Московский Государственный Технический Университет им. Н.Э. Баумана
Факультет ИУ Кафедра ИУ-8
Чашкин А.В.
Булевы функ...
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Московский Государственный Технический Университет им. Н.Э. Баумана
Факультет ИУ Кафедра ИУ-8
Чашкин А.В.
Булевы функции и преобразования
1.
1.1. 1.2. 1.3. 1.4. 1.5. 1.6.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !" . . . . . . . . . $" !! . . . . . . . . . '" " . . . . . . . . . . . . . . . . . . . .
2. 2.1. 2.2. 2.3. 2.4.
) !"!" ) " . . . . . . *" . . . . . . . . . . . . . , " . . . . . . . . . .
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/" . . . . . . . . . . . . . . . " . 4!" ! . . . . . . 4 ! . . . . . . . . . . . . . . . . . . . . . . . . . " . 15 !5!". 5!" , !7 " 28 .! 8"
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" 01 . . . . ) 2 . . . . . . . . . . ) . . . . . . . . . . . . . . ' {* . . . . . . . . . . . .
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- .! ! !" .! . - .! ! ! .! . . . 95 .! . . . . . . . . . . . . . . . " . . . . . . . . . . . . . . . . . . . . . . . 5!" .! /
" 01 . . . - .! . . . . . . . . .
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3 8 14 21 27 29
33
33 37 40 43
47
47 55 58 63 65
71
71 77 81 84 88
94
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. 94 . 99 . 103 . 107 . 110 . 111
- .! :; n . . . . . . . . . . - .! !!" :;. < !!" . - .! !!" :;. 9 !!" . . . - .! ! /" :; . . . . . 0! "".! " . . . . . . . . .
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. 115 . 117 . 123 . 126 . 128
1
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3
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6. $ % %&
% 6.1. 6.2. 6.3. 6.4. 6.5.
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5.
5.1. 5.2. 5.3. 5.4. 5.5. 5.6.
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4.
4.1. 4.2. 4.3. 4.4. 4.5.
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3.
3.1. 3.2. 3.3. 3.4. 3.5.
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115
2
7. & 7.1. 7.2. 7.3. 7.4. 7.5.
1"7! 8 . . . . . . . . . . . . . " . . . . . . . . . . . . ".! . . . . . . . . . . . . . 0! "".! " !" 8 5!" ! !5!" . . . .
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133
. 133 . 137 . 138 . 143 . 146
153 155
1.
1.1.
1. '!"" 0 1 ;"! !"". 9 . " . 0 1, 7 . , ;"! . >! "! 8 . 5 8 n ;"! .! " 1 n ! : ." 1, | n. *5!" ! n "! !" n ."! ! B n . 1 8 8".! " " 5!" B n . A! /"8 5!" !!"" /" !"" 8 n-8 !"!" Rn, " " " !, ." " ! 5 2 .8 . 4/" .!" ;"! "5 2 n-8 .8 . u = (u1 : : : un) B n ;"! . u=
j
j
n X i=1
ui 2n;i
u =
k
k
n X i=1
ui :
1 ;" 5!" B n "2 8 . 8" , ." u 2 v, ! u v . 4 , "2 , "! . *5!" ! n ! k " k- B n , . . B nk . >! k- ! n-8 .8 .! !." n /" k | !" ! ! ! n 8 5 " k , . " , n = n! : n Bk = k (n k)!k! 5 2 u v B n "! .! d(u v) = Pn i=1 ui vi , .!" ! ;7 u v. 1" " , ." !!" d "! ", ". . d | 5" !".! 8", ;7 . "8 " "8 , 8 8" ! ;", " ! !" "8 : d(u v) d(u w) + d(w v) ; " u, v w B n . 1 u v ;"! , ! d(u v) = 1. A! d(u v) = n, " ;"! . ! .;"! 5 ! " , " 5 | ! . ' % 1.1.1. - " B 3 !!" ." u1 = (000), u2 = (100), u3 = (110) u4 = (111). G ! /" ! !": u1 = 0 u2 = 4 u3 = 6 u4 = 7 u1 = 0 u2 = 1 u3 = 2 u4 = 3:
j
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j
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j j
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j
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k
k
k
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3
4
1.
4 !!" 5 !!" d(u1 u2) = 1 d(u1 u3 ) = 2 d(u1 u4 ) = 3 d(u2 u3) = 1 d(u2 u4 ) = 2 d(u3 u4 ) = 1: " , u1 u4 | " 5 , ui ui+1 i = 1 2 3 | !! . 4 !! 2 ;"! . A! u v | !! 2 , .;7! i- , " 8", ." (u v) " i- ! " /" 2 . 4!" i1 : : : in;k | . " .!, ! 7 n, u1 : : : un;k | !"" . *5!" v B n vij = uj j = 1 2 : : : n k "! k- B n . )8 " , ." k- 8 n-8 ! 5" 2k . 2. 4!" u B n . ! k ! " u "! 5!" Bnk (u) !!"7 ! " v B n , ." d(u v) k. ! k ! " u "! 5!" Snk (u) !!"7 ! " v B n , ." d(u v) = k. )8 " , ." k- ! B nk "! ! ! k ! " ! ! n k ! " . . 1 ! !" ! 2 ! ", ." ; 8 u B n ; 8 k = 0 1 : : : n k n X Snk (u) = nk Bnk (u) = : i=0 i ' % 1.1.2. !!" " B 3 , 5 ! 5 ! !. 1.1.1. H"" ! 5" ! 2, ;7 ." !. 1 ! B 30 !!"" !" (111) 2 (000). 4 ! B 31 ! 5" " 2 : (100), (010) (001). -" ! B 32 "5 ! 5" " 2 : (110), (101) (011). I, , "" (110) (101) (011) ! B 33 !!"" 2 (111). - 12 , 5 ! ", ! ;7 2 . - 5 " (100) (010) (001) " ." . - B 3 ! 5"! 2!" 8, ! 57 ." 2 . -2 /" 8 ". ! . 5. )8 " , ." ". (000) 2 5" 5!" 1 3 5 7 . < B31 (010) !. 1.1.1 !!"" 2 (010) " 2, ! ! : (000), (110) (011). J ! 2 ;" ! S31(010). 2. 4 5!" G = g1 : : : gm B n "! ! !!" d, ! ; 8 /" gi gj !!" 5 2 d. K", ." G ! " t 2 , ! 8 !!" 2 . 2t + 1. $", ." ! 8 5 8 /" g G ! !!" 2t + 1 !"" 2 ! t ! " /" /", " 2 ! " /" " !" !. G!"" , ! " v 5" !.; 2 ! " /" gi gj , " /" .", ." d(v gi ) t d(v gj ) t. 1 "8 ! !" "8 d(gi gj ) d(v gi ) + d(v gj ) 2t. 4".. ' % 1.1.3. !!" " B 3 . - ; " 5 2 ;" ! !!" ", , (000) (111) . ' ! !!" ;"! 5!" ! ."8 ! 5!" ! ."8 !. , . . m(n d) ! 5 .! /" . n, !!" "8 d. G ! ." !!" " !" ! ;7 "". f
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5
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P2t ;n
i=1 i
m(n 2t + 1)
Pt
2n ; : n
i=1 i
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j
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! 1
p
bn=2+ n log2 nc p
k=bn=2; n log2 nc
Bn k
B n
= 2n:
. , .! ! " "."! " n2 . t :
k:jn=2;kj>t
Bn k =
n = X (n=2 k)2 n 2 k:jn=2;kj>t k k:jn=2;kj>t (n=2 k) k n n 2 n 1 X n k2 n 1 X t2 k:jn=2;kj>t 2 k t2 k=0 2 k k : X
;
;
;
;
(1.1.2)
1 !, !"7; .!" !" (1:1:2). )8 " , ." n X k=0
n k
n n n2 nk + k2 = n k2 = X 2 4 k=0 k n n n n 2 X X n = 4 (n k)k: k=0 k k=0 k
;
;
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(1.1.3)
6
1.
4 ! .!" (1:1:3) n22n;2. 1 "; !: n;1 nX ;1 X k)kn! = (n k)k nk = (n k)k nk = (n k=0 k=1 k=1 (n k)!k! nX ;2 n 2 nX ;1 n(n 1)(n 2)! = n(n 1)2n;2: = (n k 1)!(k 1)! = n(n 1) k k=0 k=1
n X
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k=0
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n k2 = n2 2n;2 n(n 1)2n;2 = n2n;2: 2 ;
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p
4 !" . !" ; .!" (1:1:2) 8 t n log2 n, , ."
p
k:jn=2;kj> n log2 n
Bn k
n2n;2 = 2n;2 = o(2n): n log2 n log2 n
J , ! ! !, " ".;"! " n2 2 . n log2 n, "! o(2n) p n. " , ." ! 5" !, !!"7 2 n log2 n ! !. J . 4. ' !!"8 2 8 , 5!" B n !7!"" !"!" .!". . K", ." u ! v (u v), ! ui vi ! i = 1 2 : : : n. A! u v u = v, " 8", ." ." u ! v (u v). 1 u v ;"! , ! u v, v u. A! /" "2 "!, " ;"! . 4! " !" 2 u1 u2 : : : uk "! " , ! d(ui ui+1 ) = 1 ui ui+1 ! i = 1 2 : : : k 1. -2 uk "! 2 2 u1 u2 : : : uk , 2 u1 | 2 2 /" . >! 2 "! . K", ." ! " 2 u v " . 2 w, ! u v, ;"!, !""!", ! 2 , w 5" /" . L "! , ! "! .!" ; 2 . *5!" ! 2 "! " . 0" "! , ! "! 5!" 8 " , !!"7 28 .!" 2. ' % 1.1.4. - B 3 (!. !. 1.1.1) !7!"" 2!" ! . G 5 ! ." . '5 ! ."! 2 (000), ! " " . 2 8 2 "8 !, ."! 2 (111). - B 3 !7!";": (i) ! " !!"7 " 2, /" " !M (ii) " ! " !!"7 2, 5 " " !!"" 2 8 ! " 5 2 "8 !, , (100) (011) M (iii) ! " !!"7 2 , /" 2 (000) (111). p
6
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;
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bn=2c
n 2p + 1, p . J 5 n. 4 n = 1 "5 " . | B 1 5 " !" ; 2 (p = 0). G !", ." " n = k. - B k+1 !!" 5!" B0 B1 , !!"7 ! , " (k + 1)- !""!" "
;
1.1.
7
; . '5 5!" Bi B k , /" ! 5 5" " " !;7! ; ; ", ." ." .! k 2p + 1, 8 p = 0 1 : : : k=2 , kp p;k 1 . !!" " 5!" B1 B0 , 5 " "" ! " !" k. ,. , ." /" " !;"! !" ; ;" B k+1 . 4!" C1 C0 | !!" " 5!" B1 B0 , v1 v0 | 2 /" /" . ,. , ." v0 v1 v 1 v0 ! ! 8 ! ;". ;
b
c
;
B1
r r
B0
r r
r C1
r C0
v1
r r
r r
r r
r r
v0
r
r r
r
r r
v1
C1
v0
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!. 1.1.2 !!" :" !" .!" ! 1.1.2. 1 /" ! 2" 5 , 7 (k + 1)- . 4 C1 C0 ! ;7 : v1 C1 C0 . "" k 2p + 1 . k 2p = (k + 1) 2p 1 k 2p + 2 = (k + 1) 2(p 1) 1. 1 5 .!" ! 1.1.2. - ! " B1 B0 . ,. , ." ;" B k+1 !;"!. 4 /" (k+1) 2p 1 ."! 28 k 2(p 1) 1, 8 k 2p 1. 4/" " .! (k + 1) 2p 1 k k + k k = p p 1 p 1 p 2 k k k k k + 1 k + 1 = p + p 1 p 1 + p 2 = p p 1 : ;
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*& 1.1.1. 1" ! 5 " "8 B n . 1.1.2. > ! !!" 5 , 57: a) B n M b) B nk M c) Bnk M d) Snk ?
8
1.
1.1.3. 1" ! !!" 5 B n . 1.1.4. 1" .! . B n . 1.1.5. 1" B n : a) .! . k- 8M
b) .! . k- 8, 7 . !; 2M c) .! ! 8. 1.1.6. 2 ! ! 5" 8 n-8 ? 1.1.7. 1" .! , 7 . 2 k- 8 n-8 . 1.1.8. 1" .! , 7 . 2 , 57 k- ! n-8 . 1.1.9. 1": a) Pu2Bn u M b) Pu2Bn u M c) Pu2Bnk u . 1.1.10. - n- / ! ! ! "! 5!" , ! !!" " 5 8 " ! .! k. ! 5"! / ! n-8 ? 1.1.11. G ; ! B n " ! 8 n "! 5!" I( ) = B . 4" , ." ; " "! 8 ;, 8 | ". 1.1.12. 4" , ."1) n 2n : n n=2 k
k
j
j
j
j
2
f
2
j
b
c
g
p
1.1.13. 1" B n .! ! ! . 1.1.14. 1" .! ! 2 B n . 1.1.15. 1": a) .! . ! B n M
b) .! . ! , 7 . !; 2 k-8 ! B n . 1.1.16. 4!" T | " B n , Tk = T B nk . 4" , ." \
n X
k=0
Tk
j
j
n k
1:
1.1.17. 4" , ." " B n , !" " !" "-
1.1.3, ! n 2p + 1 .;"! p- ! , .;"! 8 (n p)- !. ;
;
1.2.
1. f(x1 : : : xn), " 5;7 B n B 1 , "! n- -
. *5!" ! ."! . P2, 5!" ! !7 " n | . P2(n). '5 " .; !" , ." " !" ; " ; f P2(n), .! ! B n . f /" . - .!"!",
f(x1 : : : xn) 5" " " !!"7 2k !", 5 " !" !""!" k, 2n;k !" , 5 " !" !""!" n k. 4" k " . " 0 n. - " " (J . 1.2.1) . f (1 : : : k k+1 : : : n) 7"! !. !", !""!";7 (1 : : : k ), !" , !""!";78 (k+1 : : : n). A! " 1.2.1 " k ;, " 8", ." "! "-!" ! ., ! k = n | "!" . J 5 /" " , ;7 ;, ;, ;
1) f (n)
g(n) a b, f (n) ag(n) g(n) bf (n).
1.2. J 1.2.1.
0 0
0 0
:::
x1
0 0
:::
k
:::
1
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:::
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0
9
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k+1 k+2
x x
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x
0 1
, " 8 " , ." .! . " , , !""!", .! . , !7 " n , 22n . >! B n , " f " . ., "! f /" : k k
f =
k k
X
u2Bn
f(u):
' % 1.2.1. $ " ; f(x1 : : : x5) ; " " = (1 : : : 5), " ;"! !" .!. J !
x1 x 2 0 0 0 1 1 0 1 1
J 1.2.2. 0 0 0 0 1 1 1 1 x3 0 0 1 1 0 0 1 1 x4 0 1 0 1 0 1 0 1 x5 0 0 0 0
1 0 1 0
1 0 0 0
1 1 1 0
0 0 0 0
1 1 0 1
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1 0 1 1
5" .!, ! 7 31, !" ;"! .! 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, " ! !!" f 12, !5 " , ." " 1.2.2 !"" " ; f. 2. 4 !!" 5!" P2(1) P2(2), !!"7 , !7 " . 4 5!" !!"" ."
. -"-!" /" , !7 " x, !" " 1.2.3. 4 ."" ;"! , x 0 1
J 1.2.3. f0 fx f: 0 0 1 0 1 0
f1 1 1
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10
1.
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J 1.2.4. f_ f f 0 0 1 1 1 0 1 1 0 1 0 1
f& 0 0 0 1
fj 1 1 1 0
f# 1 0 0 0
f! 1 1 0 1
"! 5. -" "! (' ". 1" "" , ." f& (x y) = min(x y) f_ (x y) = max(x y). ;7 f "! , 8 "5 ;" !;.;7 . >"" f "! ) , /" ! . 8" ! ;". 4" "! !* , 2!" | , ! | ". J , 2 " 1.2.4, ! !" ;". 3. 4 xi f(x1 : : : xn) "! $ , ! "! " !" u1 : : : ui;1 ui+1 ::: un, ." f(u1 : : : ui;1 0 ui+1 : : : un) = f(u1 : : : ui;1 1 ui+1 : : : un) 1!7!" "! "5 . 1" " , ." !
f " "; ;, " f | ." .!. G!"" , !" xi | " f. J8 X X f = f(x1 : : :xn) = f(x1 : : :xn ) + 6
k k
k
k
x2Bn
+
X
x2Bn xi =1
x2Bn xi =0
f(x1 : : :xn ) = 2
X
x2Bn xi =0
f(x1 : : :xn ):
' % 1.2.2. 1 .! 57 P2(n) !7!" !7 " ! ! . I! .! . . N. 4!" Ai | 5!" ! " n P2 (n) " xi "! !7!". ,. , n Ai . *7!" : 5!" Ai .! 7 ." N 22 i=1
;.-!;.. I ;j
n
i=1
Ai =
X
j
1 i n
Ai
j
X j;
+ ( 1)k+1 ;
1 i1
j
Ai1 Ai2 + : : : \
X
1 i1 <:::
j
Ai1 : : : Aik + : : : + ( 1)n+1 A1 : : : An :
j
\
\
j
;
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j
1 7!" 5!" A1 : : : Ak . H" 5!" !!"" !7, " !7!", " " ! n k xk+1 : : : xn. 4/", 8 " , ." 7!" !!"8 5!" .! P2(n k), ".. A1 : : : Akn;k= 22n;k . ,. , ." 7!" !. ; k 5!" ; Ai "5 22 . 9." , ." k !7!" 5 " nk ! ! , , ." n n X n 22n;k : k +1 Ai = ( 1) k \
\
;
;
j
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j
i=1
k=1
;
1.2. " , N =2
2n
n X ;
k=1
(;1)k+1 ;
11
n n 22n;k = X n 22n;k : k ( 1) k k k=0
;
)8 " , ." ni=1 Ai = o 22n n . 4/" ! !" n ." !
P2(n) !7!" !" " ! ! . 4. , ! !"2 : !" !"", "5 !" , " . A! f(x1 : : : xn) g(xk+1 : : : xn) ! 5 . xk+1 : : : xn ! !" f(1 : : : k xk+1 : : : xn) = g(xk+1 : : : xn) " 8" , ." g . f 1 : : : k !" x1 : : : xk . ; g " " f. ,. ! !""! !. !" !"" 1 : : : k !" xi1 : : : xik . ' % 1.2.3. 1 ! 5 f (x y). G /"8 !" !" 0 1. - "" . f (0 y) f (1 y) . " 8 "! " 1.2.4: f (0 0) = 0 f (1 0) = 1 f (0 1) = 1 f (1 1) = 0: " , f (0 y) = y f (1 y) = f: (y). 08. !" !"" !" " . x f: (x). ,. , ." !" !"" !" x y, . " !"" 0 1. J , f !" 2!" : 0, 1, x, y, f: (x) f: (y). A! f(x1 : : : xn) g(x xk+1 : : : xn) ! 5 . xk+1 : : : xn ! !" f(0 : : : 0 xk+1 : : : xn) = g(0 xk+1 : : : xn) f(1 : : : 1 xk+1 : : : xn) = g(1 xk+1 : : : xn) " 8" , ." g . f * x1 : : : xk. ' , . ! !""! !. "5 !" xi1 : : : xik . ' % 1.2.4. - f(x y z), 5 " , "5 !" ; "" ; . G /"8 !!" " " !" , " . "" ! ;". J !" "2) , "", 2!" ! . 4! /"8 /" ." !" !!" ; " . - 5 !" ", "" , !" " ;, !" !""!" t. J . .2! g(t y) ! 5 ! . j
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!!" ; f(x1 : : : xn) ! " x1 : : : xk ; g(xk+1 : : : xn) .;7;! f !" !"" 1 : : : k !" k . J /" f " , " "" !" " "" 5 ; 1 : : : k. 4/" 5 8" , ." g(xk+1 : : : xn) = f(0 : : : 0 xk+1 : : : xn): 8" , ." g . f " x1 : : : xk, f . g " x1 : : : xk. ' , . ! !""! !. ( ) " xi1 : : : xik . f g ;"! , ! f . g " . 5. f(x1 : : : xn) "! * xi xj , ! ; 1 : : : n ! !" f(1 : : : i : : : j : : : n) = f(1 : : : j : : : i : : : n) ".. . "! !" i-8 j-8 8". f(x1 : : : xn) "! * xi1 : : : xij , ! !".! "!" ; 5!" xi1 : : : xij . , !".! "!" ! ! , "! . -! , .! " 1.2.4 , ;"! !".!. I !".! 8 ! ", ." . 5 " ; x . "! ! /"8 . !".! !".! 8 , " !"!" .!" ;" . .. " n * m "! " m (x1 : : : xn), ." ( Pn 1 ! xi m m (x1 : : : xn) = Pin=1 0 ! i=1 xi < m: )8 " , ." :; :; ;"! !".! ! 8 1 2 !""!", ". . f_ (x1 x2) = 1 (x1 x2) f& (x1 x2) = 2 (x1 x2). !".! 8 ! !" ;" n(x1 : : : x2n;1), n 2, , 2!". - " 1.2.5 8!, !7 " " f
g
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14
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7 ! 1 2 3 n=2 . 1.2.2. 1" .! . , !7 " n 8" ;7 ." !. 1.2.3. , " !7!" " , " .: a) (1111 011011110110)M b) (0101 0111 11010010)M c) (1001011111111011). 1.2.4. 4!" f(x1 : : : xn) " . (f0 : : : f2n;1). G" , ." ! xi "! " , " fj = f2n;i +j ! j, 57 5!" k 2n;i+1 : : : (2k + 1) 2n;i+1 1 , 8 k = 0 1 : : :2i;1 1. 1.2.5. 4!" f(x1 : : : xn) !7!" !" " k . 4" , ." f "! 2n;k . 1.2.6. 1" P2(n) .! . !".! . 1.2.7. 1" P2(n) .! . , !".! "!" k 8". 1.2.8. 9" . !""!" 5 5!" " " 5!" !".! " ! . 1.2.9. 4" , ." ; " " 5" " . !".! " ! "5 !" . 1.2.10. 4" , ." ; 5" " . !".!
"5 !" . 1.2.11. 4" , ." ; !".! ". " !"" !7!" !" " ! ! . 1.2.12. 1" .! , " .: a) (0001 011010010000)M b) (1101 0111 01010010)M c) (1001011111111000). 1.2.13. ' .! 5" " n-!" , !7!" !7 " ! ! 8"? 1.2.14. (n n)- " "! . , ! f(x) = f(y) ; x y B n . 1" .! . (n n)- ". 1.2.15. 1" .! (m n)- " ", ." f(x) = 2 5 8 x Bn. d
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F(x1 : : : xn) = f(F1 (x1 : : : xn) : : : Fk(x1 : : : xn)):
F 5!" x1 : : : xn ( ; f(x1 : : : xn), ! F(x1 : : : xn) = f(x1 : : : xn) ! (x1 : : :xn) B n . 4!" F, ;7 ; f(x1 : : : xn), !!" ! x1 : : : xn ! f1 : : : fm . J8 8", ." F f ;"! (" f1 : : : fm . G ." " 5 ! " !., 8 /" " " . ;. ' % 1.3.1. !!" F = f: (f& (f: (x) f:(y))) 5!" x y. - /" !" x y !" . !" : F(0 0) = f: (f& (f: (0) f: (0))) = f: (f& (1 1)) = f: (1) = 0 F(0 1) = f: (f& (f: (0) f: (1))) = f: (f& (1 0)) = f: (0) = 1 F(1 0) = f: (f& (f: (1) f: (0))) = f: (f& (0 1)) = f: (0) = 1 F(1 1) = f: (f& (f: (1) f: (1))) = f: (f& (0 0)) = f: (0) = 1:
. . ! . " 1.2.4 , ." f: (f& (f: (x) f: (y))) " :;; x y. 2. -5 "!" ; ;"! !5!" 8 3). l(F) F ! B "! .! ! B 7 F. , d(F) F ": 1. A! F = xi , " d(F) = 0M 2. A! F = f(F1 : : : Fk ), 8 f B, " 2
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4 5; F1 "! !" d(F1) logk ((k 1)l(F1 ) + 1) : 4 !" ! !" (1:3:2) 8 "" !; k . logk ((k 1)l(F) + 1) 1 + logk ((k 1)l(F1 ) + 1) 1 + d(F1) d(F): J . 3. 4!" B | !, !!"7 .8 .! . , ! .! . 5!" x1 : : : xn ! B, !, ." !5!" !!" ! " . . I" !" ! ;7 "". )% 1.3.2. m | , B = f1 : : : fm , k |
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18
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!5!" L = (k;1) log2 n ; o 22n . , 8 N(L n k). I! . N(L n k) 7 " n ")2n log (cn) = (1 ")2n 1 + c < 1 " 2n: log2 N(L n k) = (k(k 1)(1 2 1) log2 n log2 n 2 " , N(L n k) = 2log2 N (Lnk) = 22n (1;"=2) = o 22n : J . - " 1.3.3 "5 "!, ." ." ! , !7 " n , ;"! . !5 . I8 "5
" !8 5!" , ! !" " " " ." !", 8 " /", 8 " ." 5 " /" 5!". 4!" An 1 n=1 | ! " !" . 5!" !";7 7!". 8" , ." /" 5!" An " !!" , ! n "2 .! /" 5!" An ;7 !!" 7!" An !""! . I! ", " 1.3.3 ! ;7 : n " ( P2(n) ( ( B , ! (k;1)2nlog2 n , k | " ( B . 4. A! ! B !!"" " !" !" , " !" F(x y) ! " 7 !{! (x y), 8 | ! !" , F(x y). 1 .!" !".;7! ! !" ! " 1.2.4 .!" 5 ! ! !""!";7 f. J . :; .7 !8 ! "! ! &, ".. f& (x y) = (x&y). I8 :; ."! " 5 . , ! !"!. 8 !" , .! " 1.2.4, ! ;"! ! ;7 : f_ (x y) = (x y) f (x y) = (x y) f (x y) = (x y) fj (x y) = (x y) f# (x y) = (x y) f! (x y) = (x y): G /"!" !" ! 8 ! "! ! . , !"; , ;7; ; ", ! " 7 8" ." ;7 8": F: (x) = (x). I! ! {! 5 ! " !5 . 1 : f: (f_ (f& (x1 x2) f& (x3 x4))) = ((f_ (f& (x1 x2) f& (x3 x4))) = (1.3.4) = ((f& (x1 x2) f& (x3 x4))) = (((x1 &x2 ) (x3 &x4))): G 7 ! !5 " !. !" ! . 1. 48, ." " "! " ! !" , !" ! 8 8" " ". J , ! "!"!";" ! , " !. "! ". 1 , (x1 x2) = x1 x2 . ! 1
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20
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6
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6
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=(11 ) : : : (mm ) f(1 ::: m m+1 ::: n) = f(1 ::: n):
J . (1:4:2) "! ( f x1 : : : xm . ,. (1:4:2) 7"! !. 5 xi1 : : : xik . -5 .!" !. "8 5 "! 5 . 5 f 8 " ! ;7 : f(x1 : : : xk) = x1 f(0 x2 : : : xk ) x1 f(1 x2 : : : xk): ' % 1.4.1. 5 ; ; x y z. J 0 y z = y z 1 y z = y z, " x y z = x(y z) x(y z): _
_
4) '" ( ! " (
x , . .
