AAEI?ONNEEE AINOAeA?NOAAIIUE OIEAA?NEOAO EIOI?IAOEEE E ?AAeEIYEAEO?IIEEE
Eaoaae?a ?YN
?aoa?ao ia oaio:
«Eiaeu aac iaiyoe. Eiaeu Oaooiaia. Eiaeu n iaiyoueth»
IEINE, 2009
Eiaeu aac iaiyoe. Eiaeu Oaooiaia
I?inoaeoeie eiaeaie, ia iniiaa eioi?uo iiaeao auiieiyoueny naeaoea
aeaiiuo, yaeythony eiaeu aac iaiyoe. A eiaea aac iaiyoe eaaeaeue neiaie
a eiaee?oaiii aaeoi?a aeaiiuo caiaiyaony eiaeiaui neiaii ec i?aoeeniiai
iiiaeanoaa aeaie/iuo iineaaeiaaoaeueiinoae eee neia.
I?aoeeniui iiiaeanoaii aeaie/iuo iineaaeiaaoaeueiinoae S iacuaaaony
eiia/iia iiiaeanoai aeaie/iuo iineaaeiaaoaeueiinoae, oaeeo, /oi ie iaeia
iineaaeiaaoaeueiinoue a yoii iiiaeanoaa ia yaeyaony i?aoeenii, eee
ia/aeii, ieeaeie ae?oaie iineaaeiaaoaeueiinoe a S.
E i?eia?o, iiiaeanoai aeaie/iuo neia S1 = 00, 01, 100, 110, 1010, 1011
yaeyaony i?aoeeniui iiiaeanoaii aeaie/iuo iineaaeiaaoaeueiinoae,
iineieueeo, anee i?iaa?eoue ethaoth ec 30 aiciiaeiuo niaianoiuo
eiiaeiaoeee (wi wj) ec S1, oi aeaeii, /oi wi ieeiaaea ia yaeony
i?aoeenii (eee ia/aeii) wj. N ae?oaie noi?iiu, iiiaeanoai S2 = 00, 001,
1110 ia yaeyaony i?aoeeniui iiiaeanoaii aeaie/iuo
iineaaeiaaoaeueiinoae, oae eae iineaaeiaaoaeueiinoue 00 yaeyaony
i?aoeenii (ia/aeii) ae?oaie iineaaeiaaoaeueiinoe ec yoiai iiiaeanoaa –
001.
Oaeei ia?acii, anee iaiaoiaeeii caeiaee?iaaoue iaeioi?ue aaeoi? aeaiiuo
X = ( x1, x2,… xn ) n aeoaaeoii aeaiiuo A ?acia?a k, oi eiaee?iaaiea
eiaeii aac iaiyoe inouanoaeyaony neaaeothuei ia?acii:
– ninoaaeytho iieiue nienie neiaieia a1, a2, aj … ,ak aeoaaeoa A , a
eioi?ii aj – j-e ii /anoioa iiyaeaiey a X neiaie, oi anoue ia?aui a
nienea aoaeao noiyoue iaeaieaa /anoi ano?a/athueeny a aeoaaeoa neiaie,
aoi?ui – ?aaea ano?a/athueeny e o.ae.;
– eaaeaeiio neiaieo aj iacia/atho eiaeiaia neiai wj ec i?aoeeniiai
iiiaeanoaa aeaie/iuo iineaaeiaaoaeueiinoae S;
– auoiae eiaea?a iieo/atho iauaaeeiaieai a iaeio iineaaeiaaoaeueiinoue
anao iieo/aiiuo aeaie/iuo neia.
Oi?ie?iaaiea i?aoeeniuo iiiaeanoa e ?aaioa n ieie – yoi ioaeaeueiay
na?ueaciay oaia ec oai?ee iiiaeanoa, auoiaeyuay ca ?aiee iaoaai eo?na,
ii ianeieueei iaiaoiaeeiuo caia/aiee ana-oaee i?eaeaony naeaeaoue.
Anee S = w1, w2, … , wk – i?aoeeniia iiiaeanoai, oi iiaeii
ii?aaeaeeoue iaeioi?ue aaeoi? v(S) = ( L1, L2, … , Lk ), ninoiyuee ec
/enae, yaeythueony aeeeiaie niioaaonoaothueo i?aoeeniuo
iineaaeiaaoaeueiinoae (Li – aeeeia wi ).
