HAYR YRPA : . . .
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R
30
1\ J\ 1989
519.72 sll 1 . .
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v: _. V- . , , , V - {0, 1 j. n R'\ n = = 1 1. -r . k: _. , -, , . dL KJJacco , /* (v: v , k (v) = k*J k*. k : _. _., , ,
ii , k, k ~ : , ~ ~. !I -
~ k (v), . . k (v) - ~ v _. , k (v) (v) = k (v) v v, .11 ,., v (t v.
, k, -,., ,., . v k (v) k (v) ,., k -
v, Jir v.
r . , . . [37-43, 70, 71, 90), . . [7, 8, 19-211, . . [84, 881, . . [44-47, 66), . . [123, 124), .~ . . , . . , . . [6), . . [51], . [89), . 3. [ 104), . . [100] 11 . , 1 /\ , r, 111 11 .
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1 1 - . v: -+ R, 1 !- R1r 1 -
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n , n , . . 1 . [761, , t~ , . n, , - ,
!2
,
. -Ii
' : Tv F (') '-+ {0, 1 }, '. v (') F (', v (')). F (Tv) .
F (Tv) - '! = {1 , 2 , ... , }, F ('), ' '9", v F (Tv, v) = F (', v (')).
'!
, -. , -, , k-, /- l > k.
- , ,
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( [761). - ,
, .
. 1. , Tv , , . v, : 1 ( ). , . . , { t : t Tv, v (t) = 1} . ,
>> - . , , ,
n , n , n . , , , npecta;'leHo : . , , n .
2. n Tv {(i, j) : 1 ~ i ~, 1 ~ j ~ n} i j. n i1 , i 2 , j 1 j 2 Tv-+ {0, 1 }, 1 { (i, j) : (i, j) Tv, it ~ i ~ i2, it ~ i ~ bl . n ,
I , n
(. 1.1). , ,
, ((i, j), (i + 1, j), (i, j + 1), (i + 1, j + + 1)). ., . 1.1. , -
n. n . Tv V
- CmiVIm
n 2 n , n = 1 Tv 1. -
21v1n. , n,
: 1 V 1 < 1 V ln. , = n - 1 n > 1 1. : 1 V Im > 1 V ln, . n . '' . [76], , , "I !l ,
15
- . , , J
.1 n , . , , , , . , - .
.
, F (Tv) v0 1. F (Tv) ~ , - . .
. , ,
v0 F(T'O, v0) =. (1.2)
, F (Tv) - '!/. F ('), ' ~.
F (Tv, V0) = 1\ F (', '00 (')). (1.3) '!
(1.2) (1.3) * ~.
F (*, V0 (*)) = .
, '00 , v*, ,
'* (*) = v0 (*), F(Tv, v*) = 1.
(1.4)
-
(1.5)
(1.6)
(1.6) F (Tv) ,
F ("', rJ* (*)) = 1. ( 1.7)
(1.7) (1.5) F (*, '00 (*)) = 1, (1.4). F (Tv) . .
. , n
16
Tv (i, J), i = 1, 2, ... , n j = 1, 2, ... , , n - .
k (i, j) Tv Tk (i, JJ : Tk (i, j)= = { (i', j') : (i', i') Tv, i ~ i' ~ i + k - 1, j ~ j' ~ j + + k- 1 }. !flk !flk = {Tk (i, j) : : (i, JJ Tv}. , tt!k - , k.
i1 , i 2 , j1 , j2 Vt,,t i .. i..
1, j = j1 , i1 ~ i ~ i 2; 1, j = j 2, i1 ~ i ~ i 2;
Vt,,t i . J. (i, j) = 1, i = i1 , j 1 ~ j ~ j 2; 1, i = i2, i1 ~i~i2; .
{vt,iol.i: (i1,ii) Tv, (i2, i2) Tv}. . 1.2
, . . , - .
,
,
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n . , n n -
2 8-2045 17
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18
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2. , - -
n F0 (Tv), , n
F0 (Tv,v)= F0 (T0 ,v(T0 )). (1.11) ~
F0 (Tv) -r- (1.11) F1 (Tv) :
F0 (Tv,v)= F0 (T0,v(T0))= F1 (T1,v(Tl))= 0~0 ~ 1~1()
- F1 (1 , v (1)) = F1 (Tv, v). 1~1
n (1.11), , - (1.10), (1.8) (1.9) . .
. , F (Tv) v0 , F (Tv, v0) = , t Tv v1, , v0 (Tv '\.._ { t}) = v1 (Tv / { t}), F (Tv, v1) = 1.
3. F (Tv) , - - .
. F (Tv), , 1 - ' 0 = { Tv '\.._ { t} : t Tv}. , 2 - . .
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20
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- - .
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. , , - (, ) .
- , . -
.
F1 (Tv) , F2 (Tv), F1 (Tv) ~ F2 (Tv), v, F1 (Tv, v) ~ F2 (Tv, v).
F1 (Tv) ~ F2 (Tv) v, , F1 (Tv, v) < F2 (Tv, v), F1 (Tv) < < F2 (Tv).
F* (Tv) - - F (Tv) ', :
1) F* (Tv) - - '; 2) F* (Tv) ~ F (Tv); 3) F' (Tv),
', , F (Tv) ~ F' (Tv) < F* (Tv). ,
- .
v ' v (v, ') v', v' (') = v (').
4. ' F (Tv) .-z Q1- - -
F* (Tv), 'tlv((F*(Tv,v))= V F(Tv,v')). (1.12)
' '!{ u'E~Iv. ') .
(1.12) F (', v (')). , F(', v(')} = V F(Tv, v'). (1.13)
v'E;(v,T')
(~ .12) , F* (Tv) -.
, F* (Tv} ~ F (Tv). F (Tv, v) = v, . v F (Tv, v) = 1. (1.13) F (', v ('))= 1 ' '!!. , (1.12) F* (Tv, v) = 1. F* (Tv} ~ F (Tv} .
, - F** (Tv), , F** (Tv} ~ F (Tv), , F** (Tv} ~ F* (Tv). , , . . - F** (Tv}, , F** (Tv} ~ F (Tv), F** (Tv) ~ F* (Tv) .
F** (Tv) - '!!, F* * ('), ' '!!,
'tlv(F**(Tv, v) = F**(', v(T')). (1.14) ''![
F** (Tv) ~ F (Tv),
'tl v (F** (Tv, v) ~ F (Tv, v)). ( 1.15)
F** (Tv) ~ F* (Tv) ,
3v(F**(Tv, v)
(1.13) **, n
v** (') = v* ('),
F(Tv, v""*) = 1. ( 1.19)
(1.20)
, v** F** (', v** ('}) =, ( 1. 14)
F**(Tv, v"'*) =. (1.21)
(1.20) (1.21) (1.15). ,
- 11 F** (Tv), , n F* * (Tv) ;;;:::. F (Tv) F** (Tv) ;;;:::. F* (Tv). .
~ !i. F (Tv) u:, '5", - '5". . i
Tv--+- (0, 11. (i1; it) (i2 , j 2) , 1 i1 - i2 l + 1 i1 - i2l = -:- 1, v , t0 Tv t11 Tv, , v (t0) = 1 v (tk) = 1, t0, t1, . , tk , , s = l, 2, ... , k v (t5) = 1, /5 , ts-l - . .
- , ~ . . . [76], . , - -, J< , , . .
, n , , . . - iiJI . ( ) , , . , , , , -
. - , ,
.
1.2.
- Tv , , V- . - Tv--+- V, ., J ~7
24
- . , . - .
, , . , , . . . , .
, Tk-+ , Tk, , . , .
. . S V U . -+ S, , - . , -.
'!! - , . . . , , . ' '!/" F (') : s-+ {0, 1}, '. Q; Q = { F (') : : ' '!/"}.
, . . (V, , Tv, Tk, '!/", Q), Q = {F(T'): : ' '!/" }.
s : -+ S ' s {') s ', t s (t) - t.
s : -+ S G = (V, , Tv, Tk, '!/", {F (') : ' '!/"}), ' '!/" F (', s (')) = = 1.
25
v : Tv -+ V G, s, , , v = s (Tv).
, G, rL (), S (G).
- ,
- F G, (F (v) = 1) (::) (v rL{G)), Tk = eJ k = eJ. .
. 7. Tv, ~ V F (Tv), - ~. { .11 -
G, , ff (G) = (v: v V1 , F (Tv, v) = 1\. . ,
. Tk ,1, ~ - F (Tv). ' ~ t (')
v ('), v (', t1) = v*, k (', v (')) = k*.
Fr ("), " '! .
Cflocmo 1. ' '! s ('). , F (', s (')) = l, s (' U ( t (')}), , " = = (t1 , t (')}, /1 ', Fr(T", s (")) = l. s (t (')) = k (', s (')).
' '! '! (') " '! " ' U { t (')}.
2. ' '! s (' U (t (')}), , Fr(T", s(T")) = l
"7'
G, s (Tv) = v st , n G. .
n. t. Tv - ,
: /1 , t2 , f3 , t4 ; ~ : 1 = { /1 , 12 , t3 } 2 = { f1 , f2 , t4 }; V = {0, 1}. F (1) , 1, :
(v1 (t1) = 1, v1 (t2) = , v1 (t3) = 1) -n 1; (v2 (t1) =, V2 (t2) = 1, V2 (/3) =)- 1
.~ 1.4. - --
F (2) . :
(v3 (f= 1, v3 (t2)= 1, v3 (t4)= 1)- 2 ;
(v4 (t1) = , V4 (t2) = , V4 (t4) = 1)- 2 v F (1 , v (1))
1\ F (2 , v (2)). G,
. ~ , Tk : t5 t6 , t5 {t1 , t2 , t3 }, t6 - {t1 , t2 , t4 }. ~ G , , : { (t1, f5), (t2, /5), (t3, /5), (tl, t), (t2, f), (t4, f)}.
F (1) F (2) , ~ : k1 k2
G . 1.4, t1 , / 2, t8, t._ n t~ t0 n!'l. ~
28
~
n , j l ~ l < n. , , , , .
, JI, . .
. 1.5. .
v: Tv-+ (0, 1 }, , t Tv, Ji v (t) = 1, . . 1.5 Ji , . ; Ji - . JI , JI -Ji. ( ) , , Ji
( ) . Ji Ji , Ji, Ji. Ji - Ji . : , , - .
1.3.
, Ji
JIOKaHOKOIIIOI\KTBIIM
30
-r. t, , , , i , . .
, , (Tv, Tk, V, , 'f/, {F(T'), ' E'ff}}, Tv, Tk, V, - , 'ff- Tv U Tk, F ('), ' :, - (V U ) -+ --+ {0, 1}. ~: (Tv U Tk)-+ (V U ) G , ' 'ff F (', s (')) = 1. v : Tv -+ V G , s, G, , v s Tv. , , . .
. 8. G G * , , ff (G) = ff (G * ). . G = (Tv, Tk, V, ,
'ff, (F ('), ' ff}}. (TtJ U Tk)--+ -+ (V U ) -l\1 ' . 7 G0 , , v (G0) = = s (G).
G0 = (Tv U Tk, Tk0 V U !(, 0 , 'f/0 , {F0 (T'), ' E'f/0 }}. G0 Tk0 ,
0 , :0 F0 (') , r
= S (G*), s* G* , Tv v. , v !f (G*).
, L (G*) : !f (G). v* !f (G*). s* S (G*),
, v* = s* (Tv). s* G0 , S (G*) = S (G,,). s* (Tv U Tk) Tv U Tk G0 , , G, v (G0) = S (G). 1' (Tv U Tk) G, s*(Tv) n G. s* (Tv) = v*. , v* {f (G). .
, , , .
t.4. -
n - .
G* - - , !f (G*)- , n ~.
n !l (G*, ) n !f (G*), n. , !f (G*, n) !f , G, :
1. Tv G n , n /1 , t2 , , tn.
2. n (v1, v2 , ... , v) !f (G*, n) n G v: Tv- V, v (t1) = v1 i = l, 2, ... , n.
3. n G v: Tv - V n, (v (/1), v (t2), ... , v (tn)) !f (G*, n).
, n GA n G, , !f (G) = !l (GA, n). , n, - . ..
;, 9. GA
32
n , , ! (, n) = ! (0). .
(V, , k,* ), V - , ; k* - , ; = 3 U 2 , 3 : V , 2 : V. . , v1, v2 , ... , Vn n , k1 , k2 , ... , k k1 = k*, (k, V) 2 , i = 1, 2, ... , n - 1 (kt, Vt, kt+t) 3 .
