Learning Automata based Algorithms for Solving Stochastic Minimum Spanning Tree Problem

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1 Lg Autot bs Algoths fo Solvg Stohst Mu Sg T Pobl Jv Ab Tost Dtt of Cout Egg Isl Az Uvsty A Bh A I b@u.. Moh Rz Mybo Dtt of Cout Egg IT Ab Uvsty of Thology Th I Isttut fo Stus Thotl Physs Mthts IPM Shool of Cout S Th I ybo@ut.. Abstt Du to th hss of solvg th u sg t MST obl stohst vots th stohst MST SMST obl hs ot v th ttto t ts sflly wh th obblty stbuto futo PDF of th g wght s ot o ow. I ths w fst oos lg utot bs slg lgoth Algoth to solv th MST obl stohst ghs wh th PDF of th g wght s ssu to b uow. At h stg of th oos lgoth st of lg utot s oly tvt ts th gh gs tht ust b sl tht stg. As th oos lgoth os th slg oss fouss o th sg t wth th u t wght. Thfo th oos slg tho s bl of sg th t of ussy slgs shotg th t u fo fg th SMST. Th ovg of ths lgoth s thotlly ov t s show tht by o ho of th lg t th sg t wth th u t wght b fou wth obblty los ough to uty. Nul sults show tht Algoth outfos th st slg tho. Sltg o lg t s th ost hllgg ssu lg utot thoy by whh goo t off b hv btw th ost ffy of lgoth. To ov th ffy.. th ovg s ovg t of Algoth w lso oos fou thos to ust th lg t Algoth th sultt lgoths ll s Algoth though Algoth 5. I ths lgoths th obblst stbuto ts of th g wght t to osto fo ustg th lg t. Sulto ts show th suoty of Algoth 5 ov th oths. To show th ffy of Algoth 5 ts sults o wth thos of th ultl g sstvty tho MESM. Th obt sults show tht Algoth 5 fos btt th MESM both ts of th ug t slg t. Kywo: Lg utot Mu sg t Stohst gh. Itouto A u sg t MST of g wght two s sg t hvg th u su of g wghts og ll sg ts. Th wght ssot wth th gh g sts ts ost tvsl t o lgth g o th ott. Mu sg t hs y ltos fft s suh s t stog ] sttstl lust lyss ] tu ossg 5] slly outo twos 6 7]. I two outg otools th u ost sg t s o of th ost fftv thos to ultst o bost th ssgs fo sou o to st of sttos. I ost sos th g wght s ssu to b f but ths os ot hol lwys tu thy vy wth t l ltos. Fo l th ls outo two y b fft by ollsos ogstos tfs. Thfo th MST obl s glz tow stohst u sg t SMST obl whh th g wght s o vbl th th ostt vlu. Th hv b y stus of th u sg t obl lg wth th tst ghs svl wll ow sutl lgoths suh s Bouv ] Kusl ] P ] whh th MST obl b solv olyol t hv b st. Futho svl fft

2 stbut 0 ] oto 5 7] lgoths hv b lso oos to solv th MST obl tst ghs. Howv wh th g wght s llow to b o vy wth t ths obl bos osbly h to solv. Ths bos o ffult f th obblty stbuto futo PDF of th g wght s ot o ow. Ul th tst ghs u to hss th MST obl hs ot v uh ttto stohst ghs. Ish t l. ] oos tho fo solvg th stohst sg t obl whh th to obl s tsfo to ts oy tst uvlt obl th olyol t lgoth st to solv th ltt obl. I ths tho th obblty stbuto of th gs' wghts s ssu to b ow. Ish Nsh 5] os stohst vso of th bottl sg t obl o th gs whos wghts o vbls. Thy show tht u sobl sttos th obl b u to u bottl sg t obl tst s. Moh 6] oos tho fo stohst sg t obl ll tvl lto tou svl oftos to th lgoth of Ish t l. ] show tht th of lgoth s bl to obt th btt sults lss t. Ish Mtsuto 7] st olyol t lgoth to solv th obl stt ] wh th obblty stbuto of th gs wght s uow. Thy l sttstl oh to stohst sg t obl os ol of th stohst sg t obls wth of go of uow stbuto ts. Alooulos Jobso 8] oos so thos to t th obblty stbuto of th wght of th MST stohst ghs whh th gs hv t st o wghts th obblty tht gv g blogs to u sg t. Dhh t l. 9] Swy Shoys 0] foult th stohst MST obl s stohst otzto obl oos so oto ohs to solv two ultstg stohst otzto obls. I ] Hutso Sh stu svl ohs to f o to otz th u sg t wh th gs ugo th wght hgs. Rt P tho ut st tho yl tg tho ultl g sstvty tho th oos ohs to f th MST of th stohst gh whh th g wght ssu ft ub of stt vlus s o vbl. To ot th t wght of th otl sg t Hutso Sh us th lgb stutu to sb th ltosh btw th fft g wght lztos of th stohst gh. Thy o fft ohs show tht th ultl g sstvty tho whh s h f to s MESM outfos th oths ts of th t olty th sz of th ostut stt s. Ktg t l. 8] tou fuzzy bs oh to ol th u sg t obl s of fuzzy o wghts. Thy th s wh th g wghts fuzzy o vbls. Fgguo Hu 9] os th obl of u sg t ut twos whh th g wghts o vbls. Thy tlly f th ot of th t u sg t thtlly ol foult th obl ogly. Th bs o th sultg ol thy oos hyb tllgt lgoth whh s obto of th gt lgoth stohst sulto thu to solv th SMST obl. I ] w lssfto s st fo MST lgoths. Ths glly lssfs th stg ohs to tst stohst MST lgoths. Dtst lgoths futh subv to ost uost lsss. I ths th uthos oos hust tho to solv th stohst vso of th u sg t obl. I 9] th Püf og sh whh s bl to st ll ossbl ts s us to o th osog sg ts th gt stto. Al t l. 0] stu th u sg t obl wth fuzzy ts oos t lgoth to solv ths obl. Th wb of th bov to SMST lgoths s tht thy fsbl oly wh th obblty stbuto futo of th g wght s ssu to b o ow whs ths ssuto ot hol tu lst ltos. I ths w fst oos lg utot bs oto lgoth ll Algoth to solv th u sg t obl stohst gh wh th obblty stbuto futo of th g wght s uow. As Algoth os th oss of slg fo th gh s ott o th gs of th sg t wth th u t wght. Ths s thotlly ov fo Algoth tht by th o ho of th lg t th sg t wth th u t wght wll b fou wth obblty los ough to uty. To show th fo of Algoth th ub of sls tht s u to b t fo th stohst gh by ths lgoth s o wth tht of th st slg tho wh th ot vlu of th g wght ovgs to ts vlu wth t obblty. Th obt sults show tht th vg ub of sls t by Algoth s uh lss th tht of th st slg tho. I