Aaeoi? (L1, L2, … , Lk), ninoiyuee ec iaoiaiueoathueony
iieiaeeoaeueiuo oeaeuo /enae, iacuaaaony aaeoi?ii E?aooa. Aeey iaai
auiieiyaony ia?aaainoai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. (1)
Yoi ia?aaainoai iacuaaaony ia?aaainoaii E?aooa. Aeey iaai ni?aaaaeeeai
neaaeothuaa ooaa?aeaeaiea: anee S – eaeia-eeai i?aoeeniia iiiaeanoai,
oi v(S) – aaeoi? E?aooa.
Eiuie neiaaie, aeeeiu aeaie/iuo iineaaeiaaoaeueiinoae a i?aoeeniii
iiiaeanoaa oaeiaeaoai?ytho ia?aaainoao E?aooa.
Anee ia?aaainoai (1) ia?aoiaeeo a no?iaia ?aaainoai, oi oaeie eiae
iacuaaaony eiiiaeoiui e iaeaaeaao iaeiaiueoae n?aaee eiaeia n aeaiiui
aeoaaeoii aeeeiie, oi anoue yaeyaony iioeiaeueiui.
Ieaea i?eaaaeaiu i?eia?u i?inoaeoeo i?aoeeniuo iiiaeanoa e
niioaaonoaothuea ei aaeoi?u E?aooa:
S1 = 0, 10, 11 e v(S1) = ( 1, 2, 2 );
S2 = 0, 10, 110, 111 e v(S2) = ( 1, 2, 3, 3 );
S3 = 0, 10, 110, 1110, 1111 e v(S3) = ( 1, 2, 3, 4, 4 );
S4 = 0, 10, 1100, 1101, 1110, 1111 e v(S4) = ( 1, 2, 4, 4, 4, 4 );
S5 = 0, 10, 1100, 1101, 1110, 11110, 11111 e v(S5) = ( 1, 2, 4, 4, 4, 5,
5 );
S6 = 0, 10, 1100, 1101, 1110, 11110, 111110, 111111
e v(S6) = (1,2,4,4,4,5,6,6).
Aeiionoei iu oioei ?ac?aaioaoue eiae aac iaiyoe aeey naeaoey aaeoi?a
aeaiiuo X = ( x1, x2,… xn ) n aeoaaeoii A ?acia?ii a k neiaieia. Aaaaeai
a ?anniio?aiea oae iacuaaaiue aaeoi? /anoio F = ( F1, F2, … , Fk ),
aaea Fi – eiee/anoai iiyaeaiee i-ai iaeaieaa /anoi ano?a/athuaainy
neiaiea ec A a X. Caeiaee?oai X eiaeii aac iaiyoe, aeey eioi?iai aaeoi?
E?aooa L = ( L1, L2, … , Lk ).
Oiaaea aeeeia aeaie/iie eiaeiaie iineaaeiaaoaeueiinoe B(X) ia auoiaea
eiaea?a ninoaaeo
L*F = L1*F1 + L2*F2 + … + Lk*Fk . (2)
Eo/oei eiaeii aac iaiyoe aue au eiae, aeey eioi?iai aeeeia B(X) –
ieieiaeueia. Aeey ?ac?aaioee oaeiai eiaea iai ioaeii iaeoe aaeoi? E?aooa
L, aeey eioi?iai i?iecaaaeaiea L*F auei au ieieiaeueiui.
I?inoie ia?aai? aiciiaeiuo aa?eaioia – aiiaua-oi, ia naiue eo/oee niinia
iaeoe oaeie aaeoi? E?aooa, iniaaiii aeey aieueoiai k.
Aeai?eoi Oaooiaia, iacaaiiue a /anoue aai ecia?aoaoaey – Aeyaeaea
Oaooiaia, – aeaao iai yooaeoeaiue niinia iienea iioeiaeueiiai aaeoi?a
E?aooa L aeey F, oi anoue oaeiai L, aeey eioi?iai oi/a/iia i?iecaaaeaiea
L*F – ieieiaeueii. Eiae, iieo/aiiue n eniieueciaaieai iioeiaeueiiai L
aeey F, iacuaatho eiaeii Oaooiaia.
Aeai?eoi Oaooiaia
A???i O???a ?y?i ????o ??th ?ath ???????i ei????y n ??????? ????? ???? e
???? ??oth?i ??a?i:
1. A??u?? a ?yae ?a ???u ???? a ??y?a ?????y ?e ???ey aa?y??? ? ?y??ey a
???.
