
aanupaathikamaaya kekku murikkal

aanupaathikamaaya kekku murikkal orutharam nyayamaaya kekku murikkalaan. oro pankaliyum thante vihita vihitam motham vihithathinte kuranjathu 1/"n" aanennu karuthunna anupatha maanadandam thripthippeduthunna oru vaividhyamaarnna vibhavathinte ("kekku") vibhajanamaanith.
aanupaathikatha charcha cheyyumbol saadhaaranayaayi rund anumanangal nadathappedunnu
aupachaarika nirvachanangal.
kekkine formula _ 1 kondu soochippikkunnu. formula _ 2 aalukalundu. oro vyakthikkum formula _ 3 nuu oru moolya function formula _ 4 undu. kekkinte oru partition, formula _ 5, "aanupathika" ennu vilikkunnu oro vyakthikkum formula _ 6 formula _ 7.
nadapadikramangal.
rund aalukalkk, vibhajikkuka, thiranjedukkuka ennathaanu classic pariharam. oru vyakthi vibhavathe thulya pakuthikalaayi vibhajikkunnu, matoraal avar eshtappedunna "pakuthi" thiranjedukkunnu. kattarinu kekku rund thulya kashanangalayi murikkan kazhiyumennu noonn-aattomitti anumaanam urappunalkunnu; adittivitti anumaanam rund pankaalikalum avarude kashanangale kuranjathu randilonnu aayi vilamathikkunnuvennu urappu nalkunnu.
eee prakriya randil kooduthal aalukalilekku vyaapippikkaan niravadhi maargangalund. oro maarggathinum athintethaaya gunangalum doshangalumundu.
lalithamaaya nadapadikramangal.
enn" aalukalkkaayi vikasippichedutha aadyakaala aanupathika vibhajana prakriyayaan avasaanathe deminishera
induction vazhi, niyamangal palikkunna oro pankaalikkum mattu pankaalikal enthu cheythaalum 1/"n" moolyam labhikkumenn urappakkaan kazhiyum. ithu marimari kalikkaan kazhiyunna oru prathyeka prakriyayaan. ettavum mosham saahacharyathil, formula _ 8 pravartthanangal aavasyamaanu oro kalikkaaranum oru aakshan. ennirunnaalum, eee pravarthanangalil bhooribhaagavum kadalaasil cheyyaan kazhiyum; yadhaarthathil kekkinte "n"-1 murivukal maathrame aavashyamulloo. athinaal, kekku aduthaanengil oro kashanavum aduthaanennu urappakkaan kazhiyum.
avasaanathe deminisherinte thudarchayaaya samaya pathippaanu doubins-spanier moving-naif nadapadikramam.
thudarchayaayi cheria "thulya" bhaagangalilekku vibhajanam thudarunna oru algorithamaanu fink protocol.
eee protocolinte prayojanam ithu oonlainil nadappilakkaan kazhiyum ennathaanu-puthiya pankaalikal partyil praveshikkumbol, muzhuvan division prakriya punaraarambhikkenda aavasyamillaathe nilavilulla division avare ulkkollunnathinayi krameekarikkunnu. oro pankaaliykkum bandhippicha orotta kashanathekkaal vichedikkappetta dhaaraalam kashanagal labhikkunnu ennathaanu poraayma.
orotta ejantu nirmmicha thulya vibhajanathe adisthaanamaakkiyulla oru prakriyayaan loonn devider. oru samamithamaaya nyayamaaya kekku murikkal nalkaan ithu saamaanyavalkkarikkam ennathaanu ithinte prayojanam.
ithum kaanuka 
aavarthana pakuthi.
oru vibhajikkukayum vijayikkukayum cheiyunna thanthram upayogichu, samayathil oru aanupathika vibhajanam nedaan kazhiyum ("n" logue "n"). laalithyathinaayi, thulya ennam pankaalikalkkaayi nadapadikramam ivide vivarichittundu, pakshe ithu ethra pankaalikalodum eluppathil poruthappedutthaam
mattu pankaalikal enthu cheythaalum niyamangalkkanusrithamaayi kalikkunna oro pankaalikkum kuranjathu 1/"n" moolyamulla oru kashanam urappunalkunnuvennu induction vazhi theliyikkaan kazhiyum.