V. . Tv n t1 , t2 , ... , tn, Tk - 1 , 2 , ... , 't'n '!! ,
{t1 }, ( t} {t, tl, Tt+t}, i = 1, 2, ... ... , n- 1.
Tv U Tk, '!!, .
' = {1 } F (') 1 s (1) = k*.
' = { t} n F (', s (')) 1 s (') = (s (), s Un)) 2
' = (t, tt, 't't+t}, i = 1, 2, ... , n, F (', s (')) 1 s (') = (s ('t't), s (tt), s (t't+t)) 8 .
. 10. - t n G, , ! (, n) = fL (0). .
(V, , k*, ), V - , k* -, . [ 10), . - , = = 2 U , 3 : , 2 : V. s* s1 , s2 , ... , sm, , ;;> , St V U . i. , s2 s* 11 ~ ~ s*, s1 ~ s2, s1 8 1-2040
s1 = As1B, s* s*, s2 :
s2 = S2s3B, (s1 , s2, s8) 8 ; S2 = As8B, (s1 , s2) 2
(1.25) (1.26)
- - - - (1.25) s1-2 ... s2, (1.26)- s1 --2 .... s2
, (1.25) s2 !_I s~, s2 sl. (1.26) sl ~ s2, s2 sl.
sl , .... s2 _ 1 _ - _ _ , _
s1 ==> s2 ~1 -2 .... _ s2 - 31 =>_Sg , , s1 - s2 - s1 ==> Sa sl ==?- Sg fl. (, ) s10 Sg, ... , Sn, , k*, , s1 V i = 1, 2, ... , .
, s fl. (. ) s,
, 1 _
k* => s ==> s. s n
'!_:, t . s2, s3, ... , S Sn ==> s, . ,
it 5; - -, , -
, St-1 _ .... St _ .... s'+' - - -
. St-1 --+ s1, St-1
34
Uk1V, S*, VES*, k1 EK. s1 - - -Uv1 V, (k, v1) 2 s1 --'!..,. St+1 ,
-Uv1 V --'!..,. St+1, :
1. U=Wk2T, WEs*, TEs*, k2 EK;
st+1 = Wk3k4Tv1V, (k2, k3, k .. ) 3 2. V=Wk2T, WEs*, TEs*, k2 EK;
St+1 = Uv1Wk3k4T, (k2, k3 , k4) 3
5;+1 = Wk3k4Tk1 V, W~Tk1V ..!:..! .... Wk3k4Tk1V ..!:..! .... Wk3k4Tv1V, . . St-1 ..!:..! .... :S; ~ .... -+ St+1
-
:S; = Uk1 Wk3k4T, Uk1 Wk2T ,_,. Uk1 Wk3k4T ,_,. Uv1 Wk3k4T, . . : St-1 ~ ... 5; ..!:.! .... St+1 ,
1 ., 1 - 8 , - 1 - 1 - 1 ff' --+ S2 --+ S --+ --+ Sn --+ Sn+1 --+ Sn+2 --+
1 - 1 - - .. S2-1 --+ S,
.
r Tv /1 , / 2, ... , tn Tk {-rti: j = 1, 2, ... , i; i = 1, 2, ... , n). , s , s (,;11) = k*, i = 2, 3, ... , n (s ('t't-1, 1), s ('t't-1, 2), ... , s (t-1,t-) (s (-rll), s ('t't2), ... ... , s (u)) St-1 s~, ,
- - St-1 ___,!_,. s1, , , (s ('t'1), s (2), ... ... , s ('t)) (s (t1), s (t2), , s (tn)) -
- - . ._ sn s, , Sn =>- s. .
Tv Tk . V. {, 3, }, - , , 3, - , , . r ,
35
k - , k, k k, . ind JUJ , , 3, . , ~ (k, ind), k- , ind {, 3, }.
1) ind1i = , kl/ = kl+l./; 2) indii = , k11 = kL+l,J+!; ) indl/ = 3, (kli kl+l./ kl+t,J+ 3
, ~ ('tll, 't12, ... , 't'u) ('t'1+11, 'ti+I,2. 'tt+t., ... , 'tt+l.:.t+t),! = 2, 3, ... , n- 1,_2_: s1 Si+I, , s1 -2. .... Si+I
F (') (-r"'' tj), j = 1, 2, ... ... , n, ~s (-r,t1), s (t1)) = = (ki idnf, VJ), (k"i vi) 2 , ('t!, 'tn2 , 't) (t1 , f2 , ... , tn), -
- . Sn s, , Sn => s.
, , n , - . .
1.5.
,
Tv V. .
. 11. 1 . Tv, V . V v Tv -+ V .. G, , V = {/, (G). .
t G 't, ,
1 V 1 . - V k (v) , v, v (k) -, k. '! G ((-r, t): t Tv). t Tv F ('), '= = (., t) , k Oll 11 1 ~, v (k) (t)). F ('),
'fl k G s, , s () = k, Tv v (k) V, G . , v (k) {L (G)
37
k . {v (k) : k ) , V k (v (k) ff (G)) V' : : {t (G). k s, , s ('t) = k, 1 S (G) 1 = / /( 1- , 1 1 = 1 V 1. , 1 S (G) 1 CZt 1 , 1 ff (G) 1 :::;;;; 1 S (G) 1. , 1 !t (G) 1:;;;;; :::;;;; 1 1. ff (G) => ff () = . .
, - ,
- - . , l\! . , . , .1 , . .
2
.I
- , , . , ,
. , . .
.
r ,
, . , , , , , . , , , ,
, : 6 , .
, , , n n , n r n n.
n , , n ( ), n, , ~. n .11 , n,r, , , . . 11 n-
9
. , , n, , . , , n . n. . u _ .
n
. . no n , . , no n, . .n,
, ii
n n n .
, , , n . n nn n
n . n . n , , n. , n n , n
n . . : - , , , . , . , , n n .
40
2.1.
Tv ( ) V ( ) - ; v -
- yTv , . . Tv-+ V; V- 2. ,
. {, 1} * - {0, 1 }. {, 1} * d () .
{~ 1}* -
Q : -+ V.
,
'lJ V {: , Q () = 'l!} =1= " -
'lJ V D ('1!) , I Q () = '1!, . .
D ('ll) = min d (). {c:cEC,Q(c)=r}
, . k = 1 V l 1 v1 - . 1 k {0, 1} * k. Q !Ir , = 1 , 2 , ... , ak i '11 = Q () , CXt = 1.
, , D ('V) = k 'V v. , . n . , , . n J . Teope.Jta 1.
V . 'lJ 1 ~, D ('11 1) < k. /tt.: '11 ii, D (IJ 2) > k.
41
. : {0, 1}*- , , , Q- -
-+ V. V , ,
(2.1)
, , . .
* : , 'V V *, Q () = , d () ~ k, 1 *, , Q (1 ) = 1 , d (1) <
(2.7)
kt < kt-1 2nio-nt-J. kt-1
kt-1 < 2ni-J.
(2.9) (2.8), . .
(2.10)
(2.10) (2.9), ki < 2nl-l 2nt-nt-l = 2ni, (2.8) i = i*. (2.8), , i = 1, 2, ... , l.
: 1 1
1 * 1 = ~ 1 tl = ~ (kt-l 2nnl-l - kt) = l=l l=l
1
= 2n' + ~ ki-1 (2nni--l-1)- k1 < 2n' + l=2
1 1
+ ~ (2ni- 2ni-l)- kl < 2n' + ~ (2'\l- 2ni-l) = 2nl ~ 211. i=2 l=2
(2.11) , 1 , 2 , ... , 1 . (2.6). . (2.8). .
(2.11) 1 * 1 < < 2k, (2.1). , , . .
, D (V') ;;;;;;.: k. cpe-rEV ,
D (V') = k. rEV
, , , .11
, ,
. .
43
, , , , . [ 134, 136-140], , ,
,
, . ,
. , .
2.2.
1. -, ,
( ) . , , n . Tv ( n, n - ) ((i, j) : 1 ::;;;; i::;;;; n, 1 ::;;;; j::;;;; j. i . 1. j . 1. {0, 1 J. .1 ((1, j 1), (2, bl. (3, i), ... , (n, i)J , 1 ji- ii-l 1::;;;; 1 i = 2, 3, ... , n. v : Tv-...+ (0, 1 J , , v (t) = 1 t . . 2.1, , n , . 2.1, , - .
, . .1 Tk ( (t): t Tvj, n Tv; (t) , t Tv .
?7 W" = {{(t), t} :tETv} U
1--
1--
~ 1 ---1 ~ ~
~ ~ ~ ~ ~
~
~ ~ ~
~ ~ ~
~ ~
~ 1--1 f--
~ ~ - -
- - ~ 1-- --~ ~ t--~ 1--f--~
. 2.1. .
U {1 (i, j), ( i + 1 , j)} : i = 1 , 2, . . . , n - 1 ; j = 1, 2, ... , n) U
U {l(i,j), (i,j+ 1)}: i= 1, 2, ... , n; j=1,2, ... ,n-1}.
Tv, Tk '!/ . 2.2,
45
' = { (i, /), (i + 1, /)}, . . t< , , -
F (') (~) (~) .
' = { (i, j), (i, j + 1)}, . . , , (, ), (, ), (, ), (, ). , , , -
nn n nnn nn nn n nnn n nn nnn nn n nn n n nnn n .II ./1 nnn n J/. n nnn " n nn i
1)1 N nn n
6 6 . 2.3. (, )
()
. . 2.1, , , . 2.3, .
, . 2.3, . , , (i, J). , s, s ('t (i, = . F (') s ('t (i, j - 1)) = , s ('t (i, j + 1)) = . (i + + 1, J) v (i + 1, J) = , s ('t (i + 1, , . , (i + 1, J) , (~) . , 't (i + 1, j) , 't (i + 1, j + 1) - . ,
47
, s, (
't (i + 1 t i + 1 >) 0 -r(i, j + 1)
(~). . , . 2.1, .
. 2.1, .
. 2.4. (, ) (, )
, , ,
, ,
,
. . [131]
. . 2.4, , , - . 2.4, , .
F (') {1i {t), t}, t Tv.
,
t , Jl 't' (t) . {, 0), {, 1), {, 0), {, 0), (, 1), (, 1). .
2. , . - , :r , . 3 ,
48
r-- -
1 1 ~ -
r-- - r-- -~ - ~ -r-- - r-- -r-- - r- -r--
~ ~ r- -
.__ - r-r-
.__ - ~ r-r-1 -- - ~ r-r-- ,-- r-r-f--~ ~ f- r--~ ~ r-r--1 ~ r-- r-~ - r-- r- - r-~ ~ '--~ r2; ~ ~
r-,__ ~ ~ r-~
~ ~ ~ ~
- ~ ~ - - ~ ~
1- '-
~ ,__ - 1- f-
~ ~ 1---~ 1---
r- - ~ ~ 1--
r- - ~ 1--
. 2.5. (, ) (, )
- . ,
T={(i,j):lj-i-I\~1, i=2, 3, ... , n}, = ((i,j):lj-i-I\~1; i=2, 3, ... , n}.
(, ) , i1 i 2 (i, -- i) = (j,- j,). , (i,- j,);;;?: 2. v , , , v (t) = 1 t U .
. 2.5, , , . 2.5, , - . 4 8-2045 49
i. (it+t - j) nr tt~ -1, , + 1. 1, 2, 3. , - : (i, j) 1, , (i + + 1, j - l) - I, 2, . .'l
((i + 1, j)) F (') ' = (i, j)
~ ~ ~ ~
.~ - r-r- r-~
~ f--~ r-~
f--
~ --~ -~ ~
~
~ ...... ~
~ ~
~ ~
~ ~ ~
. 2.7. , , 3 (), , ()
, 1 1, 2 , 2- I, 2 , - (. 2.6). 1 , 2 - , - . , 1, 2, 3 , 't (i, j), Ut+t - j), . (i+t - i).
3. . 1, , .
51
, , , ,
, ,
, .
[61 ]. - { (i, ie), i = 1, 2, ... , n} , , 1 it- it-II ~ 1
/1
I(
/(fl
~~"
llff/n
l~n
i, , i1 i 2 , , i2 - i1 -, II 1 it. -- it, 111 i2 - i1 1 ~ 2. . 2.7, , . 2.7, .
, , , ,
,
. (), ( , ), ( ,
. 2.8. , - ), (- 2.7, . ).