! " .
x
! "
22
1.
2. . (/0 % %. ! ('
( G1) f(x1 : : : xn) "! 5 f(x1 : : : xn) =
_ (1 :::n )
x(11 ) : : : x(nn ) f(1 ::: n)
! . )8 " , ." G1 f !" :; ! /" :; !! ! " , " f . ' % 1.4.2. 1 !2 :;" :; x y. J " . . (00) (01) (11), " !""!" ! G1 : x
!
y = x(0)y(0) x(0)y(1) x(1)y(1) = xy xy xy: _
_
_
_
':; x&y " ! x = y = 1 , /", ! "! ! !2 :;" . 2 :;" !" . !". - " ! !, G1 /" " ., ! . 8 !" , .!" G1 5 !" !!"" . 28 .! /" :;. J, , G1 :; x1 xn ! 5" 2n 1 /" :;. / (! ". .! ! 7 ) ;"! :;" . +(' (G1) f "! ;7 f , ;7! :; /" :;. - ". " G1 :;" f "! ., ".. f 5" " ! ;7 G1. / :;" f "! G1, ! 57 .! ! ! ! G1
f. 4." G1 5 G1 7 /" . ' % 1.4.3. 1 ; G1 . /" G1 7 7 8 !": _ _
x
!
;
y = xy xy xy = xy xy xy xy = = x(y y) y(x x) = x y: _
_
_
_
_
_
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_
_
J !7!" !" " , " . , ." ; G1 ! 5" ! . J , x y " G1 . 3. 1 /0 % %. G ; 8 x = (x1 : : : xn) ; 8 = (1 : : : n) :; _
d (x) = x(11 ) : : : x(nn ) !" 1 : : : n x1 : : : xn "! ) (' ", !! ! = (1 : : : n). A! ! i , " :; d (x) "! . G d (x) ! !"2 _
_
(
d () = 0 ! | " 5 1 !" !.: 4!" f(x1 : : : xn) | . 4 !" " f !2 :;" : f(x1 : : : xn) =
_
=(1 :::n )
x(11 ) : : : x(nn ) f(1 ::: n)
1.4. #$ $ % !
23
8 :; "! ! " , ." f( ) = 0. - " " .!" !" !"!", , ." f ! !" f(x1 : : : xn) =
_ ( ) 1
x1
: : : x(nn ) =
^ ( ) 1
x1
_
: : : x(nn ) _
(1.4.3)
" :; :; "! ! " = (1 : : : n), " f( ) = 0. , !"7 .!" !" (1:4:3), "! ! ' ( '1) f(x1 : : : xn). ' % 1.4.4. 1 !2 :;" !" 4! :; x y. J !" 4! " . (01) (10) (11), " !""!" ! (1:4:3) : x y = (x(0) y(1) )(x(1) y(0) )(x(1) y(1) ) = (x y)(x y)(x y): _
#
_
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_
_
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G:; x y ; " ! x = y = 0, /" ! "! ! !2 :;" . 0 ' ('1) f "! ;7 f , ;7! :; /" :;. ' !. :;" '1 "! , ! ! 5" .! ! ! '1 f. ' % 1.4.5. 1 ; '1 !" 4!. /" '1. )8 " , ." _
x y = (x y)(x y)(x y) = (x y yx)(x y) = x y: #
_
_
_
_
_
J !" 4! !7!" !" " , " . , ." ; '1 ! 5" ! . J , x y " '1 !" 4!. 4. $2 % %. G ; 8 x B 8 ( !" !" ;) "! 5) 2
x
(
= 1 = 0M x = 1:
G ; 8 x = (x1 : : : xn) ; 8 = (1 : : : n) p (x) = x1 1 : : : xnn
(1.4.4)
!" 1 : : : n x1 : : : xn "! ., !! ! = (1 : : : n). -! /"8 "! (1:4:4) ."! deg p (x). . " x1 : : : xn !"!" ;"! !! | . (1:4:4) "! . . G . . p (x) "5 ! " . pjj (x) !" pjj , ! "!" " " !" !!" .. G p (x) ! !"2 j
j
(
p () = 1 ! 0 !" !.:
5) $ . . 21
(1.4.5)
24
1.
' % 1.4.6. 15 ! !! ! /" . " : = (0 0) = (0 1) = (1 0) = (1 1)
j
j
j
j
j
j
j
j
= 0 = 1 = 2 = 3
p0 = x01x02 = 1 1 = 1M p1 = x01x12 = 1 x2 = x2M p2 = x11x02 = x1 1 = x1M p3 = x11x12 = x2 x1 :
)8 " , ." deg p0 = 0, deg p1 = deg p2 = 1 deg p3 = 2. )% 1.4.2. 0 " f(x1 : : : xn) (
f(x1 ::: xn) =
M
=(1 :::n)
x1 1 : : : xnn p
(1.4.6)
p 2 f0 1g. 1 (1:4:6) ( " 2 .
f
. 45 !8 5, ." 5 "! . 8. R8. /" " " "8. 4 5, ." "; ; f(x1 : : : xn) ;" . 8. R8 h1 h2 . J8 h1 (x) h2 (x) = f(x) f(x) = 0. 8 !" , " h1 = h2 , " 8. h1 h2 ! 5" " .. " , "5 !" "! 8. h1 h2 . - 8. h1 h2 . !" , ! " . ! , " ; . 4!" p | .. )8 " , ." ! . p 5" 8. h1 h2 , " , /" ! . I (1:4:5) ! ", ." p ( ) = 1 p ( ) = 0 ; 8 28 !8 ! . 4/", . 8. h1 h2 ; 2 ! ! . 4".. " , 5 "! . 8. R8. G ." 8 " !" " !"! " .! . 8. " n . J 5 ., !7 " n ( " !7!"), . "! n, " . , ." .! . 2n . '5 . n", " 8.. " , .! . 8. 22 , ". . 8. !" 5, ! , !7 " n . J 5 "! . 8. R8, " 5
"! ;7 8.. J . I", 5 f !" !""! 8. R8. f, ."! . deg f, "! ! !" ., 7 8. R8. ' % 1.4.7. *" /
" 8. R8 :; . G /"8 :;; x y !" 8.
6
_
x y = a bx cy dxy _
(1.4.7)
! !" /
" a b c d. 4 !" ; ; .!" (1:4:7) !" x y, ., ." a = 0. 48 x = 1, y = 0, a b = 1. 4 !" x = 0, y = 1 x = y = 1, ;", !""!", a c = 1 a b c d = 1. J , ." !" /
" . !!" ." . 2 /" !!", 8 a = 0, b = c = d = 1. " , x y = x y xy.
_
1.4. #$ $ % !
25
4 G1 f, G1 :;
! & . I !!"8 !" ; . /" :;, !7 " " 5 , 8 . M f(x1 : : : xn) = x(11 ) : : : x(nn ) f(1 ::: n) (1.4.8) f
g
(1 :::n)
" " !. "! , . G1 8. R8. 6. - ;. 8 !!" ! !" , ;7 . . ' % 1.4.8. G u B n 5 Su] = Pni=1 ui2i;1. ! x = (x1 : : : xn) y = (y1 : : : yn ) ; 2
(
fn (x1 : : : xn y1 : : : yn) = 0 ! Sx] Sy] 1 ! Sx] > Sy]: 1" " , ." f1 (x1 y1) = x1y1 , n > 1 fn , ! "! " ! ;7 !: xn < yn M xn = yn fn;1(x1 : : : xn;1 y1 : : : yn;1) = 1. I /" ! fn 8 ."! " : fn (x1 : : : xn y1 : : : yn ) = xn yn (xn yn )fn;1 (x1 : : : xn;1 y1 : : : yn;1): (1.4.9) I! (1:4:9) F1(x1 y1 ) = x1 y1 , ;7; ; f1 , ! " 2 F2 F3, " ;" ! ;7 ! f2 f3 : F2(x1 x2 y1 y2) = x2 y2 (x2 y2 )F1(x1 y1) = x2 y2 (x2 y2 ))x1 y1 M F3(x1 x2 x3 y1 y2 y3) = x3 y3 (x3 y3 )F2(x1 x2 y1 y2 ) = = x3 y3 (x3 y3 )(x2 y2 (x2 y2 )x1 y1 ): 08. ; . n ! fn 5 !"" ;7; Fn . 1 !5!" !" " . J !5!" F1 ", (1:4:9) ! ", ." !5!" 5 ! ;7 !; ! 7 ."! " , " n > 1 !5!" Fn l(Fn ) = 5 + l(Fn;1) = : : : = 5(n 1) + l(F1 ) = 5n 3: 2 8. "! 8 Fn . J 8 F1 , (1:4:9) ! ", ." d(Fn) = 2 + d(Fn;1), " 8 " , ." n > 1 8
Fn 2n 1. ' % 1.4.9. 1 , ;7 " s1 s2 s3 " !5 . .! x = x22+x1 y = y2 2+y1 1.2.6. G ! ! 2 !" 8" !5 .! "!" ". 1" " , ." !!" !. ! !" s1 = x1 y1 s2 = x2 y2 x1y1 s3 = 2 (x2 y2 x1y1 ): 4 !" x2, y2 x1y1 8!, ; 1.3.3 !" 20, s3 s3 = x2 y2 x2 x1y1 y2 x1y1 = x2 y2 x1y1 (x2 y2 ):
_
_
_
_
_
_
;
;
;
26
1.
' % 1.4.10. 1 , ;7 " " ." 1.2.6. 1 , ." /"" " .!" !" x y . .! x = x22 + x1 y = y2 2 + y1 . !" !""! " . r1 r2 r3 ", ." .! 2r2 + r1 ; !" x y, . r3 " !" | r3 = 0 ! x y, r3 = 1 ! x < y. r1 r2 r3 ;"! " 1.2.7 !" 13. 45 !8 ", ." r3 1.4.8. I /"8 : ;
r3(x1 x2 y1 y2) = x2 y2 (x2 y2 ))x1 y1 : _
I " 1.2.6 1.2.7 , ." r1 s1 78 . " , r1 = x1 y1 . 1 r2 . . r2 2 G1, " 7 /" :
r2 = x2x1 y2 y1 x2x1 y2 y1 x2x1 y2 y1 x2x1y 2 y 1 x2 x1y 2 y1 x2x1 y 2y1 = = x2x1 y2 (y 1 y1 ) x2 x1y2 y1 x2 x1 y 2y 1 x2x1 y 2(y 1 y1 ) = = x2y2 (x1 x1y1 ) x2 y 2(x1 y 1 x1 ) = = x2y2 (x1 y1 ) x2y 2 (y 1 x1) = x2 y2 (x1 y1 ) x2y 2 (y1 x1): _
_
_
_
_ _
_
_
_
_
_
_
_
_
J , r2 = x2y2 (x1
_
_
!
y1) x2y 2 (y1 _
!
!
_
!
x1 ).
*& 1.4.1. 5" ;:
a) x1 x2 x3 , b) 1 x1 x2 x3 , c) x1x2 x3 . 1.4.2. 4" , ." 5 " " !".!; ; . 1.4.3. 1" !2 :;" ! . 1.4.4. 1" .! . :;" 5!" x1 : : : xn. 1.4.5. 4" , ." ; G1 x1 : : : xn !!"" : a) 2n;1 /" :;M b) n2n;1 ! . 1.4.6. 1" !2 :;" ! . 1.4.7. 1" .! . :;" 5!" x1 : : : xn. 1.4.8. 4" , ." ; '1 x1 : : : xn !!"" : a) 2n;1 /" :;M b) n2n;1 ! . 1.4.9. 1" .! . !" k " n . 1.4.10. 1" .! . ." !" " n . 1.4.11. 1" .! . !" n2 " n . 1.4.12. 1" 8. R8 ! . 1.4.13. 1" 8. R8 f !: a) f = x y z, b) f = (x y) z, c) f = (x y)&(y z). 1.4.14. f(x y z t) , ! "! " " !: 1) x < yM 2) z < tM 3) .! 8x + 4y + 2z + t !"" "! " . 1 !" ! P2 (2), " " f. 4.; ! !"" . _
_
!
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#
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!
1.5. ' !
27
1.4.15. f(x y z t) , ! "! " !: 1) 2x + y 2z + tM 2) .! 4y + 2z + t | !". 1 !" ! P2(2), " " f. 4.; ! !"" . 1.4.16. f(x y z t) , ! "! " !: 1) x y z tM 2) .! x + y + z + t | .". 1 !" ! & , " " f. 4.; ! !"" . 1.4.17. f(x y z t) , ! 4x+2y+z 5t > 0. 1 !" ! & 1 , " " f. 4.; ! !"" . 1.4.18. 1 !" , ;7; .!".; ;, ; " .: a) (000 0 100 0000)M b) (11 1 111 010 0010)M c) (100 0111 1111 1 ). 1.4.19. 4" , ." ! f2n 1.4.8 5" "
, 8 " ! "".! ! " 2n.
f
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;
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1.5. # !
4!" R | 5!" . 3 5!" R "! 5!" ! , " 5 " ! R. $ 5!" R ." . SR]. *5!" R "! ( " ) ( 5!", ! ! " ! ! , ".. R = SR]. !!" " 52 " 5!" P2. >!" !!" 5 " 5!" ;"! " 5 " !!. 1. 8" , ." f(x1 : : : xn) !" , ! f(0 : : : 0) = 0: *5!", !!"7 ! !;7 , ."! . T0 . )8 " , ." 0, x, x&y, x y x y 5" T0 , 1, x, x y, x y, x y x y 5" T0 . J "5 !" !" , ; !;7 f0 f1 : : : fk ! !" _
j
#
!
f(0 ::: 0) = f0 (f1 (0 ::: 0) : : : fk (0 ::: 0)) = f0 (0 ::: 0) = 0 ".. f0 (f1 : : : fk ) f " 5 !" , " 8 " , ." 5!" T0 ". ); " 2n ! " " " .
T0 . 4/", T0 ! 5"! 22n ;1 P2 (n). *5!" T0 P2(n) ." . T0 (n). 2. 8" , ." f(x1 : : : xn) !" , ! \
f(1 : : : 1) = 1: *5!", !!"7 ! !;7 , ."! . T1 . )8 " , ." 1, x, x&y, x y, x y x y 5" T1 , 0, x, x y, x y x y 5" T1 . G" !" "!" 5!" T1 8. " !" "!" 5!" T0. J5 8 " , ." T1 ! 5"! 22n ;1 P2(n). *5!" T1 P2(n) ." . T1 (n). 3. 8" , ." f(x1 : : : xn) "! g(x1 : : : xn), ! f(x1 : : : xn) = g(x1 : : : xn): _
j
!
#
\
28
1.
; !"; f ." . f . )8 " , ." (f ) = f ; f. I !"!" ! ", ." (x&y) = x y (x y) = x&y. f "! , ! f = f . *5!", !!"7 ! ! !" , ."! . S. !" ;"! x, x, x1 x2 x3. , !7!" !7 " , " ! !" . G5 "!" 5!" ! !" . 4!" f0 f1 : : : fk | ! !" . !!" ; ; f = f0 (f1 : : : fk ). J " !"" ! !"; ; !!", " 8. 7!" 8" , ." ! fi !" " " 5 x1 : : : xn. J8 _
_
f(x1 : : : xn ) = f0 (f1 (x1 : : : xn) : : : fk (x1 : : : xn)) = = f0 (f 1 (x1 : : : xn) : : : f k (x1 : : : xn)) = = f 0 (f1 (x1 : : : xn) : : : fk (x1 : : : xn)) = f(x1 : : : xn): " , f | ! !". J , 5!" S ". J 5 ! !" " 5 " " 5 ., " ; ! !" !"". n;" . " 2n . " , S ! 5"! 1 2 2
P2(n). G 5!" ! !" , !7 " n , ." . S(n). 4. f(x1 : : : xn) "! , ! !" 8. R8 ! " , ".. f(x1 : : : xn) = 1 x1 : : : nxn 0
8 i | !" . *5!", !!"7 ! , ."! . L. ,. , ." ! P2(2) ;"! " 0, 1, x, x, x y x y. 1 ! !" ! " "!" 5!" L. J 5 . "! /
" !8 8. R8, 5 ! /
" . !" 2 ;, " 8 " , ." L ! 5"! 2n+1 P2(n). G 5!" L P2 (n) ." . L(n). 5. f(x1 : : : xn) "! , !
\
f(1 : : : n) f( 1 : : : n )
; = (1 : : : n) = ( 1 : : : n) ", ." . *5!", !!"7 ! " , ."! . M. - P2 (2) " ;"! 0, 1, x, x&y x y. G5 "!" 5!" " . 4!" f0 f1 : : : fk | " . ,. , ." " !"" "; ; ". 4/" 8. 7!" 8" , ." ! fi !" " " 5 x1 ::: xn. 4!" , | " B n , ." . !!" ; ; f = f0 (f1 : : : fk ). J fi () fi ( ), " (f1 () : : : fk ()) (f1 ( : : : fk ( )), /"
_
f() = f0 (f1 () : : : fk ()) f0 (f1 ( ) : : : fk ( )) = f( ):
" , f | ". J , 5!" M ". - ". " 5!" T0(n), T1 (n), S(n) L(n), 7!" " 8 2, ". .! " P2 (n) 2 n !". 4/"
1.6. ) ! $
29
! " !" " ! "" 8 .! " P2(n). , . 5!" " n . M(n). J8 n 2n : p log2 M(n) n n=2 2 ' % 1.5.1. 1 7!" 5!" L(n) S(n). I ! !" ! ", ." ! f L(n) S(n), " f ! !" j
j
\
2
f(x) = 0 1x1 = (0 1 x1
\
nxn = 1 0 1(x1 1) n(xn 1) = nxn) (1 1 n) = f(x ) (1 1
n ):
," ! ", ." 1 n = 1. J , " ! !" "8 " "8 , 8 !7!" !" " ."8 .! . " , L(n) S(n) = 2n.
j
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*& 1.5.1. 1" .! " P2(3). 1.5.2. 4" , ." !" " " 5 " ". 1.5.3. 1" .! " , !7 " n ;7 . . 7 . 1.5.4. 4!" f(x1 : : : xn) 5" 5!" Sx y] !7!" !" . " . G" , ." " . . . 2n;1 . 1.5.5. " 5!" !".!
" ? 1.5.6. 4" , ." log2 M(n) ;bn=n2c. 1.5.7. 4" , ." log2 M(n) ;bn=n2c log2 n. 1.5.8. 1" a) L(n) T0 (n) T1 (n) M b) L(n) S(n) T1 (n) M c) L(n) M(n) T0 (n) M d) (S(n) T0 (n)) L(n) M e) (S(n) T0 (n) T1(n)) L(n) M f) S(n) (T0 (n) L(n)) M g) S(n) L(n) T0 (n) T1 (n) M h) (L(n) S(n)) M(n) : !
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1.6. %! ! !
!" F = f1 f2 : : : fi : : : "! " , ! ; 5" " ! F. J 5 "! ! !2 :;" 8 .! , " !!" & & 1 " . G !" " 8 !!" 5 ! " ! ! ;7 ". )% 1.6.1. F G | * ", F " ( ( G. . G | . . 45, ." h 5" " ! G. /" 8 , ;7
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30
1.
H = f(H1 : : : Hk ), 8 f
F H1 : : : Hk ! F. ,. , ." d(Hi ) < d(H) 5 8 i 1 : : : k /" 5; H1 : : : Hk h1 : : : hk ;"! " 5 H01 : : : H0k ! G. 4 !; " f(y1 : : : yk ) "! F ! G. - /"
5 ; ; yi H0i . 4. H0 "
; h "! ! G. J . I " !"!" ! " " !!" & . ' % 1.6.1. 45, ." !!" x y x "! . J x y = x y, " x y = x y, ". . :; "! ! ". "., ." :; " ;" ; !!". " , ! 7 " !!" x y x "! . )% 1.6.2. + " F , * , " ( ( * T0 , T1, S , M L. . * % . 4!" !!" F P2. 4 5, ." F ! 5"! " !! R T0 T1 L S M . J8 !!" , ;. F R !" SF] = P2 ! ", ." P2 = SF] = R. 4/", R = P2. 8 !" 5 !! T0 T1 L S M ". " P2, ".. R = P2. 4".. 1 !" . +. J !!" F ! 5"! " " !! .! ! " , " /" !!" "! " ( " . ) fT0 , fT1 , fS , fM fL ", ." fT0 = T0 fT1 = T1 fS = S fM = M fL = L: A! fT0 (1 : : : 1) = 0, " fT0 (x : : : x) = x. A! fT1 (0 : : : 0) = 1, " fT1 (x : : : x) = x. A! 5 fT0 (1 : : : 1) = 1 fT1 (0 : : : 0) = 0, " fT0 (x : : : x) = 1 fT1 (x : : : x) = 0: " , ! "5 !" fT0 fT1 . (i) ", (ii) "5 !" !"" 0 1. 4! " !!" /" 5!". (i) J fS "! ! !", " "! " " 5 1 : : : k 1 : : : k , ." fS (1 : : : k) = fS (1 : : : k ): !!" ; '1(x) = fS (x(1 ) : : : x(k) ). )8 " , ." '1 (0) = fS (0(1 ) : : : 0(k) ) = fS (1 : : : k ) = = fS (1 : : : k ) = fS (1(1) : : : 1(k) ) = '1(1) ".. '1 "! !"". -" !"" ."! '1 ". (ii) J fM "! ", " "! " !! = (1 : : : k) = ( 1 : : : k ), ." fM (1 : : : k) > fM (1 : : : k ): 4 5 1 i k ; 8 > <0 ! i = i = 0M gi (x) = >1 ! i = i = 1M : x ! i = 0 i = 1: 2
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31
!!" ; '2 (x) = fM (g1 (x) : : : gk(x)). )8 " , ." '2 (0) = fM (1 : : : k) > f4 ( 1 : : : k ) = '2 (1) ". . '2 | ". J , fT0 , fT1 , fS fM . !"" ". J , !!" ; ; fL . - 8. R8 /" "! !8, ! 57 . 8. 7!" 8" , ." /" x1 x2 . K !8 , ! 57 x1, x2 x1x2, 8. R8 fL fl (x1 : : : xk ) = x1x2f0 (x3 : : : xk ) x1f1 (x3 : : : xk) x2f2 (x2 : : : xk) f3 (x2 : : : xk ) 8 f0 ". " "5 !"8 . G ! ! !" x 1 = x. 4!" (3 : : : k ) ", ." f0 (3 : : : k ) = 1. - ;
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n en) = f(v ) f(u): J . 4. !!" !"!" V 8 !"!" W. !"!" V !"!" W "! 5!" + W, !!"7 ! " + w, 8 | ! " V, w | " W. -" "! !58 !! + W. )% 2.1.3. 5 * V W : (i) M (ii) . . G !", ." !5 !! + W + W ;" 7 " . J8 W "! " w1 w2 ", ." = + w 1 = + w2 . J , = + w1 + w2, ; 8 " + w !58 !! + W + w = + (w1 + w2 + w) ".. + w + W , ! " , + W + W. 08. "! ;. + W + W. J , + W = + W. 9"5 (i) . -" "5 " 8 ! " "8, ." ! ! + w . ! . w. J . ' % 2.1.5. - !"!" P2(2), !!"7 ! , !7 " x y, , !7 " y, ;" !"!", ", . , !!"" ." : 0, 1, x x 1. , . /" !"!" . Vx. 4!" !5 !! !"!" P2(2) !"!" Vx. J P2 (2) !!"" 16 , " P2(2) " ." !5 !! Vx. 4 !5 !! " ! !"!" Vx. G "8 !58 !! .!" 8 !"" 57; Vx ; | y. J8 y Vx = y y 1 y x y x 1 . 4 !"" "" 8 !58 !! "
57 : ; . J ", , xy. J8 xy Vx = xy xy 1 xy x xy x 1 . 1 ! ."" !5 !! !!"" ." !"2! , ".. y xy 1 y xy x y xy 1 x xy .
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" fij " f. -. fij ;"! /" " (2:3:1). " m !" n !" " (m n) " F = (fij ) " m n. A! m = n, " (n n)" " " " n, !" " n. - .!"!", 8 " , ." " "5 !"8 " " " " En = (eij ), /" " "! !"
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41
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+ * ) * * " , . . A ? = B ? .
A B *
. ,. , ." !" " 8 " 8 "!. 4/" ! 8 !" (2:3:2) !" !" " " "8 !"!". " , " !" " !"". " , ." " /" !" "5 " "8 8 !"!". !!" !"!" V, 5 " v1 : : : vk , !"!" V0, 5 " v01 : : : v0k . 8" , ." " !!" . i-8 " j-, ". . v0t = vt t = j v0j = vi vj . )8 " , ." /" !. vj = v0i v0j . G5 !" V? = (V0)? . G /"8 !!" " u !"!" V?. ,. , ." (u vt ) = 0 5 8 " vt !!" . 4/", 0 = (u vj ) (u vi) = (u vj vi ) = (u v0j ): " , u (V0)? , , " , !"!" V? ! 5"! !"!" (V0)? . 8 !" , ! u (V)? , " (u v0t) = 0 5 8 " v0t " !!" . 4/", 0 = (u v0j ) (u v0i) = (u v0j v0i ) = (u vj ): " , u V?, !"!" (V0)? ! 5"! !"!" V?. J , V? = (V0)? . J . )% 2.3.3. + (m n)- " A
dim A = dim A T : (2.3.3) 8 (2:3:3) ( " A ( ( rank A. . 4!" dim A = k. 8. 7!" 8" , ." k !" " A ! , !" m k !" " ;"! . 4 7 /" !" " A /"; " B, " " k !" " . G /"8 5 ! m k !" " A " ; ; ; k !". - ! 7 " A ? = B ? . J ! " !" " B !! ". 6
2
2
2
;
;
2.4. $ %
43
k !", " 8 " , ." !"!" !" " B " " !"!" !"!" B k . 4/", dim B T k. J , 7 !!5 " 2.3.1
dim A T = n dim A ? = n dim B ? = dim B T ;
;
k = dim A :
" , dim A T dim A . 4 !!5 " AT , 8 . !" dim A T dim A . J .