vibhajanavum vijayavum thanthrathinu nandi, avasaanathe deminisher nadapadikramatthile ooo ("enn") nuu vipareethamaayi aavarthanangalude ennam ooo (logue "enn") maathramaan. oro aavarthanathilum oro pankaliyum orotta adayaalam undaakkendathundu. athinaal, aavashyamaaya motham maarkkukalude ennam o ("n" logue "n") aanu.
eee algorithathinu kramarahithamaya oru pathippund, athu maarkkukalude ennam kuraykkaan upayogikkaam; eevan-paas algoritam kaanuka.
thiranjeduppu nadapadikramangal.
oro pankaaliyeyum pankaalikalude ennathe aashrayich oru nischitha ennam kashanagal varaykkaan anuvadikkuka, "pii" ("enn"), oro pankaalikkum avar thiranjeduttha kashanangalil onnu nalkuka, kashanagal ovarlaappu cheyyappedaathirikkuka ennathaanu kekku murikkunnathinulla mattoru sameepanam.
oru selection nadapadikramatthinte lalithamaaya udaaharanamaayi, kekku oru dimensional edavelayaanennum oro pankaalikkum orotta thudar edavela labhikkaan aagrahikkunnuvennum karuthuka. thaazhepparayunna protocol upayogikkuka
oro aavarthanathilum oro pankaaliyudeyum paramavadhi oru edavela neekkam cheiyumennu ghattam #2 leeii selection niyamam urappunalkunnu. athinaal, oro aavarthanathinum sesham oro pankaalikaludeyum edavelakalude ennam ippozhum pankaalikalude ennathinu thulyamaanu, oro pankaaliykkum oru edavela labhikkunnathuvare prakriya thudaraam.
eee protocolil oro pankaliyum "n" chodyangalkku utharam nalkendathundu, athinaal avasaanathe deminisherinu samaanamaayi chodya sankeernnatha o ("n" 2) aanu.
kramarahithamaya pathippukal.
chodyangalude ennam kuraykkunnathinu kramarahithamaakkal upayogikkaan kazhiyum. oro pankaliyum report cheiyunnu ennathaanu ashayam, "enn" sthaanaarthikalude muzhuvan shekharavum allah, marichu kramarahithamayi thiranjeduttha sthaanaarthikalude sthiramaaya ennam "dii" maathramaan. anweshana sankeernnatha oo ("enn") aanu, ithu theerchayaayum saadhyamaaya ettavum mikachathaanu. pala kesukalilum, oro pankaaliykkum sthaanaarthikal ovarlaappu cheyyaathavidham orotta sthaanaarthi nalkaan ippozhum kazhiyum. ennirunnaalum, atharamoru vihitam asaadhyamaaya saahacharyangalund.
niravadhi vittuveezhchakal nadatthiyaal namukk ippozhum ooo ("enn") chodyangal upayogichu oru kekku murikkam
pothuvaaya padhathi iprakaaramaanu
oo (1 "eiii" 2) enna probability ullappol oro pankaaliykkum thante candidate kashanangalil pakuthiyenkilum labhikkumenn algoritam urappunalkunnu, ithu (moolyangal additteev aanenkil) kuranjathu 1/2 "enn" enna moolyathe soochippikkunnu. oo ("enn") candidate peasukal undu, oo ("enn") steppu #5 ll adhika divisionukal undu, avaa oronninum oo (1) samayamedukkum. athinaal algorithathinte motham rann-time o ("n") aanu.
eee schemile pradhana velluvili ghattam #4 leeii avasana bhagangal thiranjedukkuka ennathaanu. vishadaamshangalkku, edmunds-proose protocol kaanuka.
