Clustering Process to Solve Euclidean TSP


 Jonas Gardner
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1 Cluteig Poce to Solve Euclidea TSP Abdulah Faja *Ifomatic Depatmet, Faculty of Egieeig Uiveita Widyatama Badug Idoeia # Faculty of Ifomatio ad Commuicatio Techology Uiveiti Tekikal Malayia Melaka Abtact Huma i able to clute ad filte object efficietly. Cluteig poblem ha bee appoached fom divee domai of kowledge like gaph theoy, tatitic, atificial eual etwok ad o o. Thee ha bee gowig iteet i tudyig combiatoial optimizatio poblem by cluteig appoach, with a pecial emphai o the Euclidea Tavelig Salema Poblem. Claical ETSP appea a a fudametal poblem i vaiou poblem uch a tapotatio, maufactuig ad logitic applicatio. Thi tudy will focu o tou cotuctio. Mot of method focu o tou impovemet ad uig eaet eighbohood fo tou cotuctio. Thi pape will ue cluteig poce to decompoe ETSP ito malle ub poblem. Cluteig poce hieachically aage adjacecy ad vetice to fom clute. A thehold of edge weight i applied to plit oe clute to eveal ub clute. Uig thi appoach the uig time ca be cut ito half compaed to TSPLib tadad time. The mai objective i to develop bet cluteig poce to ETSP ad poduce a ea optimal olutio withi % of bet kow olutio i TSPLib. Keywod: Hieachical Cluteig, Euclidea TSP, Tou Cotuctio, Adjacecy I. INTRODUCTION Huma ha a atual ability to clute multiple object quickly. Thi capacity ait huma i makig outie deciio. Howeve makig compute act like huma beig i olvig the poblem i vey difficult ad demadig the attetio of compute cietit ad egiee aoud the wold util ow [] Cluteig poblem ha bee appoached fom divee domai of kowledge like gaph theoy, tatitic, atificial eual etwok ad o o. Thee ha bee gowig iteet i tudyig combiatoial optimizatio poblem by cluteig tategy, with a pecial emphai o the Euclidea Tavelig Salema Poblem (ETSP []. Claical ETSP appea a a fudametal poblem i vaiou poblem uch a tapotatio, maufactuig ad logitic applicatio, thi poblem ha caught much attetio of mathematicia ad compute cietit. The et of thi pape will decibe about motivatio of thi tudy egadig Euclidea TSP i ectio II, objective thi tudy will be decibed i ectio III. Related wok will be decibed i ectio IV ad ectio V decibe hieachical cluteig appoach. Sectio VI ad VII Nu Azma Abu#, Naa Suyaa Hema#, Shahi Shahib # # Faculty of Ifomatio ad Commuicatio Techology Uiveiti Tekikal Malayia Melaka #{ ua, decibe hieachical cluteig poce ad cluteig poce to Euclidea TSP. The impact of cluteig poce will dicu i ectio VIII. Fial ectio (i.e. ectio IX will decibe cocluio. II. MOTIVATION Euclidea Tavelig alema poblem (ETSP i a combiatoial global optimizatio tak of fidig the hotet tou of vetice give the weight of edge. I a moe fomal way the goal i to fid the leat weight Hamiltoia cycle i a complete gaph G. The TSP poblem i a NPhad poblem. A NPhad poblem i extemely ulikely to have a polyomial algoithm to olve it optimally although it i ot poved that the wot cae expoetial olutio uig time ae uavoidable. Good appoximatio algoithm ca poduce olutio that ae oly a few pecet loge tha a optimal olutio ad the time of olvig the poblem i a lowode polyomial fuctio of the umbe of vetice. Some appoximatig algoithm poduce tou whoe legth ae cloe to that of the hotet tou, but the time complexity i ubtatially highe tha liea. III. OBJECTIVE OF STUDY The pimay objective of thi tudy i to poduce a bette hieachical cluteig tategy that fit ito Euclidea TSP olutio method. The cuet eeach i hieachical cluteig to olve ETSP motly poducig thei bet algoithm with the uig time ea O(. Fo pactical iduty pupoe, a algoithm with uig time O( 3 i till acceptable/viable ad aveage fial ETSP olutio i maximum % above the bet kow olutio. The othe uppotig objective ae to develop efficiet algoithm which poduce o edge coig i plaa gaph; clute geeatio fo lage ize clute; poduce Hamilto path withi ad ite clute ; ite clute coectio a iitial tou to fom global tou. IV. RELATED WORK Haxhimua et. al [] fomulated a pyamid algoithm fo ETSP motivated by the failue to idetify a exitig algoithm that could povide a good fit to the ubject data. Moe ecetly, hieachical (pyamid algoithm have bee ued to model metal mechaim ivolved i othe type of viual poblem. The mai apect of the model ae (multi eolutio pyamid achitectue ad a coae to fie poce
