1. 1 7
1.1. . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2. . . . . . . . . . . . . . . . . . . . . 11
1.3. . . . . . . . . . . . . . . . . . 15
1.3.1. . . . . . . . . . 16
1.3.2. 17
I 24
25
2 26
2.1. . . . . . . . . . . . . . 26
2.2. . . . . . . . . . 30
3 39
3.1. . . . . . . . . . . . . . . . . . . . 39
3.2. . . . . . . . . . . . . . . . . . . . . . . . . 43
4 49
4.1. . . . . . . . . . . . . . . . . . . . . 50
4.2. . . . . . . . . . . . . . . . . . 53
4.3. . . . . . . . . . . . . . . . . . . . . . . . 56
? 4.4 4.1 . . . . . . . . . . . . . . . 59
4.4.1. . . . . . . . . . 59
4.4.2. . . . . . . . 63
5 71
5.1. . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2. . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

2. 2
5.5.1. . . . . . . . . . 87
5.5.2. . . 89
5.5.3. . . . . 91
6 96
6.1. . . . . . . . . . . . . . . . . . . . . 96
6.2. . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.1. . . . 103
6.3. . . . . . . . . . . . . 108
6.4. . . . . 113
6.5. . . . . . . . . . 118
6.6. . . . . . . . . . . . . . . . . . . 123
6.6.1. . . . . . . . . . . . . . 123
6.6.2. . . . . . . . . . . . . . . . . . . . 125
6.6.3. . . . . . . . . 126
II 132
133
7 136
7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2. . . . . . . . . . . 138
7.3. . . . . . . . . . . . . . . . . . . . . . 140
7.4. . . . . . . . . 143
7.5. . . . . . . . . . . . . . . . . . 143
8 148
8.1. . . . . . . . . . . . . . 148
8.2. . . . . . . . . . . . . . . . 151
8.3. . . . 155
8.4. . . . . . . . . . . . . . . . 157
8.4.1. . . . . . . . . . . . . 157
8.4.2. . . . . . . . . 158
9 166
9.1. . . . . . . . . . . . . . 166
9.2. . . . . . . . . . . . . . . . . . . 169
9.3. . . . . . . . . . . . . . . . . . . . 171
9.4. . . . . . . . . . . . . . . . 174
10 180
10.1. . . . . . . . . . . . . . . . . . . 181
10.2. . . . . . . . . . . 182
10.3. . . . . . . 185

3. 3
III 192
193
11 197
11.1. . . . . . . . . . . . . . . . . . . . . . . 197
11.2. . . . . . . . . . . . . . . . . . . . . . 201
11.3. 206
11.4. . . . . . . . . . . . 210
12 - 217
12.1. . . . . . . . . . . . . . . . . . . . . . 217
12.2. - . . . . . . . . . . . . . . . . . . . . . . . . 219
12.3. - . . . . . . . . . . . . . . . . . . . . . . . 225
12.3.1. . . . . . . . . . . . . . . . . 226
12.3.2. . . . . . . . . . . . . . . . . . . . . . 227
12.3.3. . . . . . . . . . . . 228
12.4. . . . . . . . . . . . . . . . . . . . 231
13 242
13.1. ? . . . . . . . . . . 243
13.2. . . . . . . . . . . . . . . . . . 245
13.3. . . . . . . . . . . . . 248
? 13.4 . . . . . . . . . . 252
14 - 262
14.1. - . . . . . . . . . . . 262
14.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 264
14.3. . . . . . . . . . . . . . . . . . . . 267
14.4. . . . . . . . . . . . . . . . . . . . . . . . . . . 271
15 280
15.1. . . . . . . . . . 280
15.2. 285
15.3. . . . . . . . . . . . . . . . . . . . 288
IV 296
297
16 299
16.1. . . . . . . . . . . . 300
16.2. . 307
16.3. . . . . . . 312
16.4. . . . . . 317

4. 4
17 326
17.1. . . . . . . . . . . . . . . . . . . 326
17.2. ? . . . . . . . . . . 330
17.3. . . . . . . . . . . . . . . . . . . . . . . 333
? 17.4 . . . . . . 341
17.4.1. . . . . . . . . . . . . . . . . . . . . . 341
17.4.2. 343
? 17.5 . . . . . . . . . . . . . . . . . . . 346
18 352
18.1. . . . . . . . . . . . . . . . . . . . . 353
18.2. . . . . . . . . . . . . . . . . . . . . 356
18.3. . . . . . . . . . . . . . . . . . . . . 359
18.4. . . . . . . . . . . . . . . . . . . 362
18.4.1. . . . . . . . . . . . . . . . 362
18.4.2. . . . . . . 365
V 372
373
19 - 376
19.1. - . . . . . . . . . . . . . . . . . . 379
19.2. - . . . . . . . . . . . 381
19.3. - . . . . . . . . . . . . . 388
20 394
20.1. . . . . . 395
20.1.1. . . . . . . . . . . . . . . 395
20.1.2. . . . . . . . . . . . . . . . . 397
20.2. . . . . . . . . . . 399
20.2.1. . . . . . . . . . . . . . . 400
21 413
21.1. . . . . . . . . . . . . . 414
21.2. , . . . 416
21.3. . . . . . . . . 425
21.4. . . . . . . . . . . . . . 429
22 434
22.1. . . . . 434
22.2. . . . . . . . . . . . . . 437
22.2.1. . . . . . . . . . . . . . . . . . . . . 441
23 452
23.1. . . . . . . . . . . . . 453

5. 5
23.1.1. . . . . . . . . . . . . . . 453
23.1.2. . . . . . . . . . . . . . . . . . . 456
23.1.3. . . . . . . . . . . . . . . . . . . 464
23.1.4. . . . . . . . . . . . 471
23.1.5. . . . . . . . . . . . 473
24 481
24.1. . . . . . . . . . . . 482
24.2. . . . . . . . . . . . . . 486
25 493
25.1. . . . . . . . . . . . . . 497
25.2. - . . . . . . . . . . . . . . . 507
25.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
25.4. . . . . 512
26 522
26.1. - . . . . . . . . . . . . . . . 529
27 546
27.1. . . . . . . . . . . . . . . . . . . . . . . 547
27.2. { . . . . . . . . . . . . . . . . 552
27.3. . . 562
27.4. . . . . . . . . . 565
27.5. - - - . . . . . . . . . . . . . . 574
28 586
28.1. . . . . . . . . . . . . . . . . . . . . 587
28.2. . . . . . . . . . . . . . . . . . 590
28.3. . . . . . . . . . . . . . . . 592
28.4. . . . . . . . . . . . . . . . . . . . . . . 594
28.5. . . . . . . . . . . . . . . . . . . . . 596
29 601
29.1. . . . . . . . . . 602
29.1.1. . . . . . . . . . . . 602
29.1.2. . . . . . 603
29.1.3. . . . . . . . . . . . . . . . . . . . . . 603
29.1.4. . . . . . . . . . . . . . . . . . . . 604
29.1.5. . . . . . . . . . . . . . . . . . . . 605
29.1.6. . . . . . . . . . . . . . . . . . . . . 605
29.2. C . . . . . . . . . . . . . . . . . . . 605
29.2.1. . . . . . . . . . . . . . . . 606
29.2.2. . . . . 607
29.2.3. -
. . . . . . . . . . . . 608

