hopfield ework, uliproessor sysems, Quadrai - programmig, Task assigme problem.. Iroduio The ask assigme problem play a vial role i a ompuaio sysem wih a umber of disribued proessors, where a se of asks mus be assiged o a se of proessors miimizig he sum of eeuio oss ad ommuiaio oss bewee asks. This problem has bee proved o be a P-hard problem [6]. Several varias of he ask alloaio problem have bee osidered, wih differe arhieure of disribued sysem or i sruure of oss []- []-[3]-[3]-[]. They are basially divided io hree aegories: The graph heoreial miimizes he oal ierproessor ommuiaio os by performig a pariioig algorihm o he graph suh ha, eah pariio iludes a se of asks, whih are assiged o speified sigle proessor []. Ieger programmig usig olum geeraio or brah-ad-boud ehiques a be used o solve he problem more effiiely [5]-[6]. ea-heurisi ivolvig geei algorihm [8]-[5] ad simulaed aealig [9]-[5] have bee used o derive approimae soluios wih reasoable ime [4] hey are appliable o larger dimesioal problems. The ask assigme problem wih o-uiform ommuiaio oss a be modeled as - quadrai programmig whih osiss i miimizig a quadrai fuio subje o liear osrais (QP. To solve he QP problem, may differe mehods are ried ad esed suh as ierior poi, semi defiie relaaios ad lagragia relaaios [8]-[9]. I his paper, we irodue a ew approah usig he oiuous Hopfield ework for solvig he QP problem. Hopfield eural ework was irodued by Hopfield ad Tak [6]-[8]. I was firs applied o solve ombiaorial opimizaio problems. I has bee eesively sudied, developed ad has foud may appliaios i may areas, suh as paer reogiio, model ideifiaio, ad opimizaio. I has also demosraed apabiliy of fidig soluios o diffiul opimizaio problems []. The ieresig seps for his mehod are o defie he geeralized eergy fuio for solvig ay ombiaorial problem ad o deermie he parameers seig of H [4]-[]. I his paper, our mai objeive is o propose a ew mehod o solve he TAP usig he oiuous Hopfield ework. This paper is orgaized as follows. I seio, we provide a formulaio of Task Assigme Problem wih o uiform ommuiaio os as a - quadrai programmig (QP. I seio 3, we desribe he mos ieresig seps for solvig he QP problem usig he opyrigh ( Ieraioal Joural of ompuer Siee Issues. All Righs Reserved.

2 IJSI Ieraioal Joural of ompuer Siee Issues, Vol. 9, Issue, o, arh ISS (Olie: oiuous Hopfield ework. The eperimeal resuls are preseed i seio 4.. Problem Formulaio The ask assigme problem wih o uiform ommuiaio oss osiss i fidig a assigme of asks o proessors suh ha he oal eeuio ad ommuiaio oss is miimized. This problem is saed as a wo ses ad wo parameers where: T T,..., T a se of asks. P P,..., P a se of proessors. The eeuio os e of ask i if is assiged o proessor k. The ommuiaio os jl bewee wo differe asks i ad j if hey are respeively assiged o proessors k ad l. I he followig, we wa o prese a formulaio of he ask assigme problem as a - quadrai programmig [7]. For eah ask i {,.., }, we irodue biary variables, k {,.., }, suh ha: If he ask i is assiged o proessor k Oherwise This mari is overed o a -veor: =,,...,,,,...,,...,,,..., wih = Eah ask should be assiged o ealy oe proessor: = i {,..., } The liear osrais a be rewrie as : =, i =,..., A = b The mari A IR wih ad he veor b IR of he liear osrais are: T The mai objeive is o miimize he oal eeuio ad ommuiaio oss iurred by he ask assigme subje o resoures osrai. The, we a defie he objeive fuio F ( i he followig way: F ( jl jl i k j il i k e Where is mari wih he geeral erm is deoed by jl. The firs ad seod erms i he objeive fuio represe he oal eeuio os ad ommuiaio os, respeively, iurred by he assigme (,..,,..,,..,. = Fially, we obai he followig - quadrai program (QP wih a quadrai fuio subje o liear osrais represeig he TAP problem wih variables ad liear osrais: ( QP i Subje o {,} f ( = e A = b Therefore, is mari wih he geeral erm is deoed by jl, A is mari ad b IR. Wihou los of geeraliy, we a suppose ha is symmeri ad also ha diagoal erms of are equal o. If his mari is o symmeri, i a be overed o he symmeri form ad he liear erms a be subsiued for he diagoal erms, beause = for {,}. The, we obai he followig QP problem: ( QP i Subje o f ( = {,} A = b e Q e A =, b = where Q=. Alhough he QP problem is P-hard [4]-[6], some speial isaes are polyomial ime solvable. Those isaes are solvable i O ( ime i heerogeeous opyrigh ( Ieraioal Joural of ompuer Siee Issues. All Righs Reserved.