*& 2.3.1. 4" , ." rank AB min(rank A rank B) ; " "
8 . 2.3.2. ' " " A = (aij ) "! "8 , ! aij = 1 i j aij = 0 i > j. ' " " B = (bij ) "! 5"8 , ! bij = 1 i j aij = 0 i < j. 1" rank AB, ! A | "8 , B | 5"8 " n. 2.3.3. 1" .! (m n)-" 8 k, ! k m n.
2.4. ( !
- /" 8 ; det | " !!" " u1 u2 : : : un ; ! " ! , ; ; ! " ! . G /"8 " ;"! " . - !" !, " 5 "" 5 " . - " 5, ." !, ! , "" !" , /" 5 .!;7; . 1, "" 5, ." " !"" ! " ; ! . 45 !8 . m 8"
2
Bn,
f : B| n :{z: :
m
Bn ! B }
"! i- 8", ! !" f(u1 : : :ui;1 u0i u00i ui+1 : : : um ) = (2.4.1) = f(u1 : : : ui;1 u0i ui+1 : : : um ) f(u 1 : : : ui;1 u00i ui+1 : : : um )
! ! uj u0i u00i B n . %% 2.4.1. " det : (B n )n
B n " ( , , ( . . det , det
*. !
. 4 "" !!" . . ".!" !" "8, ." !!" !!" !" " !" " !!". J" !!" ! " !!" ; !!" ! ". 1 !" " 5 8" . . 45, ." det 5 " ! 8". )!" 8 8" "! 8.. 4!" u1 u2 : : : un u0n B n . !!" " !!" " U = u1 u2 : : : un , U 0 = u1 u2 : : : u0n U 00 = u1 u2 : : : un u0n , ".;7! " ! ". 45, ." ! 2
f
f
g
g
f
g
44
2. ,! $
/" !!" ! " , ! " " ! . G8 !, ! - !!" ! ! , " "" !!" " " !. 1 8. 7!" !!5 , !!" !.: (i) !!" U U 0 ! M (ii) !!" U U 0 ! . - !. "! " w w0 B n , ." 2
n;1 M i=1
wi ui
;1 n M
wnun = 0
i=1
wi0 ui
wn0 u0n = 0
A! " .! wn wn0 ;, " . , ." /" !. U 00 "5 !. A! wn = wn0 = 1, " "8 ;1 n M i=1
(w wi0 )ui
(un u0n ) = 0:
" , U 00 !. - " !. !!" !!" U U 0 ! " !7!" " w w0 B n;1 , ." 2
;1 n M i=1
wi u i
;1 n M
un = u0n
i=1
wi0 ui
u0n = un :
4 !" ", 5 !, ." !!" U 00 !. J " " , ." det ! ! 8". J ! !!" U, U 0 U 00 ! ." .!, " ! !" det(u1 : : : un) det(u1 : : : u0n) det(u1 : : : un u0n) = 0 ", 8 " , /" !" (2:4:1) i = n. ) . - ! ;7 !""!, ." !7!"" !" , ";7 ! 2 " !. %% 2.4.2. det : (B n )n B n | ,
!
", *. . ( * f i = (ui1 : : : uin)g
u
det(u1 : : : un) =
M 1 j16=6=jn n
u1j1 : : : unjn :
(2.4.2)
. J det | ! ! 8" , " !" u1 ! ! ", n M
det(u1 : : : un ) = det
j1 =1
u1j1 ej1 u2 : : : un =
n M
j1 =1
u1j1 det(ej1 u2 : : : un):
- 8. ! " u2 : : : un, . det(u1 : : : un ) =
n M
u1j1 det ej1
j1 =1 n M n M
n M
j2 =1
u2j2 ej2 u3 : : : un =
u1j1 u2j2 det(ej1 ej2 u3 : : : un ) = : : : j1 =1 j2 =1 n n M M = u1j1 : : : unjn det(ej1 : : : ejn ): j1 =1 jn =1 =
(2.4.3)
2.4. $ %
45
J det | !".! , ;7 . . . ! ", ; ! ! 8" "! , " ! 8 !" ., ." det(u1 : : : un ) =
M
1 j1 6=6=jn n
u1j1 : : : unjn :
) . G 5, ." ! 2 ! ! ! !!" " ;"! "5 !"". , ". . !" , ";7 /" !, !"" ; ! !!" ; ; ! !!". %% 2.4.3. u1 : : : un B n . . " det, (2:4:2),
2
(
det(u1 : : : un ) = 1 u1 : : : un ( M 0 u1 : : : un ( : . 48, ." i1 : : : in | 5" ! ;7 ! 5 " ! " n. . 5, ." det(u1 : : : un) = 0, ! " u1 : : : un ! . I !!" !!" " ! ", ." "! " ; !" wi , ." n wi ui = 0. G !", ." w1 = 1. J8 u1 = n wi ui i=0 i=2
det(u1 : : : un ) = det
n M i=2
wi ui u2 : : : un
n M i=2
wi det(ui u2 : : : un):
'5 " , 7 ! ;; !, " 8". " , ! ! ;. J , , ." ! " u1 : : : un ! , " . " /" " ;. 4!" " u1 : : : un | ! " B n . 45, ." det(u1 : : : un ) = 1. J !!" " ;" B n !, " "! " !" wij , ." n M ei = wij uj : j =1
- .! det(e1 : : : en). - , 8. (2:4:3), ., ." det(e1 : : : en) =
M
1 j16=6=jn n
w1j1 : : : wnjn det(u1 : : : un ):
A! det(u1 : : : un) = 0, " . det(e1 : : : en ) = 0. 4".. " , det(u1 : : : un) = 1. ) .
*& 2.4.1. , " " " A "! " !". 4" , ." det A = 1 "8 " "8 , 8 A 5 . 2.4.2.0 1" det A, !: 1 0 1 0 1 :::::: 1 C a) A = 1: : : :0: : :1: : ::::::: : :1:C A, 1 :::: 1 0 B B @
1 1 0 :::::: 0 C b) A = :0: ::1: ::1: :: :0:: ::::: :: ::0:C A. 1 0 :::: 0 1 B B @
46
2. ,! $
2.4.3. 4" , ." det AB = det A det B ; " " A 8 . 2.4.4. 4" , ." det A = det AT ; " " A.
B
3.
- /" 8 !!";"! ! !"" 8 !"!" 2 . ., ;7 " , " " !5 !". 3.1. !
1. !!" !!" m 8 > > > <
a11x1 a12 x2 : : : a1n xn = b1 a21x1 a22 x2 : : : a1n xn = b2 > : : : : : :: : :: : : : : :: : :: : :: : :: : :: : :: : : > > : am1 x1 am2 x2 amn xn = bm
(3.1.1)
! /
" aij , ! . bi c n !" xi. H" !!" 5 !" ".8 0
10
1
0
1
a11 a12 a1 n x 1 b1 B a21 a22 C B C B C a x 2n C B 2 C B B b2 C B . CB . C = B . C . . . .. . . .. A @ .. A @ .. A @ .. am1 am2 amn xn bm
(3.1.2)
! (m n)-" A = (aij ), !!" /
" !!" (3:1:1), "-!" ! . b = (bi) "-!" !" x = (xi ). *". "! , ! " " 2. G ." 2 (3:1:2), , ! " , 2 !!" (3:1:1). 45 !8 !, !. (3:1:2) !8!. G /"8 " A " b !!"8 !!" ; " (A b), " A .!" 8 !" " b " 8 /" " A " . 4.2! " j
0
1
a11 a12 a1n b1 B a21 a a2n b2 C 22 B C : B . . . .. . . ... ... C @ .. A am1 am2 amn bm
"! ! " (3:1:2) ( !!" (3:1:1)). I" !" ! ;7 " !8!!" ".8 . 47
48
3. ,! $
)% 3.1.1.
Ax b A A Ab . A! rank A = rank (A b), " " b "! !" a1 : : : an " A, ".. 9 = (m n)- " ! , " ! " ( j ). j
b = 1a1
nan : (3.1.3) J !" (3:1:3) /" ". !" A = b, " . , ." " = (1 : : : n) " 2 Ax = b. 8 !" , ! "-!" = (1 : : : n) "! 2 Ax = b, ", . , ! !" (3:1:3), "8 ! " !" 8 " A (A b). J . )% 3.1.2. : Ax = b (m n)- " A *
j
! , ! ) ( B n " .
A
. G !", ." Ax = b " 2 x0 . - /" !. " !" " !"". " , ." ; ", 57 ! " x0 !5 !! !"!" B n "8 !"!" " A, "! 2 Ax = b, ", ; 2 !!"8 5" " 5 !5 !! !"!" B n "8 !"!" " A, ." " x0. !!" "8 !"!" A ? " A. G 5 8 " v !"!" A ? ! !" A(x0 v) = Ax0 Av = b 0 = b: " , " x0 v "! 2 Ax = b. 8 !" , ! y | 2 !!"8 , " A(y x0) = Ay Ax0 = b b = 0: 4/", y x0 A ? . " , y 5" " 5 !5 !! !"!" B n A ? , ." " x0 . J . I " ! ", ." 5 ! 2 Ax = b !"". 2" .: (1) " " 2 x0 /"8 , " 2 "! M (2) " "8 !"!" " A. H" . .;"! ! ;7 ". 2. *" A 5 8 8 " A "! ". J !" 8 5 8 (n n)- " n, ! "! . !" " /"8 ", " 8 " , ." !" 5 " ! . '5 5 " n " !"; " A;1 ";, ." A;1A = AA;1 = En: 1" " , ." " En "! !" " n ", ." ; " A "8 5 ! !" AEn = A En A = A. " , ! " " A B 8 n "! " !" AB = A BA = A, " B = En . ,"!; !; . 8 ! ", ." ! " " A B " , ." AB = En BA = En, " A;1 = B. 1" " , ." 7 /" !" ; ; 5 ; " n 5 " .; " "8 5 . 4 !""!";7 8", !!"7 n 28. 4! i-8 28 8" i !" ! " " i !" . " .
2
3.1. # ! !
49
!!" ; 5 ; " 0 1 a11 a12 a13 a1 n B a21 a22 a23 a2 n C B C B A = Ba31 a32 a33 a3 n C (3.1.4) C: @: : : : : : : : : : : : : : : : : : :A an1 an2 an3 ann J " 5 , " !" "! " . /". - " A !" !" ", ." ! !" /" !" !" . $" ! !", " /" , ; !". - "" . ; " 0 1 1 a012 a013 a01n 0 B 0 a0 a02n C 22 a23 B C 0 0 0 B 0 a a a03n C A = B 32 33 C @ : : : : : : : : : : : : : : : : :A 0 a0n2 a0n3 a0nn " !" !"."! " | !". ,"", ." ! /" a022 a032 : : : a0n2 "! " . A! /" ", " " !" " A0 " ! " ! !" , !!"" , , ! " , " A0 " 5 . 4!" " /" " a0i2. - " A0 !" "; i-; !". $" ! !", " " /" , "; !". - "" . ; " 0 1 1 0 a013 a01n B0 1 a0 a02n C 23 B C 00 0 B a03n C A = B0 0 a33 C @ : : : : : : : : : : : : : : :A 0 0 a0n3 a0nn " " !" !"."! " | " !". - "", ." ! /" a033 a043 : : : a0n3 "! " . A! /" ", " "" !" " A00 " 5" . 8 "8 !" /" " . )8 " , .", !; !"2! n 2 !" , 5 " " A .; " n. J , 5 5 " /" . " "8 5 . J ", ." 8 /"8 " ! "! 5; /" " ! .; " ! !" i- j- !", "8 /"8 | 5; ! .; " ! " , !"7 !. j- !" i-8 !" . 1 , !" ."" !" 5"8 " .""8 0 10 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 B 0 1 0 0C B 0 0 1 1C B0 0 1 1C B CB C B C @0 0 1 0A @0 1 1 1A = @0 1 1 1A 1 0 0 0 1 1 1 1 0 0 0 1 !" /" " ."" !" 1 1 0 0 10 1 1 1 0 1 0 0 1 0 0 0 1 B0 1 0 0C B0 0 1 1C B0 0 1 1C C C B B CB @0 0 1 0A @0 1 1 1A = @0 1 1 1A : 1 1 1 1 0 0 0 1 1 1 1 1
;
50
3. ,! $
J , 5 5 " A "! " B, ;7! " !""!";7 /" , ", ." BA = E. " , A;1 = B. I !8 2 ! " !" ! ! 7 5 " A. . /" " A "! .; ", , ." /", " A 5"! ! "; " B ";, ." BA = E. $" 8. . ", ".. 5 .; " " 5 " B , ! " , BE = B. ,. , ." ", .2! . " , " " " A. ' % 3.1.1. 4 ! 2 8" 7 5 " " 001 011 111 . H" 7 . " " , /" " " . )8 " ,." ! ! ;7 !"2: 0 1 0 1 0 0 1 1 0 0 1 1 1 0 0 1 @0 1 1 0 1 0A @0 1 1 0 1 0A 1 1 1 0 0 1 0 0 11 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 @0 1 1 0 1 0A @0 1 0 1 1 0A : 0 0 11 0 0 0 0 11 0 0
;1
J , 001 011 111 = 011 110 100 : 4 8" 7 5 " " 8 2" Ax = b ! 5 " A. G 2 /"8 !"". "" " A 5" ; " A;1 " b. G!"" , 5 " A;1 ; ; .!" !!"8 A;1b = A;1Ax = Ex = x: " , 2 Ax = b ! 5 " A "! " A;1 b. !!" !" . ' % 3.1.2. 2 ". 0 10 1 0 1 0 0 1 x1 1 @0 1 1A @x2A = @0A : 1 1 1 x3 1 *" /"8 7 7 . I! 8 "", 0 1 0 10 1 0 1 0 1 1 1 1 x1 @x2 A = @1 1 0A @0A = @1A : x3 1 0 0 1 1 J , 2 !!"8 "! "-!" (111). $", ." 2 Ax = b ! 5 " A " /" " 7" . G!"". " " b !", ;7 " A .; ". ' % 3.1.3. 2 Ax = b, " " A " 5 7 , " b (110). H" " A " b " , " b !" " A. - " 5 3.1.1, : 1 1 1 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 1 1 0 0 1 @0 1 1 1A @0 1 1 1A @0 1 1 1A @0 1 0 0A : 1 1 1 0 0 0 1 1 0 0 11 0 0 11
3.1. # ! !
51
" , x = (101). 3. - /" ! ;7 !!"" ". , " 8 " 2 .! !" , ".. 2 .! ! !" . 8" , ." (m n)-" A A0 -) , ( ."! . A c A0 ), ! " A0 . " A 7 /" !", !" !" !". 45, ." 5 (m n)-" A 8 k (k m n) "-/" " "
0
1
1 0 : : : 0 0 a01 k+1 a01n 0 0 B A0 = B@0: : :1: : ::: :: : :0: : 0: : :a:2 k: +1: : : : : : : a: 2:n:CCA 0 0 : : : 0 1 a0k k+1 a0mn
(3.1.5)
!!"7 | . " m ! ;7 " m (n m). *" (3:1:5) "! ". . !!" (m n)-" A 8 m " m !" ! . *" Am , !!" m !" " A, " 5 , , ! " , "! ! " !" /" R, 7 " Am .; " m. )8 " , ." /" 5 ! " !" R " " A /"; !!"".!; " (3:1:5). G !" " , ." 8 (m n)-" A k, k < m, k !" ! . J8 5 !"2! !" "! k !" . - /" !. "! ! " !" /" R, 7 " Ak , !!"; k !" " A, " A0k , !!"7; | . " k 7! (m k k)-" . 4 " A 7 ! " !" R /"; " A0 . J ! " A 5 ! n k !" "! k !" , " " A0 5 ! n k !" " k !" . 4/", " A0 k !" ;" !!"".!; (k n)-", ! (m k) !" !!"" " . 4! !" " A0 !"" !!"".!. 1 ", ." ; (m n)-" B 8 k 5" " . !" !" 7 (m n)-" A 8 k ! k ! !" . " , ; (m n)-" 8 k "/" " !!"".! (k n)-". G " "-/"; !!"".!; " ! " ; 8 7 8" 5 " .;. ,". " ! 8 8" !!"" " ", ." . 8 28 8" " 5" " !", 8 /" 28 !" " 7 !" . - /" !. !" !" !"" 5" 8" ! ! ;7 !" . $."! 8" !" (! " !" "!) !" 7 !" " . !!" 8" " "-/"; !!"".!; " ! ;7 !" . ' % 3.1.4. 4 ; " A "-/"; !!"".!; " A0 . 4 /"8 ! 2 8", ! " . " 5 ". 4 ." " /" " A, ! , ;7! !!"".!, "-/
;
;
;
;
;
52
3. ,! $
":
0
1
1 B1 A =B @0 1
0
1
0
1
1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 B0 1 1 0 1 1C B 0 1 1 0 1 1C 0 1 1 0 1C C B C B C 1 1 0 1 1A @0 1 1 0 1 1A @0 0 0 0 0 0A 0 1 0 1 1 0 1 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 0 0 1 1 1 B0 1 1 0 1 1C B0 1 1 0 1 1C c B C B C @0 1 0 1 1 1A : @0 0 0 1 1 0A @0 0 0 1 1 0A 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 28 8" !" " A " "" !". 1 " 28 " !" " !" !". 1 "" 28 8" !" !" "" ."" !" " . )8 " , ." . " "" !" ! !" . 4/" ."" 28 "" !" !" , ."" !" | "" !" " . 1 ! 28 !" "" ."" !" , ! !", !!"7 " , . 4. "" " "! !!"".!. 4. 4 8", " " ! "8 8 !"!" (m n)-" . . !!" .!" !. | 2 8" 2;7 !!"; . !!"".! " . G /"8 !!"".! " A = Em Ae !" !""!" (n m n)-" A0 = Ae T En;m , " !!"" | "! " Ae ! ;7 . " (n m). )8 " , ." AA0T ! !":
;
;
T
Ae
E A A En;m = E A En;m = Ae Ae = 0: J , !"!" A 0 , 5 !" " A0 , " "8 !"!" " A. J !" /"8 !"!" n m, !" !"!" !" " A m, ".. dim A 0 + dim A = n, " " 2.3.1 8 ! ", ." A ? = A 0 . J , .!" ! "8 8 !"!" " A 5 " !" " A0. 4 8" . "! 8" 5 ! "8 8 !"!" " . *" A " "-/"; !!"".!; " A0 . 4 /" ! " " ! !! !" !" . $" " A0 ! 2 ! ! !""! " B0 , !" " 5 ;" "8 !"!" " A0 . 1 " B0 ! !" !" 7;"! ! !". 4. " B " 5 " "8 !"!" ! " A. ' % 3.1.5. G " A !!"8 2 3.1.4 !" ", 5 ;7; "8 !"!". . " A "-/"; !!"".!; " A0. I 3.1.4 : me
eT
me
;
0
1
0 1 1 1 0 1 1 0 1 0 0 1 1 1 B1 0 1 1 0 1C A = B@0 1 1 0 1 1CA A0 = @0 1 0 1 1 1A : 0 0 1 0 1 0 1 0 1 0 1 1 4 A A0 !"! "" ."" !" . G " A0 8 5 ;7; "8 !"!" " B0: "!
3.1. # ! !
53
" 110 111 110 . " 111 111 010 , " ! ! .; " "" 8 . $" " B0 "; !" !" | ! !" "" ."" !" . "" : 0
1
1 1 0 1 0 0 B0 = @1 1 1 0 1 0A 1 1 0 0 0 1
0
1
1 1 1 0 0 0 B = @1 1 0 1 1 0A : 1 1 0 0 0 1
1 ! !" 8 " ! ", ." AB = 0. 5. J 8" 5 .!"8 2 Ax = b. >!" 2 ! !!"".! " A = (Em B) "! 8. 1 , !"
(Em B) 0b = En b
B0 = b
! ", ." "-!" (b 0) " 2 !!"8 Ax = b. G !!" ! !!"".! (m n)-" A 8 m, " m !" ! . )8 " , ." 7 " /" !" " A 5" " /"; !!"".!; " B. (!". 49), ." ; 8 /"8 !" " " ! "! 5; /" " ! "; " 8 . 08. "5 ! " . 1 , 8 " , ." 5 (m n)-" A ! "; " m, .;7;! . " "8 5 !. i- !" j-8 !" , !""!"" ; i- !" " A j- !". " , ; (m n)-" A 8 m, " m !" ! , "! " " C m 5 "; ! " " A /"; !!"".!; " B. J
Bx = CAx = Cb " !" " A " " b /" , 7 A B, " Ax = b Bx = Cb, 2 "8 ! ;" ! 2 ! 8 . J " B !!"".!, " "-!" (b0 0), 8 b0 = Cb, " .!" 2 Ax = b. ,. 8" 5 ! " 2 ! ". ' % 3.1.6. 1 ! 2 0
1 B1 B @0 1
1 0 1 0
0 1 1 1
1 1 0 0
1 0 1 1
0
1
1 0 1 0 Bx1 C 1 x B1C 2C 1C B C C . C=B @0A : 1A B @ .. A 1 x6 0
(3.1.6)
2 8 ! " 5 " (3:1:6) 7 . 4/" ! 2 (3:1:6) !"! " - 8 .!" 2. /" ! 2 ! ! . H" !" " " " " , " /
" .!" " 8 !" " ! .. )8 " , ." ! ;7 8. -
54
3. ,! $
78 : 1 0 0 1 1 0 1 1 0 1 1 1 B1 0 1 1 0 1 1 C B0 1 B C B @0 1 1 0 1 1 0 A @0 1 1 0 1 0 1 1 0 0 1 0 1 0 B0 1 B @0 0 0 0 J , .
0 1 1 1 1 1 0 0
1 0 0 1 1 0 1 0
1 1 1 0 0 1 1 0
0 1 1 1 1 1 0 0
10
1
1 0C C 0A 1 1 1 0C C 1A 0 1
0
1 B0 B @0 0 0 1 B0 B @0 0
0 1 0 0 0 1 0 0
1 1 0 0 1 1 0 0
1 0 0 1 0 0 1 0
0 1 0 1 1 1 1 0
1 1 0 0 1 1 0 0
1
1 0C C 0A 1 1 0 0C C 1 A: 0
1 Bx1C 0 x B0C 2C 1C B C C .C=B @1A 0A B @ .. A 0 x6 0 .!" 2 "8 "! 8. - " /"8 ."" !" ", !"7 8 .!". 4/" 8 " , ." " (000100) " .!" 2 ! 8 , , ! " , (3:1:6). 1 5 2 /" "! ! 8 .!"8 2 "8 " "8 8 !"!" " (3:1:6), ".. !""! ! (000100) 1 (111000) 2 (110110) 3 (110001) 8 1 , 2 3 | !"" . 0
1 B0 B @0 0
0 1 0 0
1 1 0 0
0 0 1 0
1 1 1 0
0 1
*& 3.1.1. 4!" A, B | " n ", ." AB = En. 4" , ." BA = En. 3.1.2. G" , ." 5 5 " " " -
!"; "; ". 3.1.3. 4" , ." " " 5 "8 " "8 , 8 !" ! . 3.1.4. 1" .! . 5 " n. 3.1.5. 4!" m n. 1" .! . (m n)-" 8 m. 3.1.6. 1" 8 (2n 2n)-" An, ! A0 = (1), W 2 n " !" "
: An AnM c) An+1 = An An. n M a) An+1 = A0n A b) A n +1 = An An An An An $ ! . 0 . ", ! /" " ;, . A | " .;7! A " ! /". 3.1.7.0 1" !120 Ax =0b !: 1 1 0 1 1 1 0 0 1 1 1 0 1 1 0 B0 1 1 0C B1C B 1 0 1 0 1C B0C C B C C B C a) A = B b) A = B @0 0 1 1A, b = @0AM @0 1 1 1 1A, b = @0A 1 0 0 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 B1 0 1 1C B0 C B0 1 1 0 0C B1C C B C C B C c) A = B d) A = B @1 1 0 1A, b = @0AM @0 0 1 1 0A, b = @1A. 1 1 1 0 1 0 0 0 1 1 1
3.2. .+ / 0*
55
3.2. ) * + ! ,-
1. 4!" v1 : : : vn v01 : : : v0n ! B n . J8 5 " u
2
Bn
"! " /" !, ". . u = u1v1 u2v2 unvn = u01v 01 u02v02 u0nv0n : (3.2.1) !!" . " " u B n ! v1 : : : vn , ! !" 8 " ! v01 : : : v0n. G !", ." " v0i "8 ! 5;"! . " 8 ! ! ;7 : v01 = 11v1 12v2 1nvn 0 v2 = 21v1 22v2 2nvn (3.2.2) : : :: : :: : :: : :: : :: : :: : :: : :: : :: : :: : :: : :: : : v0n = n1v1 n2v 2 nnvn: 4 !" /" ; .!" !" (3:2:1). J8
2
u=
n M i=1
u0i v0i =
n M i=1
u0i
n M
j =1
ij vj =
n M n M j =1 i=1
ij u0i vj =
n M n M j =1 i=1
ij u0i vj :
" ,
u1 = 11u01 21u02 n1u0n u2 = 12u01 22u02 n2u0n : : :: : : : : :: : :: : : : : :: : :: : : : : :: : :: : :: : :: : : un = 1nu01 2nu02 nnu0n: 4! !" 5 !" ". 0 10 1 0 1 11 21 n1 u01 u1 0 B 12 22 C B C B u n2 C B 2 C B u2 C B C (3.2.3) B . B . C = B . C: . . .. . . ... C @ .. A @ .. A @ .. A 1n 2n nn u0n un *" (3:2:3) "! " " ! v01 : : : v0n ! v1 : : : vn. A i- !" !!"" " " v0i ! v1 : : : vn. J , " " u ! v1 : : : vn 8 !" " ! v01 : : : v0n " " 5 8 " v0i !, " !!"" " , 5" " u. 2. -! ! 2 " " " ! .! /
" 8. R8
. !!" 2n- !"!" B 2n , !!"7 " . ! , !7 " n . G "!" 8" " . , !" . - !"!" B 2n !"!" ! Kn Pn. ! Kn = k0 : : : k2n ;1 !!"" ! 2n /" :; kjvj = x(1v1) : : : x(nvn) x1 : : : xn, ! Pn = p0 : : : p2n;1 | ! 2n . pjvj = xv11 : : : xvnn " 5 . - .!" !!" !"!" B 2 B 4 . 4 !"!" !!"" ." , !7 " x, " | 16 , !7 " x1 x2. - !"!" B 2 ! K1 !!"" x ": k0 = x(0) = x = (1 0) k1 = x(1) = x = (0 1)M
f
f
g
g
56
3. ,! $
! P1 | "5 !" x: p0 = x0 = 1 = (1 1) p2 = x1 = x = (0 1): - !"!" B 4 ! K2 P2 !!"" ! ;7 : p0 = 1 1 = (1 1 1 1)M k0 = x1 x2 = (1 0 0 0)M k1 = x1 x2 = (0 1 0 0)M p1 = 1 x2 = (0 1 0 1)M k2 = x1 x2 = (0 0 1 0)M p2 = x1 1 = (0 0 1 1)M k3 = x1 x2 = (0 0 0 1)M p3 = x1x2 = (0 0 0 1): , . . Pm " " ! Pm ! Km . 1" " , ." " P1 P2 ! !": 0 1 1 0 0 0 P1 = 11 01 P2 = BB@11 10 01 00CCA : 1 1 1 1 /" " ", ." P2 !!"" ." | " " P1 " , !!"7 . 45, ." 8. !" ! " Pn. G /"8 !!" " Tn , !!"7; 2n !" 2n !" , .! " 2n 1. 4!" u v B n . - " Tn u - !" !" !""!" u, v - !" | . pv = xv11 xvnn . 1 !. u - !", !""!";7 (u1 : : : un), v -8 !" , !""!";78 . xv11 xvnn , !" . . xv11 xvnn u1 : : : un. 1 " T1 T2 8 " ! ;7 : x1 x2 1 x2 x1 x1x2 x1 1 x1 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 1 0 1 0 1 1 1 1 1 1 J p(0v2:::vn ) (x1 x2 : : : xn) = p(v2 :::vn) (x2 : : :xn), ; u v B n ! !"
