I?i?noa?noai ina?oe Oe?a?ie

Aeiiaoeueeee aea?aeaaiee oi?aa?neoao

Ia i?aaao ?oeiieno

OAeE: 330.115: 368

Nieieianueeee Aeieo?i Ai?eniae/

Iiaeaethaaiiy niinoa?aaeiino?

eiioe?eoii? aeiiii?/ii? nenoaie

no?aoiaiai oeio

Niaoe?aeuei?noue 08.03.02 — Aeiiii?ei-iaoaiaoe/ia iiaeaethaaiiy

Aaoi?aoa?ao

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa aeiiii?/ieo iaoe

Aeiiaoeuee-1998

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a ?inoeooo? aeiiii?ee i?iieneiaino? IAI Oe?a?ie
(i.Aeiiaoeuee)

Iaoeiaee ea??aiee: eaiaeeaeao aeiiii?/ieo iaoe,

noa?oee iaoeiaee ni?a?ia?oiee,

Ia?oyeiaa Ieaia Aieiaeeie??aia,

?AI IAI Oe?a?ie, caa?aeoth/ee naeoi?ii;

Io?oe?ei? iiiiaioe: aeieoi? aeiiii?/ieo iaoe,

i?ioani?,

Caa?iaenueeee Ay/aneaa Aaeaiiae/,

Oa?e?anueeee aea?aeaaiee oi?aa?neoao

I?i?noa?noaa ina?oe Oe?a?ie (i. Oa?e?a),

caa?aeoth/ee eaoaae?ith aeiiii?/ii? eiaa?iaoeee;

eaiaeeaeao aeiiii?/ieo iaoe,

Iaioaea?iei Aieoi? Ieoaeeiae/,

AAO «IE «Acianoaeue» (i. Ia?ioiieue),

oiiainiaee aee?aeoi?

I?ia?aeia i?aai?caoe?y — Oa?e?anueeee aea?aeaaiee iie?oaoi?/iee
oi?aa?neoao,

I?i?noa?noaa ina?oe Oe?a?ie, eaoaae?a aeiiii?/ii?

eiaa?iaoeee oa ia?eaoeiaiaiai iaiaaeaeiaioo (i. Oa?e?a)

Caoeno aeena?oaoe?? a?aeaoaeaoueny 25 eenoiiaaea 1998 ?ieo i 14 aiaeei?
ia can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee E 11.051.01 Aeiiaoeueeiai
aea?aeaaiiai oi?aa?neoaoo I?i?noa?noaa ina?oe Oe?a?ie ca aae?anith:
340055, i.Aeiiaoeuee, aoe. Oi?aa?neoaonueea, 24, e.309

C aeena?oaoe??th iiaeia iciaeiieoenue o a?ae?ioaoe? Aeiiaoeueeiai
aea?aeaaiiai oi?aa?neoaoo ca aae?anith: 340055, i. Aeiiaoeuee, aoe.
Oi?aa?neoaonueea, 24.

Aaoi?aoa?ao ?icineaiee «23»_aeiaoiy_» 1998 ?.

O/aiee nae?aoa?

niaoe?ae?ciaaii? ?aaee A.N. Iaa/ei

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue aeine?aeaeaiiy. E?aa?ae?caoe?y aeiiii?/ieo a?aeiinei
i?ae noa’?eoaie ainiiaea?thaaiiy, a?aeiiaa a?ae aei?noeiai ieaioaaiiy ia
ei?enoue iiaeeeaoeaiiai i?aeaeuothoue ae?ia?aei?noue iaia?aaeaa/oaaieo
ianeiaeeia iiae?e, ui ni?e/eiy? ?ecee o?iainiaeo ao?ao. Oiiae
ia?ao?aeiiai ia??iaeo c eiai ianoaa?euei?noth, i?iaeaie c aeieia??th
ciaioothoue a?ae/oooy caoeuaiino? ianaeaiiy oa aeeeeeathoue
iaiao?aei?noue caeieniaiiy caoiaeia, ui ni?yiiaaii ia nioe?aeueiee
caoeno.

Iaeiei c cania?a caoenoo o?iainiaeo, iaeiiaeo oa ?ioeo ?ioa?an?a ?
no?aooaaiiy. C iioiai aieo, ia?aaaaeaiiy ia?iaeiiai ainiiaea?noaa e?a?ie
ia «?eieia? ?aeee» aeiaaa? iaeneiaeueii? cai?ie
aaei?i?no?aoeaii-eiiaiaeieo cania?a oi?aae?iiy aeiiii?/ieie. Aoaeoeaia
?aaoethaaiiy no?aoiai? ae?yeueiino?, cae?enithaaia i?aaiaie aea?aeaaiiai
no?aoiaiai iaaeyaeo, oi?aae?iiy no?aoiaeeaie ?ano?naie no?aoiaiai
oiiaeo, i?iaiic ci?ie neooaoe?? ia ?eieo no?aooaaiiy aeiaaathoue
iiiiaiaiiy, aiae?co e ocaaaeueiaiiy ciaiue i?i ?ecee, a?aeiinei iii?ae
no?aoiaeeaie oa no?aoiaeeia c? no?aooaaeueieeaie.

??ciia?/i? a?aeiineie noa’?eo?a no?aooaaiiy a ?aieao no?aoiaiai ?eieo
yaeythoue niaith neeaaeio nenoaio, oeiei ae?yeueiino? ie?aieo aeaiaio?a
yei? ia?iaeei aoaathoue a?aeiiiieie, a a aeayeeo aeiaaeeao aeyaeythoueny
i?ioeeaaeieie. O ca’yceo c oeei aeioeieueiei ? iiaeaethaaiiy
iioaioe?eii? eiioe?eoii? neooaoe?? ana?aaeei? nenoaie no?aoiaeo
a?aeiinei, ui aeei?enoiao?oueny aeey io?eiaiiy iiai? ?ioi?iaoe?? i?i
nenoaio e ooi/iaiiy aaea iayaii?. Caaea/a iiiiaiaiiy ?ioi?iaoe?? i?i
aioo??oi?e noai ia’?eoo (nenoaie), eiee ia’?eo, ui aea/a?oueny (aai ae
eiai iiaeaeue) iiaea?oueny o aeaeyae? «/i?ii? ne?eii», a ?ioi?iaoe?th
i?i eiai noai iaea?aeothoue ia i?aenoaa? cia/aiue aeo?aeieo ia?aiao??a,
? iaei??th c eeane/ieo caaea/ oai??? oi?aae?iiy — caaea/ath
niinoa?aaeaiiy.

Aeoiaey/e c oeeo i??eoaaiue iiaeia c?iaeoe aeniiaie i?i oa, ui i?iaeaia
iiaeaethaaiiy no?aoiaeo aca?iia?aeiinei, cie?aia, eiioe?eoii?
aeiiii?/ii? nenoaie no?aoiaiai oeio, a oaeiae iinoaiiaea caaea/?
niinoa?aaeaiiy noaio icia/aii? nenoaie ? aeooaeueiith.

Ca’ycie ?iaioe c iaoeiaith oaiaoeeith. ?iaioa aeeiiaia o
a?aeiia?aeiino? c oaiaoeeith ?inoeoooo aeiiii?ee i?iieneiaino? IAI
Oe?a?ie ca oaiaie: 3.1.5.42 «Aeiiii?/i? oa nioe?aeuei? i?iaeaie
iie?ioaiiy oiia i?aoe? oa a?aeii/eieo a ia??iae ia?aoiaeo aei ?eieiai?
aeiiii?ee» (iai?yiie «Aiae?c ae?th/iai iaoai?cio no?aooaaiiy a?ae
iauanieo aeiaaee?a ia ae?iaieoeoa?»), 3.1.5.09 «Aeine?aeaeaiiy iai?yie?a
? iiaeeeainoae nioe?aeueiiai ia?eao?iao» (iai?yiie «?ic?iaea iaoiaeeee
?aae?caoe?? aaoiiiiii? nenoaie no?aooaaiiy a?ae iauanieo aeiaaee?a ia
ae?iaieoeoa?»), 3.1.3.13 «?ic?iaea iaoeiai-iaoiaee/ieo iniia ia/eneaiiy
aeo?ao ia ioi?iio i?aoe?» (iai?yiie «?ic?iaea iaoeiai-iaoiaee/ieo
?aeiiaiaeaoe?e ii i?aai?caoe?? no?aoiaeo a?aeiinei»).