6. 6
29.2.4. :
. . . . . . . . . . . . . . . . . . . . . 609
29.2.5. . . . . . . 610
29.3. . . . . . . . . . . . . . . . . . . 612
29.3.1. . . . . . . . . . . . . . 612
29.3.2. . . . . . . . . . . . . . . 614
29.3.3. . . . . . 614
29.3.4. . . . . . . . . . . . . . . 615
29.3.5. . . . . . . . . . . . . . . . . . . . . 616
29.4. . . . . . . . . . . . . . . . . . . 617
29.4.1. . . . . . . . . 617
29.4.2. . . . . . . . . . . . . . . 618
29.4.3. . . 618
29.4.4. . . . . . . . . . . . . 619
29.4.5. . . . . . . . . . . . . . . . . 619
29.4.6. . . . . . . . . . . . . . . . 620
29.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
29.6. . . . . . . . . . . . . . . . . . . . . . . . 623
30 625
30.0.1. -
(PRAM) . . . . . . . . . . . . . . . . . . . . 625
30.0.2. -
. . . . . . . . . . . . . . . . . . . . . . . . . 626
30.0.3. . . . . . . . . . . . . . . . . . . 627
30.0.4. . . . . . . . . . . . . . . . . . . . . . 627
30.1. . . . . . . . . . . . . . . . . . 628
30.1.1. . . . . . . . . . . . . . . . . . . 628
30.1.2. . . . . . . . . . . . . . . . . . . . 630
30.1.3. . . . . . . . . . . . . . . . . . . . . . . . 630
30.1.4. . . 631
30.1.5. . . . . . . . . . . . . . . 633
30.2. CRCW- EREW- . . . . . . . . . . . . . . . 636
30.2.1. . . . . . . . . . . 636
30.2.2. . . . . . . . . . . . 637
30.2.3. CRCW-
EREW- . . . . . . . . . . . . . . . . . . 639
30.3. . . . . 642
30.3.1. . . . . . . . . . . . . . . . . . . . . 645
30.4. . . 645
30.4.1. 646
30.4.2. . . . . . . . . . . . 647
30.4.3. . . . . . . . . . . . . . . . . . . . . . . . 647
30.4.4. . . . . . . . . . . . . . . . . . . . . 649
30.5. ( -
) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649

7. 7
30.5.1. -
. . . . . . . . . . . . . . . . . . . . . . . 650
30.5.2. 6- . . . . . . . . . . . . . 651
30.5.3. -
6- . . . . . . . . . . . . . . 653
30.5.4. . . . . . . . . . . . . . . . . . . . . 654
30.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
30.7. . . . . . . . . . . . . . . . . . . . . . . . 657
31 659
31.1. . . . . . . . . . . . . . . . . . 659
31.2. . . . . . . . 668
31.3. . . . . . . . . . . . . . . . . . . . . 675
31.4. -
. . . . . . . . . . . 679
33 p - p 712
33.1. . . . . . . . . . . 713
33.2. . . . . . . . . . . . . . . 718
33.3. . . . . . . . . . . . . . . . . . 722
33.4. . . . . . . 726
33.5. . . . . . . . . . . . . . 729
33.6. . . . . . . . . . . . . . . . . . . . . . 732
33.7. RSA . . . . . . . 735
33.8. . . . . . . . . . . . . . . . 741
34 757
34.0.1. . . . . . . . . . . 758
34.1. . . . . . . . . . . . . . . . . . . 759
34.2. | . . . . . . . . . . . . . . . 761
34.2.1. . . . . . . . . . . . . . . . . . . . . 764
34.3. . . . 765
34.3.1. . . . . . . . . . . . . . . . 765
34.3.2. . . . . . . . . . 766
34.3.3. . . . . . . . . . 770
34.4. | | . . . . . . . . 771
34.4.1. - , 771
34.4.2. . . . . . . . . . . . . . . . . . . . 773
34.4.3. - . . 774
34.4.4. KMP . . . . . . . . . . . . 776
34.5. | . . . . . . . . . . . . . . . . 777
34.5.1. - . . . . . . . . . . . . . 778
34.5.2. . . . . . . . . 780
34.5.3. . . . . . . . . . . . . . . . . . . . . 784
34.6. . . . . . . . . . . . . . . . . . . . . . . . . . 785

8. 8
35 787
35.1. . . . . . . . . . . . . . . . . . . . . 788
35.2. ? . . . . . . . . . . . 793
35.3. . . . . . . . . . . . . . 798
35.4. . . . . . . . . . . . 804
35.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 807
36 NP- 811
36.1. . . . . . . . . . . . . . . . . . . 812
36.2. NP . . . . . 819
36.3. NP- . . . . . . . . . . . . . . . . 824
36.4. NP- . . . . . . . . . . . . . . . . . . . . . 837
36.4.1. . . . . . . . . . . . . . . . . . . . 837
36.4.2. . . . . . . . . . 839
36.4.3. . . . . . . . . . 842
36.4.4. . . . . . . . . . . . . . . 846
37 850
37.1. . . . . . . . . . . . . . 852
37.2. . . . . . . . . . . . . . . . . . . 854
37.2.1. ( -
) . . . . . . . . . . . . . . . . . . . . . 854
37.2.2. . . . . . . . . . . 856
37.3. . . . . . . . . . . . . . . 863

9. -
. -
,
.
-
.
260 , .
-
.
, -
| .
-
.
:
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.
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(
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10. 10
).
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-
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11. 11
.
,
, .
( , -
, ). :
Introduction to Algorithms
MIT Labratory for Computer Science
545 Technology Square
Cambridge, Massachusetts 02139
-
,
algorithms@theory.lcs.mit.edu , Subject:
help . ,
.
, -
( 1997), , -
, , , -
( , . ).
, , -
: , !
, algor@mccme.ru
( , 121002, ., 11,
, ).]
. -
.
(Massachusetts Institute of Technology, Laboratory for
Computer Science) -
. -
. :
Baruch Awerbuch, Sha Goldwasser, Leo Guibas, Tom Leighton, Al-
bert Meyer, David Shmoys, Eva Tardos. , -
( : Microvax, Apple Macintosh, Sun Sparc-
station) William Ang, Sally Bemus, Ray Hirschfeld
Mark Reinhold TEX, -
.
Thinking Machines