;
j
2
j
j
j
j
j
j
j
uv11 uv22 uvnn
(
v2 = u2 0
uvnn
! v1 u1 ! v1 > u1
" 7 !. (!. " 3.2.1, 8 u = (u2 : : : un), v = (v2 : : : vn), , ! " , . pv !" " x1 ) " Tn !"!" ! "! ." 2n;1 2n;1, " " ! ;" ! " Tn;1 , ."" , !" u1 = 0, v1 = 1, !!"" . I !" (1:4:8) ! ", ." ; f P2(n) " . "! " " ! /" :;. 4/" !" 2 " Tn , " j- !" !!"" " " pj ! Kn , ".! "! " " ! Pn ! Kn . " , 5 " Pn, n 2, ! " !"
2
Pn = PPnn;;11 Pn0;1
(3.2.4)
;7 !"". !" " " " ! . ! /" :;. G " " ! /" :; ! . "" " Pn .
3.2. .+ / 0*
x1 x2 : : :
0 0
x
n
J 3.2.1.
0
0
u2
u
n
pv
x1
pv
pv (u)
0
pv (u)
pv (u)
57
0 1 1 0
1 0
1
u2
u
n
1 1
%% 3.2.1.
1
+ "
Pn, n 1,
P;n 1 = Pn:
(3.2.5) . ) 5 n. 4 n = 1 "5 8 ! " . 8 !" 1 0 1 0 1 0 P 1 P1 = 1 1 1 1 = 0 1 : 4 5, ." "5 ! ! " n !7 " m. 4!" Ek | . " k k. 4 5; P2m = E2m . - .! P2m+1 ! 5 !" (3.2.4):
P2m+1 = PPmm P0m PPmm P0m = 2 = ;P2 PmP2 P02 = E02m E0m = E2m : 2 m m m +1
) . J , .! /
" 8. R8 f P2(n) 5" " Pn " . f. A! " . (f0 f1 : : : f2n ;1) " fjuj f(u1 : : : un), " " (f0? f1? : : : f2?n;1 ), !!"7 /
" 8. R8, 8 v - " " /
" . xv11 xvnn . ' % 3.2.1. 4 .; .! /
" 8. R8 :; . )8 " , ." 0 10 1 0 1 1 0 0 0 0 0 B1 1 0 0C B1C B1C B CB C B C @1 0 1 0A @1A = @1A : 1 1 1 1 1 1 " , x1 x2 = x2 x1 x1 x2. , " /
" "; 5 . 1.4.7. j
j
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! " ! " /
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58
3. ,! $
3.3. & .
1. ! 5!" D, 578 n- , "! " 5 /"8 5!" 2 !". , ", ;7 " " 5, "! " 2 5!" D. A! 2 . /" 5!" D . , " 2 "! ! . ""!", " f "! ! *! 5!" D, ! f(x) = f(y ) ; x y D. , " !28 2 !" D " 5 " :" " !" D 8" , ." /"" " !"" !" D :". X2 7 " "! . 15 !!""! . !28 8 2 8 5!" B n . G ; !" D B n . D? . 5!" ! ! . /" /" !", ". . 6
2
D? = y y = xi xj 8 xi xj D xi = xj : f
j
2
6
g
! ;7 "5 !7!" 8 " !28 2.
)% 3.3.1.
D?
* )
D
Bn
2n;m+1 > D? + 1: . $ ' D (n m n)- . j
j
;
G" !" " 3.3.1 ! ! " .!"8 !. | 5 .
%% 3.3.1.
D?
* )
D
Bn
2n > D? + 1: . $ ' D (n 1 n)- . j
j
;
. A! " f :" !"" !" D, ".. f(xi ) = f(xj ) ; xi xj D, " 6
f(x i xj ) = f(xi ) f(xj ) = 0: (3.3.1) " , xi xj = ker f. 4/" (3:3:1) ! ", ." 5!" D? " f !;"!. )8 " , ." ": ! 5!" D? !"!" H !;"!, " H "! 8 " " 5;78 ! ;7 !" D ! ;7 . G!"" , !!" !"!" H , ;7 7 ! D? , " f, "8 "! H . 4!" xi xj | D. J xi xj = H = ker f, "
6
2
2
f(x i ) f(x j ) = f(xi xj ) = 0
6
".. xi xj . . 4/" !" " 8 8 " !"". " B n !"!" H , " !"! ! 5!" D? !" "8 . 7!" "8 !"!" 8 ! " ! . J 2n > D? +1, " ! /" B n "! 57 D? . H"" !" ! " !!"" " !"!". ) . j
j
3.3. ,! 1
59
3:3:1. -! ! 3.3.1. I /" ! " !7!" "8 8 (n 1 n)- " f1 , ." f1 (x) = f1 (y) ; x y D. G 5!" D ! " . D0 . > D1 . !" D0 !" f1 . )8 " , ." 7!" 5!" D1? , !!"78 ! . /" 5!" D1 , ! " 7!" 5!" D0? . G!"" , ! /" ", " D0 5 !"!"" " x1, x2 y1, y2 , ." x1 x2 = y1 y2 f1 (x1 ) f1 (x2 ) = f1 (y1) f1 (y2 ): , . , ." " /" !"2 5" " ! . A! 2n;1 > D0? , " 2n;1 > D1? /" 5 ! ! " ! 3.3.1, 5!" D1 . I /" ! " !7!" 8 (n 2 n 1)- " f2 "8, ." f2 (x) = f2 (y ) ; x y D1 . 45 D2 = f1 (D1 ). ' 7 !., 8 " , ." D2? D1? . $", ." f2 f1 " f2 f1 " :" D (n 2 n)- ". 4 5, ." !; 7 !5!" k 1 5 8 8 i " k 1 . :" !" Di;1 (n i + 1 n i)- " fi 57 B n;i 5!" Di ", ." Di = fi (Di;1 ), Di? Di?;1 , fk;1 f1 "! :" !" D (n k + 1 n)- ". A! 2n;k+1 > Dk?;1 + 1, " 3.3.1 5 " 7 . J 5; (n k + 1 n)- " fk;1 f1 " 5" !" D Dk;1 , (n k n k + 1)- " fk !"" Dk;1 :", " 8 " , ." fk (fk;1 f1 ), . "" 3.3.1 " fi , " :" !" D (n k n)- ". 1 ", ." k m ! " ! 8 5 ! ;" !" 2n;k+1 2n;m+1 > D0? Dk?;2 Dk?;1 : 4/" . , ." 3.3.1 5 ! " ! 7 !5!" m , .2! "" (n m n)- " fm f1 " :" !" D. J . ' % 3.3.1. 1 ", !";7 :" 5!" 80 1 0 1 0 1 0 19 1 0 0 0 > > > = 0 1 0 C B C B C B0C : D = >B @0A @0A @1A @0A> > > : 0 0 0 1 ;
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J 42 + 1 < 23, " " 3.3.1 8"" !7!" :"8 D 8 (2 4)- ". H"" " !""!" ! 8", 5 " !" " 3.3.1. *5!" ! D? !!"" ! " ! , , ! " , " (1000) 5" D? . 4/" .!" (3 4)- " f1 ", "8 5 "! " (1000). -" (1000) !!"" !!"".!; " ! !" ." !" . - !""!" ! 8" !" "8 8 !"!" !!"".! " , 5 !". 52, !"!", "8 " (1000), 5 "! !" " 0 1 0 1 0 0 F1 = @0 0 1 0A 0 0 0 1 ", " " " " f1 . J (2 3) " f2 !!" 5!" D1 = f1 (D), ", 8 " , !!""
60
3. ,! $
!" " F1 . J 5 8 " , ." 5!" 8 ! D1? !!"" ! " ", ! " ! " . " , " (111) 5" /" 5!". - .!" (2 3)- " f2 ", "8 5 "! " (111). !!" " (111) !!"".!; " ! !" " !" , , ." 8 "8 !"!" 5 "! " F2 = 11 10 01 : " , " F2 " " " f2 . 1 5 " F1 F2 " F f2 f1 " f1 f2 : 0 1 0 1 0 0 F = 11 10 01 @0 0 1 0A = 00 11 10 01 : 0 0 0 1
J , :" 5!" D (2 4)- " "! " F. I " 3.3.1 8 "! 8 :"8 5!" 8 ". ""!";7; " " !". )% 3.3.2. + D B n , $ ( 2n , (m n)- ! *! , p
m 2 log2 D 1: 2. *5!" , !!"7 (m n)- ", "!n *! $ 5!" , !!"78 !" D B , ! ; !" D "! :" D " f. G 5!" Dnd , !!"78 ! d-/" 5!" n-8 , 7!" 8 ! 8 2;78 5!". I" !" ! ;7 "". )% 3.3.3. d, * $ 2n, Dnd $ ;2n *! $ nd , $ ( log2 d * ( 2 log2 d n)- . G" !" " "! ! 8" ; . 2n, m = 2 log2 D . . %% 3.3.2. D B n , D * (m n)- ' D. . G !", ." "5 . J8 5 !" D, ";7 ! , "! 21 2mn ", 5 " " 5" ;- D . J .! . /" !" D 12 D ( D 1), " D " "! x y, " " 5;"! . 2mn D ( D 1) (m n)- ". ,. , ." x y 5" 5 8 /" ". 8 !" , ! 5" 2mn;m (m n)- ". " , 2mn mn;m : D ( D 1) < 2 b
j
jc ;
F
D
D
F
p
L
d
d
e
p
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d
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j j
j j
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j j
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je
e
3.3. ,! 1
61
," ! !5 . !" 2m < D ( D 1) " . "." !; , 5 m. " , ! 5 5. ) . 3.3.3. 45 m = 2 log2 d . 4!" 8 ; ; 2n mn " T, !!"7; 2 !" d !" . '5 !" !" !""!" (m n)- ", 5 !" | !" Dnd . - " !. !", !""!";7 " f, !" , !""!";78 !" D, !" , ! " f !"" D :", " !. !" . 8" , ." i- !" " T " j- !" , ! /" tij , !"7 " !. i- !" j-8 !" , . I 3.3.2 ! ", ." !" " 5 !" ! 5" 1 2mn . 2 1" " , ." ; n" !" " !"". !"" !7!" " T " log2 2d !", " !" ;" ! !" . " !" ", ." . !" 2 .! 7 " !" . , . . k ; !" , ;2n " k !", ".. k !" ;" k d ;2n !" . ; J8 !"2! " (1 ) !" ! 5"! k d n 1 k ) 2d 2mn . " , ! !" "! !", 2 (1 " !. ! " !" ! 5"! . ;2n n 1 mn 1 2 (1 k ) d 2 > 2 (1 k ) 2d 2mn k . J8 , /" !" ", , ." .! !" , " (k + 1) !", ! !" n n n 2 2 1 k+1 d > k d + 2 (1 k ) 2d : " , k+1 > 12 (1 + k ): j
j j
j;
d
d
e
e
;
;
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; n
," .! " !" (1 k+1 ) 2d . !" n n n n 2 1 2 1 2 1 (1 k+1 ) d < 2 (1 k ) d < < 2k (1 1 ) d < 2k+1 2d : ;
;
;
; n
;
; n
48 k = log2 2d , , ." (1 k ) 2d < 1, ".. " " 8 "8 !" . J . I! " !" " " "! 8 " . 3. J 5, ." ; 8 8 m, ! 78 n2 , n- "! !" Dm , !!"7 2m+1 1 , ", ." .! " ; 8 8 " !28 2 /" !" 2 2m. 4!" m n2 e1 : : : e2m | 2m ! " !" "8 ! En. 45 Dm = e1 : : : em em+1 : : : e2m : )8 " , ." Dm !!"" 2m+1 1 . , 5!" Dm? ! Dm !" ! ;" !"!" !" 2m B n . 4/", . , ." !" ; 8 !"!", !;78! ! Dm? , ! " n 2m. " , 8 ; 8 8 " !28 2 !" Dm 2 . 2m. d
e
;
;
h
i h
;
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62
3. ,! $
J 2 log2 Dm 1 = 2m, " " 3.3.2 8"" !7!" 8 (2m n)- " !28 2 Dm . J 7 !. !" " 3.3.2 "! ". !" 8 . I"! """ , ." !7!"" !"". !" (m + 1 n) " !28 2 !" Dm ! !" ". H"" " "! ! ;7 !": b
j
jc ;
y1 = x1 xm+1 : : :: : :: : : ym = xm x2m ym+1 = (x1 : : : xm )&(xm+1 : : : x2m ): G 5, ." !" " 3.3.2 "! ! "".! ". log2 d ." ! !" Dnd !, ." log 8. !"" n . 2n n /" ! ;7 . . !" D B
; , !"
_
_
_
_
! 1
(D) = minrank f " "! ! :" !" D ". $" . (D) ! ." ! !" Dnd . * D Dnd )% 3.3.4. n d n22n. . n p
! 1
2
(D) 2 log2 d 2 log2 n 2:
;
;
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J 5, ." " !" . 2d !" Dnd ! (m n)- " "! " !28 2. J .! . (m n)- " 2mn , " ! 5 (m n)- " "! " !28 2 . ; n 2d 2;mn !" 7!" d. " , "! ", " " " !28 2 n P = 2d 2;mn . !". 8 !" , , ." . P ! M(f d). 4/" (3:3:2) ! 8 !" n m 2 2 2(n;m)d : ; mn d 2 d ," ! !5 . n m 2 2 1 mn (n;m)d : d d 2 2 )8 " , ." n m 2 2 = 2n(2n 1) : : :(2n d + 1) d 2m (2m 1) : : :(2m d + 1) d
;
;
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3.4. ,! %
63
, ! ; .!" ! 8 !". G /"8 .!" 5 ! (d 1) !5" . (2n d). , " , ! ! ", ." ! ".! 2 ! 8 8".!8. 4/" " ! (d 1) !5" (d 1)- !" ; ! 8 ".!8. 4 /" 8" , ." m 2 n 2. - " !. m n 2 2 log2 d 2 log2 n 2 P ;1 j = 2(d(dd;;1)1) = 2d , 11;;xy 1 x+y !" ! . G, " d;1 1 dj =1 1 > y x 0 m + 2 n, " ;
;
;
;
;
;
;
;
2n d
2m = 2n 2n(1 d=2n) d;1 d 2m 2m (1 d=2m+1 ) d;1 d;1 1 1 d ( n ; m ) d ( n ; m ) d 2 1 + d 2m+1 2n 2 1 + 2m+2 :
;
;
;
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d;1
1 mn 2 : n 2, " ; !" n
1 + 2md+2
J 1 + x1 x > 2 x > 1 m ! 8 !" ;
;
;
2m+2
! 1
d(d;1)
1 2mn 1 + d d 2m+2 > 2 d2(md;+21) : 2m+2 )8 . !" 2m , ! !" d n . d(d 1) d 2 : 2m > 4n(n 1) 2n 2n(n;1)
;
;
J .
*& 3.3.1. G" " 3.3.2. 3.3.2. 4!" D B n , D 2n. 4" , ." !7!"" "
Bn,
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1. 1 , ." 5!" n-8 G = g1 : : : gm "! ! !!" d, ! ; 8 /" gi gj !!" 5 2 d. J 5 8", ." G ! " t 2 , ! 8 !!" 2 . 2t + 1. , 8 " v B n " g G c B n ", ." !!" " " v " g 2 . !!" " v ; 8 88 /" G, c = g v, "! " v, , ! " v. -" c "! ! . ' G "! (n k)- , ! "! k- !"!" !"!" B n . (n k n)-" H "! " 8 G, ! Hg = 0 5 8 g G Hx = 0 5 8 x = G. f
2
2
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2
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6
2
64
3. ,! $
(k n)-" G "! $ " 8 G, ! . g1 : : : gk !" " G ! " ! G. )8 " , ." . 5 ;7 " ; 8 8 ! !" HGT = 0. )% 3.4.1. 4 G t ) : t = g6=min g: 0g2G h
i
k k
. J !8 5" , " . , ." !!" ! " ! 8 8 /". J !", ." t < min g . - /" !. G "! /" g1 g2 , !!" 5 " 2 t. " , g1 g2 = d(g1 g2) < t: 8 !" , ! g1 g2 " 5" G. 4/" g1 g2 t. 42 ".;. J . )% 3.4.2. f | (m n)- ! *! ! Bnt(0) t " . . f (n n m)- 2t + 1. . 4!" L | (m n)- ", ";7 ! " , x y | 8 . J (m n)- " ! 5" 2 . 2n;m , " " !" " !"". " , ." !!" 5 x y 2 2t + 1. A! d(x y) 2t, " B n "! x0 y0 ", ." x0 t, y0 t x0 y0 = x y. J8 L(x0 ) L(y0 ) = L(x0 y0 ) = L(x y) = L(x) L(y ) = 0: " , L(x0 ) = L(y 0). 8 !" , x0 y0 5" 2 Bnt (0). 4/", L(x0 ) = L(y0 ). 4".. " , d(x y) 2t+1. J . I 7 " " 3.3.2 8 ! " "5 !7!" !"". 2 . )% 3.4.3. $ (n m)- 2t + 1, n, m t 2t X n : n ; m +1 2 > i=0 i 2. 4!" (2n 1 2n n 1)- Hn, ! ;7 2 n 2 1. I! 7 8 . " Hn . 4!" Hn = (hij ) | (n 2n 1)-", " j- !" hj = (h1j : : : hnj ) ! " ! . 5 .! j. 1 , " H3 8 " ! ;7 : 0 1 0 0 0 1 1 1 1 H3 = @0 1 1 0 0 1 1A : 1 0 1 0 1 0 1 )8 " , ." Hn ej = hj , , ! " , Hn ei = Hn ej ! i = j. 4/" !""!";7 " Hn (n 2n 1)- " " " !28 2 2 .8 ! ! " . J , " 3.4.2 ! ", ." " Hn "! . " , ! ;78 2 2n 1. ' ! . " Hn "! X8. I! 2 " v ! X8 "! . !". )8 " , ." " Hm v !!"" , ! 2 , ! . 2 !"!"", " " Hm v "! . !" 2 .8 . k
k
k
k
k
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6
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3.5. )% %{-
65
*& 3.4.1. 4" , ." ! (n k)- " !!" d, "
n k + 1 d. 3.4.2. 4!"" 5 ;7; " X8 7. 3.4.3. 4!" G | (n k)- . G" , ." ! G " " ! ."8 !, " ! ! ."8 ! ;" (n k 1)- . 3.4.4. 4!" (k n)-" H ; t 1 !" ! . 4" , ." " H "! . " 8 (n n k)- ! !!" 2 t. 3.4.5. (K -2{K ") 4" , ." !7!"" (n k)- ! !!", 2 d, " "8 n, k d ";" P ;2 ;n !" di=0 2n;k: i ;
;
;
;
3.5. % {'
' {* RM(n k) 2n k "! 5!" " . ! n-!" , !" 8. R8 " ! " k.1) ' % 3.5.1. *5!" RM(2 0) !!"" " . !"", ".. RM(2 0) = (0000) (1111) . *5!" RM(2 1) !!"" " . ! x1 x2: f
g
0 = (0000) 1 = (1111) x2 = (0101) 1 x2 = (1010)
x1 = (0011) 1 x1 = (1100) x1 x2 = (0110) 1 x2 x1 = (1001):
1 5!" RM(2 2) ! " ! P2 (2). 1" " , ." !!" 5!" ;"! ." ! !!" , !""!", ." , .
)% 3.5.1. 0 {/ RM(n k) 2n n n;k
2
2
.
k
. J 5 .! n. 4 n = 2 "5 " ! " !!"8 2 . 4 5, ." " ! n ! 7 " m 1 2. 45, ." /"8 5 ! " "5 " n = m. J ! !" k " 5 " !" k, " . , ." 5!" RM(n k) "! !"!". 4/" ! " 3.4.1 !"". " , ." ! 5 RM(n k) 2 . 2m;k . 4!" f | m-!" !" k. A! k = m, " "5 " . , " f 1. 4/" 8, ." k < m. - 8. R8 f ! !" . ! 57 ! 57 xm . - "" . !" ;
k
k
f(x1 : : : xm ) = xm f1 (x1 : : : xm;1) f2 (x1 : : : xm;1 )
(3.5.1)
1) , 3.2, ! " ( " % # (, % # !.