F (') ' { ({), t}, t Tv (, 0), (, 1), (, 1), (, 0). , : , - . , , . . , : , - .
F (') ' l(i, JJ, (i, j + 1)1, . . , ,
(, ), (, ), (, ), (, ), (, ), (, ) , .
{'t'(i + l,j)} F (') ' (i, j) , . . , 52
,
:
(~) (I:) (~~) (~~) (~)' (~~) (~~), (~), (;:I), (~).
. ,J ,
. 2.7, , . 2.8. ,
, ,
.
4.
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. ,
, { 1, 2, ... . . . , 1 }, -
1 1 1 1 1 1 1
1 "1 J 10 10 g 1
1 1 ~ 2 2 1
1 1 4 2 2 8 1
1 1 4 2 2 8 1
1 1 4 2 2 8 1
1 1 4 2 2 8 1
.f 1 5 8 8 7 1
1 1 1 1 1 1 1
1 1 1 1 1 1 . 1
. 2.9. ,
{, 8}. , . , 3 - , 8 - . . I
.Ji , , . (6, ), , (6, > , 6 (), ().
, .
5. n ,
-
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~
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,
, . . 2.10. ,
,
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, -
. 2.10. . , , 4. ,
. , . . [25].
Tv { (i, j) : 1 ::;;:; i ::;;:; n; 1 ::;;:; j::;;:; n}, {, }. + n , . n , (i) , i- , i = \, 2, ... , n. , (j) , j- , j = 1, 2, ... , . ! (i) : : 1 ::;;:; i ~ n} U {Ji (j), 1 ~ j ~}. 131f -54
{1, 2, ... ; 7}. '! :
1) ( (1)}, {'t" (n)}, {'t"cron (1)}, ( ()}, ;
2) n- 1 ( (i), 't" (i + 1)}, i = 1, 2, ... , n-- 1, ;
3) - 1 { 't" (j)' 't (j + 1))' j = 1, 2, ... .. . , - 1, ;
4) n ((i, j), 't" (i), (j)), i = 1, 2, ... , n; j = 1, 2, ... , , .
, :
1) 't" (1) (1) 1, 't" (n) 't" () - 7;
2) . (1, 1), (1, 2), (2, 3), (3, 3), (3, 4), (4, 5), (5, 5), (5, 6), (6, 7), (7, 7);
3) ((i, j), (i), 't" (j)} (v, k. k) , k. k v (k. k), , (v (k. k), k. k) . , (v, k. k) ,
1111111111 . 2.11. : - , . , .
, .
- . n Tv {(i, j) : 1 ~ i ~ n; 1 ~ j ~ ~}.
Tr/TP (I -
56
, Tv, , ,; (t) v' t Tv. v (f , Tv, (t) v , t Tv. , v = { (t) : t Tv}, = { ( t) : t Tv} .
----- 7 W////& -G
~ W/ ~ ~ ~ ~
1
~ ~ ~~ ~
W/. ~
~ ~
~ ~
~ W ~ ~ ~
. 2.12. r
{ 1, 2, ... , 7}. ,
,
. , , 2, 4 6, - 1, 3, 5, 7, . . , , . 2.12. , 3.
f6 , , -
57
, G, . 2.13.
, G , 49 , . 2.14,
t t t t t t 1 1 2 J 4 s 6 'l . 2.13.
, .~ 1111 .
G, , , ( (i, j): 1 ~ i ~ n; 1 ~ j ~), ~ .. v v, 11 6 V6, (J
58
((i, j), ~ (i, J), ~ (i, j)}, i = 1, 2, ... , ; j = 1, 2, ... , . , f (f, G . F (') {(i, j), ~ (i, J), ~ (i, J)) ,
. 2.14. n .~ n.~r n
n , .
, , . . 2.15 .
7. , , , , ,
. n n , n
9
, . n: l!
~
. ~
~
wm ~ ~ ~ ~ ~ ~
~ w~
~
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, . . ,
, . .
Tv, n + 1 V0 , V1 , , V, , n + 1 vl: Tv- Vt. i = , 1, 2, ... , n. Z V0 V1 ... V , v0 Z- v1, v2, .... Vn, t Tv (v0 (t), v1 (t), ... ... , v,. (t)) Z.
6()-
(i1, (f2 , , (fn - n Tv V1 , V2 , ... , Vn . fi0 Tv-+ V0 Z- fi1, i = 1, 2, ... , n, v0 , V1 fi1 , V2 fi2 , ... , Vn !ln, V0 Z- V1, V2, ... , Vn. . Z-. 1. fl 0 = fl1 fl2 , fl1 fl2 , Z- fl1 !f~
Z = {(, 0'1, 2): Vo, 0'1 V1. 0'2 V2, .= 0'2 = crl
2. v1 v2 Z-
Z={(a0,0'1 ,0'2):a0 EV0 cr1 EV1 , 0'2EV2, 0'0 =max(a1 ,0'2)).
, , , V1 V2 , max .
3. Z- fl 1 , !f2 , ... , fl. fl - {1, 2, .... , n- 1), Z = {(0 , 1 , 2 , ... , ): = ). - Z- , v0 , . , , , Vn (. . 2.16). Z- !t1 U fl2 . , fl3 - v; v;, , t Tv v; (t) = 1, v; (t) = 2. .
n G1 = (Tv1, Tk1, V1, 1 , 'f/1, {F1 ('), ' 'f/1}), i = 1, 2, ... , n, Tv1, i = 1, 2, ... , n, , Tk1, i = 1, 2, ... , n. , Tv0, Tv1, i = = 1, 2, ... , n, - -
1 . Tv0 Tv . Tv', - t Tv0 Tv", -
1 . : (t) Tv'. V0 Z V0 V1 ... V" G0 , !t (G0) Z- !t (01), fl (G2), , !t (G").
61
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ::;? ~ ~ :: ~ ~ t;;;~ ~ ~ ~ %1. ~ ~
fZ:':: ~ ~ ~ ~ ~ ~ ~ :'l. ~ ~ ~ ~ ~
~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ 12 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
. 2.16. , , ; -
Tv0 , - V0
t!
U (Tvi U Tkt), i=l
t! . .
- U (V 1 U '). i=l
62
~ (Q1 ~~) U ~, ~ = = { {t, 't't (t), 't'2 (t), ... , 't' (t)} : t v}.
F ('), ' 'f/0 , . ' 'f/.;, i = 1, 2, ... , n, (') = pi ('); ' 'f/* F0 (') , , (5 (t), 5 ('t'1 (t)), 5 ('t'2 (t)), . , 5 ('t'n (t))), Z.
2.3.
. ,
, , , .
V - , S v, . . -. V. S ' : , 51 S, 52 S 51 (') =1= 52 (') 51 ( '\. ') =1= 52 ( "- ').
G Tv, Tk V ,
rl (G) = vv, . . , S (G) Tk, . . .
G, , , S (G) Tv, . . .
, , , , .
, n, 1, J, , 1 ~ , J ~ n. Tv - (i,j), -+ 2~ i ~ 1,-n + 2~j~J; V W- , , ; v - Tv-. V, , v (i, j) = , min (i, J) ~. Tv' (i, j), 1 ~ i:;::::;; /, 1:;::::;; i:;::::;; J, n , (i', j'),
63
i- + i ~ i' ~ i, J- n + 1 ~ j' ~ }, v (Tv' (i, /))- v Tv' (1, j). q> n , . v (Tv' (i, j)) . W. q> (i, fl v: Tv-+ V, q> (v (Tv' (i, j))) =1= . ::::: = q> (v (Tv' (i, j))) W '\_ {}. . v {(i,j,k):l~i~/, l~j~J, q>(v(Tv'(i,j)))=k, k=I=}.
- , . q> .
q> . -, q>, . , q>, . -, q>* , .
q> Gcp, . V, - W. Tv
{(i, j):-+2~i~l, -n+2~j~J}.
Tk , = {(i, J) : 1 ~ i ~ /, 1 ~ j ~ J}. -r (i, J) , (i, J) Tv. , Tk = ( (t) : t }.
' '1 '2 , '1 = { {(i, j) : min (i, J) ~ 0}, - '2 = {Tv' (i, J) U { (i, i)} : min (i, j) > 0).
F (') ' '1 \11 , (i, j), , min (i, J) ~, )!. . ' ' 2 , . . ' == = Tv' (i, J) U {:t (i, j) ), F (')
64
s (Tv' (l, JJ U {'t (i, J)}), s ( (l, f)) = (s (Tv' (i, j))).
, n G." n n . n , , , , G,P . 01' .
2. G , , t v1 (Tv' (i, j)) v2 (Tv' (i, i)), , v1 (Tv' (i, J) '\ {(i, j)}) = v2 (Tv' (i, J) '\. {(i, /)/, v1 (i, J) =1= v2 (i, J), (v1 (Tv' (i, J))) =1= =1= (v2 (Tv' (i, J))). . 1. n,
, , ,
G n s; s;, , s; (Tk) = s-; (Tk), s; (Tv) =1= =1= s; (Tv). v1 (Tv' (, n)) v2 (Tv' (, n)), v1 (, n) =1= v2 (, n), v1 (Tv' (, n) '\. {(, n) }) = v2 (Tv' (,- n) '\. {(, n) }),
(v1 (Tv' (, ))) = (v2 (Tv' (, ))). (2.12) nn, ,
. v~ v;, t Tv, t =1= (1, J) v; (t) = v; (t), , v; (Tv' (1, J)) = v1 (Tv' (, n)) v; (Tv' (1, J)) = v2 (Tv' (, n)). , (i, JJ =1= =1= (/, J) v~ (Tv' (i, /)) = v; (Tv' (i, j)), ,
(v; (Tv' (i, j))) = (v; (Tv' (i, j))), (i, j) =1= (/, J). (2.13) (2.12) (2.13) , q; , , v; v;, .
2. , v~ v;, , , , . v; =1= v;, D = {(i, j) : v; (i, JJ =1= v; (i, J)) . (i*, j*),
v; (Tv' (i*, j*)"\.{(i*, j*)}) = v; (Tv' (i*, j*)"\.{(i*, j*)}). -zo45
(2.14)
65
, .n. . D , i, n , j. (i*, j*) , i* > j* > , v~ (i, JJ = v; (i, JJ = = , D. , (i*, j*) , (i', JJ, (i', j') = = (i*, j*), i' :::;;;;; i*, j' :::;;;;; j*, v; (i', j') = = v; (i', j'). , v; (Tv (i*, j*) \_ { i*, j* }) = = v; (Tv (i*, j*) '\.. { i*, j* }), v; (i*, j*) ::1= v; (i*, j*), qJ (v~ (Tv' (i*, j*))) = ) Sq> S (G (t) = } =::> {t: t Tk, Sq> (t) = }. . ,
v (v*)), ' $. v (v*) ' , , $.( v' = v (v*) (v') =f=. $, ' v (v*) , .
,
' v (v*), v*. ' , , , G' . ' , v (' (i, /)) q> (v (' {i, J))) = , ' . v
{{i, j):ep'(v(Tv'(i, j))) = } ~ {(i, j):ep(v(Tv'(i, j))) = }. .
, .
V {1, 2, ... , 1 V 1} - 1 1 V 1. , , . (i, j), v {i, J) =f=. v (i, j- 1), v (i, j) - v (i, j - 1). , , , . . - . W 2 1 V 1 - 1 , , W V . , 1 V 1 = 256, , .
(v (i, j) - v (i, j- 1)) d (v {i, j), v (i, j - 1)), , , 1 V 1 . ,
d (v (i, j), v (i, j- 1)) = v (i, j)- v (i, j- 1), v(i, j)-v(i, j-1)~0,
d (v (i, j), v (i, j- 1)) = 1 V 1 + v (i, j)- v (i, j- 1), v(i, j)-v(i, j-1}
t t, - ~ , v (i, J) - 2v (i, j- 1) + v (i, j- 2) (v (i, J) --v(i,j-1))-(v(i-1,J) -v(i-1, j-1)) , , . W 4 1 V 1 - 3. , , , 1 V 1 .
, . . V = {0, 1 }, , ,
. . , (, . .), , r . . q>, , q>', , 1 V 1 = 2. , , "1, . , .
, ,
, , , , - .
, , , , , . , .
, , ,
(i, j), (i, j- 1), (i -1, J), (i -1, j- 1). 16 . . 2.17.