12. 12
.
,
. -
: Richard Beigel (Yale), Andrew Goldberg (Stanford), Joan
Lucas (Rutgers), Mark Overmars (Utrecht), Alan Sherman (Tufts,
Maryland), Diane Souvaine (Rutgers).
-
, . -
: Alan Baratz, Bonnie Berger, Aditi Dhagat, Burt Kaliski,
Arthur Lent, Andrew Moulton, Marios Papaefthymiou, Cindy Phillips,
Mark Reinhold, Phil Rogaway, Flavio Rose, Arie Rudich, Alan Sher-
man, Cli Stein, Susmita Sur, Gregory Troxel, Margaret Tuttle.
-
: (Denise Sergent),
(Maria Sensale), (Al-
bert Meyer), (Shlo-
mo Kipnis, Bill Niehaus, David Wilson), (Marios
Papaefthymiou, Gregory Troxel), (Inna Radzi-
hovsky, Denise Sergent, Gayle Sherman, Be Hubbard).
, Bobby
Blumofe, Bonnie Eisenberg, Raymond Johnson, John Keen, Richard
Lethin, Mark Lillibridge, John Pesaris, Steve Ponzio, Margaret Tuttle.
, -
Bill Aiello, Alok Aggrawal, Eric Bach,
Vasek Chvatal, Richard Cole, Johan Hastad, Alex Ishii, David Johnson,
Joe Kilian, Dina Kravets, Bruce Maggs, Jim Orlin, James Park, Thane
Plambeck, Herschel Safer, Je Shallit, Cli Stein, Gil Strang, Bob Tar-
jan, Paul Wang.
, Andrew Goldberg, Danny Sleator, Umesh Vazirani.
LATEX
( TEX). -
Apple Macintosh Mac Draw II -
(Joanna Terry, Claris Corporation Michael Mahoney, Advanced Com-
puter Graphics).
Windex, . -
BibTEX. -
-
Autologic
(Ralph Youngen,
). : Rebecca Daw, Amy Henderson ( -
LATEX), Jeannet Leendertse ( ).
-
MIT Press (Frank Sallow, Terry Ehling, Larry Cohen,

13. 13
Lorrie Lejeune) McGraw-Hill (David Shapiro)
,
(Larry Cohen).
, (Nicole Cormen, Lina Lue
Leicerson, Gail Rivest) (Ricky, William Debby Leicerson
Alex Christopher Rivest)
(Alex Rivest
( 6.6.1). ,
.
,
1990
(Thomas H. Cormen)
(Charles E. Leiserson)
(Ronald L. Rivest)
:
MIT
, ( -
).
-
( . . ).
-
, -
,
:
K. , . , . , . , .
. , . , . , . , . -
, . , . , . , . , .
( )
. , . , . , . , . -
, . , . . , A. ( -
)
. ( )
. . ( )

14. 1
,
, -
|
.
,
, -
, -
, ,
.
1.1.
(algorithm) | -
, (input),
, -
(output).
(computational problems). ,
, -
, , ,
.
(sorting prob-
lem)
-
. :
: n (a1 a2 ::: an).
: (a0
1 a0
2 ::: a0
n) -
, a0
1 6 a0
2 6 ::: 6 a0
n.
, h31 41 59 26 41 58i, -
h26 31 41 41 58 59i.
(instance) .

15. 15
-
.
-
, , ,
( , ,
).
(correct), -
( )
, .
, (solves) -
. (
) .
( , |
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33 .
, .)
,
|
, .
(pseu-
docode), -
( , , ). ,
, -
. , -
( , ),
, .
(insertion sort) -
. -
:
,
, ( . . 1.1)
Insertion-Sort, -
A 1::n] (
n, ). -
A length A].
(in place), (
).
Insertion-Sort A
.

16. 16 1
. 1.1
Insertion-Sort(A)
1 for j 2 to length A]
2 do key A j]
3 . A j] A 1::j ;1].
4 i j ;1
5 while i > 0 and A i] > key
6 do A i+ 1] A i]
7 i i;1
8 A i + 1] key
. 1.2 Insertion-Sort A = h5 2 4 6 1 3i. -
j .
. 1.2 A = h5 2 4 6 1 3i.
j ( -
). A 1::j ; 1]
( ), A j + 1::n] | . for
j . A j]

17. 17
( 2 )
( j ; 1- ) , -
. ( 4{7). 8 A j]
.
, :
1. .
, for ( 1) 2{8, -
while ( 5) 6{7, 8.
if-then-else. -
begin end . (
-
, ,
.)
2. while, for, repeat if, then, else
, .
3. . ( ).
4. i j e ( i j
e) j e i j (
).
5. ( i j key) -
( ).
6. : A i] i-
A. :: : A 1::j]
A, A 1] A 2] ::: A j].
7. (objects), -
( elds), , , -
(attributes).
]. ,
length, A
length A].
( ), .
, , -
. y x
f f y] = f x]. ,
f x] 3, f x] = 3,
f y] = 3, y x x y
.
nil, -
.
8. (by value): -
-
. -

18. 18 1
, ,
| . , x | -
, x y, ,
, f x] 3 | .
1.1-1 . 1.2, ,
Insertion-Sort A = h31 41 59 26 41 58i.
1.1-2 Insertion-Sort , -
( -
).
1.1-3 :
: n A = ha1 a2 ::: ani v.
: i, v = A i],
nil, v A.
(linear search),
A v.
1.1-4 n- ,
n- A B. (
) (n+1)- C. -
-
.
1.2.
-
, , -
( , ), -
. , , -
.
(random-access machine, RAM), -
. ( -
-
.)
:
-
: ,

19. 19
. -
. ( ,
, :
, .)
(input size)? -
.
( , ). -
, -
. -
, ( ,
).
(running time)
, |
, . ,
-
( - -
| x- ).
(call) (
) (execution),
.
, Insertion-Sort
( ) , -
. j 2 n (
n = length A] | ) ,
5, tj. ( ,
, -
, .)
Insertion-Sort(A)
1 for j 2 to length A] c1 n
2 do key A j] c2 n ;1
3 . A j] -
. A 1::j ;1]. 0 n ;1
4 i j ;1 c4 n ;1
5 while i > 0 and A i] > key c5
Pn
j=2 tj
6 do A i+ 1] A i] c6
Pn
j=2(tj ; 1)
7 i i; 1 c7
Pn
j=2(tj ; 1)
8 A i+ 1] key c8 n ;1
c, m , cm -
. (

20. 20 1
!) ,
T(n) = c1n+ c2(n; 1)+ c4(n ; 1)+ c5
nX
j=2
tj +
+ c6
nX
j=2
(tj ; 1)+ c7
nX
j=2
(tj ;1) + c8(n;1):
,
n, , n .
Insertion-Sort , -
. 5
( A i] 6 key i = j ;1),
tj 1,
T(n) = c1n+ c2(n; 1)+ c4(n ; 1)+ c5(n;1) + c8(n;1) =
= (c1 + c2 + c4 + c5 + c8)n;(c2 + c4 + c5 + c8):
, T(n), -
n,
(linear function) n, . . T(n) = an + b
a b. ( -
c1 ::: c8.)
( ) ,
:
A j] A 1]:::A j ; 1].
tj = j. ,
nX
j=2
j = n(n + 1)
2 ;1
nX
j=2
(j ;1) = n(n ;1)
2
( . . 3), ,
T(n) = c1n + c2(n ;1)+ c4(n; 1)+ c5
n(n + 1)
2 ; 1 +
+ c6
n(n ; 1)
2 + c7
n(n;1)
2 + c8(n ; 1) =
= c5
2 + c6
2 + c7
2 n2 + c1 + c2 + c4 + c5
2 ; c6
2 ; c7
2 + c8 n ;
; (c2 + c4 + c5 + c8):
T(n) | (quadratic function), . .
T(n) = an2 + bn + c. ( a, b c -
c1{c8.)