3 IJSI Ieraioal Joural of ompuer Siee Issues, Vol. 9, Issue, o, arh ISS (Olie: eworks [4]. I his work, our objeive is o solve he ask assigme problems usig he oiuous Hopfield eworks. The, i his ase, he mos impora sep osiss of represeig or mappig he TAP i he form of he eergy fuio assoiaed wih he oiuous Hopfield eworks. Aordig o he QP model, we defie he assoiaed eergy fuio ad he parameers seig. Therefore, he oiuous Hopfield eworks a be used o solve he ask assigme problem. 3. Task Assigme Problem solved by H Hopfield eural ework was irodued by Hopfield ad Tak [6]-[7]. I was firs applied o solve ombiaorial opimizaio problems. As a be oied, afer modelig he ask assigme problem io a - quadrai programmig wih a quadrai fuio subje o liear osrais, we prese a geeral mehod for solvig he TAP problems usig he oiuous Hopfield eworks. 3. Usig he oiuous Hopfield ework o solve QP problem The oiuous Hopfield eural ework is a fully oeed eural ework i.e. he euros of he H are fully oeed, whih meas ha every euro is oeed o all oher euros. Le T ij be he sregh of he oeio from euro j o euro i. Eah euro i has a offse bias i b i. The urre sae ad he oupu of he euro i are respeively represeed by u i ad [3]. The dyamis of he H are desribed by he differeial equaio: du u b = T i ( d b Where u, ad i are he veors of euro saes, oupus ad biases. The oupu fuio i = g( ui is a hyperboli age, whih is bouded below by ad above by. ui g( ui = ( ah( where u > ad i =,..., u Where u is a parameer used o orol he gai of he aivaio fuio. I order o use he oiuos Hopfiled ework, for solvig ay ombiaorial problems, we should be reformuled his laer io eergy fuio assoiaed o he H. This eergy fuio is defied by he followig epressio [3]: i b E ( = T ( i. ( Typially, i he H, he eergy fuio is made equivale o he objeive fuio whih is o be miimized, while eah of he osrais of he opimizaio problem are iluded i he eergy fuio as pealy erms. 3. Eergy fuio ad parameer-seig for he TAP problem I order o solve he ask assigme problem usig he oiuous Hopfield eworks, we defie he geeralized eergy fuio for he TAP problems basig o he model. Reall ha, he ask assigme problem are modeled as - quadrai programmig wih variables ad liear osrais. ( QP i Subje o f ( = Q e A = b {,} The geeralized eergy fuio allows represeig mahemaial programmig problems wih quadrai objeive fuio ad liear osrais. This eergy fuio iludes he objeive fuio f ( ad i pealizes he liear osrais A = b wih a quadrai erm ad a liear erm. The geeralized eergy fuio for he QP problem is defied by [4]: E ( = Q e ( A ( A (3 diag ( ( A Wih [,], IR, IR, IR ad is a symmeri mari. Here diag ( deoes he diagoal mari osrued from he veor. To defie he eergy fuio of he (QP problem, he followig osideraios eed o be ake io aou so ha he mahemaial epressio of eergy fuio (3 is simplified. Oly he mai diagoal erms of he quadrai mari parameer are osidered: kj = Where is a posiif salar. if j k if j = k opyrigh ( Ieraioal Joural of ompuer Siee Issues. All Righs Reserved.