2 of ucceive tou appoximatio. Thi algoithm wa motivated by the popetie of the huma viual ytem ha mai idea to ue agglomeative (bottom up pocee to educe the ize of the iput by cluteig the ode ito mall goup ad topdow efiemet to fid a appoximate olutio, a ame a Gaham do i []. The ize of the iput (umbe of vetice i the gaph i educed o that a optimal olutio ca be foud by the combiatoial each. A pyamid i ued to educe the ize of the iput i the bottomup pocee. I bottomup cluteig, vetice ae cloe eighbo ae put ito the ame clute uig geedy appoach. Thee cluteed vetice ae coideed a a igle vetex at the educed eolutio. Gaph pyamid tategy ue MST uig Bouuvka algoithm. MST i ued a atual lowe boud fo TSP olutio. I the cae TSP with tiagle iequality which i the cae fo the Euclidea TSP, MST ca be ued a to pove the uppe boud. Gaph pyamid olutio give good appoximatio compaed to TSPLIB. Solutio eo fo thi olutio ted to lige aoud %. Ruig time pefomace i quadatic fo thi olutio epecially wheeve umbe of vetice i moe tha 0. V. HIERARCHICAL CLUSTERING APPROACH A cluteig algoithm i expected to dicove the atual goupig that exit i a et of patte. I hieachical cluteig, the data ae ot patitioed ito a paticula clute i a igle tep. Hieachical cluteig may be epeeted by a two dimeioal diagam kow a dedogam, a how i Figue CluteIdex Figue. Dedgam of Complete Likage Method fo Datzig.tp Amog the mot popula hieachical cluteig algoithm, BIRCH [7] ca typically fid a good cluteig i a igle ca of the data ad impove the quality futhe i a few additioal ca. It i alo the fit cluteig algoithm that hadle oie effectively. CURE [] epeet each clute by a cetai umbe of poit elected fom a wellcatteed ample ad the hikig them towad the clute cetoid by a pecified factio. It ue a combiatio of adom amplig ad patitio cluteig to hadle lage databae. ROCK [9] i a obut cluteig algoithm fo Boolea ad categoical data. It itoduce two ew cocept: a poit eighbo ad lik, ad to meaue the imilaity/poximity betwee a pai of data poit. Cluteig appoach divide TSP poblem to ub gaph o ub clute ad educe poblem ize. Thee ae may cluteig appoach available but it hould be chooe a uitable oe. Hieachical i the implet algoithm that ca be choe a cluteig appoach. A agglomeative hieachical cluteig pocedue poduce a eie of patitio of the data, C, C ,..., C. The fit C coit of igle object 'clute', the lat C, coit of igle goup cotaiig all cae. At each paticula tage the method joi togethe the two clute which ae cloet togethe (mot imila. Diffeece betwee method aie becaue of the diffeet way of defiig ditace (o imilaity betwee clute. Numbe of clute that i geeated i hieachical cluteig i automatically. I thi wok, it i defied a thehold R [], whee R Aea of Clute A The umbe of Vetice Thi thehold will act a guidace to plit oe clute to eveal ad eult baic clute. Baic clute have mall ize umbe of ode. Thi thehold alo pevet clute ovelapped oe to othe. Let G=(V,E,W be a gaph, G =(V,E,W ad G =(V,E,W be ub gaph of gaph G, et ad ae umbe of vetice V ad V. Hee i eveal techique of likage betwee the clute ued i thi tudy: A. Neaet likage The ditace betwee two clute i the miimum ditace betwee two clute. Thi method ca caue chaiig clute. Clute hape potetially ovelapped to aothe clute. d( G, G mi W ( V, V Figue. Sigle Likage Ditace Figue how dedogam which illutate the fuio o diviio made at each ucceive tage of aalyi [6]. 9