21. 21
, ,
. -
(worst-case running time),
. ? .
, ,
, -
, ( ) .
( -
) . ,
( ).
. , , -
n Insertion-
Sort. 5{8?
A 1::j ; 1]
A j], tj j=2, -
T(n) n.
-
(average-case running time, expexted running time) -
. ,
( -
), -
. (
, .)
Insertion-Sort
.
, i- -
ci. an2+bn+c.
,
(rate of growth, order of growth) n2, -
( )
n2. : T(n) = (n2) (
).
: , ,
(n2), | (n3), (
| ).

22. 22 1
1.2-1 n :
,
.
, .
(selection sort). -
.
, - .
1.2-2 ( . 1.1-3).
,
( -
)?
? - ?
1.2-3 x1 x2 ::: xn. ,
(nlogn) , -
.
1.2-4 a0 a1 ::: an;1
x. -
, (n2).
(n), ?
:
n;1X
i=0
aixi = (:::(an;1x + an;2)x+ :::+ a1)x + a0:
1.2-5 n3=1000; 100n2 ; 100n + 3 -
- ?
1.2-6 , -
. ?
1.3.
, -
. -
, (incremental approach): -
.
, -
(divide-and-conquer approach),
.

23. 23
1.3.1.
(recursive algorithms):
,
.
. -
. (
| , ). -
,
.
. -
. -
.
-
. -
, (
1 ).
.
Merge(A p q r). -
A p q r,
. , p 6 q < r
A p::q] A q + 1::r] , (merges)
A p::r].
( . 1.3-2), , Merge
(n), n | (n = r;p+1).
. ,
.
?
( )
. , -
. , -
, (n).
Merge-Sort(A p r),
A p::r] A, -
. p > r
, .
q,
A p::q] ( dn=2e ) A q + 1::r]
( bn=2c ). bxc
x ( , x),
dxe | , x.

24. 24 1
sorted sequence |
merge |
initial sequence |
. 1.3 A = h5 2 4 6 1 3 2 6i.
Merge-Sort(A p r)
1 if p < r
2 then q b(p + r)=2c
3 Merge-Sort(A p q)
4 Merge-Sort(A q + 1 r)
5 Merge(A p q r)
Merge-Sort(A 1 length A]). n = length A]
,
2,
4
n (
n=2). . 1.3.
1.3.2.
? -
, -
,
(recurrence equation). ,
.
. ,
n a , -
b . ,
D(n), | C(n).

25. 25
T(n)
n ( ): T(n) = aT(n=b) + D(n) + C(n).
n,
. n,
, - -
. n ,
.
, (n)
. ( 4, -
.)
. ( )
(1), | (n). -
T(n) =
(
(1) n = 1,
2T(n=2)+ (n=2) n > 1.
4, T(n) =
(nlogn), log ( -
, , ,
). n -
,
(n2).
1.3-1 . 1.3, -
A = h3 41 52 26 38 57 9 49i.
1.3-2 Merge(A p q r).
1.3-3 ,
T(n) =
(
2 n = 2,
2T(n=2)+ n n = 2k k > 1,
T(n) = nlogn ( n, ).
1.3-4
: A 1::n], ( ) -
A 1::n ; 1], A n] -
A 1::n;1]. -
.

26. 26 1
1.3-5 ( . 1.1-3), ,
, ,
, -
. (binary search).
, -
. , (logn).
1.3-6 , while 5{7
Insertion-Sort ( . 1.1) -
A 1::j ; 1] .
( . 1.3-5),
(logn).
(nlogn)?
1.3-7? S n ,
x. (nlogn) , x
S?
, -
| .
: , , -
, : -
.
-
, .
.
| (100 -
)
(1 )? -
, n
, , 2n2 .
50nlogn.
2 (106)2
108 = 20000 5 56
50 (106)log(106)
106 1000 17
.
, |
, -
. .

27. 1 27
1.3-1
8n2 64nlogn .
n -
? ?
1.3-2 n ,
100n2 , , 2n ?
1-1
, n f(n)
. ,
t? , -
.
1 1 1 1 1 1 1
lognpn
n
nlogn
n2
n3
2n
n!
1-2
, n .
, -
. ,
.
. n k n=k. -
, (
) (nk).
. , -
(nlog(n=k)).
.
(nk+nlog(n=k)). k
n, - (nlogn)?
. k -

28. 28 1
?
1-3
A 1::n] | n . -
(inversions) , . .
i < j, A i] > A j].
. h2 3 8 6 1i.
. -
n?
.
? .
. , -
n (nlogn). ( : -
.)
.
: , 4,5], 14], -
14], 105], 121, 122, 123],
142], 144, 145, 146], 164], -
, 167], 175], 201]. -
:
24,25], 90].
1968 -
121, 122, 123],
.
.
, -
- (al-Khow^arizm^, al-Khw^arizm^).
, 4] -
-
. -
.
123] -
. -
, (
). -
, (D.L.Shell),
-
.
,
, -
1938 . -

29. 1 29
, (J. von Neumann), -
,
EDVAC 1945 .

30. I

31. , -
.
,
.
2 , -
( .), .
- , -
.
3
( -
).
4
. -
( 4.1),
. ,
, -
.
5 ,
, , , ,
.
6
. -
, (
) ,
.

32. 2
1, ,
( ) -
n2, ( ) | nlgn.
,
n .
, -
.
,
(asymp-
totic e ciency). ,
,
, . ( .)
2.1.
-
, .
-
2 , T(n) -
n (n2). -
: c1 c2 > 0
n0, c1n2 6 T(n) 6 c2n2 n > n0. ,
g(n) | , f(n) = (g(n)) ,
c1 c2 > 0 n0, 0 6 c1g(n) 6 f(n) 6 c2g(n)
n > n0 ( . . 2.1). ( f(n) = (g(n)) :
.)
!!!!!!!!! 2.1 - : ]
f(n) = (g(n)), f(n) = O(g(n))
f(n) = (g(n)).
, -
: , f1(n) = (g(n)) f2(n) = (g(n)),

33. 33
, f1(n) = f2(n)!
(g(n)) , f(n) g(n)
, . . -
n. , f g -
, n0 (
c1 c2 , n -
).
f(n) = (g(n)), , g(n) -
f(n).
: f(n) = (g(n)), g(n) = (f(n)).
1 , (1=2)n2 ; 3n =
(n2). , -
c1 c2 n0 ,
c1n2 6 1
2n2 ;3n 6 c2n2
n > n0. n2:
c1 6 1
2 ; 3
n 6 c2
,
c2 = 1=2. , ( ) n0 = 7 c1 =
1=14.
: -
, 6n3 6= (n2). , c2 n0,
6n3 6 c2n2 n > n0. n 6 c2=6 n > n0
| .
, -
,
n . -
,
( c1 c2). -
, f(n) = an2 + bn + c,
a b c | a > 0. -
, ,
f(n) = (n2). , -
c1 = a=4, c2 = 7a=4 n0 = 2 max((jbj=a)
p
jcj=a) ( ,
). ,
p(n) d -
p(n) = (nd) ( 2-1).
-
: (1) , -
. (
, .)