68
- , , 1-6, , - . , 1- , . (
EJ8EIIj 1 2
5 7 8
9 fO 11 12
13 14- 15 16 . 2.17. ~
), . , . 3 , , . t 9 15 , 14 16.
69
70
. 2.18. n
V W n 1 v /J
v (Tv' (i, j)) 1 W 11v1mn qJ , , ,
~ G!j). .1 2 1 W \\VImn-l . 3 -
1 v l'v'mn-l .
2mn-l 2 , -
= n = 2 , ~
. : = n = = 2 256 . , = n = 1 V 1 = = 2, , v ,
. ,
i< , -
1 , , 256 .
Tv' (i, J) - (i, J), (i- 1, J), (i, j- 1), (i- 1, j- 1). v (Tv' (i, j)) . 2.18. , , /{ (i, J). , q- - , (q, ). Grp. . . ,- , . 2.18, , , . Grp , , . v , .
, N (1), N (2), ... , N (8), 11 Tv 11 2 , 11 Tv 11 2 , , / Tv 1- , (i, j), i = 1, 2, ... , /; ; = 1, 2, ... , J. N (q),q = 1, 2, ... , 8, 11 Tv 11 2/2. (i, j) . v (Tv' (i, /)) (q, ) . 2.18, , (q,)=v(v' (i, j)). N (q) 1, N (q) =1= 11 Tv 11 2 , = 1, 1, N (q) =1= , = 2. N (q) . N (q'), q' =1= q, . . q = = 1, 2, ... , 8, N (q) ~ 11 Tv 11 2/2, (, 1) , , (q, 2) - . N (q) < li Tv [1 2/2, { (q, 1) , , (q, 2) - .
. 2.19 ,
l . (i, j), W (i, i) "= , , W (i, j) =1= ~~
71
1 ~
~ 1-1--Lu 1-1--1-1--~
w, ~ 171,;
1 1 1 1 1 1 1 1 1
~
J'lj ~
~ ~ ~ ~ w ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1
~ ~ ~ -~ ~ ~ ~ ~ ~ ~ ~1
~ ~ ~ ~ ~ rz;; ~ w r%1, -~ ~-
ra ~ ~ ~ ~ ~ ~ W, ~ ~ ~
~ ~ ~ 72 ~ ~ ~ ~ ~ ~
~ ~ ~ ~
1?/ ~ , 1?/ ~ ~;.
~ ~ ~ i f!;l; ~ ~
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1 1 ...
:;
; ~-
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~ ~ ,_ '---
[%/ ~;
. 2.19, - , - n
2.4.
. . , . . , ((i, j) : 1 ::;;; i ::;;; ; 1 ::;;; j ::;;; n}. , . . , , , ,- . , , , , . , , [24, 101 ], , , . , . , .
, - . , . . , . . , . : , - , . :, , . . , . .
3
, , . , ,
, .
,
. ,
G v v {f (G). 3.1 , , v {f (G), , , v !t {f (G). , . . , , , , v {f (G), v !t {f (G), , , v !t {f (G), v {f. (G}. , , , ,
, .
J
, . , ,
. , , ){ ,
- , . . .
3.3
. , . , , . . , . ,
.
, , . 3.4, , - . , . - , .
, , G = {Tv, Tk, V, . '!7, /F ('), ' '!7)) G = {Tv, Tk, V, , '!7, {Z ('), ' '!7}), Z (') s ('), F (', s (')) = 1. , r .
3.1. I JII
. . G = (Tv, Tk, V, , ;; , {Z ('), ' 'f/))- . , 'fl :::::> / [t) : t Tv U Tk), = Tv U U Tk, S = V U /(. , '!1. = '!71 U '!72 , 5"1 = = //t) : t Tv U Tk), '!72 : (Tv U Tk) (Tu U Tk).
(.'Ji 1. ' '2 t ' (Z ('), t) Z (') t {~:~ S, ~ = s (', t), s (') Z (')}. ,
75
~ S (Z ('), t), s (') : ' -+ S, . Z (') t ~.
2. . , ' t, ' ~2 , t ',
Z({t}) = P(Z(T'), t). (3.1) 3. G1 = (Tv, Tk, V, .. ~. {Z1 ('),
' ~}) 2 = (Tv, Tk, V, . ?1', (Z2 ('), ' ?!'}), 1 2 , Z1 (') Z2 (') ' ?!'. , 1 G2 , L (G1) : : L (G2) S (G1) : S (G2).
4. (G) G, (G) - , () G ', G' =1= (G), (G) : G' G.
., 1. G = {Tv, Tk, V, , ?!', {Z ('), ' ~}) . . . Z1 ()
Z2 () t
(Z1 () U Z2 (), t) = (Z1 (), t) U (Z2 (), t). (3.2) , G1 G2,
G, , , G1 =1= G2, . . 3 ' Z1 (') Z2 (') Z, G1 2 G1 , G1 - G. 2
V ' . . : G. (3.2), (3.4), (3.6), (3.8) , V ' 76
(3.3) (3.8) , 3 ' ~ (Z2 (') =1= i* Gt+! = 01 = = (01). .
i = 2, 3 .... 01 Gt-J. , 01 = Gt-J, Gt+k = 01 k. , ,
~ 1 Z (') 1 , ' ~
17
11 r 01 : Ot-1. 01 =1= Gt-1 r , I: 1 Z (') 1 + 1. ( 1'1
T'E
' '!/2 ,
Z({t})cZ({t}), Z({t})=Z({t}) U Z({t}) (3.15)
t Tv U Tk. 81 (G)
Z1 (') = Z' () (3.16)
' ~. , (3.14)
Z (') Z1 ('). (3.17)
t Tv U Tk :
zl ({t}) = z ({t}) n < n (Z ('), t> = T'E~2(t)
= Z({t}) ( n P(Z(T')) U Z(T'), t)) = T'E~1(t)
- -= (Z ({t}) U Z ({t})) ( n ( (Z ('), t) U (Z ('), t))) =
T'E~2(t)
= (Z ({t}) U Z ({t})) ( ( (Z ('), t) U Z ({t}))) ::::> Z ({t}). T'E~1 (t)
81 (0), - (3.14), - (3.15) (3.12), - (3.13). . -
i ({t}) Z1 ({t}) 3 t Tv U Tk. (3.17) , 81 (G).
,
82 (G), 82 (G) -, (3.10), . . ,
- -( G) =>( 82 (0)), (3.18)
. G
82 (G): 08 =82(0)=(Tv, Tk, V, , ~.{Z2 (T'), '~}).
79
l Tv U Tk
z ({t}) : z ({t}). (3.19) G : G.
2 (G) t Tv U Tk Z1 ({t}) = Z ({t}),
z ({t}) : Z2 ({t}). (3.20) t1 t2 , ,
{t1 , t2 } '5"2 i ({t1 , t2 }) : : (Z ( { t1 , t2 }), t1) (Z ( { t1 , t2 }), t2, ,
Z({tl, t2J> : Z({t1}) Z({t2J), (3.21) G .
(3.19) ,
- -z ({tl}) z ({t2}) : z ({tl}) z ({t2}). (3.22) , (3.21) (3.22)
G : G,
z ({tl, t2}) : z ({tl, 12}), (3.23)
(3.23)
Z ({tt, /2}) : Z ({tl, /2}) U (Z ((t1}) Z ({t2})). (3.24) G2 = 2 (G), Z2 ({ti, t2 }) = Z ({t1 , t2 }) U (Z ({t1 }) Z ({t2})), - (3.24) , Z ({/1 , t2 }) : Z2 ({t1 , 12}), , (3.18).
(3.11) (3.18) , i -- l 1
G : G - - l . 1 G : G = (G~- ). , G : G1 G*, G1 , G* G1 .
. 3. , . G, . , . . (G), :. . .
80
. ,
{f. ( (G)) {f. (G),
. Tv -= = {(i, J), i = 1, 2, ... , n; j = 1, 2, ... ,). i1 , i 2 , j1 , j2 (i1 , i2 , j1 , j 2) (i1 , i2 , j 1 , j 2): Tv- {0, 1}, ,
1, i = i2, j 1 ~ i ~ i2; (i i 1- 1. ) (i 1.) _ 1, j = j 1 , i 1 ~ i ~ i 2; 1 2 1 2 ' - . .
1, 1 = ] 2, t 1 ~ t ~ t2 ; , .
v = k
= ~ v u:. i~. j:, j~). , 11 , 12 , 11 =1= 12 -1=1
-
: i~' < ;:- 1, i~ < i1'- 1, j~' < k- 1, j~ < j{'-- 1. . 3.1 , . 3.2 - .
, -
, (. 3.3).
Tk = {1: (t), t Tv);
V = {, 1 };
Jie S = {0, 1, 2, 3, 4, 5, 6};
. 3.1. ~ - , 3.2. .'
82
2,J 2,S 2,J
1 1 1 1 ' 4,5,& lf.,5,G lf.,5,6 2,J 2,S
1
' ' 1 1 ' 1,.,5,& 1 4,5,6 2,J 2,J 2,J 2,J 2,J 2,J
4,5,6 4,5,6 4,5,6 4,5,6 1 .jr.,S, 6 1 4,5,6
2,J 2,J z,s 2,J
4,5;6 1 1 4,S,6 1 4,5,6 1 4,5,6
2,J 2,J
4,5,& 1 1 4,5,6 1 1 1 1
2,J 2,J
4,5,6 1 1 4,5,6 1 1 1 1
2,J 2,J
4,5,6 ' 1 4,5,6 1 1 1 1 2,J 2,J
4,5,6 ' 1 4,5,6 ' 1 1 1
. 3.4.
,
. , , . . . , , , , , . , , , , i-, , n
n,. , i- n n .
84
---
----
3 4 5 3 4 5 1 26 , 6 2 1 6
3 4 4 5, 6 , 6 3 4 4 5 6 1 6 2 , , 26 , 26 , 6 2 1 1 6 G 1 6
2 , , 26 , , , 2 1 1 G
2 , , 26 , , , 2 1 1 26 ,
2 , , 26 , , , 2 1 1 2G , 2 , , 26 , , , 2 1 1 26 ,
.J 4 5 3 4 5 ,
2 1 6 2 1 6
3 4 4 s 12 6 1 G J 4 * 5 2 1 6 2 ' 1 6
, 2G 1 6 2 ( 1 6 2 1 6
2 1 , IZ 6 1 2 1 1 6 2 ' 01 126 , 101 2 ( 1 8
2 1 101 26 , , 1 1 6 2 1 IO 1 26 , , 2 1 1 6
6 iJ
t - . q1 , q2 , ... , qk, q1, q2, ... , qk- , 0"'10"'10 0"'1"'0, n1 > , n2 > 1, n3 >, n4 > 3. .
i- . {(i, j), (i, j)J { (i, j), (i, j + 1)} , i- 12 ...
0"'21"'60 0"'10"'10 0"'34"'-250 Pk 1 = , q1 = , Pt = , q1 = 0"'1'1'0. i- , , , I< , . ,
85
1 1 1 1 1 1 j,
1
2,S
2,S 2,S 2,S 2,S
1 4,5,6 '1,5,8 4,5,6 4,5,6 4,5,6 1 1
2,3 2,J
1 4,5,6 1 , 1 4,5,6 1 7 2,3 2,J
1 4,5,8 1 1 1 4,5,6 1 f
2,1
2,S
7 4,5,8 1 1 1 4,5,6 1 1
2,S 2,3
1 4, 5,6 1 1 1 4,5,6 1 1
2,J
1 4,5,6 1 1 1 1 1 ' 2,J
1 4,5,6 1 1 1 1 1 1
J 4 4 4 5 , J * 4 4 s 26 , , , 26 1 z 1 1 1 6 28 , , , 26 , z , , , 26 28 , , , 28 , 2 , 01 , 26 .
28 , , 1 26 , 2 , , , 26
26 , , , , , 2 , , , ,
26 , , , , , 2 , , , ,
6 1. 3.5. 1
3 4
2 1
to 2 1 2 ' 2 1
2 ' 2 '
J 4
2 ' 2 ' 2 1
2 1
2 1
2 1
--,--
4 4 s 1 1 6
1 1 ' 1 6 , , 6
1 ,
, ,
4 * s ' 1 6 1 ' 6 1 1 6
1 1 6
7
' :
3 * 4 4 s 2 1 1 1 6
2 7 , 1 6 2 1 1 1
2 1 1 ' 6 2 7[)( rx " ~
1 (i. J), (i + 1, /))
.
, ,
, ,
1
- .
1.