3 Algoth lg utoto s ssg to h gh o th t gs of o f s ts to st. Ths lgoth s oos of ub of stgs t h stg sg t of th gh s ostut by th o slto of th gh gs by th tvt utot. At h stg f th t wght of th ostut sg t s lg th th vg wght of ll sg ts ostut so f th ostut sg t s lz t s w othws. Th fo of lg utot bs lgoth s tly fft by th lg t. Choosg th s lg t fo ll lg utot to ut th to obblty vtos y olog th ovg fo so lg utot lt th ovg to o otl soluto fo so oths. To vo ths w lso oos fou thos fo ustg th lg t ly th to Algoth. Algoth whh th lg t s out by ths fou sttstl thos s Algoth though Algoth 5. Cout sulto shows tht Algoth 5 outfos th oths ts of slg t th ub of sls ovg t. To vstgt th ffy of th bst oos lgoth Algoth 5 w o Algoth 5 wth MESM oos by Hutso Sh ]. Th obt sults show tht Algoth 5 s suo to MESM ts of ug t slg t. Th st of th s ogz s follows. Th stohst gh lg utot thoy bfly vw sto. I sto ou lg utot bs lgoths st. Sto sts th ovg oof of th fst oos lgoth Algoth stus th ltosh btw th lg t ovg t of Algoth. Sto 5 vluts th fo of th oos lgoths though sulto ts Sto 6 olus th.. Stohst Gh Lg Autot Thoy To ov th sufft bgou fo ustg th bs ots of th Stohst u sg t lgoths whh st ths th stohst ghs lg utot thoy bfly sb ths sto... Stohst Gh A ut stohst gh G b f by tl G V Ε W wh V { v v... v} s th vt st E V V s th g st t W ots th obblty stbuto futo of th o wght ssot wth th gh gs wh s th ub of os. Fo h g th ssot wght s ssu to b ostv o vbl wth obblty sty futo w whh s ssu to b uow ths. A ut stohst sub gh G V Ε W s lso ll stohst sg t of G f G s ot sub gh G hs th s g st s G E E stsfyg E - wh E ots th lty of st E. Lt T { K} b th st of ll ossbl stohst sg ts of gh G W ots th t wght of sg t. SMST s f s th stohst sg t wth th u t wght. I oth wos gv stohst u sg t T s th SMST f oly f W T { W } 7]... Lg Autoto A lg utoto ] s tv so g ut tht ovs ts fo by lg how to hoos th otl to fo ft st of llow tos though t ttos wth o vot. Lg utoto hs show to fo wll out twos ] solvg obtol otzto obls 5 7 5]. Th to s hos t o bs o obblty stbuto t ov th to st t h stt th gv to s sv s th ut to th o vot. Th vot sos th t to tu wth fot sgl. Th to obblty vto s ut bs o th fot fb fo th vot. Th obtv of lg utoto s to f th otl to fo th to st so tht th vg lty v fo th vot s z. Th vot b sb by tl wh sts th ft st of th uts ots th st of th vlus tht b t by th fot sgl ots th st of th lty obblts wh th lt s ssot wth th gv to. If th lty obblts ostt th o vot s s to b sttoy o vot f thy vy wth t th vot s ll o sttoy