kaadinyam phalangal.
kaadinya phalangal robartsan-web kwayari modalinte adisthaanathil prasthaavikkunnu, athaayath.iii.avaa ejantumaarkku rund tharatthilulla chodyangal chodikkunna nadapadikramangalumaayi bandhappettirikkunnu "vilayiruthuka", "adayaalappeduthuka".
n" ≤3 pankaalikalkulla oro nirnnayaka aanupathika vibhajana prakriya ellah moolyanirnayangalum samaanamaanengilum kuranjathu "n" anweshanangalengilum upayogikkanam.
koodaathe, oro vyakthikkum oru thudarchayaaya kashanam nalkunna oro nirnnaayakamo kramarahithamo aaya anupatha vibhajana nadapadikramavum ω ("n" logue "n") pravartthanangal upayogikkanam.
koodaathe, oro nirnnayaka aanupathika vibhajana prakriya ω ("n" log "n") chodyangal upayogikkanam, oro pankaaliykkum edavelakalude samyojanamaaya oru kashanam nalkaan nadapadikramam anuvadichittundenkilum, nadapadikramam "ekadesha neethi" urappu nalkaan maathrame anuvadichittulloovenkum. oru kalikkaaran, moolyathil sambannavum veethiyil nertthathumaaya oru kekku kandethanulla sankeernathaye adisthaanamaakkiyullathaanu thelivu.
eee kaadinya phalangal soochippikkunnathu thudarchayaaya kashanagal upayogichu poornna aanupaathikatha kaivarikkunnathinulla ettavum vegathayeriya algorithamaanu recursive hafving, koodaathe bhagika aanupaathikatha nedunnathinum vichedikkappetta kashanagal polum nedunnathinumulla ettavum vegathayeriya nirnnayaka algorithamaanithu. vichedikkappetta kashanangalkk bhagika aanupaathikatha urappunalkunna kramarahithamaya algorithangal upayogichu maathrame ithu mechappeduthaan kazhiyoo.
parimithamaaya krithyathayode maathrame kalikkaarkku murikkan kazhiyoovenkil, ω (n logue n) lover boundil kramarahithamaya protocolukalum ulppedunnu.
thaazhepparayunna pattika ariyappedunna phalangal samgrahikkunnu
vakabhedangal.
vyathyastha avakashangal.
pankaalikalkk thulyamaaya avakashangal illaatha saahacharyangalil aanupaathikatha maanadandam saamaanyavalkkarikkam. udaaharanathinu, resource rund shareholdermarudethaayirikkam, athaayath alis 8/13 kaivasham vaykkunnu, jorj 5/13 kaivasham vaykkunnu. ithu "waited proportionality" (wpr) yude maanadandatthilekku nayikkunnu 1 vare thukayulla niravadhi bharangal "vaii" undu, koodaathe oro pankaliyum "aii" avarude swantham moolyanirnnayathiloode resourcinte oru bhaagamenkilum "vaii" sweekarikkanam. oru doubleupr division kandethaan niravadhi algorithangal upayogikkaam. rund pankaalikal maathram ullappol polum vettikkuravukalude ennam valuthayirikkaam ennathaanu pradhana velluvili. vyathyastha avakashangalulla aanupathika kekku murikkal kaanuka.
suuppar aanupathika vibhajanam.
oro pankaalikkum avarude swantham aathmanishtamaaya moolyanirnnayathiloode 1/n ll kooduthal vibhavangal karsanamaayi labhikkuna oru vibhagamaanu "suuppar-proportional division".
theerchayaayum atharamoru vibhajanam ellaayppozhum nilanilkkunnilla ellah pankaalikalkkum ore moolyamulla pravartthanangal ullappol, oro pankaaliykkum krithyamaayi 1/"n" nalkuka ennathaanu namukk cheyyaan kazhiyunna ettavum nallath. athinaal oru suuppar-proportional division nilanilkkunnathin aavashyamaaya oru vyavastha ellah pankaalikalkkum ore moolyaparidhi illa ennathaanu.