2. I??????? ???iy? ?a ???a n ?????e ??y?i?y? ?y??ey a ??e ???i? ???,
??y??? ?y??ey ???i? ???? ??? ??a ??y???e ???eyth?o ?i ????. A ?i? e??a
???? ???, ??? ?? ???? ??o ????th ??y??oue ?? oc?a, ??aey??y ?? ??.
3. I?????i ?? e ???o e?? ???, ???y ?????a e ???o ?? (????, ???i – 1, ???
– 0) . I????y ????a????ue ?? ???a ??i, ?????oth?a ???o ???o (?n. 1).
I???i ???a ??? ?y ????y ? ??oth?i ??ae?i:
A B C D E
10 5 8 13 10
B C A E D
5 8 10 10 13
A E BC D
10 10 13 13
BC D AE
13 13 20
AE BCD
20 26
AEBCD
46
?en. 1
Iaaeinoaoee iaoiaea Oaooiaia
Naiie aieueoie neiaeiinoueth n eiaeaie Oaooiaia, eae neaaeoao ec
i?aaeuaeouaai ianoaeaeaiey, yaeyaony iaiaoiaeeiinoue eiaoue oaaeeoeu
aa?iyoiinoae aeey eaaeaeiai oeia naeeiaaiuo aeaiiuo. Yoi ia
i?aaenoaaeyao i?iaeaiu, anee ecaanoii, /oi naeeiaaony aiaeeeneee eee
?onneee oaeno; iu i?inoi i?aaeinoaaeyai eiaea?o e aeaeiaea?o
iiaeoiaeyuaa aeey aiaeeeneiai eee ?onneiai oaenoa eiaeiaia aea?aai. A
iauai aea neo/aa, eiaaea aa?iyoiinoue neiaieia aeey aoiaeiuo aeaiiuo
iaecaanoia, noaoe/aneea eiaeu Oaooiaia ?aaioatho iayooaeoeaii.
?aoaieai yoie i?iaeaiu yaeyaony noaoenoe/aneee aiaeec eiaee?oaiuo
aeaiiuo, auiieiyaiue a oiaea ia?aiai i?ioiaea ii aeaiiui, e ninoaaeaiea
ia aai iniiaa eiaeiaiai aea?aaa. Nianoaaiii eiaee?iaaiea i?e yoii
auiieiyaony aoi?ui i?ioiaeii.
Nouanoaoao, i?aaaea, aeeiaie/aneay aa?ney naeaoey Oaooiaia, eioi?ay
iiaeao no?ieoue aea?aai Oaooiaia “ia eaoo” ai a?aiy /oaiey e aeoeaiiai
naeaoey. Aea?aai iinoiyiii iaiiaeyaony, /oiau io?aaeaoue eciaiaiey
aa?iyoiinoae aoiaeiuo aeaiiuo. Iaeiaei e iia ia i?aeoeea iaeaaeaao
na?ueaciuie ia?aie/aieyie e iaaeinoaoeaie e, e?iia oiai, iaania/eaaao
iaiueooth yooaeoeaiinoue naeaoey.
Aua iaeei iaaeinoaoie eiaeia Oaooiaia – yoi oi, /oi ieieiaeueiay aeeeia
eiaeiaiai neiaa aeey ieo ia iiaeao auoue iaiueoa aaeeieoeu, oiaaea eae
yio?iiey niiauaiey aiieia iiaeao ninoaaeyoue e 0,1, e 0,01 aeo/aoeao. A
yoii neo/aa eiae Oaooiaia noaiiaeony nouanoaaiii ecauoi/iui. I?iaeaia
?aoaaony i?eiaiaieai aeai?eoia e aeieai neiaieia, ii oiaaea oneiaeiyaony
i?ioeaaeo?a eiaee?iaaiey/aeaeiaee?iaaiey e cia/eoaeueii ?anoe?yaony
eiaeiaia aea?aai, eioi?ia ioaeii a eiia/iii eoiaa nio?aiyoue aianoa n
eiaeii.