Iaoa i caaea/? aeine?aeaeaiiy. Iaoith aeena?oaoe?eii? ?iaioe ?
iiaeaethaaiiy niinoa?aaeiinoi aeeiai?/ieo aeiiii?/ieo nenoai no?aoiaiai
oeio, ui a?oioo?oueny ia aiae?c? eiioe?eoieo aca?iiae?e noa’?eo?a
no?aooaaiiy. Aeinyaiaiiy iinoaaeaii? iaoe aeiaaa? ae??oaiiy ianooiieo
caaea/:

?ic?iaeoe eiioeaiooaeueio noaio eiioe?eoieo aca?iia?aeiinei iii?ae
noa’?eoaie no?aooaaiiy c a?aooaaiiyi ?o aeeiai?ee;

?ic?iaeoe iiaeaeue eiioe?eoii? aeiiii?/ii? nenoaie no?aoiaiai oeio o
aeaeyae? aeeiai?/ii? eiae?oe?eii? a?e c ioeueiaith noiith;

aecia/eoe iiaeeea?noue iiceoeaii? a?aeiia?ae? ia ieoaiiy i?i
niinoa?aaeiinoue eiae?oe?eii? a?e, ui ?icaeyaea?oueny;

ca oiiae niinoa?aaeiinoi eiae?oe?eii? a?e ?ic?iaeoe aeai?eoi ae??oaiiy
caaea/? niinoa?aaeaiiy a?e;

ioe?ieoe i?eaeaoi?noue iiaeae? aei ??cieo aaeocae no?aoiaeo a?aeiinei.

I?aaeiao ? ia’?eo aeine?aeaeaiiy. I?aaeiaoii aeine?aeaeaiiy a
aeena?oaoe?? ? iiaeaethaaiiy aeeiai?ee aca?iia?aeiinei i?ae aeaiia
a?oiaie noa’?eo?a, ui iathoue i?ioeeaaei? ?ioa?ane.

A yeino? ia’?eoo aeine?aeaeaiiy i?eeiyoa nenoaia no?aoiaeo
aca?iia?aeiinei.

Oai?aoe/iith oa iaoiaeieia?/iith iniiaith aeena?oaoe?eii? ?iaioe ?
i?aoe? a?o/eciyieo ? caei?aeiiieo o/aieo o aaeocyo no?aooaaiiy;
aeiiiiiei-iaoaiaoe/ii? oai???; nenoaiiiai aiae?co e
aeiiiiiei-iaoaiaoe/iiai iiaeaethaaiiy. O aeena?oaoe?? aeei?enoiaoaaeenue
iaoiaee aeine?aeaeaiiy iia?aoe?e, cie?aia, oai??? iai?; oai???
oi?aae?iiy; aeene?aoiiai i?ia?aioaaiiy.

Iaoeiaa iiaecia ?acoeueoao?a aeine?aeaeaiiy iieyaa? a ianooiiiio:

ia i?aenoaa? aeei?enoaiiy ?aeeii? oaoiieia?? aiae?co ??ciia?/ieo
no?aoiaeo iia?aoe?e ? aca?iiae?? iii?ae aeaiaioaie no?aoiaeo a?aeiinei
cai?iiiiiaaii eiioeaiooaeueiee i?aeo?ae aei iiaeaethaaiiy aeiiii?/ieo
no?aoiaeo nenoai, yeee a?aoiao? eiioe?eoiee oa?aeoa? aca?iia?aeiinei
iii?ae noa’?eoaie, aeeiai?eo ci?ie ia?aiao??a oeeo a?aeiinei, a oaeiae
iaiiaiioo ?ioi?iaoe?? i?i noai nenoaie;

a oiae? ?aae?caoe?? aeaiiai i?aeoiaeo iiaoaeiaaia iiaeaeue eiioe?eoii?
aeiiii?/ii? nenoaie no?aoiaiai oeio o aeaeyae? aeeiai?/ii? eiae?oeieii?
a?e c ioeueiaith noiith, ui aea? iiaeeea?noue io?eioaaoe aeiaeaoeiao
?ioi?iaoe?th i?i ?icaeyaeoaaio nenoaio ? ia oe?e i?aenoaa? i?iaiicoaaoe
?? iiaaae?ieo a iaeaooiueiio;

aeey a?aeiia?ae? ia ieoaiiy i?i iiaeeea?noue io?eiaiiy ae/a?iii?
?ioi?iaoe?? i?i eiae?oeieio a?o, ui iiaeaeth? aeo?aeio aeiiii?/io
no?aoiao nenoaio, noi?ioeueiaaia iiiyooy niinoa?aaeiinoi aeeiai?/ii?
eiae?oeieii? a?e;

iieacaii, ui ?icaeyaeoaaia eiae?oeieia a?a ? niinoa?aaeiith oeyoii
aeiaaaeaiiy oai?aie, yea anoaiiaeth? ?nioaaiiy ?aeeiiai ??oaiiy caaea/?
niinoa?aaeiinoi eiae?oeieii? a?e;

aeiaaaeaii, ui niinoa?aaeaiiy eiae?oeieii? a?e iiaea aooe i?iaaaeaii ia
a?eueoa, i?ae i?ney N+1 aei??o ?? aeo?aeieo ia?ai?iieo (aea N — ?icii?
eiaeioeie);

ae??oaia caaea/a niinoa?aaeaiiy eiae?oeieii? a?e oeyoii ?ic?iaee oa
i?ia?aioaaiiy a?aeiia?aeiiai aeai?eoio, yeee ia? aoaeoeaii /aniai
oa?aeoa?enoeee (eiai neeaaei?noue a iaea??oiio aeiaaeeo aei??aith?
O(N4logN), a na?aaeiueiio — O(N3logN) i?e N(+(, ui ? eiaa?eoiii a?ae
neeaaeiino? aeai?eoio iiaiiai ia?aai?o); inoaii? aeicaiey? c aeiiiiiaith
oeueiai aeai?eoio ae??ooaaoe caaea/? niinoa?aaeaiiy noaio aeiiii?/ieo
no?aoiaeo nenoai aaeeei? ?ici??iino?.

I?aeoe/ia cia/aiiy ? ?aae?caoe?y ?acoeueoao?a aeine?aeaeaiiy, io?eiaieo
a aeena?oaoe?? iieyaa? a ianooiiiio. Iaoaiaoe/iee aia?ao, yeee
aeei?enoaii a aeena?oaoe?ei?e ?iaio?, ni?ey? ye iiaeaethaaiith
ae?yeueiino? noa’?eo?a no?aooaaiiy (no?aoiaee?a, no?aooaaeueiee?a,
oiiaeo no?aooaaiiy) ia no?aoiaiio ?eieo, oae ? ioe?ioei aeeiai?ee noaio
aaciaee i?aoe? oa i?iaiicoaaiith ?ioaineaiino? iaaeoiaeaeaiue no?aoiaeo
i?ai?e a athaeaeao oiiaeo iaia’yceiaiai no?aooaaiiy.

Ai?iaaoe?y ?iaioe. Iniiai? iieiaeaiiy aeena?oaoe?? aeiiia?aeaeeny ?
iaaiai?thaaeeny ia nai?ia?ao ?inoeoooo aeiiii?ee i?iieneiaino? IAI
Oe?a?ie, Aeiiaoeueeiai aea?aeaaiiai oi?aa?neoaoo I?i?noa?noaa ina?oe
Oe?a?ie, ?inoeoooo e?aa?iaoeee IAI Oe?a?ie, a ?iai/?e a?oi? i?iaeoo
TACIS EDUC 9501 “Nioe?aeueiee caoeno a Oe?a?i?”.

Ioae?eaoe??. Ca oaiith aeena?oaoe?eii? ?iaioe iioae?eiaaii 5
ae?oeiaaieo ?ia?o caaaeueiei ianyaii 6,5 ae?.a?., c yeeo 3,55 ae?.a?.
iaeaaeaoue aaoi?o.

No?oeoo?a ? ianya aeena?oaoe?eii? ?iaioe. Aeena?oaoe?y neeaaea?oueny c
anooio, o?ueio ?icaeieia, aeniiae?a, nieneo aeei?enoaii? e?oa?aoo?e c 75
iaeiaioaaiue, 8 iaethie?a, 21 oaaeeoe? oa aeeeaaeaia ia 170 noi??ieao
ae?oeiaaiiai oaenoo.

INIIAIEE CI?NO AeENA?OAOe??

?icae?e 1. Eiioeaiooaeueia iiaeaeue eiioe?eoii?

aeiiii?/ii? nenoaie no?aoiaiai oeio

Iaaecia/aiinoue, ui i?eoaiaiia aoaeue-yeiio aeaeo aeiiii?/ii?
ae?yeueiino?, ii?iaeaeo? ?ecee caciaiiy caeoe?a aiane?aeie ae?ia?aeiino?
io?eiaiiy iaia?aaeaa/oaaiiai ?acoeueoaoa ae?e. Iaeiei c cania?a caoenoo
o?iainiaeo, iaeiiaeo e ?ioeo ?ioa?an?a a?ae ?eceeo iaia?aaeaa/aieo ao?ao
? no?aooaaiiy.

Iniiaiee i?eioeei, ia yeiio a?oioo?oueny nenoaia no?aooaaiiy iieyaa? o
noai?aii? aoaeoo ia?aaea/? oa ?ic/eiaiiy ?eceeo (ae?ia?iiy ?ecee?a).
Ia?aeiia ?ici?uaiiy ?eceeo, eiee no?aooaaeueiee, niea/oth/e ii??aiyii
iaaaeee? ca ?icii?ii no?aoia? i?ai??, ia?aeeaaea? aa?aio?th
a?aeoeiaeoaaiiy caeoe?a ia no?aoiaeea, i?e/iio caeoee, ui
a?aeoeiaeiaothoueny iiaeooue aaaaoie?aoii ia?aaaaeaoe ieaoaae?
no?aooaaeueieea.