66
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k+j 2n;k;j , "! f(x1 : : : xk k+1 : : : n ), .;7! f !" !"" k+1 : : : n !" xk+1 : : : xn. ,. , ." 8. . x1 : : : xk ; 5!" Bi ."! ! 8 . ! 8 " , !" x1 = 1 : : : xk = 1, /" 5 8 i 0 1 : : : 2n;k 1 ! !" M M x1 : : : xk = 1: (3.5.3) x1 : : : xk =
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(3.5.4)
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1 = (1111 1111 1111 1111) x4 = (0101 0101 0101 0101) x3 = (0011 0011 0011 0011) x2 = (0000 1111 0000 1111) x1 = (0000 0000 1111 1111)
x4x3 = (0001 0001 0001 0001) x4x2 = (0000 0101 0000 0101) x4x1 = (0000 0011 0000 0011) x3x2 = (0000 0000 0101 0101) x3x1 = (0000 0000 0011 0011) x2x1 = (0000 0000 0000 1111)
;7 ! !!"8 {*. , . ! . x4 x3. ! " . ." , 5!" Bi 5!" Bi0 /" : B0 = (0000) (0001) (0010) (0011) B00 = 0 1 2 3 B1 = (0100) (0101) (0110) (0111) B10 = 4 5 6 7 B2 = (1000) (1001) (1010) (1011) B20 = 8 9 10 11 B3 = (1100) (1101) (1110) (1111) B30 = 12 13 14 15 : ""!";7 5!" . ! ! v : f
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J ! ." ;. " , /
" . x4x3 5 " v ! . ;. J 5!" Bi Bi0 . x4x2. ! "" . , : B0 = (0000) (0001) (0100) (0101) B00 = 0 1 4 5 B1 = (0010) (0011) (0110) (0111) B10 = 2 3 6 7 B2 = (0000) (1001) (1100) (1101) B20 = 8 9 12 13 B3 = (1010) (1011) (1110) (1111) B30 = 10 11 14 15 : f
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""!";7 . ! " v : S0 = v0 v1 v4 v5 = 1 S1 = v2 v3 v6 v7 = 0 S2 = v8 v9 v12 v13 = 1 S3 = v10 v11 v14 v15 = 1: J " ! ." . " , /
" . x4x2 5 " v ! . . 4 8. .! !" . " !" , x4x1: S0 = v0 v2 v4 v6 = 1 S1 = v1 v3 v5 v7 = 0 S2 = v8 v10 v12 v14 = 0 S3 = v9 v11 v13 v15 = 0M x3x2 :
S0 = v0 v1 v8 v9 = 0 S1 = v2 v3 v10 v11 = 0 S2 = v4 v5 v12 v13 = 0 S3 = v6 v7 v14 v15 = 1M
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69
x3 x1: S0 = v0 v2 v8 v10 = 1 S1 = v1 v3 v9 v11 = 1 S2 = v4 v6 v12 v14 = 0 S3 = v5 v7 v13 v15 = 1M
x2 x1: S0 = v0 v4 v8 v12 = 0 S1 = v1 v5 v9 v13 = 0 S2 = v2 v6 v10 v14 = 1 S3 = v3 v7 v11 v15 = 0: J , /
" . x4 x2 x3 x1 , . x4x3, x4x1 , x3x2 x2 x1 ;. 1 ! v0 = v x4x2 x3x1. J x4x2 x3 x1 = (0000 0101 00110110), " 8 " , ." v0 = v x1 x4 x2x3 = (1100 1110 0011 0011): J , ! " v0 , /
" . !" . . . ! x4. ! ", "" ."" 5!" Bi : B0 = (0000) (0001) B1 = (0010) (0011) B2 = (0100) (0101) B3 = (0110) (0111) B4 = (1000) (1001) B5 = (1010) (1011) B6 = (1100) (1101) B7 = (1110) (1111) 5!" Bi0 /" 5!": B00 = 0 1 B10 = 2 3 B20 = 4 5 B30 = 6 7 B40 = 8 9 B50 = 10 11 B60 = 12 13 B70 = 14 15 : G !""!";7 /" 5!" . ! : S0 = v00 v10 = 0 S1 = v20 v30 = 0 S2 = v40 v50 = 0 S3 = v60 v70 = 1 0 v0 = 0 S6 = v0 v0 = 0 S7 = v0 v0 = 0: S4 = v80 v90 = 0 S5 = v10 11 12 13 14 15 ! ! ;, ! " , /
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*& 3.5.1. 4!" g RM(4 2), c
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71
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x y, " ; /", ;7 &, , , . >"" " /" ;"! ! , /" 5 /" /" . ! | ! x y x y , . ";, . 1 2 Q 1 ZZ 2 Q - .!" ! 4.1.1 5 " 5 !?=Z Q ~ ? !. 4 5 ! & 1 2Z & Z?Z? ! ;7 . 1) 4 !8 ! 8"! 1 2 2 ", ".. ! " ! = ~ Z Z 2 . 4/" !", ;7 "; , "!"!";". 2) H" ! 5 "8 , ! 5 8 "8 1 7 ! , /". A! !" /", 58 "!. 4.1.1 8 , , " , ! ;7 /" ! 8 , ! ;"! ". !" "8 | ". ! " 8 ! 8"! ". ! " "8 . 4 " 5 /" " " !" !!" | . !!""! 5;. 3) '5 2, ;7! ! , "."! , 7 /" 2 , 7 ; 8; 2. G 5 !, , 5" ! .! 2 . 4!" ! S ;. P. G 5 2 w /" ! ; S(w), 2 w. /" ". (1) A! w | ! , " ! pi , " S(w) = pi . (2) A! 2 w ! !" f, " S(w) = f. (3) A! 2 w ! !" f 2 u | 2 w, .!"! S(u), " S(w) = f(S(u)). (4) A! 2 w ! !" f 2 u | f
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74
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!, .!;7; ; ; f(x1 : : : xn), !5!" 8 . ; f 5 ! : f(x1 : : : xn) = xnf(x1 ::: xn;1 0) xn f(x1 ::: xn;1 1): - !""!" ! /" 5 !" 5 ! 4.1.4 ! S. H" ! .!" ; f !!"" ! A B, 8 /" ", :;" 8 :;". 4 ! A .!" ; f0 = f(x1 ::: xn;1 0), ! B | ; f1 = f(x1 ::: xn;1 1), /" " .!" xn . G :;" 5;" f0 f1 , !""!" xn xn. $" :;" .!" :;; . )8 " , ." !5!" 8 ! S 5;"! . !5!" 8 ! A B ! ;7 L(S) = L(A) + L(B) + 4 D(S) = max(D(A) D(B)) + 2: 4 5; 5 ! A B !!"" . 3 2n;1 4 /", 8 ! " 2(n 1). 4/", L(S) (3 2n;1 4) + (3 2n;1 4) + 4 = 3 2n 4 D(S) 2(n 1) + 2 = 2n: 1!" (4:1:1) . 4. , ! .! . ! 5!" x1 : : : xn ! B, !, ." !5!" !!" ! ! " . . I" !" ! ;7 "". )% 4.1.1. B P2(2), N(L n) | ) * * ( B * x1 : : : xn L. . N(L n) (c(L + n))L+1 c | , ( $ ( . . !!" ; ! S, !!"7; n L /". H" ! " . !. 4 S, 5 2, " " 8 , ! ;7 . A! 2 v " k (v ui ) : : : (v uk), 7 2 u1 : : : uk , ": 1. 9 (v u2 ) : : : (v uk)M 2. G k 1 2 v2 : : : vk , ! 5 ! vM 3. '5 ; ; 2 vp , p = 2 3 : : : k, !5 (vp up ) ! 2 up . ,"", ." " ! " 2 . 4/" 8 " , ." ! " , ! L " 2. -!. 2 /"8 " . . L + n ! ( L !, !""!";7 /" S, n ! ). 4.2! " !"" "; !" ! 5!" L + n (!. !". 16). 4 /" !, .!;7
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*& 4.4.1. G" !" a) Lf_&:g (1) = 2M d) Lf_&:g (x y) = 2M g) Lf_&:g (x y) = 4.
b) Lf_&:g (0) = 2M e) Lf_&:g (x y) = 2M
c) Lf_&:g (x y) = 2M f) Lf_&:g (x y) = 4M
4.4.2. G" !" a) Lf&1g (x y) = 3M b) Lf&1g (0) = 1M d) Lf&1g (x y) = 4M e) Lf&1g (x y) = 3M
c) Lf&1g (x y) = 2M f) Lf&1g (x y) = 4:
j
!
#
#
j
_
!
4.4.3. G" !" a) L# (x y) = 2M b) L# (0) = 2M e) L# (1) = 3M f) L# (x y) = 4M
c) L# (x y) = 3M g) L# (x y) = 4M
d) L# (x&y) = 3M h) L# (x y) = 5:
4.4.4. G" !" a) Lj (1) = 2M b) Lj (x y) = 2M e) Lj (0) = 3M f) Lj (x y) = 4M
c) Lj (x&y) = 2M g) Lj (x y) = 4M
d) Lj (x y) = 3M h) Lj (x y) = 5:
_
!
j
!
#
_
4.4.5. 1" LB (xyz x y z). 4.4.6. 1" Lf&_:g (x y z). _
0
4.5. 1 ! , 4 ! .5 * 5 !
!!"" fn , .! 8" " !" " " 8 " n, ;78! " " 5" !"" !.!"3) . 1 , " " ln = x1 xn :; n = x1 xn. A!"!", ." !5!" L(fn ) 5 " fn !" " n. ' ". . !5!" " " "!. "8, 5 " !5!" "! . " ., 2 " "! " !.. 4 /" ! " !" ;" " !" 5 !5!". , " "! " ". ! "".! ". ( n ) . !5!". 15 !!";"! " . 1. !!" n-!"; :;; x1 xn. )8 " , ." :; !" , & ;" ! T0 . 4/" :;; x1 xn 5 .!" ! ! & . G !5!" "8 .!, " 5, ." ! , !;7 , " 2" !5!" .! :;.
_
__
! 1
_ _
_ _
f
g
3 &
4.5.1.
!
Lf&g (x1 : : : xn) = 3n 3 Df&g (x1 : : : xn) 2 log2 n : _
_
_
_
;
d
e
. . . 45 m = n=2 , l = n=2 . Sn , .!;7 ; x1 : : : xn , 8 !""! " !""!" ! b
_
c
d
e
_
x1 : : : xn =(x1 : : : xm )(xm+1 : : : xn) (x1 : : : xm ) (xm+1 : : : xn): _
_
_
_
_
_
_
f g
_
_
_
3) 0 % # ( f , ( n-! % # ! n n , % # ! ! .
4.5. #3 !, 6 1 +
89
48, ." m l 2, L(Sm ) = 3m 3, L(Sl ) = 3l l, , ." L(Sn ) = L(Sm ) + L(Sl ) + 3 = (3m 3) + (3l 3) + 3 = 3n 3: 48, ." D(Sm ) D(Sl ) = 2 log2 l ." , ." ; 8 ."8 n 28 log2 n = log2 (n + 1) , ; 8 ."8 | 2 n=2 = n, D(Sn ) = D(Sl ) + 2 = 2 log2 n=2 + 2 = 2 log2 2 n=2 = 2 log2 n : - . *3 3. 4!" S | ! ! & :; n . 45, ." Lf&g (x1 : : : xn ) 3n 3: (4.5.1) 45 !8, 5, ." S 5 " 5 " ;. " /", ;7 ; . G!"" , !", ." n- ;. " /" s1 : : : sp, ;7 :;. - S 5 x1 = : : : = xn;1 = 0. J ! ! S !" , " ! 2 , 7! ! 2 /" s1 : : : sp ;7! n- , " .!" "5 !" . 4/" "5 !" " .!" ! 2, " ;. " /" s1 : : : sp , , ! " , ! /" s1 : : : sp. 4! .", ." ! S .!" "5 !" ; . xn. 42 ".;. J 5, ." S " "! , " ;. /". 4 5, ." /" ". J8 ! "! , " ;. "" 5 /" s. 8. 7!" 8" , ." " " (n 1)- n- . - S 5 x1 = : : : = xn;2 = 0, xn;1 = xn = x. 4! " !" ! /" S " .!" "5 !" ; . x. 2 ".;. 4!" S n- ;. /" v u. 8. 7!" 8" , ." /" u " . - S 5 xn = 0. xn xn 45, ." ! " !" S 5 " /", ;7 . G /"8 !!" !& & 5 . /" v = v0 v1 : : : vk " 5 /" vi+1 ;. /" vi . /" . " "! ", " k /" ;" & & :;, ! | . (A! /" v0 " , " k = 0.) ! 4.5.1 5 . " /". A! " . ", " S !7!"" . :;, " ! " n- !. 4.5.1 ! . - /" !. !" !" xn " ", ." ! /" ! " .!" ! ! " . n 1 . I", !" w = vk | /", ;7 ! ! v . :;. A! w u . /" , " " ! !" xn = 0. G !" " , w u ! ;" ( !" 5 .!" ! 4.5.1). /" /" . w. )8 " , ." ! !" xn = 0 .! w " "5 !" . ' 2, !!" !5 . :;, .;7! w, /" z, ;7 . 4 ! , ;.;7! !" !" xi "5 !"8 ! ;7 ! i-8 ! 2, .!;7 "5 !" , i-8 ! . I 8 2 ! ", ." ! S 5 " " , !""! !" . " , 5" " n 1 . 4 /" S " . 2n
;
;
;
d
e
d
d
;
;
e
e
d
d
ee
d
d
e
d
ee
d
e
f
_
_
g
;
;
r
r
;
;
;
90
4. #3 + !
2 /" . ' "8, 8 " , ." S ! 5" . n 1 /" :;. H" ! " "8, ." !" :; n n. J , 7 .! /" ! S 2 . 3n 3. 1!" (4:5:1) . 9"5 !" ;. G:; 5" !! T0 , 8 "5 , !"". !" .!"! ! !, ! 57! T0 . 8 !" , 8 " , ." x1 xn = (x1 1) : : : (xn 1) 1. ," 8 ! " !" Lf&1g (x1 xn ) 2n + 1: J , !2 ! ! ! 8 P2 " 2" .! " . 2. !!" ; 2(x1 : : : xn), ; , ! ! 8" "! , ; ; " !.. ' !!" 2 :;, 2 ! 5"! T0 . , ". " :;, !5!" .! 2 ".! !. 3 &
4.5.2. n . .
;
;
__
__
! 1
L(2 (x1 : : : xn)) 2n
D(2 (x1 : : : xn)) log2 n:
9"5 ! " ." . - " !""! !5!" 8 , ."" | 5 !5!". %% 4.5.1. m | " , n = 2m . . : (i) $ ( $ * * x1 : : : xn " f0 f1 : : : fm ,
f0 = x1
xn 2(x1 : : : xn) = f1 : : : fm M " f0 f1 : : : fm * S
__
(ii)
_
_
$ $
(4.5.2)
L(S) 3n 1 D(S) = m:
;
. ) 5 m. 4 m = 1 "5 . , ! ;"! :; x1 x2 :; x1&x2 . - .!;7 ! S1 !!"" /" 8 . 4 5, ." ! ! ! 7 k 1. (i) 4 5; !7!";" !7 " x1 : : : x2k;1
f01 f11 : : : fk1 !7 " x2k;1 +1 : : : x2k f02 f12 : : : fk2 " ! !"2 (4:5:2). 1 f0 f1 : : : fk ! ;7 : _
;
fi = fi1 fi2 i = 0 1 : : : k 1M fk = f01 &f02 : ,. , ." f0 = x1 : : : x2k _
_
;
(4.5.3)
_
f1 : : : fk = (f11 : : : fk1;1) (f12 : : : fk2;1) f01 &f02 : (4.5.4) 4 !"; f1 fk | ", /" " !" "8 !" (4:5:2) !"". " !, ." /" ; ; ! 1, ; ! 2. 4!" | 2k ! 2. A8 ; . . 1 , "; | . 2. G 8" ! 8" ! (1) /"8 , (2) " , (3) 5" " ! , " | ". 4 5; _
_
_
_ _
_
_
_
_
_
4.5. #3 !, 6 1 +
r p p p p p p p p p xr
x1
2k;1
r p p p p p p p p pxr
x2k;1 +1
Sk;1
r
_
2k
Sk;1
fk1;1
f01
r
fk2;1
f02
ppppppppppppppp f
_
fk;1
0
!. 4.5.2 !. (1) (f fk;1 )(1) = 1, !. (2) (f12 1 2 f0 (1 )&f0 ( ) = 1. " , ! (4:5:4) 1 1 2
_ _
1
(f1
91
__
_ _
&
fk
fk2;1)(2) = 1, !. (3)
fk )() = 1:
J !!" 2k ! 1. 8. 7!" 8" , ." !" "! 8 , ".. 1 = 1, 2 = 0. J8 5; k
k
k
k
(f11 : : : fk1;1)( 1 ) = 0 f01 ( 1) = 1 (f12 : : : fk2;1)( 2 ) = 0 f02 (2 ) = 0: _
_
_
_
I ! !" (4:5:4) 8 ! ", ." (f1
__
fk )( ) = 0:
4 "5 . (ii) G5 " "5 . G !" ! Sk , .!;7 f0 f1 : : : fk ! ! !" (4:5:3). Sk !!"" !, 5 " "! / ! Sk;1, k :;" 8 :;". '!" ! !. 4.5.2. )8 " , ." L(Sk ) = 2L(Sk;1) + (k + 1) D(Sk ) = D(Sk;1 ) + 1: (4.5.5) 4 5; 8 !, .!;7 f01 f11 : : : fk1;1 f02 f12 : : : fk2;1, k 1. " , ;
D(Sk ) = D(Sk;1 ) + 1 = k: G !5!" ! Sk ! " (k 1) !" (4:5:5). 9." , ." L(S1 ) = 2 . ;
L(Sk ) = 2L(Sk;1) + (k + 1) = 2(2L(Sk;2) + k) + (k + 1) = : : : = = 2k;1L(S1 ) +
kX +1 i=3
2k+1;ii < 2k + 2k+1
1 X i=3
i 2;i = 3 2k :
) .
%% 4.5.2.
L(2 (x1 : : : xn) 6n + log2 n + 1 D(2 (x1 : : : xn) log2 n + log2 log2 n :
d
e
d
e
92
4. #3 + !
. 4!" k = log2 n . I 7 ! ", ." !7!"" ! !5!" 3 2k 8 k, " .!" " f1 : : : fk , ." 2 (x1 : : : x2k ) = f1 : : : f2k : G:; k 8 .!"! ! !5!" k 8 log2 k . " , "! ! S, .!;7 ; 2 (x1 : : : x2k ), !5!" 8 " ! ", !""!", . 3 2k+k k+ log2 k . -! ! /" !, !" !" !";7 (2k n) 8" . J n 2k < 2n, " 8 " , ." !" "5 ! . ) . %% 4.5.3. n . . d
e
_
_
d
d
e
e
;
! 1
;p
n L(2 (x1 : : : xn)) 2n + D(2 (x1 : : : xn)) log2 n + log2 log2 n + 3:
O
. 45 m = n . '5 k = 1 2 : : : n !" .! (k1 k2), 1 k1 k2 m, ", ." k = (k1 1)m + k2 . 45 ai = xi1 xim , bj = x1j xmj . J8 " !" !" d
p
e
;
__
__
2 (x1 : : : xn) = 2(a1 : : : am ) 2 (b1 : : : bm ):
(4.5.6)
_
- .!" !" (4:5:6) !"" " . 4/" "8, ." " ! ! !" (4:5:6) !"". " , ." !"7 ! ; ; ! 1, ; ! 2. 4!" = (1 : : : n) | ! 1. 8" , ." pq = 1. J8 ! ai() bj () " ap () bq (). " , .!" (4:5:6) ;. 4!" = ( 1 : : : n) | ! 2. 8" , ." pq = st = 1. ,. , ." p = s, q = t. A! " !" /" !", " ap ( ) = as ( ) = 1. A! ! ", " bq ( ) = bt( ) = 1. )8 " , ." !. .!" (4:5:6) . J , ! !" !" (4:5:6) !". -! ! /" !" !" ! S, .!;7 2. H" ! !!"" : (1) m ! Ai , .!;7 ai M (2) m ! Bj , .!;7
bj M (3) ! S1 , .!;7 ; 2 8" " ;"!
aiM (4) ! S2 , .!;7 ; 2 8" " ;"!
bj M (5) :;", .!;78 :;; , .! ! S1 S2 . ,. , ." !5!" 5 ! Ai Bj m 1, 8 | log2 m = 1 log n + 1. I 4.5.2 ! ", ." 5 S 1 log n i 2 2 2 2 6
6
;
d
e
;p
L(Si ) 6m + log2 m + 1 = n D(Si ) log2 m + log2 log2 m 12 log2 n + log2 log2 n + 2:
O
d
e
d
e
" , ;p
n = 2n + L(S) = 2m(m 1) + D(S) log2 n + log2 log2 n + 3: ;
O
) .
%% 4.5.4.
n 3.
.
L(2 (x1 : : : xn)) 2n 2:
;
O
;p
n
d
e
4.5. #3 !, 6 1 +
93
. 9"5 5 .! 2 . - ! 5 ; " . - "5 4.4.3 , ." L(2 (x1 x2 x3)) = 4. 4 5, ." 8 .! n, ! 78 k 1, "5 . 45, ." ! n = k. 4!" S | ! 2 (x1 : : : xk ). G !", ." ! S "! " ;. /". 8. 7!" 8, ." " " ! k- , ;. /" s1 s2 . -!" xk !" "5 !" . )8 " , ." ! !" ! " .!" ; 2 (x1 : : : xk;1), /" s1 s2 " " . !" . I "5 4.3.1 ! ", ." /" s1 s2 5 " " S, ." ! S0 " .!" ; 2 (x1 : : : xk;1) ! 5" /" 2, . ! !. G 2 (x1 : : : xk;1) ! 5 ;
L(2 (x1 : : : xk;1) 2(k 1) 2 = 2k 4:
;
;
;
" ,
L(2 (x1 : : : xk ) L(2 (x1 : : : xk;1) + 2 2k 2: J !!" !. 8 ! S 5 ;. /". 8. 7!" 8" , ." xk;1 xk ;. "" 5 /" s, ;7 ; v. G !", ." v | - . J8 "5 !" xk;1 xk , 5 x = xk;1 = xk . )8 " , ." ! "8 "5 !" , .! /" s, , ! " , , .! ! ! , !" " x. - "5 "5 !" ; 8" 2 (x1 : : : xk ) k 3 "
!7!" !7 " ! k 1 8". 42 ".;. G !!" !., 8 v | &- . J8 /" s .!"! ( )
v(xk;1 xk) = x(k;)1&x(k ) . 4 ! S !" !" xk !"; . J v(xk;1 ) = 0( ) | !"", " ! S0 .!"
;, " xk;1 "! !7!". 4".. ) .
;
;
*& 4.5.1. 4" , ." Lf&:g (x1 : : : xn) = 2n. 4.5.2. 4" , ." Lf_:g (x1& : : :&xn) = 2n. 4.5.3. 4" , ." LB (x1 : : : xn) = 3n 3. 4.5.4. 4" , ." Lf&_:g (x1 : : : xn) = 4n 4. 4.5.5. 1" Lf&1g(x1 : : : xn). 4.5.6. 4" , ." ; f(x1 : : : xn) ;" !" !" _
1
_
_
;
a) Lf&:g f(x1 : : : xn) 2Lf&_:g (f) + nM
;
_
b) Lf_:g f(x1 : : : xn) 2Lf&_:g (f) + n:
5.
- /" 8 !!";"! /
" ! , .!;7 " 5 ! "".! ".! ". " . - .!"!", ."! !5!" 8 .! ! , !" .!. ! ! !".;7! /" 8 ! !!"" ! . !" . 5.1. ) * ! *
1. !!" (n + 1 2n)- " Sn, .!;7 ! n 5" .!, !" . !!" !.!. 4!" n n nX +1 X X x = xi 2i;1 y = yi 2i;1 z = zi 2i;1 8 x + y = z . J8
i=1
i=1
i=1
Sn (x1 : : : xn y1 : : : yn) = (z1 : : : zn+1 ): , .!;7; " Sn , n- . !;7 "5 . %% 5.1.1. $ n- ( Yn,
L(Yn ) = 5n 3 D(Yn ) = 2n 1: ;
;
. G !" ! Yn ! ! 2 !" 8" !5 .! "!" ". - /" 8" j- ! zj ! j- !8 ! qj 7 j 1 qn+1 qn : : : q2 + xy n :: :: :: xy2 xy 1 , ".. n 2 1 zj = xj yj qj : (5.1.1) zn+1 zn : : : z2 z1 ;
)8 " , ." ! (j + 1)- ! ! ;7 qj +1 = xj yj xj qj yj qj = xj yj qj (xj yj ):
(5.1.2)
J ! "!"!"", " z1 q2 .!;"! z1 = x1 y1 q2 = x1 &y1 :
94
(5.1.3)
5.1. .+ % +
r
ry qr
xj
j
&
r
j
&
ry
x1
1
& q2
95
z1
zj
qj +1
!. 5.1.1 - !""!" ! (5:1:1){(5:1:3) !" ! Sj , ! 5.1.1. ! 5 ! Sj , .!;7 zj qj +1 j = 2 3 : : : n, ! | ! S1, .!;7 z1 q2. I! /" ! .!" ! !" ! Yn . Yn !!"" ! " ! ! S1 : : : Sn. - ! S1 ;. x1 y1 . 1 S1 .!"! ! " | q2 , " z1 . 4 j = 2 3 : : : n, xn yn xj yj x2 y2 x1 y1
rr? rr? rr? r r ppp S ppp S S S n
j
qj+1
2
1
q3 z2 q2 z1 zj !. 5.1.2 ! Sj ;. xj , yj , "" | .! ! Sj ;1
qj . 1 Sj .!"! ! (j + 1)- | q2, " zj . , 7 ! Yn ! 5.1.2. 5!" ! S1 . 4 j = 2 3 : : : n, !5!" 5 ! Sj ". 4/" zn+1
L(Yn ) =
zn
n X j =1
Sj = 5(n ; 1) + 2 = 5n ; 3:
J 8 ! Yn . I !" /" ! (!. !. 5.1.1 5.1.2) , ." 5 ! Yn ! " x1 y1 ! zn+1 , " . /" ! S1 . /" 5 ! Sj , j > 1. " , D(Yn ) = 2(n 1) + 1 = 2n 1: ) . - 2, " "! Y0n , ! ;7 n- .! ! 7 2n;1. " Y0n 8 ."! !" Yn . G!"". "" , ." ! 5 !8 ! " 2n;1, " (n + 1)- ! !" !. 8 !8 ".!" 2n;1. " , .! !"28 ! x + y !"".8 8 /" 5, " zn+1 = xn &yn . 4/" ! Sn !" Yn ( ! ! 5.1.1) 5 /" !5 7! " ! /" 5 . " !. !" Y0n !5!" 8 "8 n 2 ! !" L(Y0n ) = 5n 5 D(Y0n ) = 2n 3: (5.1.4) ;
;
;
;
96
5. #$ $
4 n = 1 ! Y1 Y01 ! ;". 4!" k = log2(n + 1) . (k n)- " W(x1 : : : xn) " , ! W(1 : : : n) = ( 1 : : : k ) Pn Pk 8 i=1 i = i=1 2i;1 i . G !8 " W .!" 8 ! . Cn, .!;7; (k n)- " !." n- . d
e
k
k
%% 5.1.2. $ C2n , -
L(C2n ) 6 2n D(C2n ) n2: . C2n !" !""!" ! ! ;7 8". 4 x1 : : : xn : 5 ! ! !. !" Y01. - "" ."! 2n;1 .!, 5 " ! " 2. 1 .! ! 5 ! ! !. !" Y02 . - "" . 2n;2 " .!, 5 " ! " 4. 4 ; " 7 (n 2) . 1 i- 28 " ! 2n;i !. !" Y0i !5!" 8 5 8 "
;
L(Y0i ) = 5i 5 D(Y0n ) = 2i 3: J8 !5!" ! ! C2n n 2 . ;
;
L(C2n ) =
nX ;1 i=1
nX ;1
2n;iL(Y0i ) = 2n;12 +
i=2
2n;i5(i 1) = ;
nX ;2 nX ;1 = 2n + 5 2n;1 i2i;11 2n + 5 2n;1 2jj ;
i=2
1X 1 1 X 2n + 5 2n;1 2k
j =1 k=j
j =1
6 2n :
08. 8 C2n n 2 ! !"
D(C2n ) =
nX ;1 i=1
D(Y0i ) = 1 +
nX ;1 i=2
(2i 3) = ;
= 1 + (n + 1)(n 2) 3(n 2) = 1 + (n 2)2: ;
;
;
;
) . 2. I!", ." ; !, ! ;7 n- . .!, !!"" . 5n 3 /". 4/" !" 2 ! Yn ;"! !5!" ! n. - "5 2 n 8 /" ! " 5 . 45, ." !7!";" !" 8 " 8 .! ! .!. G5 ! 8" "5 . %% 5.1.3. ( $ * b1 a2 b2 : : : a2k b2k " y2 y3 : : : y2k +1 ,
y2 = b1 yj +1 = bj aj yj j = 2 : : : 2k: . $ $ " y2 : : : y2k +1 * Pk , ;
!