. , , .~ , - ur. , ( J\ ), . , , 1, .
2. . - ~. ~ - .
87
6. (, ~ , 1 1 ~ 1, t0, tk, , t0 , tk , t0 =1= tk, t0, t1, ... , tk, , { t1, it+I} .. 1. (, ~) q (t), t t', , (t, t') ~ q (t') < q (t). .
. . (, ~) , ,
. , :
, , .
1. , . , i , , , - . i- : , , . .
t* - , t'. , t" =1= t', t*. , , t' t" (, ~). . . , t' t", , t', t*, t", t*, .. . t*, , -
6
t', n n i + 1 , .
, , i + 1. i + 1 , .
n, n (i + 1)- n
. n 1 1- , . .
n , , 4.
. 5. G = (Tv, Tk, V, . '!7, {Z ('), ' '!7}), v : Tv-+ V- , G (v) = {Tv, Tk, V, . '!7, {Zv ('), ' '!7}), Zv ((t}) = Z ({t}) (v (t)} t Tv, ' '!7 Zv (') = Z ('); (G (v)) = = (Tv, Tk, V, . '!7, {Z~ ('), ' '!7}) - G (v).
t Tk 1 z; ( { t}) 1 ~ 1 , 1 Z~ ((/}) 1 > 1, v . . * = ( t: t Tk, 1 Z~ (t) 1 >
> 1 ). * = 0, v, , , . * =1= $0, t * q (t), , k: Tk-+ t * .
.r k (t) t q (t) = 1 , k (t) z; ([/)). t, , t' Tk \. *, t' k (t'), n , (k (t'), k (t)) Z ( { t', /}). k (t') t.
, k (t) t *, q (t) ~ q0 q0 + 1 t*, t', , , q0 , t*. , k (t*) z: ({t* }). k (t*) ,
89
(k (t*), k ({}) Z~ ({t*, t'}). , k (t*) , . . Z~ ({t')). t, , t" "' *, t" k (t"), , (k (t*), k (t")) Z~ (t*, t"), k (t*) .
J(JJeo, , . n v, v. .
. : ,
Jl .
, . ,
, 5. .
, - , , , , . 110 . Ji, . . i , , .
3.2.
Tv Tk, KaJ( . Tv U Tk , V U - S. , - n,
- , { { t) : t }.
1. . G = (, S, ', {Z ('), ' '})- , s () : -+ S -, , 90
V ' ry (s (') Z (')): S (G) - : f: (S )- R- . s*, ,
~ f (s* (), t) = max L f (s (t), t). I sES(G) IET
, t
S S' S , S' . .
2. . G = (, S, '!, {Z ('), ' ~}) f: S -+ R . - . , ~ = ~1 U ~2 '!/.1 = {{t}: t }, ~2 .
-cr = ( S ) . . f, , {t0 : ii -+ Sa : -+ S), , cr* = (s*, t*) ta (cr*) = t*, Sa (cr*) = s*. cr t0 s0 - , - , (). ii cr f (cr) , ta (cr) -, S0 ( cr) - . t { cr : cr , ta ( cr) = t} \ , t, (t). t1 t2 , , {t1 , / 2 } ~2 . crcr (cr1, cr2) , , . . {t0 (cr1), ta (cr2)} ~2 , (sa (cr1), S0 (cr2)) EZ({ta (cr1), ta (cr2)}). , cr1 cr2 , (cr1 , cr2) crcr. t1 t2 , {t1, t2 } ~2 crcr {t1, t2)
{(cr1 , 0'2): (cr1 , cr2) crcr, ta (cr1 ) = 11 , ta (cr2) = t2}. G, ,
t , ii (t) , t. , , ,
. - , , t . , , , .
f : ii-+ R, i
.
, f 0 : cr-+ R, s S (G) - ~ fo (s (t), t) .
IET f f + f0 .
,
f : cr -+ R, . , f , s* (t) = arg max f (s (t), t),
s(I)ES t , , . . ,
, , . , , . .
f f0 , f + fo , , .
. t (t) lt' : t' , {t, t'} ~2 } , t.
-r (ta ()), , ,
s : ---+ S 1 ~ f (s (t), t) F (s). tf.T
6. q> - , s* - . (q>) ~ F (s*). . ,
s*, s* (/0 ()) = S0 (). , s*, 8 0 (s*). , (1 , 2) Ua (1 ), ta (2)) s*, 1 (s*) 2 8 0 (s*). , s*, Da (s*).
s* .
~ ~ q> (, -r) = L (J (1 , ta (2)) + aEB0 (s*) 'tf.M(t0 (o)) (0 1,o2)f.D0 (s*)
(3.27)
fP (, -r) ( 8 0 (s*)), ( !t 8 0 (s*)). (3.27) (3.26) fP. , . ,
~ ~ fP (, -r) ~; (3.28) Of.80 (s*) 'tf.M(t0(o))
~ f (s* (t), t)- ~ L fP (, -r) ~ F (s*). (3.29) tf.T aEB0(s) 'tf.M(I0- F (s*). (3.30) oEB0 (sJ
}: m~x /t ()~ L .h (). IET oEa(t) oEH0 (s)
fP,
., , (J);;;. ~ h(a). (3.31)
aeB0 ts)
94
(3.31) (3.30)
('!)~ F (s*). .
- , . , -
s* ~ f (), aEB0 (s 0 )
~ h () + ~ (1 , 2), q (3.27) oEB0(s) (a,.a,)ED0(s)
n ,
.
G, , Grp, . ,
, }.: f (s (t), t) IET
.
, . G ff! Grp ::_ G, ,
h () , . . h () =1= max h ('), (1 , 2),
O'Eo(l0 (a))
(1 , 2) . 7. ff! - ,
, Grp, G. . ,
G, , aq; : .
, aq,, , F (s) . (ff!) ff!
s* - G .
~ ~ ff! (, )= ~ (ff! (1 , ta (2)) + aEB0(s) (t0{)) (a,,a2)ED0(s 0 )
+ ff! (2 , ta, (1 ))), n 6.
(3.32)
95
(3.32) , , (1 , 2) =t= , O!f! O!f!, , s*, Olj). (3.32)
~ f (s (t), t)- ~ ~ J (, 't) = F (s*). (3.33) IET aEB0 (s*) -rEM(I0
, s*, Grp Grp,.
s* Grp L m~x h1 (}= L h1 ()= F (s*).
7 (3.36)
IET aEa(l) aEB0 (s*)
~2
L m~x h2 ()> ~ h2 ()> F (s*). (3.37) IET OEO(I) aEB0 (s*)
~2 ( ~) , ~1
L m~x h1 ()= L m~x h2 (). IET aEa(l) IET OEa(l)
(3.36) - (3.38)
L m~x lz2 () = L h2 (), IET aEa(t) aEB0 (s*)
~ h2 (}= F (s*). aEB0 (s*)
(3.38)
(3.39)
(3.40)
(3.39) , t h2 () = max h2 (1}, , ,
a,Ea(l(a)) s*, Grp,
(3.40) ,
~ L ~2(0, 't) =. aEB0 (s*) 'tEM(t0 (a))
L ~ ~2(o,'t)= L (!!'2(a1,ta(a2))+ aEB0 (s*J 'tEM(t0 (a)) (0 1,a2)ED0 (s*)
+ 2 (2 , ta (1 ))), 6,
~ 2 (1 , 2) =. (3.41) ra,,a2)ED0 (s*)
2 (3.4 1) , (3.41) , . , , ,
Gq,,. ( , s*) , s* G(j), .
7 8-2046 97
rr .~ v, , oopeJilc.ffo ,
~m. , , , , ,
, , , .
cr- , . , ,
cr' (), , (cr, cr') . , (cr, ) ' (cr, ) = (cr, ) + + 6., 6.- , (cr, ')
. 3.7.
. ' , cr1 , 2 , ... , crk, 1 , 2 , ... , rn. , (, ) (, ri), i = 1, 2, ... , . -,
' (, l) = (cr, t)- '~ 1 , i = 1, 2, .. , , ii ru . , \ Jr-, r
IP IP' , ,
, , ,
,
-, . .
. 3.8.
. . 3.6-3.10 . . 3.6 , 9 . , I
10, ; 3 ; 5 , 2, . -
. 3.9. n n (n )
, . . , ,
1 2 , , 2 , , ,-
1 . : 1- , (, ) , ;
- , (, ) , . , , , ,
, ,
. 1.-
104
. 3.7. , . . 3.8. ,
. 3 .10. r
q>, . 3.9. Grp , , . 3.10. . .
, , , ,
q> Grp , , . , , , . , .
105
3.3.
'ff () 'ff. .
. k ~ k- t* = (, 'f!). * =(*, ~*)- , , 1 * 1 ~ k; k- t* {t (, ~) ( U U (t*}, 'ff U {(t*, t1: tE *!), . . t*, , *.
k- i
1 * (t) 1 ::;;;;; k,
(T'\_/t}, (~'\.~~(/)) U ~2(/)).
. 10. 0 = (0 , ~0) k- , (0 , ~ 0), (1 , ~1 .), ... , (T1r.1. ~1.. , 1.1 = 0. ~ITol = !0, i= 1, 2, ... .... / 0 /(,, ~1) (Tt-1, ~t-1) k- t Tt-1 . ,
, , 0 k.
iTol , (0 U ~ t). 0 -
i=
~. , ~ - k- . , -
( ITol ITol )
. U ;,. U .~/ i=O, 1, ... ,/0 /, J=ITol-i f=iTol-1
~ k. , 0 k.
, 0 k .11, . , , 0 k t, k- 0 , k. 0 k k- , n ;, ~. ;. . .. .... ~--1. ;,. :vt ;, 0 , , i ~ ~-1 :vt 0 . , . , < , ~. k. i
k- . , 1 - k- . . , , .
, I , . 1 0 1! . . , k-I
k, t k-, k.
4. k- , tij-l ti1_ 1 ,
, , k.
5. t* tt1_ 1, t11_ 1, 11 k- , tt1 , tlm , lt;_1, lt; ... , ltm' , t* , .
6. j . j , , . 3.
7. k- -' /1 , / 2 , ... , tn, ti,-l ' /1 -' , k . k- t*. -; . ~
, ~ , -' , . -1 , - J k- , ,
, k- . k- , .
, , ,
. : , k, , k , k-, , , , k- , f. k . k, ~ k.
k,
110
, k. .1~ 1 ;I .
3.4. II
S - ,
' S - ' -+ S, -
'. S ' = 0 , , # -,., . ' s (')
,., s ' -+ S, . . s (') = 2 . 2 . , , s U ; (').
Q ED : Q Q -+ Q : Q Q -+ Q.
D : s -+ Q , :
1. ' 1 ~ (') 2 ; ('), , 1 2 = 0, D (1 U U 2) = D (al) D (2).
2. 1 2 , , 1 2 = 0, 1 ; (1) 2 ; (2) D (1 2) = D (1) D (2). 1 2 1 U 2 -+ S, 1 1 , 2 2
, Q, D U S{l}, . .
tET 1 S 1 1 1 Q, , , f: S -+ Q, D (0) , D ({ #}) , , 2 1 Q 12 Q @.
~
~ q, q1, qi Q, i l, l - iEI iEI
111
. . :
: qi = D(0); iEel
' qt = qi Et> ~ qi; tE/U{i*} IE/
qt=D({#}); l=el
q~ = ql. ql lEIU{i"} iE/
i* !{: /. f: S -+ Q,
, , v : ' -+ S
D({v}) = f(v(t), t). tET'
; - ' -+ S, "
D (v) = " f (v (t), t). VE~ tET'
, - , , G ~ , .
. 1. Tv v: Tv-+ S, . . v. D .
Q {0, 1}, . V . f : S -+ {0, 1}
f (s, t) = 1, s S t '\. Tv; f (s, t) = 1, t Tv s = v (t); f (s, t) = , t Tv s =1= v (t).
, , D (0) =, D ({#}) = 1.
S (G) v ! (G).
11~
2. Tv = 0, D (S (G)) G.
3. Q , f S - R. D (0) =- , D (/#})=, 9 max, + , D (S (G)) .
4. , . min, D ({ ::j:j: }) = , , .