Noeoii?noue aeiiii?/ieo a?aeiinei, ui oi?iothoue iiieo ? i?iiiceoe?th
ia niaoeeo?/iee ia’?eo eoi?ae?-i?iaeaaeo, a yeino? yeiai aenooia?
“no?aoiaa iie?eooy” (no?aoiaee i?iaeoeo), ui aeieea? a i?ioean?
caaeiaieaiiy noni?eueieo iio?aa o no?aoiaiio caoeno?, iaceaa?oueny
?eieii no?aooaaiiy. Iaia’yceiaith oiiaith ?nioaaiiy no?aoiaiai ?eieo ?
iayai?noue noni?eueii? iio?aae ia no?aoia? iineoae ? iayai?noue
no?aoiaee?a (i?iaeaaoeia “no?aoiaiai iie?eooy”), caeaoieo caaeiaieueieoe
oe? iio?aae.

Iniiaieie ae?th/eie iniaaie ia no?aoiaiio ?eieo ? no?aoiaee ?
no?aooaaeueiee. Ie??i oiai, o aca?iia?aeiineiao aa?ooue o/anoue e ?io?
noi?iie: aieiaea? iaeia — a no?aooaaii? iaeia, cano?aoiaaiee — a
iniaenoiio ? a?oiiaiio no?aooaaii?, aenia?oe (aaaioe) no?aoiaeea c
ioe?iee iaeia, ui no?aoo?oueny (nth?aaea?e e aaeaeanoa?e), a oaeiae
i?aaie no?aoiaiai iaaeyaeo (aeeth/ath/? ooao no?aoiaeo ?iniaeoi??a), o
ooieoe?? yeeo aoiaeeoue ?aaoethaaiiy no?aoiai? ae?yeueiino? ia no?aoiaeo
?eieao. Aaciina?aaeiuei icia/aia ?aaoethaaiiy iieyaa? o caiia?aaii?
iaaei?aaaeaiiai ?eceeo c aieo no?aoiaeo eiiiai?e, ciiae, oaiaee i?ae
ieie, ae?e ii ?icae?eo ?eieo c iaoith iaiaaeaiiy eiieo?aioe??,
aeeeth/aiiy aai iaiaaeaiiy aeinooio ia ?eiie ?ioeo o/aniee?a, ui iiaea
i?ecaanoe aei ooo/iiai i?aeaeuaiiy aai cieaeaiiy no?aoiaeo oa?eo?a,
iaeiio?ieueiaaiiai caoaeaiiy ani?oeiaioo iineoa oa ?ioeo iani?eyoeeaeo
aeey no?aooaaeueieeia iane?aeeia. Oaeei /eiii, iiaeia aiai?eoe i?i iaaio
i?ioeeaaeiinoue ?ioa?an?a aea?aeaae ? no?aoiaeo eiiiai?e, ae?th/eo ia
?eieo no?aooaaiiy.

Eiaeia no?aoiaa eiiiai?y, i?enooiy ia ?eieo no?aooaaiiy, i?aaia
aeaeiaooe ci nai?? ae?yeueiino? iaeneiaeueiee cene, o neeo /iai ?ioa?ane
no?aoiaee?a ? aca?iii?ioeeaaeieie (oaeiae ye ?ioa?ane no?aoiaee?a ?
aea?aeaae), aiane?aeie /iai i?ae ieie aeieea? noia?ieoeoai. Naiaiaea
oe?iiooai?aiiy, ui aiaeaeaa?oueny o iiaoaeia? oa?eoieo noaaie ia
no?aoia? iineoae, a?a iiieoo ? i?iiiceoe?? noai?th? oiiae, i?e yeeo
aeieea? eiieo?aioe?y iii?ae no?aoiaeeaie, i?e/iio a oeeo oiiaao ?eiie
no?aooaaiiy aoaea aeeiioaaoe ?aaoethth/o ?ieue.

I?ae aeaieeaie aeaiaio?a nenoaie no?aooaaiiy iathoue iinoea ianooii?
aca?iica’ycee: no?aoiaee — no?aooaaeueiee; no?aoiaee — Oiiae
no?aooaaiiy; no?aoiaee — no?aoiaee ?iniaeoi? (nth?aaea?, aaeaeanoa?);
no?aoiaee ?iniaeoi? — ia’?eo no?aooaaiiy (cano?aoiaaiee); no?aooaaeueiee
— aeaniee ia’?eoo no?aooaaiiy (cano?aoiaaiee).

E??i oiai, o nenoai? no?aoiaeo aca?iia?aeiinei iathoue i?noea
a?aeiineie oeio «aeaiaio nenoaie no?aooaaiiy — ?ecee». Iaaecia/aiinoue
?acoeueoaoo aeia?eueii? no?aoiai? iiae?? (yea ? aeaea?aeii aeieeiaiiy
?eceeo) oyaia ca niaith iaaecia/aiinoue a ai??i?iiio aeai?? iioeiaeueii?
no?aoaa?? iiaaae?iee oiai /e ?ioiai noa’?eoo ?eieo no?aooaaiiy (ui
caeaaeeoue a?ae aeiaaeeiaiai ianeiaeeo). C iioiai aieo, a?aeiineie aeaio
noa’?eo?a ?eieo, ui iathoue i?ioeeaaei? ?ioa?ane, i?ecaiaeyoue aei
aeieeiaiiy eiioe?eoii? neooaoe??, a oiae? yei? iioeiaeueia no?aoaa?y
iiaaae?iee aoaeue-yeiai c ieo caeaaeeoue a?ae aeai?o i?ioeeaaeii?
noi?iie. Aiae?c iecee i?eeeaae?a a?aeiinei iii?ae aeaiaioaie no?aoiai?
nenoaie aeicaiey? ocaaaeueieoe ?o a ?aeeio oeiiao noaio no?aoiaeo
a?aeiinei ? aecia/eoe ieceo caaaeueieo ?en, i?eoaiaiieo aoaeue-yeei
noi?iiai ae?yeueiino? aeaiaio?a no?aoiai? nenoaie, iacaeaaeii a?ae aeaeo
ae?yeueiino?, ?? oi?ie aai oa?aeoa?o, a naia iaaecia/aiinoue,
eiae?oeieiinoue, eiioeeeoiinoue, aeeiai?/i?noue. ?icaeyaeathoueny oae?
aa??aioe no?aoiaeo a?aeiinei:

??oaiiy no?aooaaeueieea i?i no?aooaaiiy iaeia;

aea?? no?aoiaeeii aaee/eie no?aoiaiai oa?eoo;

oeeaaeaiiy aeiaiai?o no?aooaaiiy;

aecia/aiiy no?aoiaeeii aaee/eie no?aoiaiai oa?eoo c o?aooaaiiyi iiieoo
ia no?aoia? iineoae;

aea?? no?aooaaeueieeii oi?ie no?aooaaiiy;

aea?? no?aooaaeueieeii i?aai?caoe?eii? oi?ie no?aoiaiai caaacia/aiiy;

ioe?iea iaeia (iiia?aaeiy e inoaoi/ia — i?ney ianoaiiy no?aoiaiai
aeiaaeeo) i?e aeia?iaieueiiio no?aooaaii?;

ioe?iea aaciaee ae?iaieoeoaa i?e iaia’yceiaiio no?aooaaii?;

i?ioean nieaoe aiane?a i?e iaia’yceiaiio no?aooaaii?;

??oaiiy no?aoiaeea i?i ia?ano?aooaaiiy aeiaiai?o;

aea?? no?aoiaeeii aa??aioo ia?ano?aooaaiiy.

?icae?e 2. Iaoaiaoe/ia iiaeaeue eiioe?eoii?

aeiiii?/ii? nenoaie no?aoiaiai oeio

Ae?th/eie iniaaie oeiiai? noaie no?aoiaeo a?aeiinei iiaeia aaaaeaoe
eiaeioei? no?aoiaee?a ? no?aooaaeueiee?a, oiio ui

no?aoiaee ?/aai no?aooaaeueiee aa?ooue o/anoue o a?eueoino? no?aoiaeo
iia?aoe?e;

oeiei iaio eiaeioeie ? aca?iii i?ioeeaaeieie;

iioa?ane an?o ?ioeo ae?th/eo in?a no?aoiaeo a?aeiioaiue i?eioeeiiai
iiaeooue aooe caaaeai? aei ?ioa?an?a no?aoiaeea ? no?aooaaeueieea.

A?oiiaa aca?iiae?y icia/aieo eiaeioeie iieyaa? a na???
?iaeea?aeoaeueieo aca?iiae?e i?ae eiaeiei /eaiii eiaeioei? no?aoiaee?a c
iaaiei (cao?eniaaiei) /eaiii eiaeioei? no?aooaaeueiee?a (oiaoi a?oiiaa
aca?iiae?y ? na???th eieaeueieo aca?iiae?e oeio
no?aoiaee-no?aooaaeueiee).

