.

Метод та засоби статичного планування задач у паралельних обчислювальних системах: Автореф. дис… канд. техн. наук / М.А. Сааде, Нац. техн. ун-т Укра

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IAOe?IIAEUeIEE OAOI?*IEE OI?AA?NEOAO OE?A?IE

“EE?ANUeEEE IIE?OAOI?*IEE ?INOEOOO”

E?AAI

NAAAeA IOOAIIAAe AE?I

OAeE 681.324

IAOIAe OA CANIAE NOAOE*IIAI IEAIOAAIIss CAAeA*

O IA?AEAEUeIEO IA*ENETHAAEUeIEO NENOAIAO

Niaoe?aeuei?noue 05.13.13 — Ia/enethaaeuei? iaoeie, nenoaie oa ia?aae?.

Aaoi?aoa?ao aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oaoi?/ieo iaoe

Ee?a – 1998

Aeena?oaoe?y ? ?oeiien

?iaioa aeeiiaia o Iaoe?iiaeueiiio Oaoi?/iiio Oi?aa?neoao? Oe?a?ie

“Ee?anueeee Iie?oaoi?/iee ?inoeooo”

ia eaoaae?? ia/enethaaeueii? oaoi?ee

Iaoeiaee ea??aiee —aeieoi? oaoi?/ieo iaoe, i?ioani? eaoaae?e

ia/enethaaeueii? oaoi?ee Iaoe?iiaeueiiai Oaoi?/iiai Oi?aa?neoaoo Oe?a?ie
Neiiiaiei Aaea??e Iaaeiae/.

Io?oe?ei? iiiiaioe:

Aeieoi? oaoi?/ieo iaoe, i?ioani? Ia/o?ei Ieeieae Eai?oiiiae/, canooiiee
ia/aeuei?ea aieiaiiai oi?aae?iiy I?i?noa?noaa ina?oe Oe?a?ie.

Eaiaeeaeao oaoi?/ie/ iaoe, aeioeaio Ua?aeia Ieaenaiae? Aiae??eiae/,
aeioeaio Ee?anueeiai aea?aeaaiiai Oi?aa?neoaoo a?oeoaeoo?e oa
aoae?aieoeoaa.

I?ia?aeia onoaiiaa:

?inoeooo i?iaeai ?aaeno?aoe?? ?ioi?iaoe??.

A?aeae?e – i?iaeaiii – i??aioiaaieo ?ioi?iaoe?eii-ia/enethaaeueieo
nenoai

Caoeno a?aeaoaeaoueny 28.12.1998 ?. a 14.30 aiaeei ia can?aeaii?
niaoe?ae?ciaaii? ?aaee Ae26.002.02 o Iaoe?iiaeueiiio Oaoi?/iiio
Oi?aa?neoao? Oe?a?ie “Ee?anueeee Iie?oaoi?/iee ?inoeooo”

(i. Ee?a, i?. Ia?aiiae 37, ei?i. 18, aoae. 306)

A?aeaoee ia aaoi?aoa?ao o aeaio i?ei??ieeao, caa??ai? ia/aoeith
onoaiiae,

i?ineii iai?aaeoe ca aae?anith: 252056, i. Ee?a, i?. Ia?aiiae 37,

a/aiiio nae?aoa?th IOOO “EI?”.

C aeena?oaoe??th iiaeia iciaeiieoeny o a?ae?ioaoe?

Iaoe?iiaeueiiai Oaoi?/iiai Oi?aa?neoaoo Oe?a?ie “EI?”

Aaoi?aoa?ao ?ic?neaiee ____________1998 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee I.I.I?eiaa

AIIOAOe?ss

Iaoith aeena?oaoe?eii? ?iaioe ? ?ic?iaea oa aeine?aeaeaiiy iaoiae?a e
cania?a aoaeoeaiiai ?iciiae?eaiiy ia/enethaaeueieo ?ia?o oa ?oiuei?
iioei?caoe??, aeicaieyth/eo ciaiueoeoe /aniao neeaaei?noue noaoe/iiai
ieaioaaiiy caaea/ o ia?aeaeueieo ia/enethaaeueieo nenoaiao.

Aeey aeinyaiaiiy iieeaaeaii? iaoe o aeena?oaoe?? ae??oothoueny ianooii?
ieoaiiy:

Aiae?c a?o?oaeoo?ieo iniaeeainoae ia?aeaeueieo ia/enethaaeueieo nenoai
(IIN), aeae?eaiiy oa eeaneo?eaoe?y ocaaaeueiaieo a?o?oaeoo?ieo
oa?aeoa?enoee, aecia/ath/eo i?eioeeie oi?aae?iiy e i?aai?caoe??
ia/enethaaeueieie i?ioeanaie a IIN.

Ii??aithth/ee aiae?c oa eeaneo?eaoe?y iaoiae?a oa aeai?eoi?a noaoe/iiai
ieaioaaiiy ?ia?o o ia?aeaeueieo ia/enethaaeueieo nenoaiao e aeae?eaiiy
iniaeeainoae oa aeiia aei nenoaie noaoe/iiai ieaioaaiiy.

Oai?aoe/ia iaa?oiooaaiiy e ?ic?iaea iaoiaeo no?oeoo?ii ( naiaioe/iiai
aiae?co ii/aoeiai? ?ioi?iaoe?? c iaoith i?aeaeuaiiy aoaeoeaiino?
ieaioaaiiy ?ia?o a N?IAe.