3 B. Complete likage Thi method i oppoite of igle likage ad ted to poduce compact clute. Outlie ae give moe weight with thi method. It i geeally a good choice if the clute ae fa apat i featue pace, but ot good if the data ae oiy. d( G, G max W ( V, V Figue 3. Complete Likage Ditace C. Aveage likage The ditace betwee two clute i the mea ditace betwee all poible pai of ode i the two clute. d( G, G i j dit( V, V Figue. Aveage Likage Ditace D. Media Likage Ued oly fo Euclidea ditace. The ditace betwee two clute i the Euclidea ditace betwee thei weighted cetoid. ~ ~ d ( G, G V V Whee V ~ i the weighted cetoid of. If wa ceated fom clute p ad q, the V ~ i defied ecuively a: ~ V ~ ~ ( V p V q Clute Clute Figue. Media Likage Ditace E. Cetoid likage Ued oly fo Euclidea ditace. The ditace betwee two clute i the Euclidea ditace betwee thei cetoid, a calculated by aithmetic mea ~ ~ d ( G, G V V Whee x ~ i the cetoid of, by aithmetic mea: i j Clute ~ V V i i Figue 6. Cetoid Likage Ditace F. Wad likage (Sum Squae Method The um quae (Wad method i appopiate fo the cluteig of poit i Euclidea pace. Wad' method ay that the ditace betwee two clute, ad, i how much the um of quae will iceae whe mege them. ~ ~ (, d G G V V Whee V ~ cete of clute ad i i umbe of poit i clute. VI. HIERARCHICAL CLUSTERING PROCESS I thi tudy, cluteig appoach decompoe poblem ito malle by plit a clute ito eveal ub clute. Thi algoithm will ue Joho algoithm ad plit oe clute to aothe uig R thehold. Let G(V,E,W gaph fo datzig.tp ample. The cluteig poce tat with clute C, C,, C whee each clute coit of oe vetex oly. The ecod tep i to fid the cloet d(c i,c i+ R (mot imila uig likage ditace ad le tha R pai of clute ad mege them ito a igle clute, o that ow umbe of clute become oe le tha the amout peviouly. The, ext tep, i to compute ditace (imilaitie betwee the ew clute ad each of the old clute. Fial tep i to epeat peviou tep util ditace betwee clute ot exceed tha R value. The eult uig complete likage fo datzig.tp i how i Figue Figue 7. Clute of datzig.tp Clute
4 VII. CLUSTERING APPROACH TO EUCLIDEAN TSP The fit tep comig fom cluteig poce to geeate mall compact clute a how i Figue 7. Each clute hall be coected to adjacet clute. I each clute, a middle poit hall be computed. Baed o global hamilto cycle hall be fomed a global clute tou. Poibility of ite clute coectio i how i Figue. Each clute hall be fomed Hamilto path baed o two ed poit of adjacet clute. The bet Hamilto path of each clute hall be itegated baed o global clute tou to eult global tou. Global tou hall be impoved ad efied to achieve hotet fial tou a how i Figue 9. Tou impovemet tou ca be applied by opt exchage move VIII. Figue. Itecoectio betwee Clute THE CLUSTERING PROCESS IMPACT The electio method of likage i hieachical cluteig i vey impotat. I peviou tudy, eaet likage i ued to clute the poblem, ad at cuet poge aothe likage i ued uch a complete, cetoid, media, wad ad aveage. Clute idex i ued with a ole i detemiig the aveage umbe of poit i the clute. Small clute poduce a bette path tha the clute with lage ize. The difficulty ecouteed i the lik betwee the path of each clute to a poblem with lage ize Figue 9. Ite Clute ad Fial tou of Datzig.tp TABLE I. NUMBER OF NODE PER CLUSTER OF RANDOM SAMPLES (N=6 TO N= Method Aveage Max Mi Stddev Neaet Complete 0.0 Aveage 3 0. Media Cetoid Wad 0.0 Radom example with = 6 [] gave eult betwee to 3 poit pe clute ule eaet likage. While example of poblem TSP Lib give imila eult with adom ample ize i till i the baic clute of to ode pe clute. Neaet likage give a diffeet eult i 6 ode pe clute. Table how the eult of cluteig (ode pe clute with a uifom patte fo all the likage method (except eaet likage. TABLE II. TOUR DISTANCE OF RANDOM SAMPLES (N=0 Method Aveage Max Mi Stddev Neaet Complete Aveage Media Cetoid Wad Statitically a how i table II, tou legth compaio betwee the peviou method (eaet likage with the othe method yield about 3% hote. Table ad howed that thee wa a igificat coelatio that the umbe of ode pe clute ca affect the legth of the tou olutio. TABLE III. RUNNING TIME OF RANDOM SAMPLES (N=0 Method Aveage Max Mi Stddev Neaet Complete Aveage Media Cetoid Wad Table III how that the uig time i coumed by a method that i cuetly ued i vey hot compaed with the peviou method (eaet likage method. Coumptio level of the uig time of cuet method i betwee % ad 3% oly compaed to the coumptio of the uig time of peviou method. I othe wod thee ae efficiecy ad uitability of the method i tem of uig time Thi tudy alo compae the method to TSP Lib bet kow olutio ad tadad time. Simulatio of thi tudy ue Itel Quad Coe.6 GHz with GB Memoy fo 6