5. Q , 9 + , D (0) = , D (/ ::j:j: }) = 1. D (S (G)) G. , , D (S (G)) , .
(). , n . , ,
, . q> : S - R Q n . f : S - Q ,
f(s,t)=(q>(s,t), -, -, ... , -),
n-1
D(&,) = (- , - , ... , - ),
n
D({::j:j:}) = (0, -, -, ... ' - ). n-1
(12 , ... , ) 9 (1 , 2 , , ) n 2n (1 , 2 , , ano 1 , 2 , , ), (1 , 2 , , ) (1 , 2 , , ) - n n2 ai + ;, 1 ~ i ~ n, 1 ~ ; ~ n. .
: , , .
8 1-2046 113
r = (, ~)- , n . .. ! ~ t1, t2 , , t-l, t, ft . t = (1 , ~~). i === = , 1, ... , n - 1, , 1+1 k- lt+l ; t, t t1 t t .
i j ; j . ~*
~, ;, ... , ~. , Tn U *, Tn =F U *. , t"
*:* *:* t1 , / 2 , , t,. * ~*. . {{ t} : t Tn '\_ U T*J -
*: ~*. ,~*'=~* U {{tj: tETn"- U *}.
* :*
~. R: - "- , R (tt) = ;, i = 1, 2, ... , n, - Tn Tm , t'- tw, t' R (t"). -+ * =? ~* Tn , *=? t, t' Tn, , t' =? t * = R (/'). *- t, R (t) = *. -+ :=> ,
, * * t, * =1= eJ, t' *, , t' =? t. , t' * t' => t. r t t', , t' =? t t' *, .11 *=> t.
:
Tc(t)={t':t'ET,., t=?t'} U {t}, tET,.,
(*)= {t:tET,., T*=?t}, *~*.
-(t)={T*:T*E'f!"*, T*c(R(t) U {11). tE'"J, tET,.,
(*) = 11: t ". *--~ t). ~*.
rr :>.
il , JI
114
* '!!* (*)= U TcJJ (/),
'7\, t n
(/) = ( U (*)) U /!}. T*E~(t)
(3.48)
(3.49)
, , - (3.48) (3.49) ,
.
.2. te t1, , R (tt) = = R (tt), te =1= tf, . (tt) (t1) .
G . 1. , t* : Tn, t* => t*. , , . . t*, t', t2, , t1, t*, t* -+ t1 -+ t2 -+ ... -+ -+ f-+ t*. t1 t*, . . t1 t1, t1, t2, . , t1, t* t1 , i1 > i*. t1 ti-l , , t* t1 , t* t*, , . t* 11 t* => t*.
2. , t1, t1, t1 t1 -+ t1 t1 -+ t1, te -+ -+ t1, t1 -+ te.
te-+ t1 t1 -+ t1 , k- t1 1 t1, t1 , k- t1 t1 t1 . , i > j, t1 R (ft) , , t1 -+ ft. i < j, te R (t1) t1 -+ t1
3. , tll t12 tl3 tl,n,-1 /ln, -+-+-+-+ -. t21 t22 [23 t2,n.-l /2n, -+-+-+-+ -.
(3.50)
(3.51)
, t11 :: t21, t111' = t2n,
(3.52)
115
n
(3.53)
(3.50) (3.51), t1n, = fn . 2 , t1'n'-1 -+ fn-1, fn-1 -+ t1'n'-1 . (3.52), - (3.53).
4. , , t1 ==?- t1 t1 ==?- t1, t,, s < l, t1 -+ t, t1 ==?- /5 , t1 ==?- /5 t1 -+ /5
5. . 3. , t1, t1, t1 , i1 ~ t1 t1 ==?-==?- t1, t1 ==?- t1, /5 , s < l, t1 -+ /5 t1 ==?- t5
6. . 4 . 5, , ft, t1 t1 t1 ==?- t1 t1 ==?- t1, it ==?- t,, t1 ==?- ft.
7. . 1, , * ==?- t, t /1 , t2 , ... , tn *. * '!/* t * * ==?- t.
8. , , . . t1 Tn t1 ==?- t1 t1 ==?- t1 n. 6 , t1 ==?- t1, t1 ==?- t1 , . .
tl, /2, ... , t, /1 -+ /1 -+ ~ / 2 -+ ... -+ tP-+ t1 R (tt) ==?- tP, . . n. 7 tP {t R (tt). tP-+ t1 , tP R (t1). , R (t =1= R (t1), . .
.. 3. f!J- , 1 '!/* 2 '!* - . 1 1 f!J 1 , t0 2 1 1 : R (t0) U { t0 } 2 : R (t0) U { fO} (I) (2) . . , { t : t
Tn, R (t) = 1 } {t: t Tn, R (t) = 2 ) . , fr, t1 t1, 1 -+ t1 ==?- t1 2 -+ t1 ==?- t1. , n. 6 2, , I-+ t, ==?- ti +- 2, I-+ t1 -
s ( 8JJ (t)) : (f) -+ S, , {s (*)} S3 (*, t, s (* U {t}))} ~ * U (f) . , * U U {t} s(T* U {t}).
* ,-, t , , *-+ t, s (*) Sa (*, t, s (*)) s (f) t, , {s (*)} Sa (*, t, s (*)) * U ( t* j, s (*).
n . * '9"* ~ (*)
St (*, s (*)) = S2 (*, t, s (*)). (3.56) 1(0 )
* ,-, t , , * -+ t, s (*)
S 2 (T*, t, s(T*)) = U S3 (*, t, s(T* U {t})). (3.57) s(I)ESa
(3.61) .
- , . , , . ,
+ -. - t' +, , D (S2 (*, t, s (*))) D3 (S3 (*, t, s (* U ( tj))), t = t'. , t* -, , ,
+ . . , +. , .
~
~ t* ' rF (t*) t' (') , t* -- t'. , t*, t' D (S2 (R (t'), t, s (R (t')))) D (S3 (R (t'), t', s (R (t') U ( t' }))).
,
D (S1 (', s ('))) (~.59) ' ~ (t*) s ('). D (S3 (R (t*), t*, s (R (t*) U ( t* 1))) (3.61) D (S2 (R (*), t*, s (R (t*)))) (3.60). D (S2 (*, t, s (*))) D (S3 (*, t, s (* U ( tj))) * t, * = R (t), n ,
t* - +. ,
D (S1 (0, #)). (3.60) t , * = R (t), s (*), , 1 S lk 1 1 . 1 S 1 - 1 $. , >
1 S lk+J 1 1 - 1 S 1'' 1 1 D (S2 (*, t, s (*)))
~1 (3.59) J1 . 1 S lk 1 1 .J, ,
11~
(3.59), 1 Sk 1 1 1 - N, N- (3.59), . . D (S1 (*, s (*))). (3.61) N (8), . . (8), D (S1 (*, s (*))), (3.61) . , (8)
1 S lk 1 1 . , , .
. 13. (, S, "fJ, Q} k-
1 S 1" 1 1 (8) 1 S lk (jS 1- 1) $.
3.5.
, , . ,
,
n [53, 56, 641. , , n . , :r - ,
.
. , n
. - n , . , , ,
, . , . ,
120
, , . , - . , , , ,
, , , -
- - .
-, . r, , , , . ,
, .
, . . , , , 106 . , , . : , , . . . , . . ,
. , , ,
, . - , . ,
. , , . ,
, .
, , , ( n , ),
121
. . .
., . . . , . - , . ~
n .
i
4
, .
, , , . , , , . Tai
, . . , 1 S 1 . (4.1) .
(4.1), , , W , , 1 S 1 , 1 S 1 . , (4.1). , , , , , , 1 S 1 , 2.
S - R, , 'V ('V s S ( (, s) = (s, ) (s))). , (4.1) , , . , , CJI, .
1 , 2 , ... , Xm S s1 , s2, ... , Sm . , CJI. .
1. , (xi. si), i = 1, 2, ... , , (, s), xi, si (1 , s1) i =1= j, -
.1 (xt. Si). -i=l
1 CJI, , :
.--... = arg max PXP(xt, st). (4.2)
PXPEg> i=l
11. ( ) , ,
125
, .. ~ .
!fP, (,, st} , . .
PXP=arg max minPXP(x1, st). ~ t
(4.3)
111. D = S, W , 1, W (s1 , Sj) =, s1 = s2, W (s1 , s2) = 1, s1 -=/= s2 (4.1) Q () = arg max (, s)~
sES , Q (). 111 !fP
(1 , st)>PXP(x1, s), s-=/=s1, i= 1, 2, . , , , . . '}J , IS . !fP , ~r . n . , , Q . , , , ~ 1
,
.
1 , 2 , ... , 1 s1, , ~. . . , , .
126
4.2.
G =(, S, ~. Q)- .
' , ~, ; (') . s - ' -+- S, ', G.
R- , : s ()-+- R- , . . , ,
V s s ( (s) ~) : (s) = 1. sesT
' : (')
; (')-+- R, , s (') ; (') (', ~ (')) = : (s). ,
s(T' ')~(,') (') t> ; ('), . . V s (') ; (') ( (', s ('))~ ) : ('))= 1.
s(T')E;(T)
; (') (') ' = 0, , ; (') #, , ( 0, #) .
' : , , ' " = 0, (' !) 1 ' 1 + 1 " 1 , s (' U ) :' U "-+- S, (", s (")) =F =F , ,
(T'IT" (' U ")) _ (' U ", s (' U ")) ' s - (", s (")) . , (' /")
; ('), . .
V s (") ; (") (( (", s ("))> ) ~
:'!> V s (') ; (')( (' /, s (' U ")) ~ 0)),
'V s () ; (") [ (( (", s ()) > ) 9
:> ( ~ (' /", s (' U ")) = 1 )] I(T')EI(T')
127
G = (, S, ~. Q) ' ~ = { t : t ', 3 t' (/_ ' ({ t, t'} ~)}. ~ , ' . , (' U U "). ( '\._ ') . . .
: "'s ()- R (, S, ~, Q), ' , " , ' U "= , ' n "= 0 (' /") = (' t;) , , ~ (s) =
sES(G)
= 1. , , , . 1- , . . [34, 35]. d- d ,
.
, [1, 2]. , , ,
. , , -,. : s () - R, , ('), (' /") . . , , , , .
k- 111 , , k- t .
, G1 = (1 , S1 , '1 , Q1 ) k- t G = = (, S, ~. Q), (1 , ~ 1)- , k- (, f/), Q = !F ('), ' '}. Q1 = {F1 ('), ' ~1 }, ' '1 ~
128
F1 (') = F ('), ' ~1 ' ft ~ s (') ; (') F1 (', s ('))= 1.
, S (G1) '\_ ( t} S (G). r , G1 ('\_ {t})-+S, , G. .
. 1. G =(, S, ~. Q); Q- , k- t G; - , G. ( '\_ { t}) , Q. .
' ", , ' U "= '\_ (t}, ' "= 0, (' /"). . ", , , , , G Q, . - " G (")~, Q - (")~. (' /(')~) = = (' /") :
1. (")~ = (")~ (4.4) 2. (")~ =!= (")~ (4.5)
1. s (' U ') ; (' U ")
('!", s(T' U ''))= ~ (' U {t}/", s(T)). (4.6) s(t)ES
G , s ()
(' U {t}/T", s(t)) = (' U {t}/(T")~p. s(T' U {t} U (")~)). (4.7)
(' U {t}/(T")~P s(T' U {t} U (")~)) = = (' /(")~. s (' U (")~))
((t}I(T' U (T")~p.s(T' U {t} U (")~)) (4.8) s (' U {t} U (")~)
9 8-2045 129
(4.7} (4.8)
(' U (t)/T", s(T)) = P(T'I(T")~. s(T' U (")~)) (a{t}/ (' U (")~). s(T' U {t} U (")~)). (4.9)
(4.9) s (t), , s (' U (")~) L ({ t)I(T' U (")~), s (' U { t} U (")~)) = 1.
s(t)ES (4.9) (4.6), s (' U ")
('/", s(T' U "))= ('/(")~. s(T' U (")~)). (4.10) (4.4), s (' U ")
('/", s(T' U "))= ('/(")~. s(' U (")~)). , ,
(' /") = (' /(")~). (4.11)
(4.4). 2. , (4.5)
(" u {t})~p =(")~. (4.12) (4.5) .
t*" ((3t' ' U {t}((t*, t'} ~)) ('v' t" ' ({t*, t") ~ ~1 ))). ( 4.13)
t', (4.13), , ", ', . . t. , t' ", t'E ' U U { t}. t' ', { t*, t'} ~ 1 t*, t' 1 '!! 1, 1 V t" ' (/ t*, t"l ~ ~1), (4.13). , t' = t, 'l (4.13) 3 t* " (((t*, t) ~) (V t" ' ({t*, t") ~ ft 9" 1))). , t ', k- t t* '!! 1 ', ( V t" ' ((t*, t") ~ ~1 ). , t " U {t}, . . ,
t ft (" U (tJ)~P ( 4.14)
130
, ,
V t' ' ((t, t'} ~ ~). (4.15)
(" U {tj)~p ~ ~ I
k- t G1. 'ff;- 0 1
: Tt-1 = Tt''\ (t); ; = {tE Tt: (t, t1} 'fr1 .