34. 34 2
O- -
f(n) = (g(n)) :
. . , f(n) = O(g(n)),
c > 0 n0, 0 6 f(n) 6
cg(n) n > n0 , f(n) = (g(n)),
c > 0 n0, 0 6 cg(n) 6 f(n)
n > n0. :
, .
- , f g -
.
( . 2.1-5), :
2.1. f(n) g(n) f(n) =
(g(n)) , f(n) = O(g(n))
f(n) = (g(n)).
f(n) = O(g(n)) g(n) =
(f(n)) .
, an2 + bn + c = (n2) ( a).
an2 + bn + c = O(n2). : a > 0
an + b = O(n2) ( c = a + jbj n0 = 1). ,
an + b 6= (n2 an + b 6= (n2).
( , O )
. , 1
T(n) = 2T(n=2)+ (n)
. (n)
, ,
c1n c2n c1
c2 n.
, -
. ,
nX
i=1
O(i)
h(1) + h(2) + :::+ h(n), h(i) |
, h(i) = O(i). ,
n O(n2).
-
| 2n2 + 3n + 1 = 2n2 + (n) =
(n2). (2n2 + (n) = (n2))
: h(n) = (n)
, 2n2 + h(n) (n2).

35. 35
o- !-
f(n) = O(g(n)) , n
f(n)=g(n) .
limn!1
f(n)
g(n) = 0 (2.1)
f(n) = o(g(n)) (
). , f(n) = o(g(n)),
" > 0 n0, 0 6 f(n) 6 "g(n)
n > n0. ( f(n) = o(g(n)) ,
f(n) g(n) n.)
: 2n = o(n2), 2n2 6= o(n2).
!- : ,
f(n) !(g(n)) ( ), -
c n0, 0 6
cg(n) 6 f(n) n > n0. , f(n) = !(g(n))
, g(n) = o(f(n)).
: n2=2 = !(n), n2=2 6= !(n2).
, :
:
f(n) = (g(n)) g(n) = (h(n)) f(n) = (h(n)),
f(n) = O(g(n)) g(n) = O(h(n)) f(n) = O(h(n)),
f(n) = (g(n)) g(n) = (h(n)) f(n) = (h(n)),
f(n) = o(g(n)) g(n) = o(h(n)) f(n) = o(h(n)),
f(n) = !(g(n)) g(n) = !(h(n)) f(n) = !(h(n)).
:
f(n) = (f(n)), f(n) = O(f(n)), f(n) = (f(n)).
:
f(n) = (g(n)) g(n) = (f(n)).
:
f(n) = O(g(n)) g(n) = (f(n)),
f(n) = o(g(n)) g(n) = !(f(n)).
:
f g a b:
f(n) = O(g(n)) a 6 b
f(n) = (g(n)) a > b
f(n) = (g(n)) a = b
f(n) = o(g(n)) a < b
f(n) = !(g(n)) a > b

36. 36 2
, , : -
. ,
a b a 6 b, a > b, -
, ( ) f(n) g(n)
f(n) = O(g(n)), f(n) = (g(n)). , -
,
f(n) = n g(n) = n1+sinn (
g(n) 0 2). , -
a 6 b a < b a = b,
f(n) = O(g(n)) f(n) = o(g(n)) f(n) = (g(n)).
2.1-1 f(n) g(n)
n. , max(f(n) g(n)) = (f(n) + g(n)).
2.1-2 ,
(n+ a)b = (nb) (2.2)
a b > 0.
2.1-3 A -
O(n2) ?
2.1-4 , 2n+1 = O(2n)? 22n = O(2n)?
2.1-5 2.1.
2.1-6 f(n) g(n),
f(n) = O(g(n)), f(n) 6= o(g(n)) f(n) 6= (g(n)).
2.1-7 , f(n) = o(g(n)) f(n) = !(g(n))
.
2.1-8
, . ,
f(m n) = O(g(m n)), n0, m0 c,
0 6 f(m n) 6 cg(m n) n > n0 m > m0.
(g(m n)) (g(m n)).
2.2.
, f(n) (is monoton-
ically increasing), f(m) 6 f(n) m 6 n. ,

37. 37
f(n) (is monotonically decreasing), -
f(m) > f(n) m 6 n. , f(n)
(is strictly increasing), f(m) < f(n) m < n.
, f(n) (is strictly decreasing),
f(m) > f(n) m < n.
x bxc (the oor of x)
, . . , -
x. dxe (the ceiling of x) -
, x. ,
x;1 < bxc 6 x 6 dxe < x+ 1
x. ,
dn=2e+ bn=2c = n
n. , x
a b
ddx=ae=be = dx=abe (2.3)
bbx=ac=bc = bx=abc (2.4)
( , , z
n n 6 z n 6 bzc ).
x 7! bxc x 7! dxe .
( ) d n (polyno-
mial in n of degree d)
p(n) =
dX
i=0
aini
(d | ). a0 a1 ::: ad
(coe cients) . , -
ad ( , d
| , -
).
n p(n)
(

38. 38 2
), ad > 0 p(n) -
( n) p(n) =
(nd).
a > 0 n 7! na , a 6
0 | . , f(n) -
, f(n) = nO(1), , ,
f(n) = O(nk) k ( . . 2.2-2).
m, n a 6= 0
a0 = 1 (am)n = amn
a1 = a (am)n = (an)m
a;1 = 1=a aman = am+n:
a > 1 n 7! an .
00 = 1.
n 7! an , -
(exponential). a > 1
: b,
limn!1
nb
an = 0 (2.5)
, , nb = o(an). -
e = 2 71828:::,
ex = 1 + x + x2
2!
+ x3
3!
+ ::: =
1X
k=0
xk
k!
(2.6)
k! = 1 2 3 ::: k ( . ).
x
ex > 1 + x (2.7)
x = 0. jxj 6 1
ex :
1 + x 6 ex 6 1 + x + x2 (2.8)
, ex = 1 + x + (x2) x ! 0,
( n ! 1
x ! 0).
x limn!1
;
1+ x
n
n = ex.

39. 39
:
lgn = log2 n ( ),
lnn = loge n ( ),
lgk n = (lgn)k
lglgn = lg(lgn) ( ).
,
,
lgn+k lg(n)+k ( lg(n+k)). b > 1 n 7! logb n
( n) .
a > 0, b > 0, c > 0
n ( 1):
a = blogb a
logc(ab) = logc a + logc b
logb an = nlogb a
logb a = logc a
logc b
logb(1=a) = ;logb a
logb a = 1
loga b
alogb c = clogb a
(2.9)
,
O(logn) , -
.
( , )
lg.
(
jxj < 1):
ln(1+ x) = x ; x2
2 + x3
3 ; x4
4 + x5
5 ;:::
x > ;1
x
1 + x 6 ln(1+ x) 6 x (2.10)
x = 0.
, f(n) (is poly-
logaritmically bounded), f(n) = lgO(1) n. (2:5) -
n = lgm a = 2c
limm!1
lgb m
(2c)lgm = limm!1
lgb m
mc = 0
, , lgb n = o(nc) c > 0. -
, -
.