L(Pk ) 4 2k D(Pk ) 4k 2:
;
5.1. .+ % + b2
rrrr
rrrr
n a2n b2n;1 a2n;1
An
0n a0n
i
i
b2
ppppppp
a2 b2
Ai 0i
b
i;1
a2
i;1
0i
b
ppppppp
rrrr
b2
a2
A1
b1
a1
0
0
a1
b1
a
97
B
r
Cn n+1
n
y2
y2
r
pppppp
n;1
y2
y2
r
Ci
i+1
i
y2
y2
pppppp
i;1
r y3
y2
. 5.1.3
. ) 5 k. 4 k = 1 5 .!" " y3 . H" 5 ! " ! P1, !!"7 8 /" !5 8 /" 5. ,. , ." L(P1) = 2 D(P1 ) = 2. 4 5, ." " k 1 " ! Pk !7!"". I! /" !, !" ! Pk+1. 45 !8 ", ." 5 j, 2 j 2k , ! !"
y2j +1 = b2j a2j y2j = b2j a2j (b2j ;1 a2j ;1)y2j ;1:
G ! j
2 : : : 2k
2 f
g
yj0 +1 = y2j +1 b0j = b2j a0j = a2j (b2j ;1 a2j ;1): 4!" , "8, y20 = b01 = b2 a2b1 . J8 yj0 a0j b0j ! ! ;7 !":
y20 = b01 yj0 +1 = b0j a0j yj0 2 j 2k :
(5.1.5)
-! ! /" !" .! yj . /" " /" . . .! ! a0j b0j . $" .! ." ! yj ! ." !. I !" (5:1:5), ! 5 ! ", ." /" 5 ! " 7 ! Pk , ;. .! a0j b0j . 1, 5 ; ; y2j ! ." ! .! y2j = b2j ;1 a2j ;1y2j ;1, ! ; .!; ; y2j ;1. - ;7 .! ! Pk+1 5 ! 5.1.3, 8 n = 2k . H" ! !!"" 2k ! Aj , 1 j 2k , ! B 2k 1 ! Cj , 2 j 2k . '" !!" ! Pk+1 !5!" 8 . 1. 4 ! A1 .!" b01. ,. , ." L(A1) = 2 D(A1 ) = 2. 4 j 2 ! Aj .!" ; a0j . )8 " , ." L(Aj ) = 2 D(Aj ) = 2. 2. 4 ! B "! / ! Pk . 4 5; L(B) 4 2k D(B) 4k 2. 3. 4 ! Cj .!" ; y2j !""!" ! y2j = b2j ;1 a2j ;1y2j ;1. )8 " , ." L(Cj ) = 2 D(Cj ) = 2. I !" ! Pk+1, . 1{3 5 8 ., ."
;
;
L(Pk+1 ) L(Pk ) + 4 2k 2 4 2k + 4 2k 2 = 4 2k+1 D(Pk+1 ) D(Pk ) + 4 4k 2 + 4 = 4(k + 1) 2:
) .
;
;
;
;
98
5. #$ $
)% 5.1.1.
$
n- ( Y n , -
L(Y n) 11n D(Y n ) 4 log2 n : . !!" !5 n- .! x y. G 5 8 j 1 : : : n bj = xj yj aj = xj yj : J8 (!. (5:1:2) !". 94) ! qj +1 (j +1)- ! x + y !
qj +1 = xj yj (xj ;1 yj ;1 )qj = bj aj qj : - .! . bj aj , .! ! qj +1 ! ! ! Pdlog2 ne 5.1.3. )8 " , !5!" 8 ! Qn, 7 .! ! aj , bj qj +1, ! !"2 L(Qn ) 2n + 4 2dlog2 ne 10n D(Qn) 4 log2 n 1: J .! ! x y !"". !5" .! ! Qn ! qj ! xj yj . J . 3. !" ; n- 5" .! x y, !" . !!" !.!, " (n + 1)- " r, ." 8 n ;" .! r 1, ; !" x y,
2 f
d
e
g
d
e ;
r1 = (r1 : : : rn) =
n X i=1
ri 2i;1 = x y j
;
j
8 (n + 1)- rn+1 /" !", ( rn+1 = 1 ! x < y 0 ! x y:
" Rn : 0 1 2n 0 1 n+1, .!;7 !" n- 5" .!, " . , 2 !" ! ! 5 !" .! x y. -!" P P ! .! x = ni=1 xi 2i;1 !!" 8 x = ni=1 xi 2i;1. ,. , ." x+x = 2n 1. " , " ! Sn ; n- .! x y "! !" Sn (x y) = 2n 1 x + y: G . s2 ." (n + 1)- .! Sn (x y), . s1 | .!, !!" 2 n Sn (x y), ". . Sn (x y) = s2 2n + s1. 45, ." (s1 + s2 )(s2 ) = x y : G /"8 !!" !.: s2 = 1 s2 = 0. A! s2 = 1, " Sn (x y) 2n, , ! " , x < y. - /" !. s1 = 1 x + y. J8 (s1 + s2 )(s2 ) = s1 + s2 = s1 + 1 = x + y = x y : A! s2 = 0, " Sn (x y) < 2n, , ! " , x y. - /" !. s1 = 2n 1 x + y. J8 (s1 + s2 )(s2 ) = s1 = 2n 1 (2n 1 x + y) = x y : J , (s1 + s2 )(s2 ) = x y .! s2 " !" x y: !" "" , ! s2 = 1, "" , ! s2 = 0. " , (s1 s2 ) " 8 " !" ; x y. f
g
! f
g
;
;
;
j
;
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;
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j
;
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;
j
;
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j
;
5.2. .+ +
)% 5.1.2.
$ * Yn , $ ( ( * ( * ,
99 n-
L(Yn) = 8n 3 L(Yn ) = 3n 1: . G " !" " !"". !"" !, .!;7; " ." Rn. - 2 , ." ; x y . " Rn(x y) 8 5"! . .! Sn (x y) = s2 2n + s1 ! ;7 : n- .! s1 " .! s2 5 "" !5" ! s2 ; 2. - !""!" ! /" !" ! Yn. !" ! " ! ! A B. 4 ! A " .!" Sn (x y), ! B | " s2 ! " 5 "" ! s2 ; 2. - 4.3 , ." ! ! ! ! !" .!" ; ;7;! " ! 5" ", " /"" " 5" " ! ", ." .! ! "!. 08. !!" ! ! .!;7 !!" : A! !!" F ! 5" " , " ; ! , .!;7 /" !!" ! P2(2), 5 " ! /" ". ,. , ." " !5 Sn ". 4/" .! Sn (x y) ! 7 !" Yn . )8 " , ." /" !. !5!" ! A ! " 5n 3, 8 | 2n 1. G s2 ! " !" Yn . J s2 .!, " 5 ! Sj , j > 1, !" Yn (!. ! 5.1.1 5.1.2) 5 " " /". 1 s2 " " n /". 4/", L(B) = 3n D(B) = n. J . A! !", ." !" "" , " .! 5 ! " ; !". G!"". "" 28 .! "". - .!"!" !7!"" ! Yn .!;7 !" n- .! x y, x y, !5!" 8 " ! !"2 L(Y n) 11n D(Y n ) 4 log2 n : (5.1.6) ;
;
;
;
d
e
*& 5.1.1. 4!"" !.". C8. 5.1.2. 4!"" !" Y 8. 5.1.3. 4!"" ! Snm, .!;7; ! n- 8 m- 8 -
.!. 5.1.4. 4!"" !, .;7; n- .! . 5.1.5. 4!"" !, .!;7; ! .! ; 2n. 5.1.6. 4!"" !, .!;7; ! n- .!, 5 " . 5.1.7. 4!"" ! Sn, .!;7; ! n- .!, " L(Sn ) = (n) D(Sn ) 2 log2 n. 5.1.8. 4!"" ! Sn, .!;7; ! n- .!, " L(Sn ) = (n) D(Sn ) log2 n. O
O
5.2. ) * *
1. 15 !!";"! !" 8 ! , .!;7 ! 28 .! 5" .!, ! . 5.
100
5. #$ $
%% 5.2.1. x y z | ( n- ( , c r | (n+1)n- ( ", c r = x + z y , , 1 = 0 2ci+1 ri = xi + zi yi ;
;
;
;
i 2 f1 : : : ng. . $ $
c r * Ye n
L(Ye n ) = 5n D(Ye n ) = 3: . J c + y = x + z + r c1 = 0 2ci+1 + yi = xi + zi + ri 5 8 i 1 : : : n , " 8 " (!. (5:1:1) (5:1:2)), ." ci+1 = xizi ri (xi zi ) yi = xi zi ri : I "8 !" ri = xi yi zi . 4 !" ri !": ci+1 = xi zi ri(xi zi ) = xi zi (xi yi zi )(xi zi ) = = xi zi xi zi yi (xi zi ) = (xi zi ) yi (xi zi ): 2 f
g
_
J8 .!" ! Ye n 5 " !, 5; ! 5.2.1. H" ! xi zi yi
r
Si+1
r r
_
&
Si;1
ci+1
ri !. 5.2.1 !!"" n ! ";7 ! Si . 4 ! Si " " ( Si ;"! ! ). - Si ;. i- .! x, z y. 1 Si .!"! (i + 1)- .! c, " | i- .! r. ,. , ." ! ! 5.2.1 !!"" 5n /", 8 ". ) . G Ye n , " .!;"! .! c, " 5" , , " .!;"! .! r, | "" . *%
5.2.1. Ye n " ." !!" | /" ! "!"!";" ! .! 5 8 .! c r ! "! " x, y z . >!" /" !!" " !"" !" ! ! ! .! !" - . !!", , 4n- .! x, y z ", ." x = 22nx0, y = 2n y0 z = z 0 , 8 5 .! x0 , y0, z 0 ! 5" 2n . 9! /" .! 5 ! 5.2.2 8 . $2" .!" 8 !""!";" " x, y z " 8" " ". " . -! !""!";7 2" !" ;. )8 " , ." 5 ! x + z y .! !"". " " ! 2n ! .! ( ! /" ! 5 5 " 2" ). H" ! ! ", ." .! ! Ye 4n !"2 n .! c r !"2 n .! x, 2 n .! r 2 n ;
5.2. .+ +
;;;;;;;;;; ;;;;;;;;;; ;;;;;;;;;;
101
x y z
!. 5.2.2 .! z , ! 2 n c . 4/" !!" x, y z ! x + z y 5 .!" ! Ye 2n " ;. ! ! .!, ". . !5!" .! x + z y .! !!"8 10n, 20n 7 !.. 4!" x1 x2 | n- . .!. 4 (x1 x2) n- x, . x !" x1 x2 . >! x1 "! 5" " x, x2 | "" " x. 1 5!" .! !"!" ;"! " .! ! .!: (x1 x2 ) = (x2 x1 ) (x1 x2 ) + (y1 y2 ) = (p q) 8 p q ", ." p q = (x1 + y1 ) (x2 + y2). $", ." " ! .! !" ( (0 0)+(1 0) = ((01) (00)) = ((10) (01))), . ! !8 .. G, 8 ! ; 8 .!" .! " "; (p q), !! .! ;7 "5 .. - .!"!" .! ! .! "! 5 ; 8 8 .! . "8 ! . ! .!. 45 . !!"" ! !5!" 8 !5 .! "5 , ;7! " ! !" " !" 5.2.1. %% 5.2.2. x y | ( n- ( , c r | (n + 1) n- ( ", c r = x + y. $ * Y02n $ c r ;
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;
L(Y02n ) = 2n D(Y02n) = 1: ) 5.2.2 5 !!"" "5 !5!" 8 !5 n- .!, !, ." "" "8 !5 " .!. %% 5.2.3. + n 1 c$ * Y2n, $ * n
( * * ,
L(Y2n ) = 10n 3 D(Y2n ) = 5: . 4!" (x y), (z w) | n- .!. 45 (x y) + (z w) = (p q). 4!" c r .! ! Yn 5.2.1 !, ." .! x, z y. J8 x + z y = c r (x y) + (z w) = (x + z y) w = (5.2.1) = (c r) w = (r + w c): ;
;
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I ! 8 !" , ." ! (x y) + (z w) 5 .!" 7 ! Ye n Ye n+1 . . ! Ye n "! .! x, z y. - ""
102
5. #$ $
. (n + 1)- .! c n- .! r. $" .! r, w c "! ! Ye n+1 . 4. j- ! ! Ye n+1 ;.;"! j- .! r w, "" | j- .! c. - !""!" ! (5:2:1), ! Ye n+1 .!" .! r + w c = (p q). J (p q) = (q p), " . , ." ! (x y)+(z w) .!: "" Ye n+1 .!;"! .! p, 5" | .! q. Y2n !". I 5.2.1 8 ! ", ." Y2n ! 5" 10n + 5 /", 8 ! " 2!". - !"" !" !!" ! ! 7. $", ." ! Ye n+1 5 ! Sj 8 "" 8 xj zj yj wj ;
r
;
r r r
_
Sj +1
;
&
Sj ;1
rj
_
cj +1
&
rc
j
qj +1 pj !. 5.2.3 , ! Sn+1 " !7!" . 4/" ! Y2n 5 !"" : n ! Sj , 5 " ! 5" !" /". '!" ! Sj 5 ! 5:2:3. 4 ! Sj " " " . - Sj ;. j- .! x, z, y, w, ! Sj ;1. -" "" Sj ;"! ! Y2n. 1 " Sj .!"! (j +1)- .! q, "" | j- .! p. 4 ! Sn "5 "! Y2n .!"! (n + 1)- .! q. 4 j 2 : : : n !5!" 5 ! Sj !", 8 | ". J " ! S1 "! "5 !" (c1 0), " 8 " , ." S1 " ! /" 5 " . 4/", !5!" ! Y2n 10n 3, 8 " 5 ! Sj , ".. " . ) . %% 5.2.4. + n 1 c$ * Y3n, $ * n2 f
g
;
( * * ,
L(Y3n) = 20n + 7 D(Y3n) = 8: . 4!" (x1 x2 ), (y1 y2 ) (z 1 z2 ) | n- .!. Y3n !!" ! Ye n ! Y2n+1. . 7 ! Ye n .! " p1 p2 q1 q2 , ." p1 p2 = x1 + y1 z 2 q1 q2 = x2 + y2 z 1 $" ! ! Y3n+1 .! .! r = (r1 r2 ), ! (n + 1)- .! (p1 p2) (q2 q1 ). ,. , ." L(Y3n ) = 2L(Ye n ) + L(Y2n+1) = 20n + 7 D(Y3n ) = D(Ye n ) + D(Y2n+1 ) = 8: ;
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5.3. 7 3 +
103
) .
)% 5.2.1. + * N n 1 $ * YNn, $ N
n- ( *
* ,
N
! 1
L(YNn ) . 10N(n + 1) D(YNn ) 5 log2 N : . 4!" x1 : : : xN | n- .!. YNn !" !""!" ! ! ;7 8". >! x1 : : : xN 5 .! ! ! !" 5.2.3 ! Y2n. - "" ."! 21 N (n + 1)- .!. 1 .! ! 5 .! ! ". . " " " , !""! !8 .!. , 8 !5!" ! YNn . I " 6.1.2 8 ! ", ." .! " 2 log2 N , " 5 " "! ! 8 " , " D(YNn ) 5 log2 N : J !5!" ! YNn . 45 R = log2 N . > Ni . .!" .!, !";7! ! i- ". )8 " , ." i Ni 12 Ni;1 + 1 < 12 N + 1: 1 i- " ! "! Ni ! Y2n+i;1, /"
d
d
e
d
e
e
d
e
R X
R X
i (n + i 1) 21 N + 1 i=1 i=1 R 1 i X i 1 N(n 1) + 2 Ni + (n + i 1) 10 i=1 2 10N(n 1) + 20N + 5R(2n + R): " , N , !5!" ! YNn ! !" L(YNn ) 10N(n + 1)(1 + o(1)): J .
L(YNn )
10(n + i 1)Ni 10 ;
;
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;
! 1
*& 5.2.1. 4!"" !, ;7; " n- .! .! !
" 5 !. 5.2.2. 4!"" !, ;7; ." n- .! .! ! " 5 !. 5.2.3. 4!"" !, .!;7; ! " n- .!, 8 " !. 5.2.4. 4" , ." n !7!"" !, .!;7 ! n n- .!, 8 " ! "".! 2 6 2 log2 n. ! 1
5.3. 6 1 *
!!" ! !" ! 5;7 . n- .!.
1. !" !"" ! .;"! ! 5 "
!" ". . !5" 5"! 5 "8 !5". $", 7 !" Yk , .!"! ! n .2! .!.
104
5. #$ $
.!"! log2 n 28. 1 28 .!"! n2 ! n- .!, " 28 | n4 ! (n + 2)- .!, ". . 4/" .! ! " "! ! !!"7 . d
dlog 2 ne X k=1
e
dlog 1 2 ne X X 1 5(n + 2k;1) 2nk < 5n2 21k + 5n 2 k=1
k=1
5n2 + 5n log2 n
/", /" 8 ! ! " 2n log2 n . J , " !" ! ;7 "". )% 5.3.1. $ * Mn, $ ( * n- ( * , n L(Mn ) . 6n2 D(Mn ) . 2n log2 n: - 2 " "! ! M4 . 4/" !!" . 4!" M4 5" x = (x1 x2 x3 x4) y = (y1 y2 y3 y4 ). - M4 5 x yi !"". ! " 16 /" :;. $" .!;"! ! z1 = xy1 + 2xy2 z2 = xy3 + 2xy4 . - .! 5 ! 5 !!"" !5 8 4- 8 8 3- 8 .!. 4/" 5 ! 5" " .! ! 16 /" 8 6. 1 .!"! ! 4z2 + z1 . A 5 !!"" ! 5- 8 3- 8 .!. 4/" 5" " .! ! 18 /" 8 7. J , !5!" 8 M4 ;"! !" L(M4 ) = 66 D(M4 ) = 14: J !!" ! , 5;7 .!. 08 " 5.3.1 .! 8 ! " " 5.2.1. ! ;7 "5 . )% 5.3.2. $ * M0n, $ ( * n- ( * * , n L(M0n ) . 44n2 D(M0n ) . 5 log2 n: J 5 " !" !" "5 !5!" 5 . .!. )% 5.3.3. $ * Me n, $ ( * n- ( * "* , n L(Me n ) . 7n2 D(Me n ) . 9 log2 n: ' 7 ! , ! Me n ! 5 " !" " 8 !""! ! Y02n, Y2k Y 2n. 2. 5!" ! !!" 2 ! 5 n- .! n2 . - !"7 " . 8" 5 .!, ;7 !"" ." / ! . 1 !" /" ! !!"" (n log2 n log2 log2 n) /", 8 log2 n. -!" ! ", !" " ! !"". !5 . 4/" 5 !!" !; !"; !";, ;7; !"" "!" 8 ! , !!"7 . n2 /". 45, ." " !" ! ;7 ". )% 5.3.4. n = 2k + 2. . n $ * M?n , $ ( * n- ( * "* , d
e
! 1
! 1
! 1
O
! 1
L(M?n ) . 95nlog2 3 D(M?n ) . 17 log2 n:
5.3. 7 3 +
105
" "! !" ! !" 5 5.3.1 5 .!. 4!" !, 5;7 .!, . ! !"8 .!"8 !. | !" ! M 4 , 5;7; 4- .! x y. 4!" x1 y1 5" , x2 y2 "" " !5". 4 xy .! ! ;7 . 1. 4 7 ! Y4 (!". 99) .! !" r x = x1 x2 ry = y1 y2 , sx sy . 2. 4 7 ! M4 p = r x ry !". 3. 45" ; p1 "" ; p2 " xy .!
: p1 = (sx sy ) p p2 = (sx sy ) p: )8 " , ." j
j
;
;
j
j
L(M 4 ) 2L(Y4) + L(M4 ) + 17 = 141 D(M 4 ) D(Y4 ) + D(M4 ) + 2 = 27:
(5.3.1)
%% 5.3.1.
n $ ( *
= 2k + 2. . k 1 $ * M n , n- ( * * ,
L(M 2k +2 ) 95 3k 90 2k + 41 D(M 2k +2 ) 13k + 14:
;
(5.3.2)
. ) 5 k. - ! 5 !"; 2 ! M 4 , 5;7; 4- .!. )8 " , ." . (5:3:1) ";" !" (5:3:2) k = 1. 45, ." ! !" (5:3:2) " k 1 ! " ! !" k + 1. 4!" n = 2k +2, x y| (2n 2)- .!. 4 !" x = x2 2n;1 + x1 y = y2 2n;1 + y1 8 5 .! x1 , x2, y1, y2 !!"" . n 1 . J8
;
;
xy = x2y2 22n;2 + (x2y1 + x1 y2)2n;1 + x1 y1 : ," ! !5 xy . !" xy = x2 y222n;2 + (x2y2 + x1 y1)2n;1 (x2 x1 )(y2 y1)2n;1 + x1 y1 : (5.3.3) ;
;
;
" , 5 (2n 2)- .! ! "! 5 (n 1)- .!, 5; n- .! ! !5. ! !" ! M 2n;2 ! 5.3.1. 48, ." /" ! ! .! . ! ", ." ! "! 8 5"! ! { ! . 1. 4 ! S1 "! / ! M n .!" z 1 = x2 y2 (n 1)- .!. " , ;
;
;
L(S1 ) = L(M n )
D(S1 ) = D(M n ):
2. 4 ! S2 "! / ! M n .!" (n 1)- .! p2 p1 ", ." z 2 = p22n;1 + p1 = x1 y1 . " , ;
L(S2 ) = L(M n ) D(S2 ) = D(M n ):
106
5. #$ $
x2
r r
rrx
x2 y2
x1y1
r
r r r r
y2 y1
1
x 2 x1
r
z6
y2 y1
;
;
z5
z7 !. 5.3.1 3. 4 ! S3 "! / ! Y2n;1 .!" !" z 3 = x2 x1 (n 1)- .!. " , L(S3 ) 10n 13 D(S3 ) = 5: 4. 4 ! S4 "! / ! Y2n;1 .!" !" z 4 = y2 y 1 (n 1)- .!. " , L(S4 ) 10n 13 D(S4 ) = 5: 5. 4 ! S5 "! / ! M n .!" z5 = z 3 z 4 n- .!. " , L(S5 ) = L(M n ) D(S5 ) = D(M n ): 6. 4 ! S6 "! / ! Y22n;2 .!" ! z 6 = z 1 + z 2 (2n 2)- .!. " , L(S6 ) 20n 23 D(S6 ) = 5: 7. 4 ! S7 "! / ! Y33n;3 .!" ! z7 " .!: (3n 3)- 8 .! (z 1 2n;1 + p2), .;78! " ! "" .! z 1 2n;1 p2 " !;"!M 2n- 8 .! z 5 M (2n 1)- 8 .! z 6. 9." . 5.2.1 !" 100 8 " , ." L(S7 ) 50n 33 D(S7 ) = 8: !5!" ! !, , ." " !" " !" ;
;
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;
;
;
L(M 2n;2) =
7 X
i=1
;
;
L(Si ) 3L(M n ) + 90n 82:
;
I /"8 !" 5 . ; L(M 2k+1 +2 ) 3 95 3k 90 2k + 41 + 90 2k 82 = = 95 3k+1 90 2k+1 + 41: G 8 ! M 2k+1 +2 !" 5 8 ! ", ." D(M 2k+1 +2 ) D(M 2k +2 ) + 13 ," ! !" . " !" . ) .
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5.4. #
107
*& 5.3.1. 4!"" ! 5 6- . .! ! 5 2 8 .
5.3.2. 4!"" ! Mnm, .!;7; n- 8 m- 8 .!. 5.3.3. 4!"" !, .!;7; .! ; 2n. 5.3.4. 4!"" !, .!;7; .! ; 2n 1. 5.3.5. 4!"" !, .!;7; " 8 n- 8 .!. 5.3.6. 4" , ." !7!"" ! 5 4- .! 8 " ! " 9. 5.3.7. 4" , ." n !7!"" !, .!;7 n- .!, !5!" " ! " 120nlog2 3 . ;
! 1
5.4. !
4!" x = (x1 : : : xn) | !"" .!. x "! !" 8 !" .. " !"."! .!" !!" .!" 28 .! 8" "! 5 " .. - /" 8 " !" ! , !";7 ;7 2 !5!" 8 . $" " , ." !" ! 8" " ! !" " , !"" .!. 1. 1 B n "! ! i j ! 1 i < j n. ! & ! n n "! ! $ * , ! " . u v "8 5 !. - ! ;7 " "! !" /
" !";7 !. H" ! !""! /" ! ! . 1 5 & ! .!"! :; , " | :; . J /" ! (!. ! 5.4.1) " . )8 " , ." !. 5.4.1 " "! !, !";7 . )% 5.4.1. $ * S2k , $ 2k,
r
f_
g
r
_
L(S2k ) = k(k 1)2k;2 + 2k 1 ;
;
D(S2k ) = 21 k(k + 1):
. . !" ! S2n2n, : ;7; . (u1 : : : u2n) (v1 : : : v2n) . (w1 : : : w4n). S2n2n "! (2n 2n)-! ."-."8 ! !""! ". - ! 5" ! S11, .;7 /" !!"7 !"8 ". ,. , ." L(S11 ) = 2
D(S11) = 1:
(5.4.1)
4 5, ." ! Snn !". J8 ! S2n2n, !" " !" ! 5.4.2, !""! ! ;7 . 1. I /" ! ." !!";"! . (u1 u3 : : : u2n;1) (v1 v3 : : : v2n;1), " !;"! ! Snn . (p1 : : : p2n).
108
5. #$ $
2. I /" ! ." !!";"! . (u2 u4 : : : u2n) (v2 v4 : : : v2n), " !;"! . (q1 : : : q2n) ! Snn. 3. 1 (p1 : : : p2n) (q1 : : : q2n) ;"! . (w1 : : : w4n) w1 = p1 , w2i = pi+1&qi , w2i+1 = pi+1 qi i = 1 2 : : : 2n 1, w4m = q2m . u2n;1 v2n;1 u2 v2 u2n v2n u1 v1
r rp p p p p p p p p r r
r rp p p p p p p p p r r
_
1." !
;
>" !
pi+1
qi
ppppppp
pppppppppp
w1
w2i w2i+1 w4n !. 5.4.2 45, ." ! S2n2n !"" " . n-/" . . G !", ." u = (u1 : : : u2n) !!"" k 2n k , v = (v1 : : : v2n) | l 2n l . J8 (p1 : : : p2n), .2! ! ! ." /" u v, " t = k=2 + l=2 2n t . J 5 8 " , ." (q1 : : : q2n), .2! ! ! ." /" u v, " !!"" s = k=2 + l=2 2n s . J ; 8 x !" x x 5" " " ; , " . ;
;
d
b
c
e
b
d
e
;
c
;
d e ; b
c
R = ( k=2 + l=2 ) ( k=2 + l=2 ) d
e
d
e
;
b
c
b
c
5" " " " .: 0, 1 2. A! R = 0 R = 1, " (p1 q1 : : : p2n q2n) " . . A! R = 2, " /" k + l 1 !" " !"" , (k + l)- !" " !"" , (k+l+1)- !" " !"" ! , !"2! !" | . I !" R = 2 8 ! ", ." .! k l ." , , ! " , k + l = 2h | ." .!. J8 , ;
w2h = ph+1 &qh = qh = 0
w2h+1 = ph+1 qh = ph+1 = 1 _
".. ", 7! ! ! , " !" ! ; (p1 q1 : : : p2n q2n). )8 " , ." ! /"8 !"" . . " , ! S2n2n !"" : " . . . J !5!" 8 /" ! . I !" ! L(S2n2n) = 2L(Snn) + (4n 2) D(S2n2n) = D(Snn ) + 1: ;
(5.4.2) (5.4.3)
I k 5, ." k 0 !5!" ! S2k 2k ! !"