Oai?aoe/ia iaa?oiooaaiiy e ?ic?iaea aeai?eoi?a noaoe/iiai ieaioaaiiy oa
?iciiae?eaiiy ?ia?o a IIN.

Ia caoeno aeianaia ianooiia.

Caaaeueia eeaneo?eaoe?y ia?aeaeueieo nenoai ia?iaee aeaieo, o?aoiaoth/a
iniaeeaino? a?o?oaeoo?e no/anieo ia?aeaeueieo ia/enethaaeueieo nenoai oa
i?aai?caoe?? ia/enethaaeueieo i?ioean?a a IIN.

Iaoaiaoe/ia iinoaaeaiiy caaea/? ?iciiae?eaiiy e iioei?caoe?? i?ioean?a
i?e noaoe/iii ieaioaaii? ?ia?o a IIN.

Oaa?aeaeaiiy, iieeaaeai? aei iniiae aaeaioeaiiai aeai?eoio ?iciiae?eaiiy
caaea/ o ia?aeaeueiiio ia/?nethaaeueiiio ioi/aii?.

Aaeaioeaiee aeai?eoi ni?yiiaaiiai iiooea ??oaiiy caaea/? ?iciiae?eaiiy
?ia?o oa iioei?caoe?? i?inoi?iai — /aniaeo oa?aeoa?enoee ?iceeaaeo i?e
noaoe/iii ieaioaaii? a IIN.

Neaioth/ee aeai?eoi i?inoi?iai ( /aniaiai ?iciiae?eaiiy ?ia?o a IIN c
o?aooaaiiyi aea?aieo iaiaaeaiue aei nenoaie ieaioaaiiy ?ia?o.

Aeai?eoi no?oeoo?ii ( naiaioe/iiai aiae?co ii/aoeiai? ?ioi?iaoe?? oa
ae?iaee no?aoaa?? aaeainiai? iioei?caoe?? i?inoi?iai – /aniaiai
a?aeia?aaeaiiy iia’ycaiiai a?aoa caaea/ a IIN ia iniia? ?ic?iaeaieo
i?ioeaaeo? aa?oeeaeueii – ai?eciioaeueii? eeanoa??caoe?? e aeae?eaiiy
aa?oei, iath/eo iciaee “yaii?”, “i?eoiaaii?” oa “ioeueoeieeeaoeaii?
o?aiceoiino?”.

?acoeueoaoe noaoenoe/iiai aeine?aeaeaiiy ?ic?iaeaieo iaoiaeia
aeeiai?/iiai ieaioaaiiy i?ioean?a a N?IAe oa ii??aiyiiy ?o c a?aeiieie.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie.

C?inoaiiy i?iaeoeoeaiino? IN ? iianye/aniei ia’?eoii aeine?aeaeaiue ia
i?ioyc? onueiai ia??iaeo ?icaeoeo cania?a IO.

Ia??ai? c i?aeaeuoaaiiyi i?iaeoeoeaiino? i?ioeani??a ?ioaineaii
i?iaiaeyoueny ?iaioe ii oaeineiiaeaiith ia?aeaeueieo nenoai ia?iaee
aeaieo, aeey yeeo ieoaiiy i?aai?caoe?? ieaioaaiiy e ia/enethaaiiy ua
a?eueo oneeaaeiththoueny. O?aaa a?aei?oeoe, ui oaoi?/ia ?aae?caoe?y
aenieiia?aeaeueieo IN aeia?aaeaea? ?ic?iaeo nenoaiiiai i?ia?aiiiai
caaacia/aiiy, ae??uoth/eo caaea/? iioeiaeueiiai ?ici?uaiiy ia?aeaeueieo
i?ia?ai o ia/enethaaeueia ioi/aiiy.

A?aeiii aaaaoi ?ia?o a aaeoc? oai??? ?icia?aeaethaaiiy ia/enethaaiue oa
?aae?caoe?? i?aeoe/ii ae?th/eo nenoai ?icia?aeaethaaiiy aeai?eoi?a ?
i?ia?ai, ia?aaeaa/ath/eo iiaeeeaino? aueiiaiiy iiia?aaeaeoaaeueii?
i?aeaioiaee ?ia?o a aaaaoii?ia?aiiiio ?aaeei? ?iaioe nenoaie. Aea
aeine?aeaeaiiy a aaeoc? oai??? iiaoaeoaaiiy nenoai ieaioaaiiy ?ia?o a
IIN ua ia aeaee aeinoaoiuei aoaeoeaieo i?aeoe/ieo aeai?eoi?a ? i?ia?ai
aeey IIN, oiaoi oaeeo, ye? a aeicaieeee c iacia/ieie iaeeaaeieie
aeo?aoaie aeei?enoiaoaaoe ?o aeey noai?aiiy ?IN Aiae?c ?acaeoeo IN a
inoaii? ?iee iieaco?, ui noai?aiiy oa aeei?enoaiiy noaiaea?o?a aeey
aenieioaeaee?niiai ca’yceo, oaeeo ye FDDI, HiPP, SONET, aeicaiey?
aoaeoaaoe aeiaaeuei? IN, c’?aeioth/? ??ciiiai?oi? oaeaeeiae?th/?
a?aeaeaeai? ia/enethaaeuei? iaoeie aeey nieeueii?, aoaeoeaii? oa
aeiiiiii? ia?aaee ??cieo caaea/.