moolyanirnayangal aditteevum noonn-aattomik aayirikkumbol, eee avasthayum mathiyakum ennathaanu athisayakaramaaya vastutha. aii.iii.kuranjathu rund pankaalikalenkilum moolya pravartthanatthil alppam vyathyastharaanenkil, "ellah" pankaalikalkkum 1/"n" ll kooduthal labhikkuna oru suuppar-proportional division undu. vishadaamshangalkku suuppar-proportional division kaanuka.
sameepasthathayude niyanthranam.
ellah kashanangalum bandhippikkenda saadhaarana niyanthranathinu purame, chila sandarbhangalil adhika niyanthranangalund. prathyekichum, vibhajikkaanulla "kekku" niravadhi raajyangalkkidayil tharkkapradeshamaayirikkumbal, oro rajyathinum anuvadicha kashanam athinte nilavile sthalathodu chernnathayirikkanam. eee propertyumaayi oru aanupathika vibhajanam ellaayppozhum nilanilkkunnu, avasaanathe deminisher protocol konformal mappingukal ulppedunna jyaamitheeya thanthrangalumaayi samyojippichu ithu kandethaam. hill-beck land division prashnam kaanuka.
dvimaana jyaamitheeya parimithikal.
vibhajikkenda "kekku" oru land estettu allengil print allengil electronic mediayile oru parasya idam polulla dvimaanamaakumbol, connectivitic purame chila jyaamitheeya parimithikal kashanagal niravettendathundu. udaaharanathinu, oro kashanavum oru chathuram, oru kozhuppu deerghachathuram allengil saadhaaranayaayi oru kozhuppu vasthu aayirikkendathundu. atharam kozhuppu parimithikalode, saadhaaranayaayi aanupaathikamaaya oru vibhajanam nilavililla, pakshe bhagikamaaya aanupaathikamaaya oru vibhajanam saadhaaranayaayi nilanilkkunnu, kaaryakshamamaya algorithangal vazhi ithu kandethanakum.
saambathikamaayi kaaryakshamamaya vibhajanam.
aanupaathikamaayirikkunnathu, vibhajanam palappozhum saambathikamaayi kaaryakshamamaayirikkendu, athaayath.iii.saamoohikakshemam paramavadhi vardhippikkuka (ellah ejantumaarudeyum upayogangalude aakethukayaayi nirvachichirikkunnu).
udaaharanathinu, 500 gram choklettum 500 gram vanila adangiya oru kekku pariganikkuka, rund pankaalikalkkidayil vibhajikkappettirikkunnu, avaril oraalkk choklettu maathram venam, mattoraalkk vaanila maathram venam. pala kekku murikkal protocolukalum oro ejantinum 250 gram choklettum 250 gram vanila nalkum. oro pankaalikkum avarude motham moolyathinte 0.5 labhikkunnathinaalaan eee vibhajanam aanupaathikamaayirikkunnath, athinaal saadhaarana saamoohikakshemam 1 aanu. ennirunnaalum, eee vibhajanam valare kaaryakshamamalla, kaaranam namukk ellah choklettukalum oru pankaalikku nalkaanum ellah vaanilakalum mattoru pankaalikku nalkaanum kazhiyum, ithu 2 enna saadhaarana saamoohika kshemam kaivarikkunnu.
saadhyamaaya ellah aanupathika vihithangalilum saamoohikakshemam paramavadhi vardhippikkunna aanupathika vihitam kandethunnathinulla prasnamaanu "opttimal aanupathika vibhajanam" prashnam. kekku oru dimensional interwell aayirikkukayum yoottility density functionukal leanier aayirikkukayum cheiyunna valare prathyeka kesil maathrame eee prashnathinu nilavil oru parihaaramulloo (i.iii. "u" ("x") = "" x "" + "" b "") ". pothuve prashnam enn. pii.-haard aanu. yoottility functionukal saadhaarana nilayilaakkaathappol (i.iii. oro pankaalikkum muzhuvan kekinum vyathyastha moolyam nalkaan njangal anuvadikkunnu), prashnam 1/enna gadakathinullil enpi-haard muthal ekadeshamaanu.