Eieaeueia aca?iiae?y no?aoiaee-no?aooaaeueiee. Iaoae ??oaiiy
no?aoiaeeii ? no?aooaaeueieeii i?eeiathoueny a iiiaioe /ano tk.
?icaeyiaii aca?iiae?th aeayei? ia?e no?aoiaee-no?aooaaeueiee aeey
aeia?eueiiai k: k=0,1,…. Aoaeaii aaaaeaoe, ui ye o no?aoiaeea, oae ? o
no?aooaaeueieea ? aea?? c aeaio a?aie/ieo aa??aio?a. Iicia/eii ?o aeey
no?aoiaeea a1 ? a2, aeey no?aooaaeueieea — b1 ? b2. Eiie?aoia ia?ia
no?aoaa?y aecia/a?oueny aeaieeith aea?aieo no?aoiaeeii ?
no?aooaaeueieeii aeueoa?iaoea. Onueiai ?nio? /ioe?e iiaeeeaeo aa??aioe
(aeea. oaae.1), ui iathoue ianooiii aeanoeainoi:

Oaaeeoey 1. Noaia aca?iiae?? oeio no?aoiaee-no?aooaaeueiee

No?aooaaeueiee (B)

b1 b2

No?aoiaee (A) a1 (a1, b1) (a1, b2)

a2 (a2, b1) (a2, b2)

Aeey no?aoiaeea

Aa??aio (a1, b1) a?eueo aea?aeiee, i?ae aa??aio (a2, b1);

Aa??aio (a2, b2) a?eueo aea?aeiee, i?ae aa??aio (a1, b2);

Aeey no?aooaaeueieea

Aa??aio (a1, b2) a?eueo aea?aeiee, i?ae aa??aio (a1, b1);

Aa??aio (a2, b1) a?eueo aea?aeiee, i?ae aa??aio (a2, b2).

Aeeiai?ea. An? /eaie iaio eiaeioeie i?eeiathoue ??oaiiy ia??iaee/ii,
i?eaeecii a iaeei ? oie aea i?ii?aeie /ano. Iiae?eeii /aniao a?nue ia
ae?eueieoe? oaeei /eiii, uia ??oaiiy eiaeiei no?aoiaeeii ?
no?aooaaeueieeii i?eeiaeeny a ?ioa?aaeao tk((tk (k = 0,1, …) (?en. 1).

((((..((((((..((((..(((((..(((((..(

t0((t0 t1((t1 . . . . . . . tk((tk . . . . . . .

?en. 1. *ania? ?ioa?aaee i?eeiyooy ??oaiue /eaiaie eiaeioeie

Aeey c?o/iinoi aoaeaii aaaaeaoe, ui ??oaiiy no?aoiaeeaie ?
no?aooaaeueieeaie i?eeiathoueny (i?e iaiao?aeiino?) oi/ii a iiiaioe /ano
tk, k = 0,1, ….

Aiae?c a?aeiinei a ie?aiie no?aoiaie iia?aoe?? aoaeaii i?iaiaeeoe c
aeei?enoaiiyi aia?aoo aeine?aeaeaiiy iia?aoe?e ? oai??? ia?. Oi?iaeueiee
iien iiaeae? eiioe?eoii? neooaoe??, ui ia? i?noea i?e no?aoiaeo
a?aeiineiao, iiaea?oueny o aeaeyae? aeeiai?/ii? eiaeioeieiii? a?e aeaio
in?a c ioeueiaith noiith.

C iioiai aieo noeoiiinoi no?aoiaeo a?aeiinei i?eoaiaiii aeanoeainoi
neeaaeii? aeeiai?/ii? nenoaie, cie?aia:

iayai?noue aaeeeiai /enea aca?iica’ycaieo ? aca?iiae?th/eo i?ae niaith
aeaiaio?a;

iayai?noue o caaaeueiiio aeiaaeeo i?ioe??/ i?ae oeieyie, ui
ia?ane?aeothoueny ie?aieie aeaiaioaie nenoaie;

a?aee?eoee oa?aeoa? nenoaie, aca?iiae?th/i? c ciai?oiii na?aaeiaeuai;

ooieoe?iioaaiiy a oiiaao aieeao aeiaaeeiaeo /eiiee?a.

Iaei??th c iniiaieo caaea/ aeine?aeaeaiiy neeaaeieo nenoai ? caaea/a
iiiiaiaiiy ?ioi?iaoe?? i?i aioo??oi?e noai nenoaie, eiee iiaeaeue
nenoaie, ui aea/a?oueny, iiaea?oueny o aeaeyae? «/i?ii? ne?eii», a
?ioi?iaoe?th i?i ?? noai io?eiothoue ia i?aenoaa? cia/aiue aeo?aeieo
ia?aiao??a — caaea/a niinoa?aaeaiiy.

?icaeyiaii a?o — A=(A, B, F), aea

A={a1, a2} — iiiaeanoai /enoeo no?aoaa?e ia?oiai a?aaoey;

B={b1, b2} — iiiaeanoai /enoeo no?aoaa?e ae?oaiai a?aaoey;

— iaiaaeaia /eneiaa ooieoe?y (iao?eoey) ao?ao a?e, ui aecia/a?oueny ia
aeaea?oiaii aeiaooeo A ? B (A(B):

fij — aea?ao ia?oiai (i?ia?ao ae?oaiai) a?aaoey ca oiiae aeai?o ia?oei
a?aaoeai no?aoaa?? ai ? ae?oaei a?aaoeai no?aoaa?? bj.

Iaoae E={(|0(((1} — i?ino?? ci?oaieo no?aoaa?e ia?oiai a?aaoey (( —
?iia??i?noue aeai?o ia?oei a?aaoeai /enoi? no?aoaa?? a1, oiae?
?iia??i?noue aeai?o ?i /enoi? no?aoaa?? a2 aoaea aei??aithaaoe 1-();
H={(|0(((1} — i?ino?? ci?oaieo no?aoaa?e ae?oaiai a?aaoey ((-
?iia??i?noue aeai?o ae?oaei a?aaoeai /enoi? no?aoaa?? b1, ?iia??i?noue
aeai?o ae?oaei a?aaoeai /enoi? no?aoaa?? b2 aei??aith? 1-(); Y=Y(E(H) —
iaiaaeaia /eneiaa ooieoe?y ao?ao a?e G, ui aecia/a?oueny ia aeaea?oiaii
aeiaooeo E ? H:

.

Ieaeiy oe?ia a?e G=((G) — iaeneiaeueiee aa?aioiaaiee aea?ao ia?oiai
a?aaoey:

(G = max (min (Y ((,())).

((E (( H

Aa?oiy oe?ia a?e G=((G) — i?i?iaeueiee aa?aioiaaiee i?ia?ao ae?oaiai
a?aaoey:

(G = min (max (Y ((,())).

((H (( E

Caiaeii iniiai?e oai?ai? oai??? iai? aeey aoaeue yei? e?ioeaai? a?e
aa?oiy oe?ia oei?? a?e ca?aa?oueny c ?? ieaeiueith oe?iith, oiaoi a?a
ia? oe?io, ? o eiaeiiai c aeaio a?aaoeia ? i?eiaeii? iaeia iioeiaeueia
no?aoaa?y.

Oe?ia a?e G — aaee/eia yG, aea yG=(G=(G.

Iioeiaeueia no?aoaa?y a?e G — aeaieea no?aoaa?e xG=((0, ( 0), aea

Y (xG) = Y((0, ( 0) = yG.

Iioeiaeueia a?a rG: rG=(xG, yG), aea yG — oe?ia a?e G (yG ( Y), Y —
i?ino?? oe?i a?e; xG — iioeiaeueia no?aoaa?y, ui caaacia/o? aeinyaiaiiy
oe?ie yG (xG ( X), X — i?ino?? no?aoaa?e a?e.

Aeae? iioeiaeueia a?a aeaio a?aaoeia aoaea eiaioaaoeny eieaeueiith
a?ith.

Eiaeioeiy — a?oia a?aaoeia, ui ia?ane?aeothoue noiae? oeiei. ?ici??
eiaeioei? (N) — /enei a?aaoeia a eiaeioei?.

No?oeoo?a eiae?oe?eii? a?e — oea aoeaaa iao?eoey q: q = (qij,
(i({1,…,N}, j({1,…,N})), oaea ui qij=1 (i,j({1,…,N}) oiae? ?
o?euee? oiae?, eiee a ?aieao eiae?oe?eii? a?e ia? i?noea eieaeueia a?a
i?ae i-i a?aaoeai ia?oi? oa j-i a?aaoeai ae?oai? eiae?oe??. Aoaeaii
aaaaeaoe, ui aeey eiaeiiai a?aaoey ia?oi? eiae?oe?? ?nio? ?aeeiee
a?aaaoeue ae?oai? eiae?oe??, oaeee ui i?ae oe?ie a?aaoeyie iayaia
eieaeueia a?a oa iaaiaee. Iaoae i?ae i-i a?aaoeai ia?oi? oa j-i a?aaoeai
ae?oai? eiaeioei? iayaia eieaeueia a?a rL (i, j)=(x L (i, j), y L (i,
j)), aea

L: {1, …, N}, {1, …, N} ( {1, …, N} —

ooieoeiy, ui anoaiiaeth? aiaeiiaiaeiinoue iiae ii?yaeeiaei iiia?ii
eieaeueiie a?e oa a?aaoeyie ia?oi? i ae?oai? eiaeioeie, yei aa?ooue a
iie o/anoue. O iiaeaeueoiio ?icaeyaeathoueny ia?e, ui caaeiaieueiythoue
ianooii?e aen?ii?.