4 vot. Th vots g o th tu of th fot sgl b lssf to ol ol ol. Th vots whh th fot sgl oly t two by vlus 0 f to s ol vots. Aoth lss of th vot llows ft ub of th vlus th tvl 0 ] b t by th fot sgl. Suh vot s f to s ol vot. I ol vots th fot sgl ls th tvl b. Lg utot b lssf to two fls 9]: f stutu lg utot vbl stutu lg utot. Vbl stutu lg utot st by tl wh s th st of uts s th st of tos s lg lgoth. Th lg lgoth s u lto whh s us to ofy th to obblty vto. Lt ot th to slt by lg utoto th obblty vto f ov th to st t stt stvly. Lt ot th w lty ts t th out of ss ss of th to obblts stvly. Lt b th ub of tos tht b t by lg utoto. At h stt th to obblty vto s ut by th l lg lgoth gv Euto f th slt to s w by th o vot t s ut s gv Euto f th t to s lz. If th u utos ll l w lty lgoth f th gv utos ll l w lty flly f 0 thy ll l w Ito. I th ltt s th to obblty vtos uhg wh th t to s lz by th vot... Vbl Ato St Lg Autoto A vbl to st lg utoto s utoto whh th ub of tos vlbl t h stt hgs wth t. It hs b show ] tht lg utoto wth hgg ub of tos s bsolutly t lso otl wh th fot sh s. Suh utoto hs ft st of tos. ots th st of to substs s th subst of ll th tos b hos by th lg utoto t h stt. Th slto of th tul to substs s oly by tl gy og to th obblty stbuto f ov th ossbl substs of th tos wh. ots th obblty of hoosg to oto o th vt tht th to subst hs ly b slt too. Th sl obblty s f s wh s th su of th obblts of th tos subst. Th ou of hoosg to utg th to obblts vbl to st lg utoto b sb s follows. Lt b th to subst slt t stt. Bfo hoosg to th obblts of ll th tos th slt subst sl s f Euto. Th utoto th oly slts o of ts ossbl tos og to th sl to obblty vto. Dg o th sos v fo th vot th lg utoto uts ts sl to obblty vto. Not tht th obblty of th vlbl tos s oly ut. Flly th to obblty vto of th hos subst s sl s fo ll. Th bsolut y ε-otlty of th tho sb bov hv b ov ].

5 . Lg Autot bs SMST Algoths Lt G V Ε W ots th ut ut stohst gh wh V s th vt st E s th g st t W ots th obblty stbuto futo of th o wght ssot wth th gh gs. A two of lg utot sooh to th stohst gh s fo by ssgg to h o of th gh lg utoto. Th sultg two b sb by tl < A α W > wh A { A A... A} ots th st of th lg utot α { α α... α} ots th st of to sts whh α { α α... α... α } fs th st of tos tht b t by lg utot A fo h α α. S th stohst gh s ut g osos th to th to α of th lg utot A o to th toα of th lg utot A. Wght w ots th o wght ssot wth g.. W. I ths t s ssu tht W b ostv o vbl wth uow obblty stbuto. I ths lgoth whh s hft f to s Algoth h lg utoto b o of two stts: tv ssv. Eh lg utoto s tlly st to ssv stt. Th oos lgoth ossts of ub of stgs. Stg of th oos lgoth s bfly sb th followg sts.. SPANNING TREE FORMATION Rt th followg utl th th ub of th slt gs s gt th o ul to o th o o ssv utot to b tvt - Choos o of th ssv utot t o. M t s tv. - Th tvt utoto hooss o of ts tos og to ts to obblty vto. - Th o wght ssot wth th slt to s to th wght of th sg t tht s bg fo th ut stg. - Eh ssv lg utoto hgs ts ub of tos o sls ts to obblty vto by sblg th tos tht y us yl ug th foto of th sg t Th oss of sblg th tos s sb lt.. DYNAMIC THRESHOLD COMPUTATION Lt us ssu tht sg t s slt t stg. Th vg wght of ll th ostut sg ts utl stg s out o wth y thshol T. At h stg > y thshol T s out s T W wh s th ub of sg ts ostut so f W ots th vg wght of sg t whh s f s W W 5 wh ots th ub of ts sg t hs b ostut W ots th wght of sg t t stg whh s f s follows W W s t s t s t wh W s ostv o vbl wth uow obblty stbuto W ots ts wght t stg.. UPDATING ACTION PROBABILITY VECTOR 6 s t At h stg f th vg wght of th ostut sg t.. W s lss th o ul to y thshol T th th tos hos by ll tvt utot w 5

6 lz othws. I ths lgoth h utoto uts ts to obblty vto by usg LR I fot sh. Th sbl tos bl g th obblty vtos ut s sb Substo... TERMINATION Sts t utl th ho obblty of sg t PMST bos gt th t thshol T PMST o th stg ub s sf thshol T. PMST s f s th out of th ho obblts of th gs of th ostut sg t. Th sg t whh s ostut ust bfo th lgoth stos s th sg t wth th u t wght og ll sg ts of th stohst gh. Algoth Th fst oos SMST lgoth 0:Iut: Stohst Gh G < V E F > Thshols T T PMST 0:Outut: Th Mu Sg T 0: T 0 0 0:Lt ots th ostut sg t ots ts wght 05:Bg Algoth 06: ϕ w 0 07: Assg utoto to h o tlly st t to th ssv stt 08: Rt 09: Lt A ots th st of tvt utot whh s tlly ull 0: Whl < V o A < : Slt o of th ssv utot t o ll t A : If α φ : Autoto A hooss o of ts tos sy : A A A 5: 6: w w w 7: Eh ssv utoto us ts to st fo yl vo 8: Els 9: A A A 0: E If : E Whl : If V Th : Cout th vg wght of th slt sg t s w : If w T Th 5: Rw th slt tos of tvt utot 6: Els 7: Plz th slt tos of th tvt utot 8: E If 9: T T w ] 0: Els : Do othg : E If : : Ebl ll sbl tos 5: Utl PMST > PPMST o > T 6:E Algoth Fgu. Th suo o of Algoth As to l t h stg of Algoth th ub of tos tht b t by ssv utoto hgs o us wth t suh wy tht o yl s th sg t s L 7 of th Algoth Fgu. To vo th foto of yl th sg t Algoth 6