sathyasandhamaya vibhajanam.
sathyasandhatha oru vibhajanathinte swathalla, marichu protocolinte swathaanu. aanupathika vibhajanathinulla ellah protocolukalum "durbalamaaya sathyasandhatha" ullavayaan, kaaranam oro pankaliyum thante yadhaartha moolyanirnayathinu anusrithamaayi pravartthikkumbol mattu pankaalikal enthu cheythaalum kuranjathu 1/"n" (allengil bhagika aanupathika protocolinte kaaryatthil 1/"an") labhikkumenn urappaan. matellaa pankaalikalum thanne upadravikkuka enna uddeshyathode oru sakhyamundaakkiyaalum, addehathinu urappulla anupatham labikkum.
ennirunnaalum, mikka protocolukalum "shakthamayi sathyasandhamalla", chila pankaalikalkk urappulla vihithathekkaal kooduthal labhikkunnathinaayi nuna parayaan prachodanam undayirikkaam. lalithamaaya vibhajanathin protocol thiranjedukkunnathinum polum ithu sariyaanu kattarinu chosarinte mungananakal ariyaamenkil, chosar randilonnil alppam kuravaanennu kanakkaakkunna oru kashanam murikkan kazhiyum, ennaal cutter thanne randilonninekkaal kooduthal vilamathikkunnu.
oru thikanja vibhajanam nedunnathinu sathyasandhamaya samvidaanangalundu; oru thikanja vibhajanam aanupaathikamaayathinaal, iva aanupathika vibhajanathinulla sathyasandhamaya samvidaanangalum koodiyaanu.
eee samvidaanangal nilavilirikkumbol oru suuppar-proportional vibhajanam nalkunnathinu vipuleekarikkaan kazhiyum
oru suuppar-proportional division nilanilkkumbol, athu randaam ghattathil thiranjedukkappedaanulla nalla saadhyathayund. athinaal oro sathyasandha pankaaliyudeyum prathekshikkunna moolyam 1/"n" ll kooduthalaanu. samvidaanam sathyasandhamaanennu kaanunnathin, moonnu kesukal pariganikkuka (eiii) thiranjeduttha partition sarikkum suuppar proportional aanenkil, nunayude eaka phalam athu anganeyallennu chinthikkaan samvidaanathe thettidharippikkuka ennathaanu; ithu samvidaanathe oru thikanja vibhajanam nadappilakkaan sahayikkum, ithu nunayanmaar ulppede ellah pankaalikalkkum moshamayirikkum. (bii) nuna parayunnavarkk maathram 1/"enn" allengil athil kuravu moolyam nalkunnathinaal thiranjeduttha partition suuppar proportional allengil, nunayude oreyoru phalam partition "suuppar proportional" aanennu samvidaanathe chinthippikkukayum athu nadappilaakkukayum cheyyuka ennathaanu, ithu nunayan thanne dosham cheiyunnu. (sii) thiranjeduttha partition yadhaarthathil suuppar proportional allengil athu mattoru pankaalikku 1/"n" allengil athil kuravu moolyam nalkunnathinaal, partition oru saahacharyathilum nadappaakkaathathinaal nuna parayunnath oru phalavum undakkilla.
jolikalude aanupathika vibhajanam.
vibhajikkaanulla vibhavam abhikaamyamallaathappol (coor divishanilennapole), oro vyakthikkum vibhavathinte "paramavadhi" 1/"n" nalkunna oru divisionaanu aanupathika vibhajanam ennu nirvachikkappedunnu (i.iii. asamathvathinte adayaalam vipareethamaanu).
aanupathika vibhajanathinulla mikka algorithangalum coor vibhajanathinu nerittu anuyojyamaakkaan kazhiyum.