Aen?iia. Aaee/eia

ia caeaaeeoue a?ae j, a aaee/eia

ia caeaaeeoue a?ae i.

Ci?noiaii oey aen?iia icia/a?, ui aaee/eia

? noaeith aeey i-ai (i ( {1, …, N}) a?aaoey ia?oi? eiaeioei? e ia
caeaaeeoue a?ae oiai, c eei ic a?aaoeia ae?oai? eiaeioei? a?i a?a?.
Aiaeia?/ii, aaee/eia

? iino?eiith aeey j-ai (j ( {1, …, N}) a?aaoey ae?oai? eiaeioei? ? ia
aoaea caeaaeaoe a?ae eiai eiie?aoiiai ia?oia?a c ia?oi? eiaeioei?.

Aaee/eie ci ? dj aoaeaii iaceaaoe eiao?oe?aioaie oe?iiino? no?aoaa?e,
a?aeiia?aeii, i-ai a?aaoey ia?oi? eiaeioei? oa j-ai a?aaoey ae?oai?
eiaeioei?. I?yi?

oa

iacaaii iaeaiaaie oe?iiino? no?aoaa?? (aai nei?i/aii — iaeaiaaie) i-ai
a?aaoey ia?oi? ? j-ai a?aaoey ae?oai? eiaeioei?.

Aoaeaii aaaaeaoe, ui ?ioi?iaoe?y i?i a?o iaaeoiaeeoue a aeene?aoii
iiiaioe /ano t0, t1, … .

Iaeaiae oe?iiino? no?aoaa?e eiaeioeieii? a?e C ? D ooai?ththoue
(ca?aeii nai?io aecia/aiith) 2 io/ea i?yieo, ui noiaeyoueny a oi/eao:
iaeaiae a?aaoeia ia?oi? eiaeioei? — o oi/oe? (0, 0), iaeaiae a?aaoeia
ae?oai? eiaeioei? — o oi/oe? (1, 0) — oeaio?ao ia?oi? ? ae?oai?
eiaeioeie. O?eniaaia a aoaeue-yeee iiiaio /ano iieiaeaiiy oeaio??a
eiaeioeie a?aeiia?aea? neooaoe??, i?e yeie inoioii oa?aeoa?enoeee
eiaeioeie caeeoathoueny iino?eieie. ?oiy ci?ia iiaea aooe a?aeia?aaeaia
a ?on? oeaio??a eiaeioeie i?ioyaii /ano.

?icaeyiaii ocaaaeueiaiiy eiae?oeieii? a?e ia aeiaaeie ?ooiieo oeaio??a
eiaeioeie. Iaoae oeaio? ia?oi? eiaeioei? ?ooaaony ca caeiiii

U1(t), aea U1(t0) = (0,0),

oeaio? ae?oai? — ca caeiiii

U2(t), aea U2(t0) = (1,0).

O ca’yceo c oei, ui nooue eiaeioeieii? a?e i?e oeueiio ia ci?ith?oueny,
c?o/ii aaanoe a ?icaeyae ?ooiio nenoaio eii?aeeiao, a yeie iieiaeaiiy
oeaio??a ia?oi? oa ae?oaee eiaeioeie cae??ieaii a oi/eao (0, 0) ? (1,
0).

Iieaaeaii, ui aeey aeia?eueii caaeaieo U1(t) ? U2(t), aecia/aia ieie
ciiiaiiy aeaiii?iiai i?inoi?o ? e?i?eiei.

I?eionoeii, ui ?oo eiaeiiai c oeaio??a iieno?oueny ??aiyiiyi:

U(t) = aU((t0) + b.

Iaea?aeo?ii nenoaio ??aiyiue:

U1(t) = a((0,0) + b,

U2(t) = a((1,0) + b,

ae??oeaoe yeo, ia?ii

b = U1(t),

a = U21(t) — U11(t),

aea U11 oa U21 — ia?o? eiiiiiaioe a?aeiia?aeii aaeoi??a U1 oa U2.

Oaeei /eiii, ?oo niinoa??aa/?a ca caeiiaie U1(t) ? U2(t) a?aeiia?aea?
ciiiaiith aeaiii?iiai i?inoi?o ca caeiiii

z(t) = (U21(t) — U11(t))(z0 + U2(t), (1)

aea z0=z (t0) — iieiaeaiiy aeayei? oi/ee ieiueie a iiiaio /ano t0, z(t)
— ?? iioi/ia iieiaeaiiy. Cai?ioiuei, iieiaeaiiy aeayei? oi/ee ((t) a
?ooii?e nenoai? eii?aeeiao iiaea aooe ia?aaeaia oi?ioeith:

(2)

i ca?aa?oueny c? cia/aiiyi z0, ai (1) ? (2) ? aca?iicai?ioiiie.

Eieaeueia a?a rl iaceaa?oueny noaa?eueiith, yeui

inio? v=(u, w) oaea, ui

Eiaeioeieia a?a r iaceaa?oueny noaa?eueiith, yeui

inio? v=(u, w) oaea, ui

aeey aoaeue-yeeo l1, l2( {1, …, N}.

Oaeei /eiii, i?e a?aeia?aaeaii? eiaeioeieii? a?e a iao?e/iiio i?inoi??
iiiyooy noaa?eueiino? icia/a?, ui an? eieaeuei? ia?e, ye? aoiaeyoue a
eiaeioeieio a?o (?oi? iioeiaeuei? no?aoaa?? ? oe?ie), aaiethoeiiiothoue
c iino?eiith oaeaee?noth. O oeueiio aeiaaeeo aaee/eia v iaceaa?oueny
oaeaee?noth aaiethoe?? a?e (aai oaeaee?noth a?e), aaee/eia u —
oaeaee?noth aaiethoe?? iioeiaeueii? no?aoaa??, a aaee/eia w —
oaeaee?noth oe?ie a?e.

Eieaeueia a?a rl (l ( {1, …, N}) iaceaa?oueny noaa?e?co?iith, yeui

inio? oaea v=(u, w) ? oae? ooieoe?? U1(t), U2(t), ui ni?aaaaeeeaa
??aiyiiy

Eiaeioeieia a?a r iaceaa?oueny noaa?e?co?iith, yeui

?nio? oaea v=(u, w) ? oae? ooieoe?? U1 (t), U2 (t), ui ??aiyiiy

ni?aaaaeeeai aeey aoaeue-yeeo l1, l2 ( {1, …, N}.

sseui eiaeioeieia a?a noaa?e?co?ia, iiaeia i?ae?a?aoe e?i?eiee caeii
ciiiaiiy nenoaie eii?aeeiao, a ye?e eiaeioeieia a?a ia? iino?eio
oaeaee?noue aaiethoe??. ?ioeie neiaaie,

yeui aeayea eiaeioeieia a?a noaa?e?co?ia a oieniaai?e nenoai?
eii?aeeiao, oi ?nio? e?i?eia ia?aoai?aiiy nenoaie eii?aeeiao, a yeiio
aeaia eiaeioeieia a?a ? noaa?eueiith.

Iaoae ia? i?noea noaa?eueia eiaeioeieia a?a ?ici??iino? N ?c
iacaeaaeiith a?ae /ano no?oeoo?ith q.

?ioi?iaoe?y i?i a?o, ui iaaeoiaeeoue a iiiaioe /ano t0, t1,…, yaey?
niaith k-aei?? a?e: (Ck, Dk).

I/aaeaeii, ui k-aei?? eiaeioeieii? a?e iiai?noth aecia/a? k-i?iaeoeeth
iioaioe?eii? ea?e pk:

,

aea

i ia i?noeoue aeiaeii? ?ioi?iaoe?? i?i no?oeoo?o a?e.

Iacaaii i?ioean a?aeiiaeaiiy no?oeoo?e eiaeioeieii? a?e ia i?aenoaa?
?acoeueoao?a ?? aei???a niinoa?aaeaiiyi, a iiaeeea?noue a?aeiiaeaiiy
no?oeoo?e — niinoa?aaeiinoth eiaeioeieii? a?e. ?icaeyiaii ianooiio
caaea/o.

Caaea/a 1. Iaoae ia? i?noea noaa?eueia eiaeioeieia a?a ?ici??iino? N c
iacaeaaeiith a?ae /ano no?oeoo?ith q. Aeiaaa?oueny:

Aecia/eoe, /e i?eoaiaiia caaeaiie eiaeioeieiie a?i aeanoea?noue
niinoa?aaeiinoi;

sseui a?a ? niinoa?aaeiith, oi cae?enieoe ?? niinoa?aaeaiiy.