7 fos s follows: Lt π ots th th otg o v to ov ot th ho obblty of g th π stvly. Now t h stg th oos lgoth ovs vy g s o s toly sbls ts osog to th to st of th ssv utot A A s f g s hos t stg both ths π π s hs b ly ostut by th tvt utot. Ths sblg oss s t utl th sto oto of st s t. By ths th yl fss of th oos MST foto lgoth s gut. At h stg Algoth uts th to obblty vtos tw. Th fst t s wh th slt tos w o lz th so t s wh th sbl tos bl g t th of h stg. I both ss th to obblts ut s sb Substo. o vbl to st lg utot... Iovts I Algoth ll lg utot us th s lg t g uhg ug th uto of lgoth. Suh lg t gvs th ul ot to ll gs of th stohst gh to bg th u sg t. Ths y olog th ovg to th otl soluto fo so lg utot lt th ovg to o otl soluto fo so oths. I ths sto w suss fou sttstl thos fo ustg th lg t so s to ov th s ovg ts of Algoth. Ths thos t to out th obblty stbuto ts of th g wght fo ustg th lg t th ous of lgoth. Mtho I Th ub of sls tht s u to b t fo gv g to h ut stto of th vlu of ts o wght s tly ootol to th v of th o vbl ssot wth th g wght. Tht s th slg t ss th lg t ss s th v of th o wght bos lg. Thfo t s t tht th ovg s of Algoth s ov f th lg t of h gh g s ust bs o th v of ts o wght s gv Euto 7. I ths tho th v ust b tlly ot to tvl 0 fo vy g. Algoth whh th lg t s ust o th bss of Mtho I s ll Algoth. 7 ] σ wh ot th ub of ts g s sl ot th wght of g t stg ts vg wght ots th ostt gl lg t. Mtho II A g wth low vg wght s o lly to b t of th fl u sg t. Tht s th gs hvg sll wght low slg t th th hvy wght gs. Thfo t s t tht th ovg t of Algoth ss f th lg t of vy g s ust by Euto 8. Ths uto ls tht th lg t of g s vsly ootol to ts vg wght. Th g wth th lowst vg wght s w o th th oths so th utoto ovgs to t by fw sls. L Mtho I bfo lyg Euto 8 th vlu ust b ot to tvl 0 fo vy g. Algoth whh Mtho II s us to ust th lg t s ll Algoth. µ Mtho III As th lgoth os th ho obblty of th gs of th sg t wth th u t wght s gully ss s tht of th oths ss. Th gowth of th ho obblty of to s tht t s w by th vot. I ft th ho obblty of to sts ts otlty. I Euto 9 th lg t of g s tly f o th bss of ts ho obblty. Ths uto ss th ovg s o o s th ho obblty 8 7

8 ss o s th lgoth ohs to th. Algoth whh th lg t s out by Euto 9 s ll Algoth. P 9 Mtho IV Euto 8 b us ltos tht sf th ug t fvo of th soluto otlty whl Euto 9 s ot fo ltos tht gv sgft ot to th ug t. Dfft obtos of th vous thos Mthos I III b lso us fo ltos whh t off btw th ug t soluto otlty s s. W ob Mtho II Mtho III o to ov both th ovg t ovg s of Algoth. Ths tho s ll Mtho IV outs th lg t s gv Euto 0. Algoth whh Mtho IV s us to ust th lg t s ll Algoth 5. P P µ 0. Covg Rsults I ths sto w ov two sults of th. Th fst sult os th ovg of Algoth to th otl soluto wh h lg utoto uts ts to st by l wto fot sh Tho. Ths sult sts tht by hoosg o lg t fo Algoth th ho obblty of th otl soluto ovgs to o s uh s ossbl. S Algoth s sg fo stohst vots wh th votl ts vy ov t th tho tht s us to ov th ovg of th Algoth tlly follows th tho gv ] to lyz th bhvo of th lg utot otg o sttoy vots. Th so sult os th ltosh btw th ovg o t ε.. th o t volvs th st slg tho th lg t of th Algoth Tho. Th so sult s t tg lg t ε u whh th obblty of ostutg th sg t wth th u t wght s ε. Tho Lt b th obblty of ostutg sg t t stg. If s ut og to Algoth th th sts lg t ε 0 fo vy ε > 0 suh tht fo ll 0 w hv Pobl ] ε Poof Th sts of th ovg oof bfly outl s follows. At fst t s ov tht th lty obblty of h sg t ovgs to th ostt vlu of th fl lty obblty f s slt lg ough. Ths oty s show L. Th t s show tht th obblty of hoosg th sg t wth th u t wght s sub Mtgl oss fo lg vlus of so th hgs th obblty of ostutg th u sg t s lwys ogtv. Ls show ths sult. Flly th ovg of Algoth to th sg t wth th u t wght s ov by usg Mtgl ovg thos. Thfo th followg ls to b ov bfo sttg th oof of th Tho. L If sg t s lz wth obblty t stg.. ob W > T ] > L. Th fo vy ε 0 > Kε w hv ob 0] < ε Poof Lt ots th fl vlu of obblty wh s lg ough. Usg w lw of lg ubs w olu tht L ob > ε] 0 H fo vy ε 0 th sts ε 0 Kε < suh tht fo ll hv Pob > 0] < ε th oof of th L s olt. < > Kε w 8