40. 40 2
n! ( , n factorial")
1 n. 0! = 1, n! =
n (n; 1)! n = 1 2 3 :::.
, n! 6 nn ( -
n). (Stirling's
approximation), ,
n! =
p
2 n n
e
n
(1+ (1=n)) (2.11)
,
n! = o(nn)
n! = !(2n)
lg(n!) = (nlgn):
:
p
2 n n
e
n
6 n! 6
p
2 n n
e
n
e1=12n: (2.12)
log n (
) : (iter-
ated logarithm). . -
i- , lg(i), :
lg(0) n = n lg(i)(n) = lg(lg(i;1) n) i > 0. (
, lg(i;1) n .) -
: lgi n lg(i) n ,
.
lg n i > 0, -
lg(i) n 6 1. , lg n | ,
lg, n , -
1.
lg n :
lg 2 = 1
lg 4 = 2
lg 16 = 3
lg 65536 = 4
lg 265536 = 5:
-
1080, 265536, n,
lg n > 5, .

41. 41
(Fibonacci numbers) -
:
F0 = 0 F1 = 1 Fi = Fi;1 + Fi;2 i > 2 (2.13)
,
0 1 1 2 3 5 8 13 21 34 55 :::
.
(golden
ratio) ' ^':
' = 1 +
p
5
2 = 1 61803:::
^' = 1 ;
p
5
2 = ;0 61803:::
(2.14)
,
Fi = 'i ; ^'i
p
5
(2.15)
( . 2.2-7). j^'j <
1, j^'i=
p
5j 1=
p
5 < 1=2, Fi
'i=
p
5, .
Fi ( ) i.
2.2-1 , f(n)
g(n) f(n) + g(n) f(g(n)) -
. f(n) g(n) n,
f(n)g(n) .
2.2-2 , T(n) = nO(1) ,
k, T(n) = O(nk) ( ,
T(n) > 1).
2.2-3 (2.9).
2.2-4 , lg(n!) = (nlgn) n! = o(nn).
2.2-5? dlgne! ?
dlglgne! ?
2.2-6? ( n): lg(lg n) lg (lgn)?

42. 42 2
2.2-7 (2.15).
2.2-8 : Fi+2 > 'i
i > 0 ( ' | ).
2-1
p(n) = a0+a1n+:::+adnd | d,
ad > 0. ,
. p(n) = O(nk) k > d.
. p(n) = (nk) k 6 d.
. p(n) = (nk) k = d.
. p(n) = o(nk) k > d.
. p(n) = !(nk) k < d.
2-2
, A O, o, , ! B
(k > 1, " > 0, c > 1 | ).
A B O o !
. lgk n n"
. nk cn
.
pn nsinn
. 2n 2n=2
. nlgm mlgn
. lg(n!) lg(nn)
2-3
. 30 -
( O( )) , -
( ):
lg(lg n) 2lg n (
p
2)lgn n2 n! (lgn)!
(3=2)n n3 lg2 n lg(n!) 22n
n1=lgn
lnlnn lg n n 2n nlglgn lnn 1
2lgn (lgn)lgn en 4lgn (n+ 1)!
plgn
lg lgn 2
p2lgn n 2n nlgn 22n+1
. f(n), -
gi (f(n) O(gi(n))
gi(n) O(f(n))).

43. 2 43
2-4
f(n) g(n) -
n. ,
. f(n) = O(g(n)), g(n) = O(f(n))?
. f(n) + g(n) = (min(f(n) g(n)))?
. f(n) = O(g(n)) lg(f(n)) = O(lg(g(n))), lg(g(n)) > 0
f(n) > 1 n?
. f(n) = O(g(n)) 2f(n) = O(2g(n))?
. f(n) = O((f(n))2)?
. f(n) = O(g(n)) g(n) = (f(n))?
. f(n) = (f(n=2))?
. f(n) + o(f(n)) = (f(n))?
2-5
- -
. 1 ,
. f(n) g(n) | -
. , f(n) = 1(g(n)),
c, f(n) > cg(n) > 0 -
n.
. , f(n) g(n), -
n, f(n) = O(g(n)),
f(n) = 1(g(n)),
(g(n)) .
. -
1 ?
f(n) = O(g(n)) , -
f(n) . -
O0 , f(n) = O0(g(n)),
jf(n)j = O(g(n)).
. -
2.1?
:
, f(n) = ~O(g(n)),
c k, 0 6 f(n) 6 cg(n)lgk n
n.
. ~ ~
2.1.
2-6
,
. f(n) |
, f(n) < n. f(i)(n), f(0)(n) = n
f(i)(n) = f(f(i;1)(n)) i > 0.
c fc (n) -

44. 44 2
i > 0, f(i)(n) 6 c. ( , fc |
f, n
, c.) , fc (n)
.
f(n) c
fc :
f(n) c fc (n)
. lgn 1
. n ;1 0
. n=2 1
. n=2 2
.
pn 2
.
pn 1
. n1=3 2
. n=lgn 2
121], O- -
(P. Bachmann, 1892). -
o(g(n)) (E. Landau) 1909 -
.
- - 124]: -
O- ( -
). .
121, 124] 33].
-
, , , .
, -
n ( ).
, -
: 1], 27].
( ., , 12]
192]). 121] -
, -
.

45. 3
(for, while),
. , -
1.2, j-
, j.
nX
j=1
j
(n2).
( , 4 ).
3.1 .
3.2 -
-
.
3.1.
a1 + a2 + :::+ an
nX
k=1
ak:
n = 0 0.
, |
. ( , .)
.
(series)
a1 + a2 + a3 + ::: 1X
k=1
ak

46. 46 3
limn!1
nX
k=1
ak:
, , (di-
verges) .
P1
k=1 jakj -
,
P1
k=1 ak (absolutely
convergent series) ,
. -
.
,
nX
k=1
(cak + bk) = c
nX
k=1
ak +
nX
k=1
bk
c
a1 ::: an b1 ::: bn.
,
.
nX
k=1
k = 1 + 2+ :::+ n
,
(arithmetic series). -
nX
k=1
k = 1
2n(n + 1) = (3.1)
= (n2): (3.2)
x 6= 1 (geometric ex-
ponential series)
nX
k=0
xk = 1 + x + x2 + :::+ xn

47. 47
nX
k=0
xk = xn+1 ;1
x;1 : (3.3)
(
jxj < 1)
1X
k=0
xk = 1
1; x: (3.4)
1+1=2+1=3+:::+1=n+::: n-
(nth harmonic number)
Hn = 1 + 1
2 + 1
3 + :::+ 1
n =
nX
k=1
1
k = lnn + O(1): (3.5)
-
, . , -
(3.4) x,
1X
k=0
kxk = x
(1;x)2: (3.6)
a0 a1 ::: an -
nX
k=1
(ak ; ak;1) = an ; a0 (3.7)
( ). - -
telescoping series. ,
n;1X
k=0
(ak ; ak+1) = a0 ; an:

48. 48 3
Pn;1
k=1
1
k(k+1). -
1
k(k+1) = 1
k ; 1
k+1, ,
n;1X
k=1
1
k(k + 1)
=
n;1X
k=1
1
k ; 1
k + 1
= 1 ; 1
n:
a1 ::: an
nY
k=1
ak:
n = 0 defn (product) -
1. :
lg
nY
k=1
ak =
nX
k=1
lgak:
3.1-1
Pn
k=1(2k ; 1).
3.1-2? , -
,
Pn
k=1 1=(2k ; 1) = ln(
pn) + O(1).
3.1-3? ,
P1
k=0(k ;1)=2k = 0.
3.1-4? P1
k=1(2k + 1)x2k.
3.1-5 ,
O-
.
3.1-6 , -
-
.
3.1-7
Qn
k=1 2 4k.
3.1-8? Qn
k=2(1 ;1=k2).