??oaiiy naii? caaea/e ieaioaaiiy oa ?iciiae?eaiiy ?ia?o c o?aooaaiiyi
on?o aeiia aei nenoaie ieaioaaiiy, e ia?aiao??a IN ?/aai aoiaeiiai
iioieo caaaeaiue ? e?eoe/ii neeaaeiith. Aiia yaey? niaith noeoii?noue
aeae?eueea neeaaeii ae??ooaaieo caaea/, iaeaaea/eo aei eeano NP-iiaieo.

Oaeei /eiii, aeine?aeaeaiiy e ?ic?iaea iaoiae?a ? cania?a i?aai?caoe??
ia/enethaaeueieo i?ioeanea a iai?aeaeueieo nenoaiao ia?iaee aeaieo ? ?o
iiaeaethaaiiy, i?aeoe/ia ?aae?caoe?y ? ai?iaaoe?y ? aeooaeueieie,
iathoue oai?aoe/io cia/ei?noue e i?aeoe/iee ?ioa?an.

Ia’?eoaie aeine?aeaeaiue ? iaoaiaoe/i? iiaeae? aeai?eoi?a noaoe/iiai
ieaioaaiiy, i???ioiaai? ia aeei?enoaiiy aaeaioeaieo iaoiae?a
iai?aaeaiiai iiooeo ??oaiue a ia?aeaeuei?o IN.

Iaoiaee aeine?aeaeaiue a?oioothoueny ia aeei?enoaii?: oai??? a?ao?a,
oai??? iiiaeei, oai??? iiaeaethaaiiy, iaoiae?a eiia?iaoi?ii?
iioei?caoe??, a oaeiae oaa?aeaeaiue, ye? aeiaaaeai? a aeena?oaoe??.
Caaaeuei? iieiaeaiiy e oai?aoe/i? ioe?iee i?aeoaa?aeaeai? ?acoeueoaoaie
?i?oaoe?eiiai iiaeaethaaiiy ia YII.

Iaoeiaa iiaecia. Cai?iiiiiaaii iiaee i?aeo?ae aei ??oaiiy caaea/
noaoe/iiai ieaioaaiiy a ia?aeaeuei?e ia/enethaaeuei?e nenoai? ia?iaee
aeaieo, nooue yeiai o iiia?aaeiueiio no?oeoo?iiio aiae?c? ii/aoeiai?
?ioi?iaoe?? ? a?cia/aii? aoaeoeaii? aaeaioeaii? no?aoaa?? ??oaiiy
caaea/? noaoe/iiai ieaioaaiiy, aa?aiooth/i? io?eiaiiy iioei?c??iaaiiai
??oaiiy ii e?eoa??yi iioei?caoe?? i?e cia/iiio ciaiueoaii? /ano
ieaioaaiiy. I?e oeueiio io?eiai? ianooii? ?acoeueoaoe:

?ic?iaeaia caaaeueia iiaeaeue noaoe/iiai ieaioaaiiy aeey nenoai
ia?aeaeueii? ia?iaee aeaieo, aeicaieeaoa ?iciiae?eeoe ?? ia i?aecaaea/?
e cae?enieoe ?o iaoaiaoe/io iinoaiiaeo.

Noi?ioeueiaai? oaa?aeaeaiiy e ne?aenoaa, aeiaiaey/? ei?aeoi?noue e
aenieo aoaeoeai?noue ?ic?iaeaieo iaoiae?a e cania?a ieaioaaiiy.

?ic?iaeai? aeai?eoie iai?aaeaiiai iiooeo aa??aioo ?iceeaaeo aeeiiaiiy
?ia?o a IIN, ye? a?oioothoueny ia no?aoaa?? iioaaiaiai eiino?othaaiiy
?iceeaaeo aeey caaea/? noaoe/iiai ieaioaaiiy e iioei?caoe??
?iciiae?eaiiy ?ia?o a IIN.

I?aeoe/ia oe?ii?noue ?acoeueoao?a aeena?oaoe?eii? ?iaioe ? a oiio, ui
aeei?enoaiiy cai?iiiiiaaiiai iaoiaeo ? cania?a aeicaiey? i?aeaeueoe
aoaeoeai?noue ieaioaaiiy a IIN.

E??i oiai, iia? aeai?eoie e i?ia?aie aaeaioeaieo ieaioaaeueiee?a
i?ecia/ai? aeey ??oaiiy oe?ieiai eeano caaea/ noaoe/iiai ieaioaaiiy
?ia?o a IIN c o?aooaaiiyi ye iniaeeainoae ?icia?aeaeaieo aeai?eoi?a
??oaiiy i?eeeaaeieo caaea/, oae e iniaeeainoae oiiieia?? ? oa?aeoa?enoee
ia/enethaaeueiiai ioi/aiiy iai ia?aae?. Iia? aeai?eoie oa i?ia?aie
a?ae??ciythoueny a?ae a?aeiieo /eiaei iaiueoith /aniaith neeaaei?noth.

O aeena?oaoe?eio ?iaioo aeeth/ai? ?acoeueoaoe, io??iai? aaoi?ii c 1995
ii 1997 ??. Ia eaoaae?? ia/enethaaeueii? oaoi?e? Ee?anueeiai
iie?oaoi?/iiai ?inoeoooo i?e aeeiiaii? aeiaiai?o i?i iaoeiaa -oaci?/ia
ni?a?ia?oieoeoai c iie?oaoi?/iei ?inoeoooii i. *ainoioiai (?anioae?ea
Iieueoa).