9 L Lt ob W > T ] b th obblty of lzg wg sg t fo ll... t stg stvly. If volvs og to Algoth th th otol tto of s f s E ] wh δ δ ; ; ] wh ots ll ostut sg ts. Poof S th fot sh tht s us to ut th obblty vtos Algoth s LR I t h stg th obblty of hoosg th sg t.. s uhg wth obblty fo ll... wh th slt sg t s lz by th o vot. O th oth h wh th slt sg t s w th obblty of hoosg th gs of th sg t whh th slt sg t ss by gv lg t s tht of th oth gs ss. To llustt th oof of th l o tl w ov t fo th u sg t of th gh show Fgu. As show ths fgu gh G hs os 5 gs 8 sg ts s follows: } } } } } { { 5 { { 5 5 { 5 6 { } 7 { 5} 8 { 5}. It s lso ssu tht s th sg t wth th u t wght of th gh gv fgu. Fgu. Sl gh G ts u sg t Lt b th ho obblty of sg t t stg. Thfo w hv

10 0 wh ots th obblty of hoosg to α of utoto A t stg. Th otol tto of ssug s ut og to Algoth s f s ] ] ] ] ] ] ] ] } }{ }{ { } }{ }{ { } }{ }{ { } }{ }{ { } }{ }{ { } }{ }{ { } }{ }{ { } }{ }{ { ] E Aft slfyg ll ts th ght h s of th uto bov so lgb ultos w obt E δ δ 8 ; ; ] ] h th oof of th l. L Th t th otol tto of s lwys o gtv subt to s ut og to Algoth. Tht s 0 >. Poof. Df ] E. Fo L w hv ] ] E δ wh δ ; ; wh s th obblty of hoosg g t stg. S th obblty wth whh th sg ts ostut w o lz s f s th out of th obblty of hoosg th gs log th sg ts w hv δ wh δ s f s gv Euto s th obblty of lzg g t stg. At h stg Algoth tly hooss gs of th stohst gh fog o of ossbl sg ts.

11 ] E Th ulty bov b wtt s E ] s s s 0 0 fo ll S wh { : 0 ; 0 S } S ots th to of S. H 0 fo ll. S g s th g wth th u t wght whh b slt by utoto A t s show tht s > 0 fo ll s. It follows fo L tht fo lg vlus of s > 0. Thfo w olu tht fo lg vlus of th ght h s of th uto bov ossts of th ogtv utts so w hv s s s fo Euto w hv s s 0 s whh olts th oof of ths l. Coolly Th st of ut vtos S 0 0 S fos th st of ll bsobg bs of th Mov oss { } wh S { : 0; } Poof L ltly ovs tht { } s sub Mtgl. Usg Mtgl thos th ft tht { } s o gtv ufoly bou futo t s olu tht L ovgs to wth obblty o. H fo Euto t b s tht wth ozo obblty f oly f {0 } wth obblty o f oly f {0 } wh h th oof s olt. L Lt Γ b th obblty of ovg of Algoth to ut vto wth tl obblty vto. Γ s f s follows Γ ob 0 ] ob 0 ] Lt C S : S R b th stt s of ll l vlu otuously fftbl futos wth bou vtv f o S wh R s th l l. If ψ. C S th otou s f s U ψ E ψ ] wh E.] sts th thtl tto.

12 It hs b show ] tht otou s l t svs th o gtv futos s th tto of ogtv futo s ogtv. I oth wo 0 Uψ fo ll S f 0 ψ. If th otou s l fo ll > ts tly w hv ] E U ψ ψ Futo ψ s ll su gul sub gul f oly f U ψ ψ U ψ ψ fo ll S. It hs b show ] tht Γ s th oly otuous soluto of U Γ Γ subt to th followg bouy otos. Γ Γ ; 0 Df ] φ wh 0 >. ] S C φ stsfs th bouy otos bov. Tho Lt. S C ψ b su gul wth ψ 0 ψ fo th Γ ψ fo ll S. If. S C ψ s sub gul wth th s bouy otos th 5 Γ ψ fo ll S. Poof. Tho hs b ov ]. I wht follows w show tht ] φ s sub gul futo ] φ ulfs s low bou o Γ. Su sub gul futos los u to ultlto by ostv ostt f. φ s su gul th. φ s sub gul. Thfo t follows tht ] φ s sub gul f oly f ] s su gul. W ow t th otos u whh ] s su gul. Fo th fto of oto U gv Euto w hv E U ] ] ] ]