L(S2k 2k ) = k2k+1 + 2:
(5.4.4)
5.4. #
109
G!"" , k = 0 !" (5:4:4) ! " (5:4:1). G !", ." (5:4:4) ! k m 1. J8 /"8 5 !" (5:4:2) L(S2m 2m ) = 2L(S2m;1 2m;1 ) + 2 2m 2 = = 2((m 1)2m + 2) + 2 2m 2 = m2m+1 + 2: " , (5:4:4) ! ! k 0. 08. , (5:4:3) (5:4:1) ! k 0 8 S2k 2k D(S2k 2k ) = k + 1: (5.4.5) J , "5 ", !" ! S4n, !";7; 4n /". - ! 5 ! S2 , !";7; /" !!"7; 8 ". ,. , ." L(S2 ) = 2 D(S2 ) = 1: (5.4.6) G !", ." ! S2n !". J8 ! S4n !!" !, !";7 2n-/" , ! ."-."8 ! 2n-/" . '!" ! !" ! 5.4.3. x2n;1 x2n y1 y2 y2n;1 y2n x1 x2
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110
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" , (5:4:9) ! ! k 1. I (5:4:5) (5:4:8) 8 S2k2k k X D(S2k ) = D(S2k;1 ) + k = j = 12 k(k + 1): j =1
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L(Sn ) = n2n;1 D(Sn ) = n: . Sn, !5!" 8 " ";" ! " !" n. S1 !""! " , !!"" 8 /" !5. 4 5, ." ! Sn;1 !". -! ! /" ! !" ! Sn , 5;7 " Pn " x = (x0 : : : x2n;1 ). 45 x1 = (x0 : : : x2n;1 ;1 x2 = (x2n;1 : : : x2n;1 ). J8 (5:5:1) , ."
Pn;1 0 x1 = Pn;1x1 Pn;1 Pn;1 x2 Pn;1x1 Pn;1x2 ".. .! Pnx ! "! .!; Pn;1x1 Pn;1x2 ! ;7 !5; 2n;1. 4/" ! Sn !"
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L(Sn ) = 2L(Sn;1) + 2n;1 = 2 2L(Sn;2) + 2n;2 + 2n;1 = = 4L(Sn;2) + 2 2n;1 = : : : = 2n;1L(S1 ) + (n 1)2n;1 = n2n;1: G 8 Sn !. 5.5.1 , ." D(Sn ) = D(Sn;1 ) + 1 = n: J .
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5.6. .+ $
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!"" ". '5 !!" ! " n , i- ;. . " ei !" "8 Rn ! En = e1 : : : en . 4 7 " ! 5 " !!" Z = z1 : : : zn .! " !"!" Rn ! " ! En. - !. ! !"" ! " !5 ." n- ". * .! " , !"". 5 !!" Z, "! !5!" ; /" !!" ."! . Lf+;g (Z). 1 , !!" Z2 = (1 1) (1 1) 5 "! !5 z 1 = e1 + e2 ." z 2 = e1 e2 . J 5 5 8 ", ".8 " " !, , , , ", . , ." !5!" !!" Z2 . G !5!" 5 !!" Z ! ! ;7 "5 . %% 5.6.2. + Lf+;g(Z) " * Z = z1 : : : zn $
Lf+;g (Z) log2 det(z 1 : : : z n) : . G !", ." !7!"" ! " !" .!, 5;7 !!" " Z t 28, 5 " !!"" .! ! !" ". 45 y1 = e1 : : : yn = en 5 8 j 1 : : : t . yn+j . ", .! j- 28. > rk . ! . ", .!8 " !!" y1 : : : yn+k , ".. rk = 1 i maxi n+k det(yi1 yi2 : : : yin ) : f
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, ! !" L(S) log2 det Hn : J " Hn "8 !"., ! " ; !" 2n, " 8 " , ." Hn Hn = (2n) 2n E2n . 4/" det Hn = 2n2n , , ! " , L(S) log2 2n2n = n2n;1: J , .8 !" 2 5.6.1, . ! ;7 "". j
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118
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,. , ." G !!"" :; y1 y2 y2 y3 , ! ! !!" F ! ;7 !" F(x1 x2 x3 x4 x5 x6) = G(x1 x2 x3 x4 x5 x6): -! ! !" ! , .!;7 !!" F. . !" ! S1 .!;7; !!" :; G ! S2 .!;7; y1 y2 y3 . $" i- ! S2 ;. i- ! S1 . )8 " , ." .2! ! !!"" " :;" .!" !!" F. !!" 8 7"! !., 8 !!" :; !". "!" ! 5!" . G!"" , !" M1 : : : Mk " !;7! 5!" 5!" x1 ::: xn , ." 8 5!" ", " 5; :;; !!" F, /" 5 xi 5" /" 5!". 4 x1 : : : xn ! ", ." xj 1 : : : xjij !!" 5!" Mj . 4! yj = xj 1 xji xjij 8 j = 1 2 : : : k, !!" F . ; !!" :; G(y1 : : : yk ) ";, ." F(x1 : : : xn) = G(x11 x1i1 : : : xk1 xkik ): , .!;7; !!" F, !!" , !!" !. 4 /" ! ! 5" n , ;. x1 : : : xn, k . 1 j- .!"! :; , 7 5!" Mj . -" ! .!" !!" G i- ;. i- ! . -! ! !" " " !" ! ;7 . %% 6.2.1. m log2 n 2 log2 log2 n. . n ( (' " F (m n) * S,
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!. 6.2.1 ! ;7 !" !!" " Un: uij = uij +2n;1 ! 2 i n 1 j 2n;1 1: 1 !5!" L(Un ) 8 D(Un ) ! Un. ,. , ." L(U2) = 2 D(U2 ) = 1. I !" ! Un , ." L(Un ) = L(Un;1) + L(D2n;1 ) + 2n;1 1 = L(Un;1) + 2(2n;1 1) D(Un ) = max(1 + D(Un;1 ) D(D2n;1 )) = max(1 + D(Un;1) n 1): 4 5; L(Un;1) = 2(2n;1 (n 1) 1) D(Un;1) = n 2: " , L(Un;1) = 2(2n;1 (n 1) 1) + 2(2n;1 1) = 2(2n n 1) D(Un;1) = max(1 + (n 2) n 1) = n 1: - !5!" 8 . 45 . " 5 !5!" !!" Un , 5 5 , ;7 " 5 !!"" "!. 4!" S | !. -2 v ! S ! xi, ! v xi ! ;, 7 . 2 v0 = xi v1 : : : vk;1 vk = v, " 5 2 vj , 8 j = 0 1 : : : k 1, " 8 " vj +1, 2 v | ". G , !!" ! , 5 ! ! 4.4.3 !" 87. - ! x y ;" 2 ", z ! "! ! 2 8 ". 1 " ! x y ;" 7; 2 8 " | /", . , z " 2 8 ". %% 6.2.3. 4 * S, $ (' " F , *
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(' " F , " (m n) " : (i) " M (ii) k ( * " , ( * * ". .
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!"7 ! ; 57 V. 4" , ." k : a) ! FV "! " f, ." L(f) & 2kk M b) 5 f FV! !" L(f) . 2kk : 6.5.2. 4!" V !"!" B n !" k, FV| 5!" P2(n), !!"7 ! , !" !5 !! !"!" B n V. ," maxL(f), 8 ! "! ! FV !, ." n k . 6.5.3. 4!" F(n N) | 5!" P2(n), !!"7 ! ; ! 2 N, ".. f() = 0 5 f F(n N) ! > N. 4" , .": a) ! F(n N) "! " f, ." L(f) & logN2 N M b) 5 f F (n N) ! !" L(f) . logN2 N : 6.5.4. 4!" F(n) | 5!" P2(n), !!"7 ! ; ."8 !. 4" , .": a) ! F(n) "! " f, ." L(f) 2nn;1 M b) 5 f F (n) ! !" L(f) . 2nn;1 : 6.5.5. 4!" G(n m) | 5!" P2(n), !!"7 ! " f, ." f() = 0, ! = 0 (mod m). ," maxL(f), 8 ! "! !
G(n m) !, ." m = o(n). ! 1
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2
j
j
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j 6
! 1
132
6. 0 $+ % $
6.5.6. 4" , ." n 5 f P2(n) ! !": b) Lf&:g (f) . 2nn M c) Lf&1g (f) . 2nn M a) Lf_:g (f) . 2nn M e) Lf#g (f) . 2nn . d) Lj (f) . 2nn M ! 1
6.5.7. 4!" B P2(2) SB] = P2. 4" , ." n 5 f P2(n) ! !" LfBg (f) . 2nn : 6.5.8. ," maxL(f), 8 ! "! ! ! !" n . 6.5.9. ," maxL(f), 8 ! "! ! " f : B n n n B ", ." f(x) = 1 ! x B . 6.5.10. 4" , ." 5 f P2(n) ! N ! !" L(f) . logNn2 N . 6.5.11. G" " 6.5.2. 6.5.12. G" 5;; " 6.5.3. 6.5.13. f(x1 : : : x2n) !". "!" n . ," L(f). 6.5.14. ," ! !5!" !".! , !7 " n , !: a) n = 3, b) n = 4, c) n = 8, d) n = 16.
! 1
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7.
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1. 4!" X = x1 : : : xn | 5!" ! . - 5!" Y = y1 : : : yl 5!" Z = z1 : : : zm . 4 5!" Y ", 5!" Z | f
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a = f(b c):
4; a * 8 " p, b, c | * /"8 ". 4!" " a X Y Z. 7 p 5 p : Stop(a): 4; a " !" p. 4! " !" P = p1 : : : pi : : : pL , !!"7 " " !", "! $ , ! ; j 1 2 : : : L 5 " pj !" ! , "8 8 " pi , 8 i < j. 1"7! 8 "" !" " t = 0 1 2 : : :, " . ! " . " . $. yi (xM t) " yi . zj (xM t) zj 8 P " t ! x = (x1 : : : xn) ": - . " t = 0 . ! " !." M A! " yi ( zj ) "! " pt , " 5 2
2 f
g
yi (xM t) = yi(xM t ; 1)
zj (xM t) = zj (xM t ; 1)M
A! " yi ( zj ) "! " pt , b(xM t 1) c(xM t 1) | . " pt " t 1, " 5
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yi (xM t) = ft (b(xM t ; 1) c(xM t ; 1)) zj (xM t) = ft (b(xM t ; 1) c(xM t ; 1)):
$. " pt 8 P ! x = (x1 ::: xn) . 8 " t . . pt (x). > n(p) . " p 8 P, ". . n(pi ) = i. 4!" pt1 : : : ptr | ! " !" P, . t1 < < tr . J8 . sj ." j- " !" 8 P, ".. sj ptj . " pi ( ; xl ) " !" sj , n(sj ) = r, . . qj , !: (i) " pi ( xl ) "! " sj . (ii) ! " pt , i < t < r, " ", "8 ! " ! " pi . 8" , ." k- " !" sk !"" .! 8 P x, ! q1 (x) = = qk;1(x) = 0 qk (x) = 1: "" !" 8 P x . . P(x) 8 l-; " Pl (x) ! ;7 :
Pl (x) =
(
zl (xM tk ) zl (xM L)
! q1 (x) = ! q1 (x) =
= qk;1(x) = 0 qk (x) = 1 = qk (x) = 0
7.1. * 6 $
135
".. Pl (x) .; l- zl " !" 8 . )8 " , ." Pl (x) = q1 (x)zl (xM t1) q1(x)q2 (x)zl (xM t2) : : : : : : q1(x)q2 (x) qk;1(x)qk (x)zl (xM tk) : : : (7.1.1) : : : q1(x)q2 (x) qr;1 (x)qr (x)zl (xM tr ) q1 (x)q2(x) qr (x)zl (xM L): ' % 7.1.1. !!" " 8 , .!;7 :;; ." . , " 5 /" 8 ! 5 ! ;7 " !" : p1 : z = 1 z = x1 x2 y1 = x1 x2 p2 : Stop(x1 ) Stop(z) y2 = x3 x4 p3 : Stop(x2 ) z = x3 x4 z = y1 y2 p4 : Stop(x3 ) p5 : Stop(x4 ) p6 : z = 0 4 8 !!"" 2!" " "" ! ;7 . . !"! . . $" ! " ;"! ! !" xi. A! , " " !" 7" " 8 . A! x1 = 0, " ." " " " !", " !; . 7" " 8 , ! x2 = 1. A! x2 = 0, " 8. "" "" " !", ", ! x3 = 0, | ."" . A! " !" " " 8 , ".. ! ! ;, " "! ! " 8 , " !" .. I! (7:1:1) 5 !, ." 8 !"" .!" :;; ." : P(x) = x1 1 x1 x2 1 x1x2 x3 1 x1x2x3 x4 1 x1x2 x3 x4 0 = = x1 x1 x2 x1 x2 x3 x1 x2x3 x4 = x1 x1x2 x1 x2(x3 x3x4 ) = = x1 x1 x2 x1 x2 (x3 x4) = x1 x1(x2 x2(x3 x4)) = = x1 x1 (x2 x3 x4) = x1 x2 x3 x4 : -" 8 !!"" " ". 08. . P(x) = (x1 x2)(x1 x2 ) (x1 x2 )(x3 x4) = x1 x2 x3 x4: J" 8 !!"" " " , /", !7!" "! ! /". ,. , ." " 5 .!" :;; x1 , x2, x3 x4 . 2. C(P) 8 P .! " /" 8 . 4 TP (x) 8 P x n(sj ) ", ." qj (x) = 1, ". . /" .! ", !" 8 . A! ! qj (x) = 0, " ;"! ! " 8 /" !. TP (x) = C(P). -. X T(P) = 2;n TP (x) 8 ! "! ! . n, 8 P. A! "8 8 " f ; 8 .8 x ! !" f(x) = P(x), " 8" , ." 8 P .!" " f. -. T(f) = minT (P) _
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136
7. # % 3 !
8 "! ! 8, .!;7 f, ( ) " f. 48 P, .!;7; " f, " ! !" T (P) = T(f), 8. -. C(f) = minC(P) 8 "! ! 8, .!;7 f, " f. -. C(f) "" , .! f 2 !., /" C(f) " 5 " !5!" ; 2 !.. ,"", ." "7! 8, ! 57 " !" .!;7
;, ".; " ! , "! . ! /", ! " !!"" ! . !"
. 4/" ! !5!" ; f(x1 : : : xn), !7!" !7 . " , 2 ! !5!", ".. T(f(x1 : : : xn)) L(f(x1 : : : xn)):
' % 7.1.2. !!" 8 P1 P2, .!;7 !!"
| :; :; ." . - /" 8 p1 : z1 = x1 x2 z1 = x1 x2 p2 : z2 = 0 z1 = z1 x3 p3 : Stop(z1) z1 = z1 x4 p4 : z1 = x3 x4 z2 = x1 & x2 p5 : Stop(z1) z2 = z2 & x3 p6 : z1 = x1 x3 z2 = z2 & x4 p7 : z2 = x1 & x3 :; .!"! z1, :; | z2. )8 " , ." !5!" " 8 , !""!", ! 2!". 1 ! " . - 8 " !" 7" " ! : (0100), (0101), (0110), (0111), (1000), (1001), (1010), (1011)M " " !" | ." : (0001), (0001), (1101), (1111)M , !"2! ." (0000), (0011), (1100) (1111) ;"! ! " 8 . 4/" ! 8 " 8 P1 1 3 8 + 5 4 + 7 5 = 9 : T(P1 ) = 16 2 -" 8 !!"" " ", , ! " , ! " /" 8 ! " ! !5!" ;, ". . T (P2) = 6. 3. 4!" a "! " pi 8 P, b | 8 " pj /" 8 . 8" , ." " pi .!"! f(x), ! a(xM i 1) = f(x) ! . ! , " . a . 08. !5, ." " pj .!"!
h(x), ! b(xM j) = h(x) ! . ! , " . b . 8" , ." " pi pj 8 ;" 7 , ! .!;"! .
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7.2. $
137
! 8 5!" 8 5, ." ; 8 . !5!" ! 8 " 5" " ; 8. 48 P , !: 5 8 " 8 P "! "5 !" !"M " !" 8 P ;" 78 .
%%0 7.1.1.
( P ( P , : (i) T(P0) T (P), C(P0) C(P)M (ii) P0(x1 : : : xn) = P(x1 : : : xn) * (x1 : : : xn). . 4!" P | 8, x = (x1 : : : xn) | ! , pt : a = g(b c) | ", "8 .!"! -
!" , ".. b(xM t 1) = const. - /" !. a(xM t) !" " " c(xM t 1), , ! " , !7!"" h ", ." h(c(xM t 1)) = g(b(xM t 1) c(xM t 1)). $ P " pt " p0t : a = h(c), . 8, ";7; ! (i) (ii). G !" " , ." 8 P "! " !" pi pj , i < j, ! " 5 a. A! a(xM i 1) = a(xM j 1) = 1, " " pi !"" 8 , " pj "!. A! a(xM j 1) = 0, " " pj !"" 8 . " , " pj 5" " 8 P, /" !5"! ". ) . )8 " , ." ! 8 , !!" 7.1.1 7.1.2, ;"! . G !!"" " 8 . ;
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;; ." . 7.1.3. 4" , ." ! !5!" !!" , !!"7 n-!" :; :;, !7 " " 5 , ! " ". 7.1.4. 4" , ." ; 8 ! n .! " !" ! " 21 (L + n), 8 L | !5!" 8 .
7.2. !
H
" , ! ! 5!" ; !.8 7 .!, .;" " ! 5 .! " . ( 8 4.2, !". 77) , ." ! !5!" 5 " ! " ." , . !7!";" , 8! 2 (x1 x2 x3) (!. 8 4.4, !". 86), !5!" " ." . G 5, ." ! !5!" ; " ! " 2 21 . 4!" f | " . 5 f f(x1 x2 x3) = x1f1 (x2 x3) x1f2 (x2 x3): )8 " , ." 8 P p1 : z = f1 (x2 x3) p2 : Stop(x1 ) p3 : z = f2 (x2 x3) _
138
7. # % 3 !
.!" ; f. G!"" , !""!" ! , .! "7! 8 ! ! !", P(x) ! !" P(x) = q1 (x)z(xM 1) _ q1 (x)z(xM 3) = x1 f1(x2 x3) _ x1f2 (x2 x3):
1 ." | (100) (101) (110) (111) | 8 P " !", !" ." | ". 4/" T (f) T(P) = 18 (2 4 + 3 4) = 2 21 : J 5, ." ! !5!" 8! 2 (x1 x2 x3) 2 12 . 4!" P | 8, .!;7 2 (x1 x2 x3) ! ! . A! " !" s1 "! "" " P, " . , ." T(P) 3. J ; 8 !" 5 !"" ", " 8, ." s1 "! " " P. J " !" !" T(2 (x1 x2 x3)) 2 21 !"". " , ." P " !" 7" .! . ." ! , " /" !. T (P) 81 (4 2 + 4 3) = 2 21 . )8 " , ." 8" " !" 5" " , ! , ".. . 8 P " ! ;7 :
p1 : p2 :
z = f(x1 x2) Stop(z)
z = f(x1 x2) Stop(xi )
8 i 1 2 3 . - !. ! TP (x) = 2, " P(x) = 1. " , " p2 7" .! " : (110) (111). - " !. . , ." p2 7" .! ." . 2 f
g
*& 7.2.1. 1" ! " ! !5!" " 2 21 . 7.2.2. 1" ! 5 ., " 5" " ! !5!"
" . 7.2.3. 1" ! " f(x y z), !7!" !7 " ! ! , " T(f) = L(f). 7.2.4. 1" T(x1 x2 x3 x4). 7.2.5. 1" T(x1&x2&x3&x4). 7.2.6. 1" T(x1 x2 x3 x4). _
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7.3. !*
- 7 8 6.5 , ." ! ".!" ; !"8 5" !5!" ; !".! .! !7!" 8". - !. ! !5!" !" | ! !5!" !" " .! 8" , ! ".!" ; !"8 5" ! " ! . n (f) + 2, 8 . (f) ."! ! .! ! " !, " f " .. 1 , x1 xn . , 8! n | (n + 1)=2 , n-!" :; :; | n. )% 7.3.1. + " f(x1 : : : xn) n ;
d
e
! 1
T(f) n (f) + 2:
;
7.3. # +
139
. *3 . 4!" f | !".! , !7 " n 8". !!" !.: (f) > n 2 (f) n 2. (1) - !. 1 n (f) + 2 < 4. 4/" 5 " ! " . 8 !" T (f) 1, ! 8 ; . (2) - " !. n (f) 1 41 (n (f)+2). 4/" " !" 5 " !"". " , ." T(f) n (f) 1. 4!" P | 8, .!;7 f, s1 | " !" /" 8 , = (1 : : : n) | , " " s1 !"" .!. A! " s1 "! k- " P k < n (f) 1, " !!" q1 (x) z(xM k 1) , .! k " P, !7!" !" . " m = n (f) 1 . J f !".! , " 8. 7!" 8, ." /" ;"! x1 : : : xm . - /" !. !" 8 P 8 ! " !7!" " !" 1 : : : m , ." " s1 !"" " P !!" " . xm+1 : : : xn. " , ; . m+1 : : : n /" "! !" f(1 : : :m m+1 : : : n) = 8 | !"". - "5 , "! " m+1 : : : n, ." P(1 : : :m 0 : : : 0) = P(1 : : :m m+1 : : : n ): A! " !7!"", " f " . n m = (f) + 1 ! " !, ." "." ; . (f). 42 ".;. " , T (f) k n (f) 1. . . J 5, ." ; !".! f, !7 " n , ! !5!" T (f) !" (n (f) + 2). !!" !.: (f) n2 (f) > n2 . (1) - !. n (f) n2 , .! f !"". ! " .; ! /". ,. , ." T(f) L(f) = (n) = (n (f) + 2): (2) !!" " !.. G !", ." ! ! " !" !, " !"8"! . (f), ."! ! h-8 !, . f /" ! . 45 m = n (f) + 1. J8 f(1 : : : n) = ; (1 : : : n), ! 57 h n (h + (f) 1) = m h . -! ! /" !!" f .!. , 2 8 P, .!;7; ; f. 45 n = (2m 1)t + k, 8 0 k < 2m 1. 48 P !" t + 1 ! " 8 P = P1 : : : Pj : : : PtPt+1 ";7 ! ;7 . 4 8 P1 !" . , .!" ! (2m 1) !"" .! ! /" ! 2 h 2 2m 1 (m h) = m + h 1. 4 5 j 2 3 : : : t 8 Pj .!" ! ;
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i=(2m;1)(j ;1)+1
xi
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2 f
O
O
2 f
g
O
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140
7. # % 3 !
, ! " 8 P. 1" " ! ", ." 8 P1 !"" .! A1 = 2n;(2m;1)
m+ h;1 2m ; 1 X
i
i=h
2n;1
(7.3.2)
, 5 8 Pj , ! j 2 ! 7 t, |
Aj =
! j ;1 ! mX +h;1 X 2m ;1 n ; (2 m ; 1) 2 ; Ai 2 i i=1 i=h
j ;1 ! 1 2n X Ai 2 i=1 ;
(7.3.3)
. I j 5, ." j X i=1
Ai 2n 2n;j
(7.3.4)
;
!8 j 1 2 : : :Pt . - ! (j = 1) 5 !" (7:3:2). G 5, ." si=1 Ai 2n 2n;s ; 8 s 1 : : : j 1 . J8 (7:3:3) 5 2 f
g
;
2 f
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j ;1 ! X 1 n Ai = Aj = Ai + Aj Ai + 2 2 i=1 i=1 i=1 i=1 j ;1 ! X 1 1 ;2n + 2n 2n;j +1 = 2n 2n;j : n = 2 2 + Ai 2 i=1
j ;1 X
j X
j ;1 X
;
;
;
P J ji=1 Ai 2n , " (7:3:4)
j X i=s
Ai =
j X i=1
Ai
sX ;1 ;
i=1
Ai 2n (2n 2n;s+1) = 2n;s+1:
;
;
(7.3.5)
J5 (7:3:4) 8 ! ", ." !" 8 P1 : : : Pt 7;" .! . 2n (1 2;t) , /" 8 Pt+1 "" . 2n;t . " , ." !" (7:3:5) !" (7:3:1), ;
1
0
j t X X 1 @ T(P) = 2n Aj C(Pi ) + 2n;tC(P)A = j =1 i=1 0
1
t X
0
t X t X
1
@ @ = 2(m) Aj j + t 2n;tA = 2(m) Ai + 2n A = n n j =1 j =1 i=j 0 1 0 1 t 1 X X @ = 2(m) 2n;j +1 + 2nA = (m) @ 21;j + 1A = (m): n j =1 j =1 O
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7.3. # + 8. 7!" 8, ." n | .": p1 : z = x1 x2 p2 : Stop(z) : : : : : :: : :: : :: : : pj : z = xj xj +1 pj +1 : Stop(z) : : : : : :: : :: : :: : : pn;3 : z = xn;3 xn;2 pn;2 : Stop(z) pn;1 : z = xn;1 xn )8 " , ."