Iaoiaeieia?/i? oa iaoeia? aniaeoe aeena?oaoe?eieo aeine?aeaeaiue iaeoee
naith ?aae?caoe?th a o/aiai-iaoiaee/iiio caaacia/aii? eaeoe?eieo eo?n?a,
i?iaaaeaii? eaai?aoi?ieo ? ea??aieoeoa? eo?niaeie ?iaioaie ii eoinai
«I?aai?caoe?y ia/?nethaaeueieo i?ioeann?a a AII, eiiieaenao, ia?aaeao oa
nenoaiao» ? «Iia?aoe?ei? nenoaie» nooaeaio?a oaeoeueoaoo ?ioi?iaoeee oa
ia/enethaaeueii? oaoieee Iaoe?iiaeueiiai oaoi?/iiai oi?aa?n?oaoo Oe?a?ie
(«EI?»)

A??ia?aei?noue oai?ai, oaa?aeaeaiue ? aeniiae?a i?aeoaa?aeaeo?oueny
oi/ieie oai?aoe/ieie aeiaaaeaiiyie e ?acoeueoaoaie noaoe/iiai
iiaeaethaaiiy.

Ai?iaaoe?y ?iaioe. Caaaeuei? iaoeia? ?acoeueoaoe aeena?oaoe?eii? ?iaioe
aoee iiaeai? e iaaiai?thaaeeny ia XXIV i?aeia?iaei?e eiioa?aioe?? ? a
aeeneon?eiiio iaoeiaiio eeoa? ”Iia? ?ioi?iaoe?ei? oaoiieia?? a iaooe?,
ina?o? e a?cian?: New Information Tecnology in Science, Education and
Business, IT+SE`97” (Oe?a?ia, E?ei, sseoa-Ao?coo, 15-24 o?aaiy 1997
?ieo), ia iaoeiaeo nai?ia?ao a IOOO(EI?).

Ioae?eaoe??. Ii iaoa??aeai aeena?oaoe?? iioae?eiaaii 4-e iaoeiaeo
?iaioe.

No?oeoo?a ? ia’?i ?iaioe. Aeena?oaoe?eia ?iaioa neeaaea?oueny c
anooio, /aoe?ao aeaa ? aeniiaeo, aeeeaaeaieo ia 148 noi??ieao
iaoeiiieniiai oaenoo, ia? 97 iaethie?a ? oaaeeoeue ia 10 noi??ieao,
nienie e?oa?aoo?e c 140 iaeiaioaaiue oa aeaa aeiaeaoee.

O anooi? iaa?oioiao?oueny aeooaeuei?noue oaie aeena?oaoe?eii? ?iaioe,
oi?ioeth?oueny iaoa ? caaea/? aeine?aeaeaiiy, caaaeuei? iieiaeaiiy, ye?
i?iiiiothoueny aei caoenoo.

O ia?oiio ?icae?e? ?icaeyaeai? ieoaiiy i?aai?caoe?? ia/enethaaiue a
ia?aeaeueieo ia/enethaaeueieo nenoaiao, aeine?aeaeaii aieea a?o?oaeoo?e
ia/enethaaeueieo nenoai ia?aeaeueii? ia?iaee ?ioi?iaoe?? (IIN) ia
i?aai?caoe?th ? oi?aae?iiy ia/enethaaiiyie.

O ae?oaiio ?icae?e? cae?eniaii aiae?c i?eioeei?a iiaoaeiae ia?aeaeueieo
iia?aoe?eieo nenoai aeey IIN, iaoiae?a, no?oeoo? oa aeai?eoi?a
noaoe/iiai ieaioaaiiy ? aeeniao/a??caoe??, aecia/ai? aeiiae ? e?eoa???
iioei?caoe?? ??oaiiy caaea/ ieaioaaiiy oa ?iciiae?eaiiy, aecia/aia
caaaeueia iiaeaeue nenoaie noaoe/iiai ieaioaaiiy.

O o?aoueiio ?icae?e? iiaeaii iaa?oiooaaiiy iaoiaeo no?oeoo?ii –
naiaioe/iiai aiae?co ii/aoeiai? ?ioi?iaoe??, i?eaaaeai? oaa?aeaeaiiy,
iieeaaeai? a iniiao i?ioeaaeo? eieae?caoe?? ciie iiooeo ? aeai?eoio
neaioth/iai i?inoi?iai – /aniaiai ?iciiae?eaiiy aca?iica’ycaieo ?ia?o ii
?ano?nai i?e ae??oaii? caaea/ noaoe/iiai ieaioaaiiy aeey nenoai c?
ni?eueiith iai’yoth ? ?iciiae?eaieo nenoai. ?icaeyaeaii aa?enoe/iee
aeai?eoi iioei?coth/iai iaeoaaiiy ?ia?o i?e ae??oaii? caaea/?
ai?eciioaeueii? eeanoa?ecaoe??. ?icaeyaeaii aeai?eoi noaoe/iiai
ieaioaaiiy c o?aooaaiiyi eiioi?eaoe?eieo aeo?ao.

O /aoaa?oiio ?icae?e? iiaeaii iaa?oiooaaiiy ia?aii? nenoaie
iiaeaethaaiiy, aecia/ai? e?eoa??? ii??aiyiiy oa i?eaaaeai? ?acoeueoaoe
noaoenoe/iiai ii??aiyeueiiai aiae?co ?ac?iaeaieo aeai?eoi?a ? i?ia?ai c
iaea?eueo a?aeiieie, /ei i?aeoaa?aeaeothoueny ?acoeueoaoe oai?aoe/ieo
aeine?aeaeaiue.