13 wh ots th fl vlu to whh th w obblty s ovg fo lg vlus of s th tto of wh th u sg t s w by th vot. U ρ ρ wh 0 > ρ s f s φ ρ ; ; o U ρ ρ ρ ρ s su gul f ρ ρ ρ ρ U If s su gul. Thfo w hv U Aft ultlyg vg th ght h s of th ulty bov by so lgb slftos w hv

14 U 0 ; 0 ; ] u u u u V u ] ] G V V U wh G s f s 6 ] ] V V G Thfo s su gul f 7 0 G fo ll S. Fo Euto 6 t follows tht s su gul f w hv 8 ] ] V V f Th ght h s of th ulty 8 ossts of th ogtv ts so w hv Substtutg by th bov ulty w wt t s Fo Euto 8 t follows tht s su gul f w hv f Fo futh slfto lt loy logths. Lt l f It hs b show ] tht

15 0 H u u H u u H u H u H u lv u u Thfo w hv 0 V ] V ] V ] V ] V ] Lt b th vlu of fo whh Euto 9 s tu. It s show tht th sts vlu of > 0 u whh th Euto 9 s stsf f s sll th fo ll. By hoosg Euto 9 hols tu. Cosutly Euto 7 s tu s su gul futo. Thfo φ ] s sub gul futo stsfyg th bouy otos gv Euto. Fo Tho Iulty 5 w olu tht ] Γ φ Fo fto of φ ] w s tht gv y ε > 0 th sts ostv ostt < suh tht ε ] Γ φ fo ll 0 <. Thus w olu tht th obblty wth whh Algoth ostuts th sg t wth th u t wght s ul to s ovgs to fty so Tho s ov. Tho Lt b th obblty of ostutg u sg t t stg ε b th obblty wth whh Algoth ovgs to sg t. If s ut by Algoth th fo vy o t ε 0 th sts lg t ε so tht wh ε 0]. Poof It hs b ov 7] tht th lwys sts > 0 u whh Euto 9 s stsf f < fo ll. H t s olu tht φ ] Γ wh s th tl ho obblty of th otl sg t. Fo Tho w hv fo h 0 < < th obblty of ovgg Algoth to th sg t wth th u t wght s ε wh ε 0. Thfo w olu tht ε 9 0 5

16 It s show tht fo vy o t ε 0 th sts vlu of u whh Euto 9 s stsf so w hv It s olu tht fo vy o t ε 0 th sts lg t ε u whh th obblty of ovgg Algoth to th sg t wth th u t wght s gt th ε h th oof of th tho. 5. Nul Rsults To stuy th ffy of th oos lgoths w hv out th st of sulto ts o fou wll ow stohst bh ghs boow fo ]. Th fst st of ts s to vstgt th ltosh btw th lg t of Algoth th o t st slg tho. Ths ts show tht fo vy o t th sts lg t suh tht fo lgoth ostut th sg t wth th u t wght. Ths st of t s out o stohst ghs Al B Al B ]. Th so st of sulto ts o th fo of th SMST lgoths oos ths Algoth to Algoth 5. Ths ts out o Al B Al B too. Th th st of ts s to show th ffy of Algoth 5 whh s th bst oos lgoth oso wth MESM ultl g sstvty tho]. I ll th st of ts out ths th fot sh by whh th to obblty vto s ut s LR. Eh lgoth stos f th I obblty of th ostut sg t PMST s ul to o gt th 0.95 o th ub of ostut sg ts s f thshol Fo h t th sults vg ov 00 fft t us. 5.. Et I Ths st of ts s out o two ss ghs ll Al Al. Al oss 9 os 5 gs Al hs 0 os gs. Th st o vbl ssot wth th g wght of Al Al hs two th stts o A B stvly. Th obblty stbuto of th o wght ssg to th gs of Al Al ts tow th sll g wghts. I oth wos th hgh obblts ssg to th gs wth sll wghts. Suh bs stbuto s o gt fo olg th two ys th sl ufo stbuto. Th fst st of t os th ffy of Algoth wth th st slg tho ts of th slg t. To o so by Tho th lg t osog to h of lvl tlly out s Tbls. Th Algoth s u wth th obt lg t ts slg t s su. Flly th slg t of Algoth s o wth tht of th st slg tho to obt th s of lvl. Th ub of sls tht th st slg tho ts fo Al B Al B to obt of lvl hs b out show Tbls stvly. To o ths w v fo 0.0 to 0.5 out th u u ub of sls fo vy g of th gh st slg tho subt toob µ < δ ] ε whδ ]. Aog to th st slg tho st 6] to obt of lvl ε fo th MST w to bul of wth lvl ε fo h g suh tht. W ssu tht th gs of th stohst gh ll hv th s of lvl ε0. Thfo sltg ε 0 ε wh ots th ub of vts of th stohst gh guts of lvl ε fo th MST0 6]. Tbl. Th totl ub of sls t fo Al B fo st slg tho Eg Cof Lvl Fo MST ٠ ٩۵ ٠ ٩٩ ۴٣۴ ۴٧٢ ۴٩٩ ٣٨۴ ۶١١ ٧۶٧ ٣٢٩ ۴۵٧ ۴٢٩ ۴۵٠ 6