5 imulatio ad u Cocode oftwae fom TSP Lib. Thi imulatio i witte uig Matlab a developmet tool oftwae. TABLE IV. TOUR DISTANCE GAP COMPARED TO TSP LIB BEST KNOWN SOLUTION FOR N=9 TO N=00 NODES Method Aveage Max Mi Stddev Neaet 7.0%.7% 0.0% 3.% Complete 6.66%.% 0.0% 3.% Aveage 6.33% 6.% 0.9% 3.% Media 6.6%.6% 0.9% 3.9% Cetoid 6.3% 6.3% 0.9% 3.6% Wad 6.%.7% 0.0% 3.0% All method alo ae how that tou legth i ot have eough diffeece oe to aothe (ee table IV but i cotadicted to table V. Complete ad Wad likage method give bet impovemet compaed to othe method. Both method coume oly 9% ad 6% uig time compaed to TSP Lib Stadad time. The maximum uig time that i coumed by two method i about time compaed to TSP Lib Stadad Time o beli.tp. I that ample (beli.tp tadad time oly 0.0 ecod while Complete ad Wad Likage Method coume 0. ad 0. ecod. Othe method coume uig time loge tha TSP tadad time. The fatet uig time i how by complete likage method almot % compaed to Cocode uig time o at7.tp ad the gap about 7%. Oveall eult fo tou ditace le tha 7% o aveage ad thi eult awe the objective. TABLE V. RUNNING TIME COMSUMING COMPARED TO CONCORDE FROM TSP LIB FOR N=9 TO N=00 NODES Method Aveage Max Mi Stddev Neaet 37.% 99.%.7% 99.% Complete.7% % 0.9% 3.0% Aveage 6.% 9.%.03% 3.7% Media.% 69.6%.%.% Cetoid 67.00% 73.7%.% 9.7% Wad 6.07% 39.%.09% 39.% IX. CONCLUSION Cluteig techique hould be able to poduce ea optimal olutio. Cluteig techique that ued hould coide umbe clute limitatio. Smalle clute ize ted to eult i moe optimal to Hamilto path fo each clute. The umbe of ode of each clute ifluece i uig time. The ix method peeted i thi tudy povide eult that ae divided ito diffeet eult. Complete ad Wad likage method poduce malle uig time with 6% loge tou ditace compaed to TSP Lib. A cocluio obtaied fom thi tudy i the ole of the cluteig poce ca acceleate the uig time by electig the appopiate likage method. The ext challege i to miimize the fial aveage eult by fidig the appopiate method fo coectig betwee the clute. The method ued to coect betwee the clute i thi tudy i the cheapet ietio method. Thi ietio method ha a wot cae eult up to time tha the bet eult, o othe method eed to be the wot cae eult ca be malle. ACKNOWLEDGMENT Thi eeach i fuded by Sciece ad Iovatio Fud fom Miity of Sciece ad Iovatio Techology Malayia. REFERENCES [] Y. Haxhimua, W. G. Kopatch, Z. Pizlo, ad A. Io, "Appoximative gaph pyamid olutio of the ETSP," Image ad Viio Computig, 00. [] N. A. Abu, S. Sahib, ad N. Suyaa, "A Novel Natual Appoach to Euclidea TSP," i The 3th Iteatioal Cofeece o Mathematic ad Statitic (ICoMS3, Bogo, Idoeia, 00. [3] J. N. MacGego, T. C. Omeod, ad E. P. Choicle, "A model of huma pefomace o the tavelig alepeo poblem," Memoy ad Cogitio, vol., pp. 3, [] D. Vicke, M. Butaviciu, M. Lee, ad A. Medvedev, "Huma pefomace o viually peeted tavelig alema poblem," Pychological Reeach, vol. 6, pp. 6, 00. [] S. M. Gaham, A. Johi, ad Z. Pizlo, "The tavelig alema poblem: A hieachical model," Joual of Memoy & Cogitio, vol. 7, pp. 9, 000. [6] S. Da, A. Abaham, ad A. Koa, "A Oveview Metaheuitic Cluteig," i Metaheuitic Cluteig. vol. 7, P. J. Kacpzyk, Ed. Beli: Spige, 009, pp. 6. [7] Zhag T, Ramakiha R, ad L. M., "BIRCH: a efficiet data cluteig method fo vey lage databae," i ACM SIGMOD Iteatioal Cofeece o Maagemet of Data, Moteal, Caada, 996. [] S. Guha, R. Ratogi, ad K. Shim, "CURE: A Efficiet Cluteig Algoithm fo Lage Databae," i ACM SIGMOD Iteatioal Cofeece o Maagemet of Data, Moteal, Caada, 99. [9] S. Guha, "ROCK: A Robut Cluteig Algoithm fo Categoical Attibute," Joual of Ifomatio Sytem, vol., pp. 6,
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