\\
(, s(T)) = P({t1}/Tj_1, s(Tt)). i=l
,
\\
(, s(T)) = P({t1)/Ti, s({t} U Ti)). (4.20) 1=1
, k- t1 , t2, , itrt, . . ; i 1 ; 1 ~ k. , ({t1)!T;), (4.20), 1 S lk (( 1 S 1 - 1) ( 1 1 - k) + 1) - 1 , (4.20) .
4.3. 3 3
. 1. - ,
. : -+ , . . (t), t , (t) - s t. , (t) , (t, s (t)) : -+ R, t s (t), . , (t) t s (t) I
-+ S, ; - ; (t, s) : -+ R - , t s S; W: sr s-+ R- ; : -+ - .
s*, , . .
s* = arg min ~ (s') ( (t, s' (t), (t))) W (s, s'). sesT s'EST IET
11. 11 , , s : -+ S , s (') '. r , .
: sr-+ R - -+ S. (, s ()), , (, s ()) =1= , . (, s (')) ,,1 1 s
51, (s) = , s () =1= s (); . (4.21)
,51 1 (s) = (\_ / , s ()), s ( ) = s ( ).
. : G = (, V, S, Q)- ; : S (G)-+ R- -+ S, ; W: S S-+ R- ; ( : , s ()).
s*, , . .
s* = arg min ~ ,511 (s') W (s, s'). sesT s'EST
, , . ,
. ~~ 9ii -
~33
,s11 , .
, s () , ,511 (4.21).
.. 1. : s- R, ' : " : , , ' n " = f2J' SH (' u ") ; (' u U "), , (' U ", s (' U ")) -=/:=,
'UT",sH('UT") = ( T',sH(T'>)T",sH(T")'
. ,
Pu",s(T'UT"I (s) = (.s
,s> (s) = ( (' U ", s'f (' U ")))=
(', sH (')) .s (' /;. s (' U ;)). ( 4.28) , , P{t}.s({t}) (' /",
s ()) s (" "' ;). :
1) t '. s (t) =F SH (t);
2) t '. s (t) = SH (t); 3) t ". s (t) = SH (t).
, t ", s (t) =1= s (t), .J:, P{t};s ( / , s) s, (4.28) .
2. P{t},sl{t})(T'!T", s) = (' '\. (1)/" U {t), s). (4.29)
, ('"\, {t) '(" U {il)r s). , (4.29) s ((" U {t}) "\.(" U U(t})rp) ~".(" U (t}), ~"'\;.(" U
135
U {t})"-{T" U {t}) , , P{t}.s({t}) {' IT", s) . s (" "- ~);
.
P{t},sH({t}> ('!", s) = ('/", s). (4.30) (4.30) s (""- ;), s ("" ;). .
. 2. - , - G; , s ()- , , (, s ()) =1= . ,s
, , , ({t)IR (t), s (R (t) U {t})) t Tn s (R (t) U U {t}) sRU{t}.
. ,
, s
P(s) = P({t}/R(t), s(R(t) U {t})). (4.29) tETn
. , "
. , (4.29), (4.29) -. (4.29) . .
,s'
(, s ()). ,..
' : s (') s (') S (G, ', s (')) G s*, , s* (') =s(T'). S(G, I, s1 (1), 2 , s2 (2), ... ) S (G, I, s1 (1)) S (G, 2 , s2 (2)) n ...
(4.21), ,s
R (t), (R (t)) '\_ '\_( (R (t)) U R (t)). S0 , , , s (R (t)), s' (R (t)) = s (R (t)) s' () = ~ () . S s' : (R (t))--+ S, s' (t') = s (t') t' (R (t)) , s' (t) = s (t). S (s'), (4.30), :
(s') = (R (t), s' (R (t)) ( (R (t))!R (t), s' ( (R (t) U R (t)))
('\_ ( (R (t)) U R (t))l R (t), s' ('\_ (R (/)))). (4.30)
,5,> ((t} U R(l), s((t} U R(t)) =
= ( ~ (R (t), s' (R (t)))) s'(R(I))ESo
~ P(Te(R(t))IR(t), s'(Tc(R(t)) U R(t))) s'(T e(R(IJ>U R(I))ESe
~ ('\_ ((R (t)) U R (/))/ R (t), s'(T'-.T e(R(/)))ES
s' ('\_ (R (/)))).
s (R (t)) s (t), s (R (t)), s (t). .s,> ((t}IR (t), s (it} U U R (t)) . ,
,s,1 ({t} U R(t), s(R(t) U {t})) = = cq>1 (t, s(R(t))) D1 (t, s({t} U R(t))) q>2 (t, s (R(t))).
,
. ({t}/R(t), s({t} U R(t)))= D1 (t,s(ft\ U R(t))) 7 ,s ~ D1(t, s({t) U R(t)))
s(IJES
(4.31)
,
,,,>
JI~ ~JIfi D~ (t, s ({ t} U R (t))) . .~ l
t s ({t} U R (t)),
D1 (t, s({t} U R(t))) =
= ~ ( (R (t))!R (t), s' ( (R (t)) U R (t))), s'(TeUR(I))ESe
, s ({ t} U R (t)) s (), . . s' : (R (t)) U R (t) ~ ~ S, s' ({ t} U R (t)) = s (( t} U R (t)), t' (R (t) U (R (t))) n s' (t') = s (t').
D2 (t, s (R (t))) = ~ D1 (t, s' (t), s (R (/)), t , (4.32) s'(f)ES
D3 (T*, s(T*)) = .!1 D~(t, s(T*)), T*E?I*. (4.33) t()
D1 (t, s ({ t} U R (t)))
D1 (t, s ({t} U R (t))) = ( f"! D 3 (*, s (*))) T*E'!J(t)
P((t}!R(t), s({t} U R(t))), trf.T s(t) = = s (t); D1 (t, s ({ t} U R (t))) = - . (4.34)
(4.31), (4.32), (4.33), (4.34) . .
4.4.
W W (s, s*) = , ecJI s = s*, W (s, s*) = 1, s =1= s*. , s*, J, . .
s*=argmax P({t}/R(t), s({t} U R(t))). (4.35) s~s t~;,T
139
({t}IR (t), s ({t} U R (t))) ,
, . (4.35)
, [15].,
D1 (t, s({t} U R(t))) =
= max P({t'}/R(t'), s({t'} U R(t'))) (4.36) s(Tc(I)"{I})Es(Tc(l)" {/}) I'ET (l)
:
D 2 (t, s (R (t))) = max D1 (t, s' (t), s (R (t))), ( 4.37) s'(I)ES
s* (t, s(R (t))) = argmax D1 (t, s' (t), s(R(t))), (4.38) s'(I)ES
D8 (T*, s(T*)) = D 2 (t, s(R(t))). (4.39) IET(T*)
:
D1 (t, s({t} U R(t))) =( ... D3 (T*, s(T*))) T*Ert{(t)
P({t}IR(t), s({t} U R(t))). (4.40)
(4.37) - (4.40) , , s* (t, s (R (t))) t.E;.T s (R (t)). s* . s* (t1) s* (t1, =#=), R (t1) = 0, , s (R (t1)) = :ff. ,
ti, i, i*.
R (ti), , i*. , s* (ti) s* (ti) = s* (ti, s* (R (ti)).
, . - . , 2-3, . , . . (5
140
,
, . , ,
- , . . . ,
,
. , ,
, . .
. S = {s1 , s2 , s3 ) - , S S- , . , . . , , (s1 , s1), (s2 , s1), (s3 , s2), (s3 , s3) 1/ 4 , . , . . , , 3/ 4 , , (s3 , s1 ,) . , , . . , ,
. (s2, s1) , . .
, ,
,
,
.
w : S S -+ R W : s s -+ R W (s, s*) = = I: w (t, s (t), s* (t)). w
tET
{ 1 s (t) =1= s* (t), w (t, s (t), s* (t)) = .s (t) = s* (t) (4.41)
141
. , w (t, s (t), s* (t))
s* = arg min : P(s) ~ w(t, s(t), s' (/)). (4.42)
* ~ (t), t (*, s (*))
(*, s(T*)) = ~ P({t} U R(t), s({t} U R(t))). (4.46) s(({I}UR(I)I'- J
(4.44) - (4.46) .
(;, s (;)), ; = 0, (0, =#=) = 1.
, (;, s (;)) i ~ i* ({tt), s ({tt})) i ~ i*. : ( { ti} U U R (tt), s (1 it} U R (ti))) (4.44), ({tt}, s ({ti))) (4.45) (*,
s (*)) * f!J- (ti) (4.46).
4.5. 1
1 .
G = (, S, ?!, Q) s s1 , s2 , , sm, , ({t)IR (t),
s ({t} U R (t))), P({t}IR (t), 1=1 IET
si ({t) U R (t))). . ,
P({t}IR(t), s({t) U R(t))) =(t~'s
- ({ t}/ R (t), s (( t} U R (t)))
~ (s')
' ({t}/R (t), S ({/} U R (/))) = _s'..;;,.E;-'-(t~,.s(:=;{t}=U....:.R
4.8. 11
,
S ({t}l !R (t), s ((t} U R (t))), t , s ((t} U R (t)) s
4. l : S -+ R .. 3, , ...... ~ P(s) log (s)
sES . 1. ,
~ (s) log (s) ~ log (s) (4.56) sEs
s S, ' : S -+ R
~ P(s) \g (s) ~ ~ P'(s) log ' (s). ES S
(4.56)
' (s) ~ P(s) \g (s) ~ P'(s) log (s) f4.57) sES
s S P'(s) ~ . ... (4.57) s S,
~ P'(s') ~ (s) log (s) ~ ~ P'(s) log (s), ~~ sEs sEs
' (s), , ~ P'(s) = 1, sES
~ (s) log (s) ~ ~ P'(s) \og (s). (4.58) ,8 sES
(4.50) '
~ P'(s) log (s)::;;;; ~ P'(s) \og ' (s). (4.59) sE;:S sS
(4.58) (4.59) '
~ ' (s) log ' (s) ~ ~ (s) 1og (s). sES sES
2. , s S, ~ P(s) log (s) > log (s), (4.60) sS
146
1'0 tt ',
~ P(s) log (s) > ~ ' (s) log ' (s). (4.61) S ses
S S1 U S11 , ,
S1 = {s: s S, 1og (s) = mi (s')}, S2 = s '\ S1 s'ES
1 : S-+- R 8 : . ~ R, ,
1 (s) = ~ , s S11 1 (s) = , s S2 ,/.,J (s)
ses, 8 (s) =
(s) ~ , s S2, 2 (s) = ,
(s)
ses. : k1 = ~ (s), k2 = ~ (s),
ses, sES. k1 + k1 = 1. , k1 =1= , (4.60); k1 . k1 = , 1 -~, S2 . : S1 . S2 , k2 =1= , . . , (4.60).
() () : S ~ R, ~ k1 ~ 1 () = = 1 + (1 -) ,.. , (k1) = .
, k1 ~ ~ 1, F () :
F () = ~ 1 (s) Iog (* ()) (s)- ~ 8 (s) log (* ()) (s). ~ ~
F- . , F (k1) k1, F (*)
( - k1) () < , . . (*- k1) L 1 (s) log (* (*)) (s) -
ses,
- (*- k1) L 2 (s) !og (* (*)) (s) * L 1 (s) log ("' (*)) (s) + ses sES,
+ (1 -*) L 2 (s) log (* (*)) (s) = ses,
= L * (*) (s) log (* (*)) (s), ses
(4.61). . , ~ 11
: S-- R, - L (s) log (s),
ses n , n
nn n .
L (s) log (s), ses
. , * n
~ (s) log (s), Ji ses 148
, , I: ** (s) ses
Jog (s) < I: * (s) log (s). ses
, .
1. 0 , 1 , 2 , ... , aq, ... , .
2. 8 >, .