17 ۴٣٢ ۴٩٠ ٣١۵ ۵٢١ ٣۵٨ ۴۵٧ ٣۶٣ ۴١٩ ۶۵۵ ٧١٢ ۵۶۴ ۴٢۶ ۴٠٧ ٣۴۵ ۴۴٨ ۶١١ ٧٠۵ ۵٠١ ٣٢٧ ۴٣۵ Totl ۶٨٨۴ ٧۴۵۵ Tbl. Th totl ub of sls t fo gh fo st slg tho Cof Lvl Fo MST Eg ٠ ٩۵ ٠ ٩٩ ۴۴٩ ۵٢۴ ۴٩٩ ۶٠٨ ۴٣۴ ۴٨١ ٣٧۶ ۴٧٩ ٨۶٩ ۶٧٠ ۴۵٢ ۴۶۴ ٣٨۴ ۴۴١ ٣٨۵ ۴١٢ ٣٩٢ ۵١۶ ۴۴۶ ۶۵٨ ۴٩١ ۴٢۴ ۵۵٠ ۴۶٣ ٣٨٨ ۴۵۶ ٣٨٨ ۵٠٠ ٣٧٨ ۴۵٩ ۵۵۵ ۵٩٣ ۵٣٣ ۴۶٣ ۴۴٢ ٣٩٣ ۴۵۴ ٣٨۵ ۵٣۶ ۵۴۶ ۴٢١ ۴٠٠ Totl ٩٨٢٩ ١٠٣۴٣ Usg th ul thos t b t by solvg uto 0 v fo Euto 0 fo vy o t. Th lg t s lult fo th obt t by solvg uto. I ths t w hg o t fo 0.5 to 0.0 o ovg t fo 50% to 99% out th osog lg t fo Al B Al B. Th w Algoth ov ths two bh ghs su th slg t of lgoth. Th obt sults show Tbls. Ths tbls lso lu th slg t of th st slg tho to hv th s of lvl t fo Tbls. Fo th sults gv Tbls t b s tht th slg t of Algoth s uh lss th tht of th st slg tho fo th s of lvl both Al B Al B. Tbl. Slg t of Algoth th st slg tho fo Al B Covg Rt Slg Rt Slg Rt Algoth St Slg Mtho Tbl. Slg t of Algoth th st slg tho fo Al B Covg Rt Lg t Slg Rt Algoth St Slg Mtho

18 Et II Ths st of ts s out to o th fo of th lg utot bs SMST lgoths st ths. I ths ts th lg t hgs fo to 0.07 h lgoth s ut o Al B Al B. Th obt sults show Tbls 5 6. I ths tbls fo h lgoth th ot sults lu th totl ub of th sls t fo th gh ST th totl ub of th sls t fo th sg t wth th u t wght MST th tg of th ovg us.. th tg of us ovg to th u sg t PC. Fo th sults of Et II th followg ots b. Th ts show tht fo ll oos lgoths th totl ub of ovg us ss th slg t s s th lg t of lgoth ss. Fo l th ovg t s 78% th ub of sls s 55 f Algoth s ut o Al B wth lg t 0.07 whs thy 98% 05 wh th lg t ss to s Tbl. Cog Algoth Algoth w obsv tht th slg t th ovg t of Algoth uh lss th thos of Algoth. Th so fo suh uto slg t of Algoth s th ft tht ths lgoth th lg t of g s ust bs o th v of ts o wght s gv Euto 7. Tht s th ub of sls tht Algoth ts fo th gs hvg sll v s sgftly lss th tht of Algoth. Th sults lso show tht fo th s ovg t th slg t of Algoth s low th Algoth. Fo th sults gv Tbls 5 6 fo Algoth t s olu tht ths lgoth s bl of ovg th ovg t of Algoth. Ths s bus Algoth ts to osto th of th o wght of g to ust ts lg t s sb Mtho II. Ths s tht th gs wth sll hv lg lg t. O th oth h t s l tht th g wth low vg wght s o lly to b t of th fl u sg t. Thfo th gs of th u sg t w o wth lg lg t so Algoth ovgs to th otl soluto wth hgh t th Algoth. Cog th sults of fft lgoths t b s tht th ovg s of Algoth s sgftly hgh th tht of th oth lgoths s t. Ths us th slg t of Algoth s o to th oths. I ths lgoth th lg t of th g ss s ts ho obblty ss. Thfo th lg t of th gs blogg to th u sg t so th ovg s of lgoth to th otl soluto ss s lgoth os. As obto of Algoth Algoth Algoth 5 hts both hgh ovg t ovg s fo ths lgoths. Fo th sults show Tbls 5 6 t s obvous tht th ovg t of Algoth 5 s uh hgh th tht of Algoth. Th obt sults lso show tht fo th s ovg t th slg t of Algoth 5 s low th tht of Algoth. Lg t Lg t Tbl 5. Coso of th oos lgoths o Al B fo fft lg ts Algoth Algoth Algoth Algoth Algoth 5 ST MST PC ST MST PC ST MST PC ST MST PC ST MST Tbl 6. Coso of th oos lgoths o Al B fo fft lg ts Algoth Algoth Algoth Algoth Algoth 5 ST MST PC ST MST PC ST MST PC ST MST PC ST MST PC PC