O aeniiaeo i?eaaaeai? caaaeuei? oai?aoe/i? oa i?aeoe/i? ?acoeueoaoe
?iaioe, ?aeiiaiaeaoe?? ii ?o aeei?enoaiith.

INIIAIEE CI?NO ?IAIOE

Nenoaia ia?aeaeueii? ia?iaee aeaieo (NIIAe) – oea noeoii?noue oaoi?/ieo
cania?a (OC) ? i?ia?aiiiai caaacia/aiiy (IC), i?ecia/aia aeey
?ioi?iaoe?eiiai ia?aeaeueiiai ianeoaiaoaaiiy ei?enooaa/?a oa oaoi?/ieo
ia’?eo?a. ?niothoue ??ci? i?aeoiaee aei eeaneo?eaoe?? NIIAe aai
ia?aeaeueieo ia/?nethaaeueieo nenoai (IIN).

A IIN (aaaaoii?ioeani?ieo, aaaaoiiaoeieo eiiieaenao, ia?aaeao) caaea/a
ieaioaaiiy aeey ?iciiae?eaiiy cayaie (caaaeaiue, i?ioean?a…) ia
i?aeoiaeeo ?ano?nao nenoaie ? iaei??th c iaea?eueo neeaaeiiae??ooaaieo a
i?aai?caoe?? ia/enethaaeueieo i?ioean?a ? oai??? ?iceeaae?a.
Ia/enethaaeueio nenoaio oa ia/enethaaeueiee aocie IIN a caaaeueiiio
aeaeyae? iiaeeeai ?icaeyaeaoe c ??ciei nooiaiai aeaoae?caoe?? ye
noeoii?noue ?ano?n?a, a ia’?eoe ieaioaaiiy ia/enethaaiue aai “caaaeaiue”
ia ?ano?ne iiaeooue aooe iiaeai? ye i?ia?aie, i?ioeaaeo?e, ia?aeaeuei?
/anoee i?ia?ai, ie?ai? aeiee eiiaiae, eiiaiaee ? ie?ai? iae?i aai
i?e?iiia?aoe??.

?iciiae?eaiiy caaaeaiue (?ia?o, i?ioean?a) ia ?ano?ne a nenoaiao
ianiaiai ?icia?aeaethaaiiy (IIN-iaeii??aeieo nenoaiao) aai o
?iciiae?eaieo nenoaiao (?N-iaiaeii??aeieo nenoaiao) ia? oaeo caaaeueio
iinoaiiao [1,2]:

Nenoaia ?ano?n?a caaeaia a?aoii nenoaie GR=(VR,ER,WVR,WER), aea:

VR= {R1,R2, …,RN}- iiiaeeia aa?oei, eaaeiee aeaiaio yeiai iiaea? iaeei
c N ?ano?n?a nenoaie ? Ri((( , i=1..N.

ER={E1,E2,…,Ed}- iiiaeeia aeoa iienoth/eo ca’ycee iaeiiai ?ano?no c
?ioei Ei={Ri,Rj}, aea Ri,Rj(VR e 0 ( d ( N2.

WVR={WVR1,WVR2,…,WVRN}- iiiaeeia aaa aa?oei, aea WVRi={REi,RTi}. Aeey
(i=1..N, REi((+ – oa?aeoa?enoeea ?ano?no Ri, RTi({0 e (+} – noai
?ano?n?a.

WER={WER1,WER2,…,WERp }- iiiaeeia aaa aeoa. Oeth iiiaeeio iiaeia
iiaeaoe o aeaeyae? aeayei? iao?eoe? RC=RC[i,j]((+, aea i=1..N e j=1..N.

JPi((+ – i??i?eoao aeaiiai caaaeaiiy.

A i?aoeyo, i?enay/aieo ??oaiith caaea/ noaoe/iiai ieaioaaiiy
a?aei?/a?oueny, ui aiie ? neeaaeiiae??oaeueieie ? iaeaaeaoue eeano
NP-iiaieo. I?aoe?, i?enay/ai? ??oaiith oeeo caaea/ [2, 4, 5,], iiaeeeai
?acae?eeoe ia ianooii? eaoaai???:

??oaiiy caaea/? ieaioaaiiy;

??oaiiy caaea/? ?iciiae?eaiiy.

O iaaeao caaea/ ia?oi? eaoaai??? aeeiio?oueny ??oaiiy eeane/ii? caaea/?
ieaioaaiiy i?ae iacaith “ Aaaaoii?ioeani?iee ?iceeaae”. Iaeaaeeoue
a?aecia/eoe, ui i?e i?i?i?caoe?? /enea i?ioeani??a ?/aai /ano ??oaiiy,
ye i?aaeei, ia aecia/a?oueny a yeee naia ia/enethaaeueiee aocie aoaea
?ici?uaia eiaeia i?aecaaea/a aai caaea/a. Oea aecia/a?oueny o iaaeao
caaea/ ae?oai? eaoaai???. Ia’?aeiaiiy oeeo caaea/ i?eaiaeeoue aei
aeaoa?i?iiaaiiai i?inoi?iai-/aniaiai ieaio ?ici?uaiiy caaea/ o eiie?aoia
ia/enethaaeueia ioi/aiiy ? aeei?enoiao?oueny a nenoaiao ianiaiai
?icia?aeaethaaiiy. Oey no?aoaa?y i???ioiaaia ia ae??oaiiy caaea/ aeey
niaoe?ae?ciaaieo ia/enethaaeueieo nenoai, aea ieoaiiy aeei?enoiaoaaiiy
eiaeiiai aeaiaioo ?ano?n?a oa i?i?i?caoe?? /ano ae??oaiiy caaea/?
iaeii/anii ? e?eoe/ii aaaeeeaeie.