4 IJSI Ieraioal Joural of ompuer Siee Issues, Vol. 9, Issue, o, arh ISS (Olie: All liear osrais are equally weighed, where is he assoiaed parameer. The parameer pealizig he o-ereme values of is. osequely, he followig geeralized eergy fuio of he QP problem is proposed: E( = Q e ( A ( A (4 diag ( ( A So, he followig geeralized eergy fuio i algebrai form of QP problem mus also be defied by: E( = ijkl i= j= l= i= l= i= ( il jl i= i= To deermie he weighs ad hresholds, we use he assimilaio bewee equaio ( ad he algebrai form of he geeralized eergy fuio. The, he weighs ad hresholds of he oeios bewee euros are: Tjl = jl ij ij kl b (6 i = e Where ij is he Kroeeker dela suh ha: ij if i j if i j I his way, he quadrai programmig has bee preseed as a eergy fuio of oiuous Hopfield ework. To solve a isae of he QP problem, he parameer seig proedure is used. This proedure assigs he pariular values for all parameers of he ework, so ha ay equilibrium pois are assoiaed wih a valid affeaio of all variables whe all osrais are saisfied. The weighs ad hresholds of he quadrai program deped o he parameers,, ad. The parameer-seig proedure is based o he parial derivaives of he geeralized eergy fuio: e (5 E ( = E ( = j = l = l = il jl jl e ( Thus, here is o ommuiaio i he same ask o wo proessors: il = ( i, k, l {,.., } {,.., } The he derivaive of he eergy fuio beomes: E ( = jl j=, ji l= ( jl e il l= To solve he QP problem, he followig ses are eeded: H is a se of he Hammig hyperube : H { [,] } H is a se of he Hammig hyperube orers : H { H : i {,}, i =,..., } H F is a se of feasible soluios : H { H : A = b}. F This proedure uses he hyperplae mehod, so ha he Hammig hyperube H is divided by a hyperplae oaiig all feasible soluios. Based o his hyperplae ad he assoiaed half-spaes [4]. The hyperplae mehod is briefly eplaied below. I order o guaraee he isabiliy of he ierior pois H H, some iiial odiios are imposed o some parameers: T = Where i {,..., } ad k {,..., }. The QP problem has oly oe family of liear osrais: d i ( = The pariio of he se H = H, i i {,..., F } H H is defied as : F = W W, W { i: d ( >} { d ( } = d i ( i Where d (. I his ase oe ask has bee eued by wo differe proessors so ha = il =, i.e., he ask i is goig o eue i wo differe proessors, whih is illogial. To avoid his, he followig odiio should be imposed:, (7 (8 opyrigh ( Ieraioal Joural of ompuer Siee Issues. All Righs Reserved.

5 IJSI Ieraioal Joural of ompuer Siee Issues, Vol. 9, Issue, o, arh ISS (Olie: E d (9 ( mi Where d mi = ( mi emi wih = i / ( i, j {,..., } ad ( k, l {,..., } e mi mi jl = i e / i {,..., } ad k {,..., } W, { i: di ( <} { d ( < } I his ase oe ask has o bee assiged o ay proessor, suh ha = k {,..., },i.e., ay proessor ha o reserved o eue he ask i, whih is oradiory. Therefore, he followig odiio should be imposed: E ( d ( ma Where d ma = ( ma ema wih = a / ( i, j {,..., } ad ( k, l {,..., } ma jl e ma = a e / i {,..., } ad k {,..., } osequely, we a deermie he parameers seig by resolvig he followig sysem: αd mi φ β γ ε d ma ( These parameers seig are deermiae by fiig, ad ompue he res of parameers, ad : = ( ( d ma d mi / = d ma =. Fially, he weighs ad hresholds of H followig: Tjl b i ( d jl ma ( e ij kl ( ( d ma d mi as he Where ij is he Kroeeker dela. Fially, we obai a equilibrium poi for he H usig he algorihm desribed i [3], so ompue he soluio of ask assigme problem. 