49. 49
3.2.
, -
( ).
,
. : ,
Sn =
Pn
k=1 k n(n+1)=2. n =
1 . , Sn = n(n + 1)=2
n n + 1. ,
n+1X
k=1
k =
nX
k=1
k + (n+ 1) = n(n+ 1)=2+ (n+ 1) = (n+ 1)(n+ 2)=2:
. , -
,
Pn
k=0 3k O(3n).
, ,
Pn
k=0 3k 6 c 3n
c ( ). n = 0
P0
k=0 3k = 1 6
c 1 c > 1.
n, n + 1. :
n+1X
k=0
3k =
nX
k=0
3k + 3n+1 6 c3n + 3n+1 = 1
3 + 1
c c3n+1 6 c3n+1:
, (1=3+ 1=c) 6 1, . . c > 3=2.Pn
k=0 3k = O(3n), .
, -
, -
. ,
Pn
k=1 k = O(n). ,P1
k=1 k = O(1). -
n, n. ,
n+1X
k=1
k =
nX
k=1
k + (n+ 1) = O(n)+ (n+ 1) !]
= O(n+ 1):
, ,
O(n), n.
,
( ,

50. 50 3
). , -Pn
k=1 k
nX
k=1
k 6
nX
k=1
n = n2:
nX
k=1
ak 6 namax
amax | a1 ::: an.
|
. ,Pn
k=0 ak , ak+1=ak 6 r
k > 0 r < 1. ak 6 a0rk,
:
nX
k=0
ak 6
1X
k=0
a0rk = a0
1X
k=0
rk = a0
1
1 ;r:
P1
k=1 k3;k.
1=3,
(k + 1)3;(k+1)
k3;k = 1
3
k + 1
k 6 2
3
k > 1. ,
(1=3)(2=3)k,
1X
k=1
k3;k 6
1X
k=1
1
3
2
3
k
= 1
3
1
1; 2
3
= 1:
, 1,
r < 1,
. , (k + 1)-
k- k
k+1 < 1.
1X
k=1
1
k = limn!1
nX
k=1
1
k = limn!1 (lgn) = 1:
r,
.

51. 51
. Pn
k=1 k. -
,
Pn
k=1 k > n.
O(n2) , -
:
nX
k=1
k =
n=2X
k=1
k +
nX
k=n=2+1
k >
n=2X
k=1
0 +
nX
k=n=2+1
n
2
= n
2
2
:
(
, , -
).
-
, ( k0)
nX
k=0
ak =
k0X
k=0
ak +
nX
k=k0+1
ak = (1)+
nX
k=k0+1
ak:
, 1X
k=0
k2
2k
(k + 1)2=2k+1
k2=2k = (k + 1)2
2k2
1. ,
, .
(k > 3) 8=9,
1X
k=0
k2
2k =
2X
k=0
k2
2k +
1X
k=3
k2
2k 6 O(1)+ 9
8
1X
k=0
8
9
k
= O(1)
|
.
-
. , ,
Hn =
Pn
k=1
1
k. 1 n blgnc -
1+(1=2+1=3)+(1=4+1=5+1=6+1=7)+:::.
1 ( ), -
blgnc + 1 ( ). ,
1+ 1=2+ 1=3+ :::+ 1=n 6 lgn + 1: (3.8)

52. 52 3
fPn
k=m f(k) :
nZ
m;1
f(x)dx 6
nX
k=m
f(k) 6
n+1Z
m
f(x)dx: (3.9)
, . 3.1, ( -
) ( -
) ( ). -
f:
n+1Z
m
f(x)dx 6
nX
k=m
f(k) 6
nZ
m;1
f(x)dx: (3.10)
:
nX
k=1
1
k >
n+1Z
1
dx
x = ln(n + 1) (3.11)
( )
nX
k=2
1
k 6
nZ
1
dx
x = lnn
nX
k=1
1
k 6 lnn + 1: (3.12)
3.2-1 ,
Pn
k=1
1
k2 -
( n).
3.2-2
blgncX
k=0
dn=2ke:

53. 53
. 3.1
Pn
k=m f(k) . -
.
.
. ( ) ,
Rn
m;1
f(x)dx 6 Pn
k=m f(k). ( ) -
, ,
Pn
k=m f(k) 6 Rn+1
m f(x)dx.

54. 54 3
3.2-3 , , n-
(lgn).
3.2-4
Pn
k=1 k3 .
3.2-5 -
-
(3.10)?
3-1
( r > 0 s > 0 ):
.
Pn
k=1 kr.
.
Pn
k=1 lgs k.
.
Pn
k=1 kr lgs k.
121] |
.
( -
12] 192]).

55. 4
, -
,
-
. (recur-
rences). , 1, -
Merge-Sort
T(n) =
(
(1) n = 1,
2T(n=2)+ (n) n > 1.
(4.1)
( ), T(n) = (nlgn).
,
, . . (
O ) . - , ,
, -
( , substitution
method). - , ,
, (
, iteration method). , -
T(n) = aT(n=b)+ f(n)
a > 1, b > 1 | , f(n) | -
. -
(master theorem). -
,
.
-
, .
, Merge-

56. 56 4
Sort (T(n) -
n):
T(n) =
(
(1) n = 1,
T(dn=2e)+ T(bn=2c)+ (n) n > 1.
(4.2)
, ,
, T(n) = (1) n. , -
(4.1)
T(n) = 2T(n=2)+ (n): (4.3)
, ( T(1)) -
,
T(n).
, , -
, ( ) ,
.
, ,
( . 4.1 4.5).
4.1.
: .
, ,
.
, -
. ,
T(n) = 2T(bn=2c)+ n (4.4)
(4.2) (4.3). , T(n) =
O(nlgn), . . T(n) 6 cnlgn c > 0. -
. bn=2c, . .
T(bn=2c) 6 cbn=2clg(bn=2c). ,
T(n) 6 2(cbn=2clg(bn=2c))+ n 6
6 cnlg(n=2)+ n = cnlgn ;cnlg2 + n = cnlgn ;cn+ n 6
6 cnlgn:
c > 1.
-
, . . n. -
, n = 1 -
, c ( lg1 = 0). -
,