Aeo?aeieie aeey ae??oaiiy i?aecaaea/? 1) caaea/? 1 ? ianooii?
oaa?aeaeaiiy ? oai?aia.

Oaa?aeaeaiiy 1. Ne?ae eieaeueii? a?e rl (l ( 1, …, N), ui aoiaeeoue o
eiaeioeieio a?o r, ia? ianooiio aeanoea?noue aeey aoaeue-yeiai k (k = 0,
1, …):

. (3)

Ia i?aenoaa? (3) i?iaeoei? eieaeueii? a?e aeae?athoueny na?aae
aeaiaio?a i?iaeoeie iioaioe?eii? eiaeioeieii? a?e p, aeoiaey/e c oiiae:

E??i oiai, a neeo (3), aeey aecia/aiiy noaio eiaeioeieii? a?e r (a
oaeiae aeia?eueii? eieaeueii? a?e rl, l( {1,…, N}, ui aoiaeeoue o
eiaeioeieio) aeinoaoiuei ciaoe aoaeue-ye? aea? ?? i?iaeoei?.

Oai?aia 1. Iaoae caaeaia eiaeioeieia a?a r, ui neeaaea?oueny c N
eieaeueieo iai? rl = (xl, yl) (l ( {1, …, N}), aei?? yei? a iiiaio tk
(k = 0, 1, …) yaey? niaith oii?yaeeiaaio ia?o (Ck, Dk)

),

i?e/iio aeey eiaeiiai l (l( {1, …, N}) ?nio? ?aeeia i (i( {1, …, N})
oaea, ui ni?aaaaeeeai ??aiyiiy

;

aeey eiaeiiai l (l( {1, …, N}) ?nio? ?aeeia j (j( {1, …, N}) oaea,
ui ni?aaaaeeeai ??aiyiiy

;

aeey eiaeiiai i (i( {1,…, N}) ?niothoue ?aeei? j ? l (j( {1,…, N},
l( {1,…, N}) oae?, ui ni?aaaaeeea? ??aiyiiy

aeey eiaeiiai j (j( {1,…, N}) ?niothoue ?aeei? i ? l (i( {1,…, N},
l( {1,…, N}) oae?, ui ni?aaaaeeea? ??aiyiiy

Oiae? niinoa?aaeaiiy eiaeioeieii? a?e r iiaea aooe cae?eniaia ia
i?aenoaa? ?ioi?iaoe?? ia a?eueo i?ae N+1 aei??o.

Oaeei /eiii, aeinoaoiueith oiiaith oiai, ui aeayeee ne?ae ? ne?aeii
eieaeueii? a?e aoaea aeeiiaiiy oiiae (3) aeey aoaeue-yeiai k (k( {0,
…, N}).

Aeiaaaeaiiy oaa?aeaeaiiy 1 ? oai?aie 1 aeicaiey? noaa?aeaeoaaoe, ui
?icaeyaeoaaiie eiaeioeieiie a?i i?eoaiaiia aeanoea?noue niinoa?aaeiinoi.

Ieae/aiaaaaeaia oai?aia anoaiiaeth? oa?aeoa?i? ?ene, i?eoaiaii? ne?aeo
aeia?eueii? ?aaeueii? eieaeueii? a?e ? aeei?enoiao?oueny i?e
aaniina?aaeiueiio ae??oaii? caaea/? niinoa?aaeaiiy eiaeioeieii? a?e, ui
caaeaia.

Oai?aia 2. Iaoae rl (l( {1, …, N}) — aeayea eieaeueia a?a, ui
aoiaeeoue o eiaeioeieio a?o r, aea

0-aei?? rl,

1-aei?? rl,

k-aei?? rl.

Oiae? ca oiiae noaa?eueiino? eiaeioeieii? a?e r aeey aoaeue-yeiai l (l(
{1, …, N}) aeey ne?aeo ?aaeueii? eieaeueii? a?e rl iiaeeeaa iaeia c
o?ueio ianooiieo neooaoe?e, eiaeia c yeeo aeeeth/a? aeai iioi.

Ceooaoeiy 1

Ceooaoeiy 2

Ceooaoeiy 3

?icae?e 3. Ae??oaiiy caaea/? niinoa?aaeaiiy eiaeioeieii? a?e

Caaea/a niinoa?aaeaiiy eiae?oeieii? a?e ae??oaia oeyoii ?ic?iaee oa
i?ia?aioaaiiy a?aeiia?aeiiai aeai?eoio. Aeai?eoi a?oioo?oueny ia
oaa?aeaeaiiyo ? oai?aiao, ye? aoee noi?ioeueiaai? a iiia?aaeiueiio
?icae?e?.

Aeiaaaeaii, ui aeai?eoi, iiaoaeiaaiee c o?aooaaiiyi anoaiiaeaieo aeua
caeiiii??iinoae, a iaea??oiio aeiaaeeo ia? /aniao neeaaei?noue, ui
aei??aith? O(N4logN) i?e N(+(. Iaicia/eii TA — /aniao neeaaei?noue
icia/aiiai aeai?eoio, T — /aniao neeaaei?noue aeai?eoio iiaiiai
ia?aai?o. Oiae? iiaeia iieacaoe, ui TA ( log T.

Aiae?c o?eaaeinoi ?iaioe aeai?eoio niinoa?aaeaiiy aeeiai?/ii?
eiaeioeieii? a?e i?e ??cieo ?ici??ao eiaeioeie (aeea. oaae.2 oa ?en.2)
aeicaiey? noaaaeaeoaaoe, ui o na?aaeiueiio /aniaa neeaaei?noue aeai?eoio
niinoa?aaeaiiy eiaeioeieii? a?e noaiiaeoue O (N3logN) i?e N(+(, ui
i?eaeecii a N ?ac oaeaeoa, i?ae a a??oiio aeiaaeeo.

Oaaeeoey 2. Ii??aiyeueia oa?aeoa?enoeea /ano ae??oaiiy caaea/?

niinoa?aaeaiiy aeeiai?/ii? eiaeioeieii? a?e c? cia/aiiyie ooieoe?e

y = (N/12)3 log2(N/12)/4 oa y = (N/12)4 log2(N/12)/4

(aea N — e?euee?noue eieaeueieo iai?)

?

i/i ?ici?? eiaeioe?e (N) *an niinoa?aaeaiiy y = (N/12)3 x

x log2(N/12)/4 y = (N/12)4 x

x log2(N/12)/4

1 1 — 14 0 — 1 n 0 — 1,5 0 — 1,8

2 15 — 16 1 — 2 n 1,9 — 2,4 2,4 — 3,2

3 17 — 18 2 — 3 n 2,9 — 3,5 4,1 — 5,3

4 19 — 21 3 — 4 n 4,2 — 5,9 6,8 — 9,3

5 22 — 24 5 — 8 n 6,9 — 9,2 12,6 — 18,3

6 25 9 — 10 n 10,5 21,9

?en. 2. Ii??aiyeueia oa?aeoa?enoeea c?inoaiiy /ano aeeiiaiiy

aeai?eoio niinoa?aaeaiiy aeeiai?/ii? eiaeioeieii? a?e c? cia/aiiyie

ooieoe?e y = (N/12)3 log2(N/12)/4 oa y = (N/12)4 log2(N/12)/4

(aea N — e?euee?noue eieaeueieo iai?)

I?aeoe/ia canoinoaaiiy ?ic?iaeaiiai aeai?eoio niinoa?aaeaiiy iieacaia
ia i?eeeaae? ae??oaiiy caaea/? niinoa?aaeaiiy aeeiai?/ii? eiaeioeieii?
a?e, ui aeieea? a neooaoe?? aca?iiae?? a?oie nth?aaea?ia c a?oiith
no?aooaaeueiee?a (iia’ycaii? c a?oiith ia’?eo?a no?aooaaiiy) i?e
ioe?ioe? noaio aaciaee ia’?eo?a no?aooaaiiy. ?ici?? iaio eiae?oe?e
aei??aith? ainueie. I?ae eiae?oe?yie ?nio? ianooiiee ca’ycie.
No?aooaaeueieee iiaeathoue nth?aae??ai ?ioi?iaoe?th i?i ??aaiue aaciaee
ia a?aeiia?aeieo i?aei?e?inoaao (ia’?eoao). ?nio? aeaa a?aie/i? aa??aioe
?o iiaaae?iee:

cayaeoe aaniethoio aaciaeo ia’?eoa no?aooaaiiy;

cayaeoe ae?eniee ??aaiue aaciaee.

Aei ooieoe?? nth?aae??a iaeaaeeoue i?iaaaeaiiy ia?aa??ee aaciaee
i?aei?e?inoa, i?e/iio aea?? no?aoaa?? eiai iiaaae?iee ciaoiaeeoueny
iii?ae oaeeie a?aie/ieie cia/aiiyie:

ia ia?aa??yoe aeiaeiiai c i?aei?e?inoa;

ia?aa??eoe on? i?aeeaae? eiio ia’?eoe.