Iaeiiaoe, eiae Oaooiaia iaania/eaaao n?aaeithth aeeeio eiaea,
niaiaaeathuoth n yio?iieae, oieueei a oii neo/aa, eiaaea aa?iyoiinoe
neiaieia enoi/ieea yaeythony oeaeuie io?eoeaoaeueiuie noaiaiyie aeaieee:
1/2 = 0,5; 1/4 = 0,25; 1/8 = 0,125; 1/16 = 0,0625 e o.ae. Ia i?aeoeea
aea oaeay neooaoeey ano?a/aaony i/aiue ?aaeei eee iiaeao auoue nicaeaia
aeiee?iaaieai neiaieia ni anaie auoaeathueie ionthaea iineaaenoaeyie.
Eiaeu n iaiyoueth
Iau/ii ?anniao?eaatho aeaa oeia eiaeia n iaiyoueth:
– aei/iua eiaeu;
– eiaeu n eiia/iie iaiyoueth.
Aei/iue eiae aeaeeo aaeoi? aeaiiuo ia aeiee caaeaiiie aeeeiu e caoai
eaaeaeue aeie caiaiytho eiaeiaui neiaii ec i?aoeeniiai iiiaeanoaa
aeaie/iuo neia. Iieo/aiioth iineaaeiaaoaeueiinoue eiaeiauo neia
iauaaeeiytho a ?acoeueoe?othuoth aeaie/ioth no?ieo ia auoiaea eiaea?a. I
aei/iii eiaea aiai?yo, /oi ii – aei/iue eiae k-ai ii?yaeea, anee ana
aeiee eiatho aeeeio, ?aaioth k.
A ea/anoaa i?eia?a niaeiai aaeoi? aeaiiuo X = (0, 1, 1, 0, 1, 0, 0, 1,
1, 1, 1, 0, 0, 1, 0, 1), eniieuecoy aei/iue eiae aoi?iai ii?yaeea.
Nia/aea ?aciaueai aaeoi? X ia aeiee aeeeiu 2: 01, 10, 10, 01, 11, 10,
01, 01.
Aoaeai ?anniao?eaaoue yoe aeiee eae yeaiaiou iiaiai “aeia?aeoaaeoa” 01,
10, 11. *oiau ii?aaeaeeoue, eaeie eiae iacia/eoue eaeiio eo neiaieia
yoiai iiaiai aeoaaeoa, ii?aaeaeei aaeoi? /anoio iiyaeaiee yeaiaioia
aeiiieieoaeueiiai aeoaaeoa a iineaaeiaaoaeueiinoe aeieia. Iieo/ei aaeoi?
/anoio (4, 3, 1), oi anoue iaeaieaa /anoi ano?a/athueeny aeie 01
iiyaeyaony 4 ?aca, neaaeothuee ii /anoioa ano?a/aaiinoe aeie 10 – o?e
?aca e iaeiaiaa /anoi ano?a/athueeny aeie 11 – eeoue iaeei ?ac.
Iioeiaeueiue aaeoi? E?aooa aeey aaeoi?a /anoio ( 4, 3, 1 ) – yoi aaeoi?
(1, 2, 2). Oaeei ia?acii, eiaea? aeey iioeiaeueiiai aei/iiai eiaea
aoi?iai ii?yaeea i?eiaieoaeueii e caaeaiiiio aaeoi?o aeaiiuo X
ii?aaeaeyaony oaae. 1.
Oaaeeoea 1
Oaaeeoea eiaea?aAeieEiaeiaia neiai01010101111
Caiaiyy eaaeaeue aeie i?enaiaiiui aio eiaeiaui neiaii ec oaaeeoeu
eiaea?a, iieo/ei iineaaeiaaoaeueiinoue eiaeiauo neia:
0, 10, 10, 0, 11, 10, 0, 0.
Iauaaeeiyy yoe eiaeiaua neiaa aianoa, eiaai auoiaeioth
iineaaeiaaoaeueiinoue eiaea?a:
B(X) = ( 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0 ).
Iiaeii i?iaa?eoue, /oi yio?iiey H(X) enoiaeiiai aaeoi?a X ?aaia 0.9887
aeo/aoeao, a noaiaiue naeaoey, iieo/aaiay a ?acoeueoaoa eniieueciaaiey
aei/iiai eiaea R(X) = 12/16 = 0.75 aeo ia auai?eo aeaiiuo, ieacaeanue
iaiueoa ieaeiae a?aieoeu yio?iiee. Iieo/aiiue ?acoeueoao, eacaeinue au,
i?ioeai?a/eo oai?aia Oaiiiia, ooaa?aeaeathuae, /oi iaaiciiaeii aeinoe/ue
n?aaeiae aeeeiu eiaea, iaiueoae yio?iiee enoi/ieea. Iaeiaei ia naiii
aeaea yoi ia oae. Anee aieiaoaeueii iiniio?aoue ia aaeoi? aeaiiuo X, oi
iiaeii caiaoeoue caeiiiia?iinoue: ca neiaieii 0 /aua neaaeoao 1, oi
anoue oneiaiay aa?iyoiinoue ?(1/0) nouanoaaiii aieueoa, /ai i?inoi ?(0).