Iniiaiith caaea/ath noaoe/iiai ieaioaaiiy ? iaoiaeaeaiiy iioeiaeueiiai
aai iioei?ce?iaaiiai ieaio ?ici?uaiiy AIA ia caaeaio e?euee?noue
i?ioeani??a. A i?aoe? icia/aii, ui iiia?aaei?e no?oeoo?iee aiae?c
ii/aoeiai? ?ioi?iaoe?? ? cia?aaeaiiy AIA o y?onii – ia?aeaeuei?e oi?i?
(ssIO) noi?nii c eiia?iaoe??th no?aoaa?e ?ai?oiueiai ? i?caeiueiai
ieaioaaiiy aeicaieythoue iie?aueoe aeai?eoie iiooeo on?o aa?oei, ye?
aoiaeyoue a e?eoe/i? oeyoe Oeieaea- Oi?oaea ? Aeaeeno?e oa io?eiaoe
iiaee aeai?eoi c /aniaith neeaaei?noth I(2A+2N). Aeey aa?oei, ye? ia ?
e?eoe/ieie, aaaaeaii iiiyooy “o?aiceoieo”. O?aiceoia aa?oeia iiaea
iaeaaeaoe o?ueii eaoaai??yi:

yaia o?aiceoi?noue” — ci?iaiiy ??aiy iaeaaeiino? aa?oeie ia ca?eueoo?
e?eoe/iee oeyo ;

Oiiaa aecia/aiiy aa?oeie “yaii? o?aiceoiino?”:

iiiaeeia aa?oei ??aiy “i”

“iayaia o?aiceoi?noue” — neeathaaiiy (eeanoa??caoe?y) aa?oei oe???
eaoaai??? ia iaeiiio ??ai? ia ca?eueoo? e?eoe/iee oeyo ;

Aa?oeia xi “iayaii? o?aiceoiino?” aieiae?? aeanoea?noth:

iiiaeeia aa?oei ??aiy “i”

iiiaeeia aeoa i?ae (Vi-1 ,Vi)/

“ioeueoeie?eaoeaia o?aiceoi?noue” — neeathaaiiy (eeanoa??caoe?y) aa?oei
oe??? eaoaai??? ia ca?eueoo? e?eoe/iee oeyo, i?e oeueiio aa?oeie
iaeaaeaoue ??ciei ??aiyi.

— aa?oeie aieiae?thoue aeanoea?noth ioeueoeie?eaoeaii? o?aiceoiino?
i?ae ??aiyie i oa m,

Aaaaeaiiy o?aiceoiino? ? eeaneo?eaoe?y aa?oei ii aeaeyaeo o?aiceoiino?
ia aoai? iiia?aaeiueiai no?oeoo?iiai aiaeeco AIA aeicaiey? cai?iiiioaaoe
neaioth/ee i?inoi?iai – /aniaee aeai?eoi iaoiaeaeaiiy iioei?ciaaiiai
ieaio ?ici?uaiiy AIA a caaeaiia /enei i?ioeani??a IIN c ni?eueiith
iai’yoth. Eeth/aaa ?aeay iiaiai i?aeoiaeo i?noeoueny ia o iie?ieiaiio
aaaaoii?io?aeiiio neeaaeaii? ?iceeaaeo c aeei?enoaiiyi no?aoaa??
?aiiueiai ieaioaaiiy ? ioe?ii/ieo ooieoe?e, a a oi?ioaaii? nii/aoeo
aaciaiai ??oaiiy i?e ciyoeo iaiaaeaiiyo ia /enei i?ioeani??a ? iio?i
ai?eciioaeuei?e oa aa?oeeaeuei?e eeanoa??caoe?? aa?oei AIA. Iaoa
eeanoa??caoe?? – aaeainoaaiiy AIA ii ??aiyi i?e /ano ieaioaaiiy, ??aiiio
Oe?. I?e eeanoa??caoe?? a?aoiao?oueny eeaneo?eaoe?eia oa?aeoa?enoeea
eiaeii? aa?oeie ii “o?aiceoiino?”. Iaeaaeeoue a?aecia/eoe, ui
aeei?enoaiiy ia?e??oaaiiy aa?oei a?aoa ii iciaoe? o?aiceoiino? ?
aeeiiaiiy ia eiai iniia? eeanoa??caoe?? aeicaiey? iiaoaeoaaoe aaciaee
ieai ??oaiiy c iie?auaieie oa?aeoa?enoeeaie ii e?eueeino?
aeei?enoiaoaa?ieo i?ioeani??a oa ?o iaaaioaaeaiino?. Oea niinoa?aaeaiiy
iniiaaii ia ne?aeoth/iio oaa?aeaeaii?.

Oaa?aeaeaiiy 1.

E?euee?noue i?ioeani??a, iaiao?aeieo aeey ?ici?uaiiy i???ioiaaiiai
aoeeee?/iiai a?aoa, iaiueo aai aei??aith? iaeneiaeuei?e oe?ei? a?aoa.