A?aeiii, ui eiaeiee nth?aae?? aca?iiae?? c ?aeeiei no?aooaaeueieeii (?
?aeeiith a?oiith ia’?eo?a no?aooaaiiy), i?e/iio oe?e ca’ycie ? iino?eiei
i?ioyaii onueiai i?ii?aeea /ano, ui aeine?aeaeo?oueny. Iioaioe?eii
aoaeue yeee nth?aae?? iiaea aca?iiae?yoe n aoaeue yeei ?c
no?aooaaeueiee?a, aea ae?eni? aea?eee nth?aae??-no?aooaaeueiee ai??i??
ia a?aeii?.

Ao?aeiith ?ioi?iaoeeae ? iao?eoe? an?o iioaioe?eieo eieaeueieo ?ai? a
eiaeiee iiiaio /ano. ?acoeueoaoe, io?eiai? i?ney canoinoaaiiy
?ic?iaeaiiai aeai?eoia niinoa?aaeaiiy eie?oe?eii? a?e, iaaaaeai? a
oaae.3, a ye?e aeei?enoai? ianooii? oiiai? iicia/aiiy:

K1, ? K2 — eiae?oe?? nth?aae???a ? no?aooaaeueiee?a, o a?aeiia?aeieo
eieiieao iaaaaeai? ii?yaeeia? iiia?e i?aaenoaaiee?a oe?o eiae?oe?e;

( — cia/aiiy iioeiaeueii? no?aoaa?? nth?aae??a ( eiia??i?noue oiai, ui
nth?aae?? aeaa?a /enoo no?aoaa?th «i?iaiaeeoe aeanio ioe?ieo aaciaee
ae?iaieoeoaa» (aeey a?oie ia’?eo?a no?aooaaiiy oea aei??aith? a?aeiini?e
aaee/ei? ia’?eo?a, ye? nth?aae?? ia?aa??y?);

( — cia/aiiy iioeiaeueii? no?aoaa?? no?aooaaeueieea ( eiia??i?noue oiai,
ui a?i aeaa?a no?aoaa?th «cayaeoe aaniethoio aaciaeo ia’?eoa
no?aooaaiiy» (?ioeie neiaaie ( — oea na?aae? cia/aiiy aaciaee
ae?iaieoeoaa aeey an?o ia’?eo?a no?aooaaiiy, aei yeeo oeae
no?aooaaeueiee ia? a?aeiioaiiy);

«oe?ia» — aaee/eia «aea?aoo» nth?aae??a;

t0, t1, t2 — iiiaioe /ano, a ye? ?iaeeenue aei??e ia?aiao??a.

Oaaeeoey 3. Iioeiaeuei? no?aoaa?? oa oe?ie an?o ?aaeueieo

eieaeueieo ?ai? a iiiaioe /ano t0, t1, t2

? i/i t=t0 t=t1 t=t2

E1 E2 ( ( oeaia ( ( oeaia ( ( oeaia

1 1 0.35 0.78 0.05 0.30 0.82 0.15 0.20 0.84 0.81

2 2 0.56 0.60 0.10 0.52 0.65 0.20 0.30 0.76 1.05

3 3 0.54 0.63 0.22 0.49 0.67 0.32 0.47 0.59 0.30

4 4 0.52 0.65 0.24 0.47 0.69 0.34 0.61 0.45 0.37

5 5 0.63 0.55 0.25 0.58 0.59 0.35 0.34 0.71 0.36

6 6 0.45 0.71 0.40 0.40 0.75 0.50 0.35 0.71 0.60

7 7 0.55 0.63 0.43 0.50 0.67 0.53 0.68 0.36 0.21

8 8 0.30 0.83 0.61 0.25 0.86 0.71 0.42 0.64 0.44

Aeey niinoa?aaeaiiy eie?oe?eii? a?e a aeaiiiio aeiaaeeo aeyaeeinue
aeineoue aeaieo o?ueio aei???a. *an ?iaioe aeai?eoio ( 0,17n.

AENIIAEE

A aeena?oaoeieiie ?iaio? iinoaaeai? oa ae??oai? caaea/a aecia/aiiy
niinoa?aaeiino? oa caaea/a niinoa?aaeaiiy aeeiai?/ii? eiaeioeieii? a?e,
ui iiaeaeth? aca?iia?aeiineie aeaiaio?a a aeiiii?/i?e nenoai? no?aoiaiai
oeio. Iniiai? iaoeia? ?acoeueoaoe, io?eiai? a aeena?oaoe??, iieyaathoue
a ianooiiiio.

Ia i?aenoaa? aiae?co i?inoi?o no?aoiaeo a?aeiinei iiaeia c?iaeoe
aeniiaie i?i oa, ui aoaeue-ye?e aeaiaioa?i?e no?aoiaie iia?aoe??
i?eoaiaii? aeanoeaino? iaaecia/aiinoi oa eiioeieoiinoe. ?icaeyaeaiiy
ie?aieo iia?aoe?e o eiioaeno? no?aoiaiai ?eieo aeicaiey? aeiaeaoe aei
oeeo aeanoeainoae oaeiae aeanoeaino? eiaeioeieiinoi oa aeeiai?/iino?. C
oeueiai i?eaiaeo aecia/aiiy aa?oino? no?aoiaeo iineoa ia i?aenoaa?
ia/eneaiiy eiia??iino? no?aoiaiai aeiaaeeo, ye oea i?eeiyoi a aeooa?iie
oai???, ia ? aeinoaoi?i.

O ca’yceo c oeei aeey a?eueo iiaiiai aeine?aeaeaiiy no?aoiaeo
aca?iia?aeiinei (cie?aia a?aeiinei, ui ii?ooothoue ieoaiiy anoaiiaeaiiy
oa?eo?a) ?ic?iaeaia eiioeaioe?y ? oaoiieia?y eiioe?eoii? aeiiii?/ii?
nenoaie no?aoiaiai oeio, ui aeicaieeoue cae?enithaaoe iiaeaethaaiiy
icia/aii? nenoaie.

Ia i?aenoaa? ?ic?iaeaii? eiioeaioe?? iiaoaeiaaia iiaeaeue aeiiii?/ii?
nenoaie no?aoiaiai oeio o aeaeyae? aeeiai?/ii? eiaeioeieii? a?e c
ioeueiaith noiith.

E?ioeaaith iaoith i?e ?icaeyae? aeiiii?/ii? nenoaie no?aoiaiai oeio ?
cae?eniaiiy oi?aae?iiy i?ioeanii no?aoiaeo a?aeiinei iii?ae noa’?eoaie
no?aoiai? ae?yeueiino? (aeaiaioaie nenoaie). Aoaeoeaia oi?aae?iiy iiaea
aooe caeieniaia o?eueee ca iayaiino? ae/a?iii? ?ioi?iaoe?? i?i nenoaio,
oiio aaciina?aaeiueith iaoith aeine?aeaeaiiy aeo?aeii? no?aoiai? nenoaie
ne?ae aaaaeaoe aeyaeaiiy aeiaeaoeiai? ?ioi?iaoe?? oa i?iaiic ia oe?e
i?aenoaa? iiaaae?iee nenoaie a iaeaooiueiio.

Aeey ae??oaiiy caaea/? iiiiaiaiiy ?ioi?iaoe?? a nenoai?, ui
aeine?aeaeo?oueny, aoee noi?ioeueiaai? iiiyooy noaa?eueiino?,
noaa?eico?iinoi oa niinoa?aaeiinoi eiaeioeieii? a?e, ia i?aenoaa? yeeo
aeiaaaeaia niinoa?aaeiinoue (iiaeeea?noue ciaeoe ??oaiiy caaea/?
niinoa?aaeaiiy) ye noaa?eueii?, oae ? noaa?e?co?ii? eiaeioeieii? a?e ia
a?eueo, i?ae i?ney N+1 aei??o ?? aeo?aeieo ia?ai?iieo (aea N — /enei
eieaeueieo iai?, ui aoiaeyoue o eiaeioeieio).

Ae??oaiiy iinoaaeaii? caaea/? cae?eniaia oeyoii ?ic?iaee oa
i?ia?aioaaiiy aeai?eoio niinoa?aaeaiiy eiaeioeieii? a?e.

Aiae?c /aniai? neeaaeiino? icia/aiiai aeai?eoio iieaco?, ui o iaea??oiio
aeiaaeeo aiia aei??aith? O (N4logN), a a na?aaeiueiio — O(N3logN) i?e
N(+( (ui ? eiaa?eoiii a?ae neeaaeiino? aeai?eoio iiaiiai ia?aai?o,
iaiao?aeiiai aeey ae??oaiiy caaea/? niinoa?aaeaiiy).

Io?eiai? oaeaeeinoii oa?aeoa?enoeee aeicaieythoue aeei?enoaoe
?ic?iaeaiee aeai?eoi i?e ae??oaii? caaea/ niinoa?aaeaiiy aeeiai?/ieo
iai? aeey eiaeioeie aeinoaoiuei aaeeeiai ?ici??o.