Neaaeiaaoaeueii, yio?iieth yoiai enoi/ieea ioaeii n/eoaoue eae yio?iieth
neiaeiiai niiauaiey, a iia, eae ecaanoii, anaaaea iaiueoa, /ai aeey
enoi/ieea i?inouo niiauaiee.
Eiae n eiia/iie iaiyoueth i?e eiaee?iaaiee aaeoi?a aeaiiuo ( X1, X2,
…, Xn ) eniieuecoao eiaeiaoth eieao, ninoiyuoth ec ianeieueeeo
?acee/iuo eiaeia aac iaiyoe. Eaaeaeay auai?ea aeaiiuo eiaee?oaony eiaeii
aac iaiyoe ec eiaeiaie eieae, ii?aaeaeyaiui cia/aieai iaeioi?iai /enea
i?aaeuaeoueo auai?ie aeaiiuo.
A ea/anoaa i?eia?a eiaee?iaaiey eiaeii n iaiyoueth ?anniio?ei oaea
oiiieiaaoeeny ?aiaa eeanne/aneee i?eia? X = ABRACADABRA. A yoie
iineaaeiaaoaeueiinoe niaa?oaiii i/aaeaeia neeueiay noaoenoe/aneay
caaeneiinoue iaaeaeo aa i/a?aaeiuie neiaieaie, eioi?ay aeieaeia
iaycaoaeueii o/eouaaoueny i?e auai?a iaoiaea eiaee?iaaiey.
Eiaea? n iaiyoueth i?e eiaee?iaaiee oaeouaai neiaiea o/eouaaao cia/aiea
i?aaeoanoaothuaai aio neiaiea. Oaeei ia?acii, eiaeiaia neiai aeey
oaeouaai neiaiea A aoaeao ?acee/iui a ni/aoaieyo RA, DA e CA (eiuie
neiaaie, eiae iaeaaeaao iaiyoueth a iaeei neiaie enoi/ieea) (oaae. 2).
Oaaeeoea 2
Eiaea?Neiaie, i?aaeuaeouee neiaieEiaeiaia
neiai(A,-)1(B,A)0(C,A)10(D,A)11(A,R)1(R,B)1(A,C)1(A,B)1
?acoeueoao eiaee?iaaiey – aaeoi? B(X) = (10111011111011) aeeeiie a 11
aeo e nei?inoueth naeaoey R = 13/11=1,18 aeoa ia neiaie aeaiiuo, oiaaea
eae i?e eiaee?iaaiee ?aaiiia?iui o?ao?ac?yaeiui eiaeii n R = 3
iiiaaeiaeeinue au 33 aeoa.
EEOA?AOO?A
1. Eeaeianeee A.E. Oai?ey eioi?iaoeee. – I., «Aunoay oeiea», 2002a. –
120n.
2. Iao?ieiaey e ?aaeeiecia?aiey a oaeaeiiioieeaoeeiiiuo nenoaiao.
O/aaiee aeey AOCia. / A.E.Iaoaaeia, A.E.Oaeeei, A.A.Oaaei?ia e ae?. –
I.: Aunoay oeiea, 2001 a. – 383n.
3. Oeaiaiei I.I. Ecia?eoaeueiua eioi?iaoeeiiiua nenoaiu. – I.:
Yia?aiaoii ecaeao, 2005. – 440n.
4. Cthei A.A. , Eeianeee Ae.Ae., Iaca?ia I.A., Oeie E.I. Oai?ey
ia?aaea/e neaiaeia. I: ?aaeei e naycue, 2001 a. –368 n.
5. A. Neey?. Oeeo?iaay naycue. Oai?aoe/aneea iniiau e i?aeoe/aneia
i?eiaiaiea. Ecae. 2-a, eni?.: Ia?. n aiae. – I.: Ecaeaoaeueneee aeii
«Aeeueyin», 2003 a. – 1104 n.
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