Oeae i?aeo?ae aeicaieea caanoe NP-iiaio caaea/o a?eueoi? ?ici??iino? aei
NP-iiaii? caaea/? iaei? ?ici??iino?, o.y. i?e aeyaeaii? “o?aiceoiino?”
aa?oei iiooe aa??aioo ?o eeanoa??caoe?? aeeiio?oueny o?eueee a iaaeao
aeae?eaieo ??aiae a?aoo. Aeey aeeiiaiiy i?ioeaaeo? ai?eciioaeueii?
eeanoa??caoe?? a i?aoe? cai?iiiiiaaii aa??noe/iee aeai?eoi
iioei?coth/iai iaeoaaiiy ?ia?o. A aeai?eoi? aeei?enoiaaii noi?uaiiy
no?aoaa?e “iaea?eueo” ? “iaeiaiueo” i?aeoiaeyuee aeey ?ic?aooieo
i?ei?eoao?a ?ia?o, i?aeeyaath/eo ai?eciioaeuei?e eeanoa??caoe??.

Iaoith ?iaioe ieaioaaeueieea — aaeainoaaiiy a?aoo.

Aeeiiaoue aaeainoaaiiy a?aoa – oea caaeainoaaoe aaae ??aiae a?aoa.
Aaeainoaaiiy aeeiio?oueny oeyoii ci?iaiiy aeey aa?oei: i?eiaeaaeiino?
??aiy, aai ?o a?oioaaiiyi (eeanoa??caoe??th).

Iniiaia iaoa eeanoa??caoe?? aai a?oioaaiiy aa?oei – ciaiueoaiiy oe?eie
a?aoa ?, ioaea, ca?aeii oaa?aeaeaiith 1, ciaiueoaiiy e?eueeino?
i?ioeani??a, iaiao?aeieo aeey ae??oaiiy caaea/? a oe?eiio. Ciaiueoaiiy
oe?eie a?aoa oeyoii neeathaaiiy aa?oei ia iiaeiii ca?eueooaaoe e?eoe/iee
oeyo.

Aeey oeueiai aeei?enoiaothoueny ne?aeoth/? oeie eeanoa??caoe??:
ai?eciioaeueia, aa?oeeaeueia, a?oiiaa.

Ai?eciioaeueia eeanoa??caoe?y.

Oea neeathaaiiy aa?oei, iath/eo iciaeo “iayaii? o?aiceoiino?” aac
ca?eueoaiiy e?eoe/iiai oeyoo. I?e aeeiiaii? oe??? eeanoa??caoe?? o?aaa
a?aoiaoaaoe ianooiia:

— on? aa?oeie ia aeae?eaiiio ??ai? ni?oothoueny ia iniia? ianooiii?
oiiae :

(1)

?ioeie neiaaie m = Iaeneiaeueia aaaa ia ??ai? l .

.Aea: X ? iiaith aa?oeiith i?ney eeanoa??caoe?? aa?oei, iath/ith aaao
ia a?eueo m=max(l). Oea ?iaeoueny aeey oiai, uia a?oioaaoe iayai?
o?aiceoi? aa?oeie ? oaeei /eiii ciaiueoeoe oe?eio a?aoa, ioiae,
e?euee?noue i?ioeani??a.

Oaa?aeaeaiiy 2:

Iaoae ie i?aoeth?ii ia iaeiiio c ??ai?a ,a?aoa G, “i”. ( |Ai| = ni =
oe?eia a?aoa ia oeueiio ??ai?. Uia ae??oeoe ia?aeaeueii on? caaea/? ia
oeueiio ??ai? iai o?aaa aeae?eeoe eiaei?e aa?oei? iaeiiai i?ioeani?a, ?
oae aeoiaeeoue, ui e?euee?noue i?ioeani??a aei??aith? e?eueeino? aa?oei
= ni .

( xij ( Ai , if ( xik ,xim ( Ai / Txim (Txik+Txij / m ( Txim

ionoue Xil = (xik + xij ) ; ( A’i = Ai – xik – xij +Xil ( A’i = Ai –
1(aa?oeia)

ni = ni – 1

|A’i| = ni – 1 . Oea aeiaiaeeoue , ui e?euee?noue i?ioeani??a noaea
iaiueoa .

?iaeoa, oiaoi ia ( xik ,xim ( Ai / Txim (Txik+Txij / m ( Txim , oi
oiae? e?euee?noue i?ioeani??a aei??aith? ni = oe?ei? a?aoa.

Aeey ae??oaiiy caaea/? noaoe/iiai ieaioaaiiy c o?aooaaiiyi
eiioi?eaoe?eieo aeo?ao o ?iaio? cai?iiiiiaaiee aeai?eoi ?iciiae?eaiiy
caaea/ ii i?ioeani?ai a IIN, iniiaaiee ia aeei?enoaii? no?aoaa??
aeae?eaiiy, ia eiaei?e ?oa?aoe?? aeai?eoia, e?eoe/iiai oeyoo ?
?ici?uaiiy aa?oei e?eoe/iiai oeyoo ia iaeei i?ioeani?.

I?e ieaioaaii? c a?aooaaiiyi ia?aneeie iaiao?aeii aeeiiaoe ia?aoai?aiiy
ii/aoeiaiai a?aoa c cai?iith ia?aneeie aa?oeiaie a?aoa. Oaeei /eiii
i?ney ia?aoai?aiiy ii/aoeiaiai a?aoa n ia?aneeeaie, io??io?ii aeaaaeaiee
aoeeee?/iee a?ao c ioeueiaeie aaaaie ?aaa?..

Iaoae G(V,E,e,c) aoeeee?/iee i???ioiaaiee a?ao, (a.i.a) aea :

— aa?oeie a?aoa G.