IA?AE?E IIOAE?EIAAIEO ?IA?O, A ssEEO

A?AeAEO? INIIAI? IIEIAEAIIss AeENA?OAOe??

Nieieianeee Ae.A. I?eiaiaiea iaoiaea iiyoaiiiai eiiaeiaoi?iiai iienea a
caaea/a i?iaiice?iaaiey ninoiyiey iaeioi?ie aeeiaie/aneie nenoaiu. —
Aeiiaoee: EYI AI Oe?aeiu, 1992. — 23n.

Aiioa A.E., Ia?oyeiaa A.A., Nieieianeee Ae.A. Iiaeoiae e auaeaeaieth
aaciauo oaeoi?ia i?e enneaaeiaaiee aeene?aoiie yeiiiie/aneie nenoaiu. —
Aeiiaoee: EYI IAI Oe?aeiu, 1995. — 22n. (Iniaenoee aianie: oi?iaeueia
iinoaiiaea, ae?ioaiiy e ocaaaeueiaiiy aeoiaeii? caaea/e)

Eiioeaioeey e oaoiieiaey enneaaeiaaiey iaaethaeaaiinoe yeiiiie/aneeo
nenoai (Aiioa A.E., Ia?oyeiaa A.A., Nieieianeee Ae.A., Eunaiei TH.A.,
Neiaaeaa A.A.) — Aeiiaoee: EYI IAI Oe?aeiu, 1998. — 42n. (Iniaenoee
aianie: aeieac niinoa?aaeiinoi nenoai cacia/aiiai aeaeo; ae?ioaiiy
caaea/e niinoa?aaeiinoi; aeai?eoi niinoa?aaeaiiy)

Nieieianeee Ae.A. Ia iaeiii aeene?aoiii iaoiaea iioeiaeueiiai iienea
aeaeaeoueony iauaeoia // Caaea/a Noaoaia ni naiaiaeiuie a?aieoeaie. —
E.: EI AI ONN?, 1990. — N.30-37.

Nieieianeee Ae.A. Oi?aaeaiea oa?aeoa?enoeeaie ninoiyiey iaeiie
aeeiaie/aneie nenoaiu // Oi?aaeaiea ia aaca ia?niiaeueiuo eiiiuethoa?ia.
— Aeiiaoee: EYI AI ONN?, 1990. — N.107-117.

AIIOAOe?ss

Nieieianueeee Ae. A. Iiaeaethaaiiy niinoa?aaeiinoe eiioe?eoii?
aeiiii?/ii? nenoaie no?aoiaiai oeio. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa aeiiii?/ieo
iaoe ca niaoe?aeueiinoth 08.03.02 — Aeiiiiiei-iaoaiaoe/ia iiaeaethaaiiy
— Aeiiaoeueeee aea?aeaaiee oi?aa?neoao I?i?noa?noaa ina?oe Oe?a?ie,
Aeiiaoeuee, 1998.

Aeena?oaoeiy i?enay/aia ieoaiith iiaeaethaaiiy aca?iiaiaeiinei iiae
aeaiaioaie aeiiiii/ieo nenoai no?aoiaiai oeio. Iiaoaeiaaia iiaeaeue
eiioeieoii? aeiiii?/ii? no?aoiai? nenoaie o aeaeyae? aeeiai?/ii?
eiaeioeieii? a?e c ioeueiaith noiith. Aaaaeaii iiiyooy niinoa?aaeiinoi
aeeiai?/ii? eiaeioeieii? a?e i aeiaaaeaii, ui cacia/aia a?a ?
niinoa?aaeiith. Ae??oaia caaea/a niinoa?aaeaiiy oeyoii ?ic?iaee oa
i?ia?aioaaiiy aiaeiiaiaeiiai aeai?eoio, yeee ia? aoaeoeaii oaeaeeinii
yeinoi. Io?eiaii ?acoeueoaoe ia i?aeoeoei iiaeooue aooe aeei?enoaii,
cie?aia i?e ae?ioaiii caaea/ aiae?co aca?iiaiaeiinei no?aoiaeo eiiiai?e
c aea?aeaaieie i?aaiaie, ioe?iee aeeiai?ee noaio aaciaee i?aoe?,
i?iaiicoaaiiy ?ioaineaiino? iaaeoiaeaeaiue no?aoiaeo i?ai?e o athaeaeao
oiiaeo iaia’yceiaiai no?aooaaiiy oiui.

Eeth/ia? neiaa: iiaeaethaaiiy aca?iiaiaeiinei iiae aeaiaioaie
eiioeieoieo aeiiiii/ieo nenoai, eiioeieoia aeiiii?/ia no?aoiaa nenoaia,
eiioeieoia aeiiiii/ia nenoaia, aeeiaii/ia eiaeioeieia a?a,
niinoa?aaeiinoue.

AIIIOAOeEss

Nieieianeee Ae. A. Iiaeaee?iaaiea iaaethaeaaiinoe eiioeeeoiie
yeiiiie/aneie nenoaiu no?aoiaiai oeia. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea iao/iie noaiaie eaiaeeaeaoa yeiiiie/aneeo
iaoe ii niaoeeaeueiinoe 08.03.02 — Yeiiiieei-iaoaiaoe/aneia
iiaeaee?iaaiea — Aeiiaoeeee ainoaea?noaaiiue oieaa?neoao Ieienoa?noaa
ia?aciaaiey Oe?aeiu, Aeiiaoee, 1998.

Aeenna?oaoeey iinayuaia aii?ino iiaeaee?iaaiey acaeiiioiioaiee iaaeaeo
yeaiaioaie yeiiiie/aneeo nenoai no?aoiaiai oeia. Iino?iaia iiaeaeue
eiioeeeoiie yeiiiie/aneie no?aoiaie nenoaiu a aeaea aeeiaie/aneie
eiaeeoeeiiiie ea?u n ioeaaie noiiie. Aaaaeaii iiiyoea iaaethaeaaiinoe
aeeiaie/aneie eiaeeoeeiiiie ea?u e aeieacaii, /oi oeacaiiiay ea?a
yaeyaony iaaethaeaaiie. ?aoaia caaea/a iaaethaeaiey ioo?i ?ac?aaioee e
i?ia?aiie?iaaiey niioaaonoaothuaai aeai?eoia, eiathuaai yooaeoeaiua
nei?inoiua oa?aeoa?enoeee. Iieo/aiiua ?acoeueoaou ia i?aeoeea iiaoo
auoue eniieueciaaiu, a /anoiinoe, i?e ?aoaiee caaea/ aiaeeca
acaeiiioiioaiee no?aoiauo eiiiaiee n ainoaea?noaaiiuie i?aaiaie, ioeaiee
aeeiaieee ninoiyiey aaciianiinoe o?oaea, i?iaiice?iaaiey eioaineaiinoe
iinooieaiee no?aoiauo i?aiee a athaeaeao oiiaea iaycaoaeueiiai
no?aoiaaiey e o.i.

Eeth/aaua neiaa: iiaeaee?iaaiea acaeiiioiioaiee iaaeaeo yeaiaioaie
yeiiiie/aneeo nenoai, eiioeeeoiay yeiiiie/aneay no?aoiaay nenoaia,
aeeiaie/aneay eiaeeoeeiiiay ea?a, iaaethaeaaiinoue.

SUMMARY

D.B. Sokolovskij. Modelling of observability for conflict economic
system of insurance type. — Manuscript.

Thesis for the degree Candidate of economic science by speciality
08.03.02 — Economic-mathematical modelling — Donetsk state university of
Ministry of education of Ukraine, Donetsk, 1998.

The thesis is devoted to a problem of modelling of relationships
between elements of economic systems of insurance type. The model of
conflict economic insurance system as a dynamic coalition zero-sum game
is elaborated. The concept of observability of dynamic coalition game is
introduced and it is proved, that determined game is observable. It is
solved the problem of observation by means of development and
programming of appropriate algorithm, that have the effective temporaty
characteristics. In practice obtained results may be used, in
particular, for solving of problem of the analysis of relationships
between insurance companies and state institutions, evaluation of
changes in condition of sabor safety, forecasting of intensity of
receipts of the insurance premiums in the budget of fund of mandatory
insurance etc.

Key word: conflict economic insurance system, modelling of
relationships between elements of economic systems, dynamic coalition
game, observability.

Nieieianueeee Aeieo?i Ai?eniae/

Aaoi?aoa?ao

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa aeiiii?/ieo iaoe

I?aei. aei ae?oeo 19/10/98. Oi?iao 60o90/16. Iai?? ae?oe. ?3

Ion. ae?oe. Iae.-aeae. a?e. 1,0. Oe?aae 100 i?ei. Caiia. ?

?inoeooo aeiiii?ee i?iieneiaino? IAI Oe?a?ie.

340048, Aeiiaoeuee, Oi?aa?n?oaonueea 77. Aaceiooiaii.

?ioai?eio ?AI IAI Oe?a?ie

Похожие записи