141
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T (P_ )
1
n=2 X
0 1 X @
1
1@ j A= n;2j A < 6 j 2n j =1 2j 3 2 4 j =1 0 1 0 1 1X 1 1 1 1 1 X X 4 1 4 8 A @ A = 6@ i =6 j 1 1 = 6 3 4 3 = 3: 4 4 4 j =1 i=j j =1
;
45, ." 8 ! "".! . "., ! ! ;7 ". )% 7.3.2. n
T (x1 : : : xn) 83 : . J T(P_ ) 83 , " " !" " !"". " , ." T (x1 : : : xn) & 83 . G /"8 .! Dn = x1 xn 5, ." n 2 ; 8 Pn, .!;7 ; Dn , " !" !" T(Pn) 83 2n1;4 : (7.3.6) 4 n = 2 3 !" (7:3:6) ! . G !" " 5, ." "8 n 3. 1 , ." " ; 8 , .!;7 !" ; ; ;, 5" " " !"M " !" " ", "8 "! . A! " !" 8 Pn "! "" " Pn, " . , ." T(Pn) > 3. 4/" !"". !!"" !., 8 8 Pn " !" !"" " !". ".!" ; 5 " . !. ": p1 : z = '(x1 x2) z = '(x1 x2) z = '(x1 x2) p2 : Stop(x1 ) Stop(x3 ) Stop(z) 8 ' | " !" . 4! " !!" /" !.. 1. 48 Pn ; 8 Pn;1, !" !" x1 " !", " ! !" x1 = 0 8 " !"" .!. 4/" X X; TPn (0 x2 : : : xn) = TPn;1 (x2 : : : xn) + 1 : ! 1
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142
7. # % 3 !
$ ! ! "! ! 5 . xi. ,. , ." 8 Pn;1 .!" :;; (n 1) , /" 8 ! !" 1 X T (x : : : x ) T (D (x : : : x )): Pn;1 2 n n;1 2 n n 2 ;1 4/", n 3 7 !"2 5 X X T (Dn ) = 21n TPn (1 x2 ::: xn) + TPn (0 x2 ::: xn) 1 2n;1 2 + 2n;1;1 + T (D (x : : : x ) n;1 2 n 2n 3 + 1 T(D ) 3 + 1 8 1 2 2 n;1 2 2 3 2n;5 = = 83 2n1;4 + 61 > 38 2n1;4 : ;
;
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;
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" pt , .!;7 "5 !" 8 2, . " !" pj : Stop(a), ! t < j. I 7.1.1 ! ", ." "8 " 5" " " . A! " "! z1) , " 8 " , ." a(xM j 1) 8 " !" 5 !7!" !" " ! n . - " !. . :; n " ; . . , ;7! !7!" 8" , .! " !". " , T (P) n. 4". ! . ,. , ." !!" !. '(0 0) = 0, " . Pn(0 : : : 0) = 1. 4 8 Pn ; 8 Pn;1, !" !" x1 x2 , Pn !"2 5 " !", .2;! 8 . I 8 2 !!" 8 .!;7 :;; ! ", ." 8 Pn;2 ! 5" " (p1 p2 ) 2 . 8 Pn , /",
;
X
TPn (0 0 x3 : : : xn)
X;
TPn;2 (x3 : : : xn) + 2 :
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1) " ! " , % # % # , ! ! ! ! z.
7.4. 0 $+ % $ $ 4/", n 4
143
X X T(Dn ) = 21n TPn (1 2 x3 ::: xn) + TPn (0 0 x3 ::: xn) 1 _2 =1 1 3 2n;2 2 + 2n;2;2 + T(D (x : : : x ) n;2 3 n 2n 2 + 41 T(Dn;2) 2 + 41 83 2n1;6 = 83 2n1;4 :
;
;
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n 8" ! !5!" ! " ! . !5!" ;, ".. L(x1 x2 xn) = T(x1 x2 xn) = n 1: ! 1
;
*& 7.3.1. 4!" f P2(n), n 3 f !7!" !" " ! ! 8". 2
4" , ." T(f) > 2. 7.3.2. 4!" n . 4" , ." ! 1 : : : n ! ! "".! !": b) T(x(11 ) & : : :&x(nn ) ) 113 : a) T(x(11 ) : : : x(nn ) ) 38 7.3.3. 9" ; fn, !7!" !7; " n 8", " T (fn ) < T(Dn ). 7.3.4. 4" , ." T (x1 x2 xn) = n 1. ! 1
_
_
;
7.4. , !!* ! ! 5
1. !!" . ! !5!" " ." !" n . 45, ." ! !5!" ." 5 ! ".!" ; !"8 5" ! " ! . !5!" ;. )% 7.4.1. n . . : (i) " f , ( $ n * n;4 T (f) & 2 n M (ii) " f , ( $ n * n;1 T (f) . 2 n : . (i) 4!" f | n , P | 8, .!;7 f. '5 . x n, !!" . ! " 8 .!, !" !""!" 8 NP (x) ", ." 1 NP (x) 2nM NP (x) < NP (y ), ! TP (x) < TP (y)M NP (x) < NP (y), ! TP (x) = TP (y) x < y. , .! ! !5!" 5 " ! " . 2nn;4 . 4!" f | " , P | 8, .!;7 f. !!" x0 ", ." NP (x0) = 2n;1. J8 ! !5!" ! ", ." X X T(P) = 2;n TP (y) > 2;n TP (y) 21 TP (x0 ): (7.4.1) y y j N (y)>N (x0 ) ! 1
144
7. # % 3 !
4/", TP (x0 ) < 2T(f). J T(f)
2n;4
n , " 8 2n;4 2n;3
" , ."
TP (x 0 ) < 2 n = n : (7.4.2) '5 . "! TP (x0 ) " ! 8 P . " . 2n;1, !!"7 .
f " 8", " P " 2 " /" 8 x0 . , . . N0 .! . 8, !!"7 . n;T4 P (x0 ) ". J8 .! , ! !5!" " ! " n;1 2 2 . , N0 . n , 8. ! . N0 2 ); 8 P "! ! ! ! " pi , 5 " . "! ! ;7 : " " | 5 !8 ", " 5" " , " !"M !" fi , .! " ( " !" /" !"!) | !7!"" !8 16 . !" M , ", ;7! 8 " ( " !" /" !"!) | ! 8 P !!"" L ", " 7 .! " ! " L 8. 7!" 8, ." " ;"! .! " 1 L 1, !"! LM , ! ", ;7! " | 8, ." ! ;"! .! " L+1 L+n, /" 7 .! ! " (L + n)2 . J .! N, 8 .! . 8, !!"7 L ", ! !"
;
N 2 16 L (L + n)2 L (4(L + n))3L : (7.4.3) 4 !" (7:4:3) !" L . TP (x0 ) ." !" (7:4:2), ., ." n 5 " !" !"
;
3TP(x0 )
N0 (4 (TP (x0 ) + n))
32n;3 =n
n;3 4 2n +n
n ;3
232 :
" , .! , ! !5!" " ! " 2nn;4 , 2 . 232n;3 22n;1 = 2 87 2n = o 22n : J , ! !5!" ." 5 , !7 " n , 2 . 2nn;4 . 4 !" " . (ii) 1 , ." 5 . (1 : : :k ) !""!"" 8 Pk (1 : : :k ) = i=1 i 2k;i. 45 s = n log2 n . ; f 5 n s : j
j
b
;
c
;
f(x1 : : : xn) =
_
1 :::n;s
f(1 : : : n;s xn;s+1 : : : xn)x1 1 & : : :&xnn;;ss :
48, .!;7; ; f, !" ! ;7 P = P0 : : : Pj : : : P2n;s ;1 8 j = (1 : : :n;s) , Pj | 8, .!;7 ; fj (xn;s+1 : : : xn) = f(1 : : : n;s xn;s+1 : : : xn) j
j
7.4. 0 $+ % $ $
145
7;7 " 8 P, ! x1 1 & : : : &xnn;;ss = 1. J ! !5!" , !7 " s , ! "".! ! " 2ss , " C(Pj ) . 2ss . 4/" n;s
n;s
!
j j 2 X;1 2 X;1 X X 1 1 s s T(P) 2n 2 C(Pi) = 2n 2 C(Pi) . j =0 i=1 j =0 i=1 ;s ;1 s 2nX 2s 2(n;s) 1 2 2n;1 2n;1 : s j . 21n 2s 2 2 . 2n 2 s s n j =0
J . " !" 8 " 7.4.1 ". )% 7.4.2. n m , m = nO(1). . : (i)
(m n)- f n;2 T(f) & 2 n m M (ii) (m n)- f n;2 T(f) . 2 n m : 2. 8" , ." 8 P ! " " : d, ! .! " /" 8 d. ;; !5!" f .! 8, : " " ! " d, . . Td (f). , " 5 ; 5 .!" 8, ! ;7 : ". 4 /" ! " 8 5" !7!" !" " : " | . 2 " , " 2 ! . ' % 7.4.1. !!" 8 , .!;7 :;; 2!" : ! 1
p1 : p2 : p3 : p4 : p5 : p6 :
z=0
z = x1 &
y = x1 &x2 Stop(y)
z=z z=z
y = x3 &x4 Stop(y) z = x5 &x6
z=z z=z
x2 & x3 & x4 & x5 & x6
- 8 : ! " , " | . " 8 3 169 , " | 5. 1" " , ." ! ; 8 , .!;7 :;; 2!" , 2 ", ! : ! 8 " . - 7 !. " !" ! ;7 "". )% 7.4.3. n , d n. . : (i) " f , ( $ n * 2n;4 M Td (f) & log d ! 1
2
(ii) " f , ( $ n * 2n;1 : Td (f) . log 2d
146
7. # % 3 !
/" " ." ! "" " !" " 7.4.1. - " !" 5 " 7.4.3 !" (7:4:2) " !" "n;3 2 7.4.1 !"!" 7"! !" TP (x0 ) < log2 d , !" (7:4:3) " .! 8, !!"7 L " ! ;7 " : d: Nd 2 16 d (d + n)2 L (4(d + n))3L (8d)3L: - " !" !" ". !!"" " s | !!" !. 5" s = log2 d log2 log2 d .
;
b
;
c
*& 7.4.1. 4" , ." 5 f P2(n), ;7 . n;
" 5 , n ! !" T(f) . 2 n 2 . 7.4.2. 4" , ." 5 f Pn2;(n), ; ."8 3 2 !, n ! !" T (f) . n . 7.4.3. G" "5 (ii) " 7.4.2. 7.4.4. 4" , ." T (Un) n. 7.4.5. 4" , ." T1(x1& : : : &xn) = n 1. 7.4.6. G n-!" f " : a) T1 (f)M b) T2 (f)M c) T3 (f). ! 1
! 1
;
7.5. 1 ! 1 !
1. , ! ! 8" ." ! ! .! " .! 2 !.. 45 (f) = C(f) T(f) (n) = max(f) 8 ! "! ! , !7 " n . I " 7.4.1 78 8 " 6.5.1 ! ", ." ! !5!" !5!" 2 !. ." ! .;"! . !" .! , ".. (f) = const ." 5 . - "5 !".! 8 ", ." "2 !5!" 2 !. ! !5!" 5" !" !" ! !" .! 8"
. 45, ." " /" "2 5" " /! 2. )% 7.5.1. $ c1 c2,
n 1=2 n 1=2 2 (n) c2 2n : c1 n . 4!" k = (n+log n)=2 g | ; " ." !" ! !5 " k , ". . ! !5!" g ! ".!" ; q n 2 !"8 5" n . !!" ;
d
e
f(x1 : : : xn) = xk+1& &xn&g(x1 : : : xk ): 8 P(g), .!;7; ; g. 1" " , ." ! ;7 8 P .!" f: p1 : z=0
7.5. #3 % 3 ! p2 : p3 :
Stop(xk+1) Stop(xk+2)
pn;k+1 : P(g):
Stop(xn)
: : :: : :
147
: : :: : :: : :: : :
4! !5 .! ! !5!" 8 P , ." 0
T(P)
1
nX ;k
1@ n;j kA 2n j =1 (j + 1)2 + (n k + 1 + C(g))2 = (1): ;
O
. "2 !5!" f J C(f) C(g) 2kk;1 , ", q n 2 ! !5!" 2 . n . " ,
(n) c1
2n 1=2 n
8 c1 | " !". J 5, ." "! !" c2 " !"". 2 n "! !" n 1=2 (n) c2 2n (7.5.1) 4!" f | " n , P | 8, " .!" f, ! " . 45 k = (n + log n)=2 . !!" x ", ." NP (x) = 2n 2k . $ ! NP (x) x "! " 5, " !" " 7.4.1. J
b
c
;
T (P) = 2;n " 8 " , ."
X
y
TP (y ) > 2;n
X
y j N (y)>N (x)
TP (y) 2;n2k TP (x)
2k;nTP (x) < T (f):
(7.5.2)
G, !" f~ | .!". , ! " yi , 2n n, " ." NP (yi ) > NP (x), ! ;7 /" ! f. J 2k " 6.5.3 ! " !7!" 8 Pf~, .!;7 f~ ", ." p
2n 1=2 : (7.5.3) n J 2 8 P0 , .!;7; ; f. . ! ! 8 P, " ! .!" f. 7 ; q 2n , .!" . f y ", ." NP (y) NP (x). J 2n;k n " (7:5:2) ! ", ." .! f /" " "! " H1 ", 8 (Pf~) =
O
H1 =
O
! 2n 1=2 T (f) : n
(7.5.4)
G .! f !"2! ! ! 8 Pf~, .!;7 ; f.~
148
7. # % 3 !
J , (7:5:4) (7:5:3) ! ", ." !5!" C(P0) 8 P0 ! " . n 1=2 n 1=2 n 1=2 T (f) + 2n 2 2n T (f): H1 + C(Pf~) = 2n J C(f) C(P0), " "! !"" c2 ";7 !" (7:5:1). J . 2. X" ! !5!" !!" " 7.5.1 f ." 2 . !5!", !" f !" !"". 2 !" (!!"7 , ";7 !" xk+1 & &xn = 1), " f "! " . !"" " ." !" . 4/" ! " 7.4.1 ." 5 f, ! 2 ! ! , ! /" !" !5!" " "." ! " C(f) " !" 5". 45, ." /"" /
" ! ! ! ! f, "! "5 7 !": ; !7 " n !7!"" !" " ! !5!" "."! " !5!" 2 !. . . n . G " ;"! . 4!" f P2(n), P | 8, D B n . P D "! . X TD (P) = D1 TP (x): x2D
2
j
j
f !" D "! . TD (f) = minTD (P) 8 "! ! 8, .!;7 f !" D. )% 7.5.2. + " f(x1 :n: : xn), $ ( $ * * *, D B ,
1 C(f): TD (f) 14n 4 " !" " 5 ; . 4!" P | 8. '5 . x D, !!" . ! " 8 .!, !" !""!" 8 NPD (x) ", ." 1 NPD (x) D , ; y D ! !" NPD (x) < NPD (y), ! TP (x) < TP (y), ! TP (x) = TP (y) x < y. X"!".!; ; !" D B n ." ! D . %% 7.5.1. D B n . + f : D B $ D0 D " h , : (a) fDnD0 = hDnD0 , (b) C(h DnD0 ) 10TD (f), (c) D0 21 D + 1. . 4!" P | .!;7 f 8 " !"8"! ! , !" x0 ", ." NPD (x0) = D 12 D . J8
2
j
j
2
!
j
j
j
j
j
TD (f)
j
1 D
0 @
j
X
xjNP (x)NP (x0 )
1
TP (x)
A
j
1 D
$ j
%
j;b
D + 1 T (x ) P 0 2
j
j
j
jc
TP (x0 ) : 2
7.5. #3 % 3 !
149
" ,
TP (x0 ) 2TD (f): 4!" q1 : : : qk | " , ;7! 8" " !" 8 P, ! " !"" " /" 8 x0 . V 45 D0 = x TP (x) > TP (x0 ) . J8 D0 21 D , D0 = ki=1 qi . h(x) = q1 (x)z(xM t1) q1(x)(q2 (x)z(xM t2) : : : qk;2(x)(qk;1(x)z(xM tk;1) qk;1(x)qk (x)z(xM tk)) : : :) ! ;" D D0 ! !""!";7 . f ; /" !". ,. , ." C(h D0 ) k + 3k + TP (x0) 5TP (x0 ) 10TD (f): ) . 7:5:2. G5 " " " "8. 45 1 C(f) D = B n : T = 14n 0 4 5, ." ; !" D0 D0 ! !" T (fD0 ) < T: -! ! 7.5.1. - ! /" !7!";" !" D1 D0 h1 , " ." fD0 nD1 = h1D0 nD1 C(h1 D1 ) 10T D1 21 D0 + 1: ! 7.5.1, fD1 . - ! /" !7!";" !" D2 D1 h2 ", ." fD1 nD2 = h2D1 nD2 C(h2 D2 ) 10T D2 21 D1 + 1: 4" ; 7 n 2 . - "" 5 8 i, 0 i n 1 . !" Di+1 , Di+1 Di hi+1 ", ." fDi nDi+1 = hiD+1i nDi+1 (7.5.5) i +1 C(h Di+1 ) 10T (7.5.6) Di+1 21 Di + 1: (7.5.7)
f
j
g
j
j
j
_
j
_
_
_
n
j
j
j
j
j
j
j
j
;
;
j
I (7:5:5) ! ", ."
j
j
j
fDi = hiD+1i Di+1 fDi+1 Di+1 : _
4/"
f = h1 D1 D1 (h2 D2 : : :(hn Dn Dn fDn ) : : :): " , ! (7:5:6) ! , " !" !" C(f) 13nT + C(fDn ): , 7!" 5!" Dn . - ! !" (7:5:7) Dn 12 Dn;1 + 1 12 12 Dn;2 + 1 + 1 : : : 1 1 : : : 1 D + 1 : : : + 1 < 1 2n + 2 2: 2 2 2 0 2n _
_
_
j
j
j
j
j
j
j
j
(7.5.8)
150
7. # % 3 !
,. , ." "! .!". fDn , " !7!" !" . " , /" C(fDn ) 1. 4 !" /" !" (7:5:8), . 13n C(f) + 1 < C(f): C(f) < 13nT + 1 14n 42 ".;. J , ! 5 . J . 3. 9!" "5 , ! ;7 ! ;; !5!" !5!" 2 !.. . 5, ." 5 f !7!"" 8, !5!" ! " " !""!" !5!" 2 !. ! !5!" /" . $" 8 " 2 8", ;7 .!;7; 8 "7;! 8 ! /". )% 7.5.3. + " f $
P,
T (P) 2T (f) C(P) 4C(f): . 4!" P | 8, .!;7 ; f ! . 4!" x | ! ", ." TP (x) 2C(f) (! "8 ", " "5 " " , " "8 T(f) = T(P) C(P) 2C(f)). U!, ." TP (x) 3C(f), ! /"8 !" ." !7!" 8 P ! 5 8 8 C(f) + 1 " 8 " , ." "." !" 8 P. " 8 P 5 !"" ! ;7 :
T (P) = 2;n
X
NP (y)
T P (y ) +
X
NP (y)NP (x)
TP (y) =
= T1 + T2 T1 + 2C(f)(2n NP (x) + 1)2;n:
;
4 8 P, " ! 2 TP (x) 8 " !" | ! /" .!;7 f. )8 " , !5!" 8 P0 ! !" C(P0) TP (x) + C(f) 4C(f)
! 8 " /" 8 | !": T (P0) 2;n
X
TP0 (y) +
X
TP0 (y) =
NP (y)
J T10 = T1 , " T(P0 ) 2T(P). J . 1 . ! ! 8 P, .!;7 (m n) ", !, .!;7; " 5 ", !!"" 8 " !" ", .!;7 " Pl !""!" ! (7:1:1). A! " !" !!";" 2 2; .!" 8 , ! 5 ; !! " !" 5 .!"! , ".. 5 8 l 1 : : : m 5 " si si+1 "! " p : zl = f(a b), " !5!" !" ! " ! " ! !5!" ; ! 8 . , 5 !", 8 ." " P "! " !" 5 2!" !! " !"
2 f
g
7.5. #3 % 3 !
151
7 .!;"!, ! " " ! !". /" !. !5!" !" ! " m 2 !5!" ! 8 . - ! ;7 " ! "! ! ! 8 ! " !5!" ! . ." ! " !5!" 8 . )% 7.5.4. ( P, $ (m n)- , ( * ( " * ) S , : (i) L(S) 4C(P)M (ii) S(x1 : : : xn) = P(x1 : : : xn) * (x1 : : : xn). . 4!" P = p1 : : : pL | 8, s1 : : : sr | ! " !", q1 : : : qr | 8" " !". ' (7:1:1) 8, ." i- " !" si 8 P "! ti - ". 4 5, ." l- 8 P .!"! " " !" ! " 5" 5!" i1 : : : ik , ".. zl (xM ti) = zl (xM tis ) ! i is : : : is+1 1 . 45
f
2 f
h00 (x) 1
;
h0k (x) =
g
g
l ^
qi (x) k 2 f1 2 : : : rg i=1 hr+1 (x) = h0r (x) hk (x) = h0l;1 (x)qj (x) k 2 f1 2 : : : rg: I! hi , (7:1:1):
Pl (x) = h1(x)zl (xM t1) _ : : : _ hr (x)zl (xM tr ) _ hr+1 (x)zl (xM L):
J ! i j ", ." 1 i < j r + 1,
hij (x) =
j _
k=i
hk (x):
J zl (xM ti) = zl (xM tis ) ! i is : : : is+1 1 , " 7 !" ! !5 "! Pl (x) =h1i1;1 (x)zl (xM t1) hi1 i2 ;1(x)zl (xM ti1 ) : : : (7.5.9) : : : hik;1 ik ;1(x)zl (xM tik;1 ) hik r+1 (x)zl (xM tik ): hi h0j ", ." ( i j 0 hi (x)hj (x) = 0 hi (x) i > j: 4/" i < j h1j (x)h0i (x) = hi+1 (x) hj (x) = hi+1j (x) 0 ".. .! h1i hi .! 5 hij , !".;7! (7:5:9), !"". 8 8 ". G !", ." 8 P ! 5" L1 ". J8 P ! S !!"" ! ;7: (a) - .! ! hi , h0i h1i. G /"8 " "! 3r 2 ". (b) - .! ! hij . G /"8 " "! !" ", ! 8 P .!;"! . J 5 5 .!"! ! !" ", " " "! L1 ". (c) - !""!" ! !" (7:5:9) .! ! " Pl . G /"8 " "! 2L1 ". (d) 9 ! " !". )8 " , ." 7 .! " " ! " 3L. " , !5!" ! S ! " 4L. J . 2 f
;
g
_
_
_
_
__
;
152
7. # % 3 !
*& 7.5.1. ," ! maxC(f)=T (f), 8 ! "! ! n
!5!" L. 7.5.2. ," maxC(f)=Td (f) 8 d, W 28 n. 7.5.3. 8" , ." f : B n B ! "! n g:B B , ! "! " h : B n+1 B , ." !
!
!
f(x1 : : : xn) = h(x1 : : : xn g(x1 : : : xn)) C(g) = (nc ) O
8 c | !"". 4" , ." "! n g ! 7! n f ", ." C(g) C(f)
O
;p
2n=n T(g)=T(f) =
O
;p
2n=n :
7.5.4. 4" , ." 8 P, .!;7 ; f, 5" " ! /" S, .!;7; f, ", ." L(S) 2C(P).
1] . ., . . , . | .: ! "#, 2000. 2] . % & '( ()*+ n , . | ).: !'&)+ '). -. .. ,.5. | .: , 1968, . 53{63. 3] ., ., . 3 4 &5 . | .: , 1979. 4] . ., . . 89& (,:. , )(( 9)+ ). | 2- 49. .: -(), 1992. 5] . !. <)* , ++ '. | 3- 49. .: -(), 1966. 6] "# !. . % + : , ()*+. | '.: =)+ 4. ,. 52. | -'), 1992, . 31{110. 7] $ . ., . ., % !. . )'). | .: -(), 1977. 8] &# .'. >)( ,. 9. . ?. 2: 3(& . | .: , 1977. 9] &# .'. >)( ,. 9. . ?. 3: @) ,). | .: , 1978. 10] & . . 9 '(. | .: -(), 1977. 11] . ., . . %' 9 9 :. * : 4* &) ()*+. | >4. #8, ), 1988, A 7, 11{19. 12] # (. . % 4 ) ) (,.BC . | ).: 3' )'). ,. 10. | .: D44, 1963, . 63{97. 13] # (. . %' 9 ,9 9 ) 4( (,.BC | ,*, )5 )9.. | ).: 3' )'). ,. 14. | .: D44, 1965, . 31{110. 14] # (. . E,&) *) : (,.BC . | .: "#, 1984. 15] )* # +. . @:5 '( ()*+. | .: -(), 1991. 16] ., , '. !9, ,.BC F'). | .: , 1976. 17] + ). . =)45 5 ) 4 ()*5 G. | ).: 3' )'). ,. 23. | .: -(), 1970, . 83{101. 18] + ). . % 5+ 4* 9& (. | ).: 3' )'). ,. 38. | .: -(), 1981, . 181{216. 19] , . '. @:5 &+. | .: D), 1998. 20] - . . . % : '( *, (BC '( ()*+. | =). ), 1994, A2, c. 43{73. 21] - . . . % 9 &. 4&+ '( ()*+. | =)+ 4 9 ,*+, 1997, A1, c. 60{78. 22] - . . . % 9 &. '( ,. | =)+ 4 9 ,*+, 1998, A1, c. 88{103. 23] - . . . @9 . &. 4&+ G '( ()*+. | =). ), 2000, A4, c. 109{120. 24] - . . . @9.. :5 '( ()*+. | ).: =). ) ,:.: ') )*+. | . .- . "# 2001, . 145{170.
153
154
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25] / . . % 4* 9,9 '( ()*+ 4 ()*5 G. | ).: 3' )'). ,. 21. | .: -(), 1969, . 215{226. 26] %0 1 . . 9 B ()*+ k-4&+ ). | ).: =). ) &) , )'), . 1, ,9 9. @. . H') %. I. <(,. | .: -(), 1974, . 9{66. 27] %0 1 . . 9 9)(B )(. | 3- 49. .: F. F), 2001. 28] Akl S. J. Parallel Computation: Models and Methods. | Prentice-Hall, 1997. 29] Ajtai M., Komlos Ja., Szemeredi E. Sorting in O(n log n) parallel steps. | Combinatorica, 1983, v. 3. A 1, pp. 1-19.
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"(' { ()*+ 73 { 73 { ( 15
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8) : ()*+ 27
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155
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3. { (C. 10 { ). 10 3 :9 11 3 ) 10 39( 15 39()*. '(+ ()* 11 39) F 72 395 : ) 36 3'4. G) 19 3 .C.. 134 { 5. 136 { ,9. 137 3 4 36 3 { 5 42 { '* * 41 { ) * 41
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#5 F(BC : 60
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157