— aeoae, oiaoi, A — oea noeoii?noue aeoa i?ae aa?oeiaie .

oea noeoii?noue aaa aa?oei.

n oea noeoii?noue aaa aeoa.

Ii/aoeiaee aeaeyae a?aoa G iiaeaiee ia iaethieo 1.

A?ao, io?eiaiee i?ney cai?ie eiioi?eaoe?e aa?oeiaie, iieno?oueny
o??eeith Gc (U,E’,e’). Gc (U,E’,e’) aoeeee?/iee i???ioiaaiee a?ao,
(a.i.a) aea :

— aa?oeie a?aoa Gc.

–aeoaa,

oiaoi, A’ — oea noeoii?noue aeoa i?ae aa?oeiaie / |E’|=2*|E| + |V|.

oea noeoii?noue aaa aa?oei.

Aeey aeeiiaiiy ae?e, ia?aaeaa/aieo cai?iiiiiaaiei aeai?eoiii oa
ia/eneaiiy e?eoe/ieo oeyo?a ? aecia/aiiy e?eoe/ieo aa?oei iaiao?aeii
aeeiiaoe aaciaa, iiia?aaei? ?iciiae?eaiiy caaea/ ii i?ioeani?ai. I?e
aaciaiio ?iciiae?eaiith i?eionea?oueny, ui e?euee?noue i?ioeani??a
aei??aith? e?eueeino? aa?oei (caaea/) ii/aoeiaiai a?aoa. I?e aoaeoaaii?
aaciaiai ?iciiae?eaiiy a?aoiaothoueny eiioi?eaoe?ei? aeo?aoe ia
ia?aaea/o ?ioi?iaoe?? i?ae aa?oeiaie. I?e aaciaiio ?iciiae?eaii?
i??ionea?oueny, ui an? ia?aneeee aeeiiothoueny iiai?noth. Ii aaciaiio
?iciiae?eaiith iiaeeeaee i?inoi?iaa-/aniaee iien eiaeii? aa?oeie
(caaea/?) ? aecia/aiiy Tb(xi)l, Tf(xi)l aea:

Tb(xi)l – /an ii/aoeo ??oaiiy caaea/? xi ia i?ioeani?? l.

.

Tf(xi)l – /an caaa?oaiiy caaea/? xi ia i?ioeani?? l.

.

Iiooe ieaio caa?ocee aeeiio?oueny a aeae?eueea yoai?a. Ia?oei yoaiii ?
aeeiiaiiy iia?aoe?e ca’ycaieo c aecia/aiiyi e?eoe/ieo oeyo?a ?
?ici?uaiiy (eeanoa??caoe?y) aa?oei ye? aoiaeyoue aei e?eoe/iiai
oeyoo/oeyoo ia iaeei i?ioeani?.

I?ney oiai, ye aecia/aii e?eoe/iee oeyo ? aa?oeie ye? aoiaeyoue a
iueiai, aeeo/a?ii oe? aa?oeie c a?aoo G, a caaea/?, a?aeiia?aei? oeei
aa?oeiai ?ici?uaaii ia iaeei i?ioeani?. sseui, i?ney ?aaeoeoe?? a?aoa
io?eio?i G1= G – Npc1, yeee ? oe?eei a?aoii, oiaoi G1 o naith /a?ao, ?
a?aoii (a.i.a.), oiae? i?iaeiaaeoaii ooeaoe ianooiiee e?eoe/iee oeyo
Cpc2.aei oeo i?? iiee noa-a?ao Gk = Gk-1 – Cpck ia? iciaee a.i.a.

Ieaioaaiiy aeeiiaiiy caaea/ ia iaeiiio i?ioeani?? i?eaiaeeoue aei
iaioeaiiy /ano eiioi?eaoe?? i?ae aa?oeiaie, ye? aoiaeyoue o e?eoe/iee
oeyo. Oaeei /eiii caa?ocea on?o aa?oei e?eoe/iiai oeyoo ia iaeei
i?ioeani? ciaiueoo? aai aeiaaeeio.

Oaa?aeaeaiiy :

I?e caa?ocoe? e?eoe/iiai oeyoo ia iaeei i?ioeani? ( T’Cpc0 .

sseui aeeiiothoueny 1) e 2) oiae? xk caa?oaea?oueny ia i?ioeani? aea
caa?oaeaiee Cpc.

.

3- sseui pred(xk) = A={ x1,x2,…,xm}/A=A1+A2 ; A1( Cpc , A2(Cpc.

Oiaoi yeui /anoeia ?c pred(xk) iaeaaeeoue N?n a ae?oaa /anoeia i?.

A oaeiio aeiaaeeo aecia/a?oueny pred (xk) oa succ(xk).

I?eionoeii, ui :

A1={x1,x2,…,xi} ,

A2={y1,y2,…,yj}/pred(xk)=A1+A2;

B1={X1,X2,…Xm},

B2={Y1,Y2,…,YK} /succ(xk)=B1+B2.

;

( >0 ( xk ( xi (pred(xk)

( 0 , oi xk(Cpc1.

sseui d2( Txk e Tbsucc(xi+1)-Tbsucc(xi2+1)(0 , oi xk(Cpc2.

sseui di=0 aai di( Txk , oi xk( max ( Tbsucc(xk)i) .

Oaia? yeui succ(xi) (B1(xi(G ,oiae? iiaeeea? aeaa aa??aioe :

( |Tb(xi)-Tb(xi+1)| ( T(succ(xi)) ( succ(xi )(
Cpc

xi(Cpc ((

( |Tb(xi)-Tb(xi+1)|

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