3. q = . 1
4. Pq (s) = -,..- s S. 1 SJ
5. s* S,
~ Pq (s) Jog (s)- log pqM (s*);;;;;:, 8. ses
6. , . 7. Qq : s-+ R, Qq (s*) =
= 1, Qq (.s) =, s =1= s. 8. pq+J = (1 - aq) pq + aq Qq. 9. q = q + 1. 10. . 5. .
5. 5 > S Pq, q = , 1, 2 ... , , , , , L pn (s) log " (s) - log pnM (s) < 8
sES ,.. s S. . ,
~ (s) log (.s) sS
: S-+ R. .J 3 (
(4.!>4)), : S-+ R ' ':1' - L (s) log (s) ~ min \og ' (s). o-
s{s ses
I:,.. (s) log (s) :" , s~.S
149
S ~ ' 'JI, ' (s) =/= s S. ' , , , -
,.. 1 ,. " : S-+ R, , " (s) = -,..- s S.
ISI , ..' . . -
q s S, ,
~ Pq (s)log q (s)- log P'.w (s) ~. ses
: Fq = ~ Pq (s) log q (s) q () = ses
= ~ ((1 -) ~ + aQq) (s) log (( 1 -) Pq + aQq)M (s). ses
dq (a)lda = . q () , 1 () 11 (), 1 () = 2 () = ..
q () = '1' q (1 (), 2 ()) =
= ~ ((1- 1 ())~+ 1 () Qq) (s) log ((1- 2 ()) Pq + ses
(4.66)
1
dq _ ' (.1 (.), .2 (.)) drx 1 + ' (. 1 (.), .2 (.)) drx2 da. - .r da. .2 lia" .
' ~ \1 .= --= ~ (-Pq + Qq) (s) log Pq. (s), rxl ,.
-es 'l'
. = , I
Qfq) s S, , s*,
~ fYl (s) \og f>'l (s) - log f>'l (s*) ;> , ES
, , -d
- da. q , = -
d::;,q ~-. qfO> = Fq, q (cxq) = Fq+l Fq- Fq-l ~ cxq + q, q , , qlcxq . cxq, q = 1, 2, ... , , , q1 q2 Fq, - Fq, , F . .
TeopeJ/Ul 6. , : S -+ R >
~ (s) \og (s) - log (.s) ~ sES
(4.67)
s S, max i log ' (s)- min log (s) ~. '9' sES sES
. ,
in ; (s) log (s) = max min log ' (s). 9' sES' '9' sES
~ (s) log (s) ~ min ~ (s) log (s), sES 9' ES
(4.68)
max min ' (s)- ~ (s) log (s) ~. P'Efl' sES sES
(4.67)
~ (s) log (s) -- ~n log (s):::;;.; ~::. .E;;s s~s
(4.68)
(4.69)
(4.70)
(4.69) (4.70), max min log ' (s)- min log (s) ~. '9' ses ses
. ,
. 6. q, , , min log pqM (s) o-
s{s
max min log (s) . !f ses J , 11 .
, ,
. . 5 s* = = arg min q (s), . 6 . ,
ses , . . ( ),
lim min pqM (s) = min (s), -q-+ ~es PEff ses .
.
, q-
max min log piM (s) min L pi (s) log plM (s). i t;;,q ses t:;;;,q ses
, , , . , , n
152
n . n n .
11 . , 3-6 n , '[/' - n , . '[/' n n ~ P(s) log (s)
sES
S-+ R. , .'l, . , .'I,
,
I I I, .
4.7. III
G = (, S, '!/', Q}. -n '[/' - : s -+ R, s S (G)
P(s) = P((t)!R(t), s((t} U R(t))), ~ P(s) = 1. ~ ~~~
G '[/' n (G, ). r , s1t () n s"', , s* () = s (). (G) , . . (G) = {~ (G, ): '[/').
S {(s1 , 1), (s2 , 2), ... , (s, TnJJ. III, (G) S, (G), , (, s (11)) = s
(s, ) S. ff
l{ (s, ) S S' (s, ) ,
153
s, s. ,
S' (s,) = {s': s' S (0), s' :::/= s, s' () = s()}. . (0), ,
'f!J, , (s) > (s') - , (s, ) S s' (s, ). r , , ({ t}/ R (t), s ({ t} U U R (t))), t , s ({t} U R (t)) S{t}UR P({t}IR(t), s'({t} U R(t))) ~r ~
(4.71) ,.. ,..
I (s, ) S s' S' (s, ). - (t, s ({t\ U R (t))) = log ((t}IR (t), s ({t} U U R (t))),
~ (t, s ({t} U R (t))) > tET
>~ (t, s' ({t} U R (t))), s' S' (s, ), (s, ) S, (4.72) tET
(t, s ({ t} U R (t))). , . . [60], , .
(t, s ({ t} U R (t))), t , s ({ t} U R (t)) s{t}UR
.'l. . (4.72) ., .* = arg in .2 .
4
. , (4.72) min .2:::!=
4
::1= . , . = arg min .2 , 4
~ ~
(s, ) S S' S (s, ) -
.* . ( (s) - (s')) ~ .2 > . (4.73) , (4.73) , . . s, s', (s, ) S s' S' (s, ),
.2 >.* ( (s)- (s')). (4.74)
F(y) =(.* (1- )+ (x(s)- x(s')). )2
, F- , , F () = .* 2 , F' () = -2.2 + 2 ( (s)- (s')) .*. (4.74) . *, < *~ ~ 1, (.* . (1 -*) + ( (s) - (s')) *)3 < .* 2 .* .1, l
, f . , , . 111.
. 8. ( 4.72) 1 n, f" () ( 4.72) . . . 4.1, IIOTO
,
, (s ()) - (s' ()) f (). (f ())2 = = min ( (1 -) + ( (s ()) - (s' ())) . ())2 , ,
'1' f ()
,
,
( (s ()) - (s'(a)). ,
.'l ( (s ())) ~ x(s(}-x(s'(a)) (s' ()),
. 4.1. (s ()) _ 8 . 4 -x(s' (:))- .
, (f ())2/2= = w2/(w2 + 2) = 1/(1 + a 2/w2) w2 ~ ( (s) - (s'))2 , 1 (s) - (s') 12 ~ 21 1.
(f ())2/2 ~ 1/(1 + 2/(21 /)). 2 ~, (f ())2/2 ~ 1/(1 + /(2 1 1)). , 2 , (f ())2 , ({2 ())2 , ... , (fq ())2 , ... 21 1 /(21 1 + ). , i ~ n f" () =t= =1= f"- 1 (:), ({'1 ())2 ~ 2 (21 1 /(2 1 1 + ))" ([" ())2 , 2 (2 1 1 /2 1 1 + )" = , , . .
4.8.
1. ~ . , , , , . . . , ,
, , , , . .
. ,
, , ,
. - , , , , , ,
. , , , , , ,
. , ,
. , , , ,
. , , .
, , . , .
. , , , . , s
151
, f , , 111.
., 8. ( 4.72) , t n, fn () (4.72). . . 4.1, I
4.8.
1. . n n , , , i
rt1 2 , , 1 , 1 () 2 () . 1 , 2 1 , 2 , ~ , ~ , 1 1 () > 2 2 (), .
, .
1 , 11 1 , 1 , , , :
- , - k {1, 2} . k1 , 1 , k2, 2 , ... . .. , k1, 1 1 , 2 1 , 2 , k , . , , , 1 , 2 1 , 2 , .
, k , . ,
, , .
, q1 () q2 () - . , , 1 2 , , q1 (1) =1= , Q2 (yt) = , ql (2) = , q2 (2) =1= , . , , , i-, i = 1 ,2, '\1 ~ Pt () q1 ().
, = 1 , 1 , 11 , 2 , , 1 , ~, ... k , .
l58
, . , , , , , 1 , . - , - .' , : i , , . .
, 'rP = ((: _. R) : }, Pv () - ,
. , , , Pv . { (pg : _. R) : } , ', ", "', , , ' 'f!J, " 'f!J, "' 'rP, ' =1= ", , "' = ' + (l -) ". r , Pv = al () 1 + () 1 () 11 () - ~;: k = l k = 2 :
1 () = 2.1 q1 () i = 1, 2. ~ a.t. ql () 1=1
1 2 ' " 'rP, , ' =F =1= ", 'rP , * =* ' + (l -*) ", ** =** ' + (1 --**) " *
= 1 1 + 2 2 , 1 2 ( ) , 1 + + 2 = 1, 1 2
q1 q2 , Q = {(Q.c: -+ R) : }, Qx ()
, . Q , ':.
1 2 , , Qx, ::/= =1= Qx, :
Qx, = tx1 (xl) 1 + tx2 (xl) q2, Qx, = tx1 (2) ql + t%2 (2) q2, (4.75)
1 (1) k = i 1 , , 1 2 , 1 2 .
(4.75) . r q1 q2, tx1 (xl) =1= tx1 (2)). at (1) tx2 (2) =1= tx1 (2) 2 (1).
1 , 2 1 , 2 q1 , q2 , k1 1 , Yt U. , tx1 1 (xi) 1 (yt) > tx2 2 (xt) q2 (xt), kt = 1, k1 = 2. .
, , , . , ,
. 2. . -
, ; - i . , k Pk : -+ R ,
Q : -+ R . Pk, k , Q . , , ':Pk, k , , Pk ':Pk, k . ':Pk, k , .
, U = (k1, 1), (k2, 8), .. ,
160
... , (km, ) (k1, , Q (k Pk (;.).
no = (k1 , 1), (k2 , 2), , (k, ) Q Pk, k , , . .
~ log Q (kt) Pk1 (xt) l=l
(4.76)
Pk ':Pk, k , ~ Q (k) = 1. k
: ( ) -+ N (N - ), ,
"' , , c:t (, k) (, k) . , (4.76)
~ ~ (, k) log Q (k) Pk (), kEK
1 1
~ (, k) log 11 ()
11 ':, k ,
~ ~ (, k) log Q (k) kEK
n : Q (k) = 1. kEK
(4.77)
(4.78)
= 1 , 2 , , Xm , -
: Q (k) Pk (). kEK
, , ;., i = 1, 2, ... ... , rn.
= (1 , 2 , , ) Q Pk, k , , . . .'I
!f () = : log : Q (k) Pk (1) (4.79) i=l kEK
11 8-2045 161
nph Pk f!Jk, k , ~ Q (k) = 1. kEK
(4.79) n Q Pk, k .
, Ji , . . (4.77) (4.78), , . . (4.79), 11 .
. 0 = (Q0 , ~, k )- , , ; k Q (k) Pk (;) .
n
0 , At, ... , 1 , ... , . . n A = (Q, ~+11 , k) A
Q1i+11 .
, , . ,
.rr Pk, Q '[!>k , :-.~ 1 , , , ,
{/, () u
: log : Q (k) Pk (xt) i=l kEK
9. Q(i), P~l, k , Q, ~+'>, k ,- , (j - 1)- j- . .
~ log : QU+I) (k) p~+t> (,) >: log : Q (k) ~> (xt). (4.83) i=l I.EK l=l kEK
.t i = 1, 2, ... , k
Q (k) p~+l> (xt)
: Q(f+IJ (k) p~J+I) (xt) =1= kEK
: Q(J) (k) ( = kEK i=l kEK
= : : au' (1 , k) log q (k) + : : a (1 , k) log ~> (1)-kEK l=l kEK i=l
~ \.-, Q1i> (k)
: (1 , k) log PV' (xt) ~ ~ a (;, k) log Q (k) ~ ~ ~ r:x.11) (xi, k) log Q (k); (4.87) kEK i=l kEK i=l
'\""" r:x.U> (xi, k) log QU> (k) PV' ~xt) :::;;:, fk ~ Q(/) (k) ~) (Xt) :::-
kEK
(1) ~ ~ r:x. (xi, k) log ~ 1.1
"1, (4.90) .
(4.87), (4.89), (4.90) (4.84), (4.85) .'! {f (A 1i+11) ~ {f (Au1) , , r
{f (A) > {L (Aii>). . , r
'*, , .r, !l0 () ~ () ~ .
!f0 () * /ik
n cx.u" = L /ik < \; CX.;k = 1, a;k > ~/(
k > , n n, n 9, rL0 () ~, . .
'\' '\' Yik ,., '\' Yik L log ;L ~ .z.. log L , i=l ~ i=l kEK
n
Y"k
~ ~ CX.;k Yik - ~ ~ cx.,k log ., LY .k ~ kEK i=l i=l kEK : L
kEK
(4.91)
~ ~ ~ a;k yik- ~ ~ a;k logau,;
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