4. ompuaioal eperimes For evaluaig ad showig he praial ieres of our approah, we have used he isaes provided i [7], where oeffiies e ad jl are radomly geeraed i he ierval [ 5,5]. These eperimes are effeuaed i persoal ompuer wih proessor Iel ore i3,53 GHz, ad 3 Go of RA. The performae has bee measured i erms of he PU ime per seod. This solver is implemeed by java laguage. I his eperimeaio, some isaes used as [7]. The sarig pois are geeraed radomly. 3 =.99 u Where i =,...,, k =,..., ad u is a radom uiform variable i he ierval [.5,.5]. The value of eah parameers is deermiaed by solvig 4 he sysem ( where = ad. =, wih is he umber of asks. Table ( summarizes he resuls of he eeuios of our approah o hese isaes. For eah size of isaes, we ru he algorihm imes ad he qualiy of he soluio obaied by our approah was evaluaed by he followig performae epressio: Objeive Opimal value value obaied by H Fially, he ieresig resuls are obaied by his approah. The resoluio imes obaied by our approah are reasoable. From a heoreial poi of view, our approah is very powerful. I a happe o solve a TAP of large size. 5. olusios I his paper, we have irodued a ew mehod o fid a soluio of he ask assigme problem. This problem has bee preseed as - quadrai problem subje o liear osrais (QP. To solve his problem, we have used he oiuous Hopfield ework. Some umerial eamples assess he effeiveess of he heoreial resuls are show i his paper, ad also he advaages of his ew approah. Several direios a be ivesigaed o ry o improve his mehod, suh as reduig he arhieure of Hopfield eural ework [], ad mehods of fiaio ad evaluaio. opyrigh ( Ieraioal Joural of ompuer Siee Issues. All Righs Reserved.

6 IJSI Ieraioal Joural of ompuer Siee Issues, Vol. 9, Issue, o, arh ISS (Olie: Isaes PLEX Opimal Value Table : The ypial isaes of TAP solved by H bes objeive value H miimum mea mode ea ieraios ea ime (ms Tassu 3_ ,8,6 75,87 5,4 Tassu 3_ ,,3,7 9,89,7 Tassu 3_ ,,69,97 73,36,7 Tassu 3_ ,3,5,9 9,57,6 Tassu 3_ ,5,33,4 93,36,97 Tassu 3_ ,,3,5 63,68,9 Tassu 3_ ,4,47 87,3, Tassu 3_ ,3 5, 5,56 66,99,57 Tassu 3_ ,3,6,3 8,,73 Tassu 3_ ,8,89 4,3 85,5,4 Tassu_5_5_ ,4,9,77 4,46,34 Tassu_5_5_ ,4,3,5 9, 8,38 Tassu_5_5_ ,8,85,94 6,67 8,74 Tassu_5_5_ ,,45,37 6,9,86 Tassu_5_5_ ,8,65,6 5,6 6,69 Tassu_5_5_ ,8,88,75 5,7 6,67 Tassu_5_5_ ,5,99, 7,3 6,58 Tassu_5_5_ ,9,68 7 6,54 Tassu_5_5_ ,6,,89 5,5 6,56 Tassu_5_5_ ,3,9,9 5,89 6,43 Referees [] R. K. Arora ad S. P. Raa, "Aalysis of he module assigme problem i disribued ompuig sysems wih limied sorage", Iformaio Proessig Leers, Vol., o. 3, 98, pp. 5. [] A. Billioe,.. osa, ad A. Suer, " A effiie algorihm for a ask alloaio problem", Joural of he A, Vol. 39, o. 3, 99, pp [3] A. Billioe ad S. Elloumi, "Plaeme de âhes das u sysème disribué e dualié lagragiee", Revue d'auomaique, d'iformaique e de Reherhe Opéraioelle (R.A.I.R.O., série vere, Vol. 6, o., 99, pp [4] S. H. Bokhari, "A shores ree algorihm for opimal assigmes aross spae ad ime i disribued proessor sysem", IEEE T. Sofware Eg, vol. 7, 98, pp [5] W. W. hu, " Opimal file alloaio i muliple ompui sysem", IEEE Tras. ompu, Vol. -8, 969, pp [6] O. I. El-Dessouki ad W. H. Hua, "Disribued euméraio o ework ompuers", IEEE Tras. ompu, Vol. -9, 98, pp [7] S. Elloumi, "The ask assigme problem, a library of isaes", hp://edri.am.fr/o/tap/tap.hml, 4. [8]. Eaouil ad. Loqma, "A ew Opimizaio odel for Solvig he osrai Saisfaio Problem", Joural of Advaed Researh i ompuer Siee, Vol., 9, pp [9]. Eaouil ad. Loqma, "osrai Saisfaio Problems Solved by Semidefiie Relaaios", WSEAS TRASATIOS o OPUTERS, Vol. 7, 8, pp []. Eaouill ad Y. Ghaou, "eural arhieures opimizaio ad geei algorihms", WSEAS rasaios o ompuers, Vol. 8, o. 3, 9, pp []. Eaouill,. Loqma ad K. haddouh, "Job Shop Shedulig Problem Solved by he oiuous Hopfield eworks", Vol., o.,, pp [] D. J. Evasi ad.. Sulaima, "Solvig opimisaio problems usig euomp-a eural ework ompiler", Ieraioal Joural of ompuer ahemais, Vol. 6, o., 996, pp. -. [3] W. Feradez de la Vega ad. Lamari, "The ask alloaio problem wih osa ommuiaio", Disree Applied ahemais, Vol 3, o., 3, pp [4] V. B. Gylys ad J. A. Edward, "Opimal pariioig of workload for disribued sysems", I Dig. OPO, Fall 976, pp opyrigh ( Ieraioal Joural of ompuer Siee Issues. All Righs Reserved.

7 IJSI Ieraioal Joural of ompuer Siee Issues, Vol. 9, Issue, o, arh ISS (Olie: [5] Y. Hamam ad K. S. Hidi, "Assigme of program asks o proessors: a simulaed aealig approah", Europea Joural of Operaioal Researh, Vol.,, pp [6] J. J. Hopfield, "eural eworks ad physial sysems wih emerge olleive ompuaioal abiliies", Proeedigs of he aioal Aademy of Siees of he Uied Saes of Ameria, vol. 79, 98, pp [7] J. J. Hopfield ad D. W. Tak, "eural ompuaio of deisios i opimizaio problems", Biologial ybereis, Vol. 5, 985, pp. -5. [8] E. S. H. Hou,. Asari ad H. Re, "A geei algorihm for muliproessor shedulig", IEEE Trasaios o Parallel ad Disribued Sysems, Vol. 5, 994, pp. 3. [9] F. T. Li ad.. Hsu, "Task assigme shedulig by simulaed aealig", Proeedig of oferee o ompuer ad ommuiaio Sysems, 99, pp [] S. aw-sheg her, G. H. he, ad Pagfeg Liu, "A L brah-ad-boud algorihm for he module assigme problem", Iformaio Proessig Leers, Vol. 3, o., 989, pp [] F. Roupi, "O approimaig he memory-osraied odule Alloaio Problem", Iformaio Proessig Leers, Vol. 6, o. 4, 997, pp [] H. S. Soe, "uliproessor shedulig wih he aid of ework flow for algorihms", IEEE Tras. Sofware Egrg, SE, Vol. 3, o., 977, pp [3] P.. Talavá ad J. Yáñez, "A oiuous Hopfield ework equilibrium pois algorihm", ompuers ad Operaios Researh, Vol. 3, 5, pp [4] P.. Talavá ad J. Yáñez, "The quadrai assigme problem. A euroal ework approah", eural eworks, Vol. 9, 6, pp [5] A. K. Tripahi, B. K. Sarker ad. Kumar, "A GA based muliple ask alloaio osiderig load", Ieraioal Joural of High Speed ompuig,, pp [6] J. D. Ullma, "P-omplee Shedulig Problems", JSS, Vol., 975, pp opyrigh ( Ieraioal Joural of ompuer Siee Issues. All Righs Reserved.