19 Et III Fo th sults of Et II w obsv tht Algoth 5 osbly outfos th oths both ts of th ovg t slg t. Th of th th st of ts Et III s to vstgt th fo of th bst oos lgoth Algoth 5 s oos to ultl g sstvty tho MESM oos by Hutso Sh ]. I Et III to to th ss ghs Al Al ll lgoths lso tst o two olt ghs wth 5 6 vts ll ]. Th g wght stbuto of olt ghs t stts stvly. Th obblty stbuto futos of f two fft os. I o th v of th o g wght s sll whl o t s lg. Th lg t of Algoth 5 s st to 0.07 by whh ths lgoth lwys ovgs to th sg t wth th u t wght. I ths t th ts of tst th vg ug t of lgoth so tht s show Tbl 7 th slg t of lgoth.. th totl ub of sls tht ust b t fo th gh whh s show Tbl 8. Gh Al A Al B Al A Al B E E E E Tbl 7. Th vg ug t of lgoths so Vts Egs MESM Algoth Tbl 7 shows th vg ug t of Algoth 5 MESM fo bhs Al Al so. Fo th sults t b s tht o t osug otst wth Al Al. Ths s bus of th lg stt s of th olt ghs s o to Al Al. I ft to f th otl soluto lg stt s lg ub of sls ust b t. Cog th sults of Algoth 5 MESM gv Tbl 7 t s obvous tht th ug t of Algoth 5 s gfully shot th tht of MESM sflly fo olt ghs. It ust b ot tht by hoosg lg t 0.07 th tg of th ovg us s 00 fo Algoth 5 whl th ovg t of MESM to th otl soluto s t ost 90 t. Howv th ug t of MESM s vy log th tht of Algoth 5. Ths s bus MESM ostuts th lg stt s of th obl ths tslf ts log t fo ot sz o v sll stohst ghs. Futho fg th otl stt fo tly lg stt s us lot of t. Whl Algoth 5 oly ostuts th ts of th obl stt s whh th obblty of fg th otl o otl solutos s hgh. Bss s Algoth 5 os th ostut obl stt s bos f f s th ho obblty of th o otl solutos ovgs to zo. Gh Al A Al B Al A Al B E E E E Tbl 8. Th vg slg t of fft lgoths Vts Egs MESM Algoth

20 Tbl 8 lus th sults of th sulto ts out to su th slg t of Algoth 5 MESM. Fo th obt sults w obsv tht th slg t of Algoth 5 s uh lss th tht of MESM. Ths s bus Algoth 5 fouss o th slg fo th gs of th otu sg t vos ussy sls by lug th o otl gs th gs tht o ot blog to th otl sg t fo th slg oss t h stg. I MESM slg fo th vy hug stt s of th stohst obl sults ush hgh slg t o to Algoth 5. Bfly sg th hgh ovg t low slg t of Algoth 5 of ts suoty ov MESM. 6. Coluso I ths w fst st lg utot bs lgoth ll Algoth fo fg otl soluto to th MST obl stohst ghs wh th obblty stbuto futo of th g wght s uow. Th ovg of th oos lgoth ws thotlly ov. Covg sults of tht by o ho of th lg t th obblty of hoosg th sg t wth th u t wght ovgs to o. Algoth ws o wth th st slg tho ts of th slg t th obt sults show tht ths lgoth outfos t. Th to ov th ovg t ovg s of Algoth w oos fou thos to ust th lg t of ths lgoth. Th lgoths whh th lg t s t by ths fou sttstl thos w ll s Algoth though Algoth 5. Sulto ts show tht Algoth 5 fos btt th th oth lgoths ts of th slg ovg ts Algoth hs th wost sults. To show th fo of th bst oos SMST lgoth Algoth 5 w o ts sults wth thos of th ultl g sstvty tho MESM oos by Hutso Sh ]. Etl sults of th suoty of Algoth 5 ov MESM both ts of th ug t slg t. Rfs ] O. Bouv O st Poblu Mlı bout t l obl P Movs Poov Solost 96 Vol ] J. B. Kusl O th Shotst Sg Sub T of Gh th Tvlg Sls Pobl Po. A. Mth. So 956 Vol ] R. C. P Shotst Coto Ntwos So Glztos Bll. Syst. Th. Joul 957 Vol ] H. Ish S. Sho T. Nsh Y. Nsuy "Stohst Sg T Pobl" Dst Al Mthts 98 Vol ] H. lsh T. Nsh "Stohst Bottl Sg T Pobl" Ntwos 98 Vol ] I. B. Moh "Itvl Elto Mtho fo Stohst Sg T Pobl" Al Mthts Coutto 99 Vol ] H. Ish T. Mtsuto "Cof Rgol Mtho of Stohst Sg T Pobl" Mthl. Cout. Mollg Vol. No ] C. Alooulos J. A. Jobso "Stt S Ptto Algoths fo Stohst Systs wth Altos to Mu Sg Ts" Ntwos Vol. 5 No ] K. Dhh R. Rv M. Sgh "O Two Stg Stohst Mu Sg Ts" Sg Vlg Bl

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