57. 57
n, . c ,
T(n) 6 cnlgn n = 2 n = 3. n
, n = 1
( bn=2c > 2 n > 3).
?
. -
.
.
T(n) = 2T(bn=2c+ 17) + n:
(4.4) 17
. , , -
: n
bn=2c + 17 bn=2c .
, T(n) = O(nlgn) ,
( . . 4.1-5).
. -
, . , -
(4.1) T(n) = (n) (
n), T(n) = O(n2) (
). -
,
, .
,
. ,
.
.
T(n) = T(bn=2c)+ T(dn=2e)+ 1:
, T(n) = O(n). -
. , , , T(n) 6 cn -
c.
T(n) 6 cbn=2c+ cdn=2e+ 1 = cn+ 1
c T(n) 6 cn.
,
O(n2), :

58. 58 4
, ,
.
: T(n) 6 cn;b -
b c.
T(n) 6 (cbn=2c;b)+ (cdn=2e;b)+ 1 = cn; 2b+ 1 6 cn;b
b > 1. -
c .
:
, ( ) , -
? |
,
.
:
T(n) = O(n)
(4.4). , T(n) 6 cn, -
T(n) 6 2(cbn=2c)+ n 6 cn+ n = O(n):
, , , -
, cn c
c, O(n).
. , -
T(n) = 2T(bpnc) + lnn
,
. m = lgn,
T(2m) = 2T(2m=2) + m:
T(2m) S(m),
S(m) = 2S(m=2)+ m
(4.4). : S(m) = O(mlgm).
T(n) S(m),
T(n) = T(2m) = S(m) = O(mlgm) = O(lgnlglgn):

59. 59
, , , -
T(n) n,
. ,
S(m) T(n) n, 2m.
4.1-1 , T(n) = T(dn=2e)+1 , T(n) =
O(lgn).
4.1-2 , T(n) = 2T(bn=2c) + n T(n) =
(nlgn), T(n) = (nlgn).
4.1-3 n = 1
(4.4), -
, ?
4.1-4 , (4.2)
(nlgn).
4.1-5 , T(n) = 2T(bn=2c + 17) + n T(n) =
O(nlgn).
4.1-6
T(n) = 2T(
pn) + 1, ( , ,
| ).
4.2.
, ? , -
( ), ,
.
T(n) = 3T(bn=4c)+ n:
, :
T(n) = n + 3T(bn=4c) = n + 3(bn=4c+ 3T(bn=16c))
= n + 3(bn=4c+ 3(bn=16c+ 3T(bn=64c)))
= n + 3bn=4c+ 9bn=16c+ 27T(bn=64c)
, , (2.4), bbn=4c=4c = bn=16c
bbn=16c=4c = bn=64c. ,
? i- -
T(bn=4ic), T(1), bn=4ic = 1, . .

60. 60 4
i > log4 n. , bn=4ic 6 n=4i,
( , -
3log4 n ):
T(n) 6 n+ 3n=4+ 9n=16+ 27n=64+ :::+ 3log4 n (1)
6 n
1X
i=0
3
4
i
+ (nlog4 3)
= 4n + o(n) = O(n)
( log4 n + 1
, 3log4 n nlog4 3,
o(n), log4 3 < 1.)
-
.
:
, .
,
( ).
( -
). , -
, (
n = 4k ). , -
,
,
. 4.3, -
( . 4-5).
(recursion tree). ,
. 4.1
T(n) = 2T(n=2)+ n2:
, n | .
( ) ( ), T(n), -
T(n), T(n=2), T(n=4) . . -
T(n), -
. n2, | (n=2)2 +(n=2)2 =
n2=2, | (n=4)2 + (n=4)2 + (n=4)2 + (n=4)2 = n2=4. -
,
-
. , T(n) = (n2).
. 4.2 | -
T(n) = T(n=3)+ T(2n=3)+ n

61. 61
Total=
. 4.1 T(n) = 2T(n=2) +n2
. -
( ) lgn ( lgn+ 1 ).
. 4.2 T(n) = T(n=3) +T(2n=3) +n.

62. 62 4
( ).
n. ,
1. -
,
n ! (2=3)n ! (2=3)2n ! ! 1 k = log3=2 n
( k (2=3)kn = 1). T(n)
O(nlgn).
4.2-1 ( ) T(n) =
3T(bn=2c) + n,
T(n).
4.2-2 , , T(n) =
T(n=3)+ T(2n=3)+ n T(n) = (nlgn).
4.2-3 T(n) = 4T(bn=2c) + n
T(n).
4.2-4 T(n) = T(n ;
a) + T(a) + n, a > 1 | .
4.2-5 T(n) =
T( n)+T((1; )n)+n, | 0 < < 1.
4.3.
T(n) = aT(n=b)+ f(n) (4.5)
a > 1 b > 1 | , f |
( ) .
,
.
(4.5) ,
n a n=b,
( T(n=b)) -
.
f(n) ( 1.3.2 f(n) = C(n)+ D(n)).
, Merge-Sort a = 2, b = 2,
f(n) = (n).
, (4.5) -
: n=b .

63. 63
T(n=b) T(bn=bc) T(dn=be). ,
, ,
.
4.1. a > 1 b > 1 | , f(n) | ,
T(n) n
T(n) = aT(n=b)+ f(n)
n=b dn=be, bn=bc. :
1. f(n) = O(nlogb a;") " > 0, T(n) =
(nlogb a).
2. f(n) = (nlogb a), T(n) = (nlogb a lgn).
3. f(n) = (nlogb a+") " > 0 af(n=b) 6
cf(n) c < 1 n,
T(n) = (f(n)).
? -
f(n) nlogb a -
, T(n) ( 1 3).
( 2), -
(nlogb a lgn) = (f(n)lgn).
. -
, f(n) , nlogb a: -
n" " > 0.
f(n) nlogb a ,
af(n=b) 6 cf(n)
, , .
, -
: , , f(n) ,
nlogb a, , -
.
2 3. , -
.
, -
.
T(n) = 9T(n=3)+ n:
a = 9, b = 3, f(n) = n, nlogb a = nlog3 9 = (n2).

64. 64 4
f(n) = O(nlog3 9;") " = 1,
, T(n) = (n2).
T(n) = T(2n=3)+ 1:
a = 1, b = 3=2, f(n) = 1 nlogb a = nlog3=2 1 = n0 = 1.
2, f(n) = (nlogb a) = (1), ,
T(n) = (lgn).
T(n) = 3T(n=4)+ nlgn
a = 3, b = 4, f(n) = nlgn nlogb a =
nlog4 3 = O(n0 793). ( " 0 2) , -
. n af(n=b) =
3(n=4)lg(n=4) 6 (3=4)nlgn = cf(n) c = 3=4.
T(n) = (nlgn).
, : T(n) =
2T(n=2)+nlgn. a = 2, b = 2, f(n) = nlgn, nlogb a = n. ,
f(n) = nlgn , nlogb a, -
: f(n)=nlogb a = (nlgn)=n = lgn
n" " > 0. -
2 3
. 4.4-2.
4.3-1 ,
. T(n) = 4T(n=2)+ n
. T(n) = 4T(n=2)+ n2
. T(n) = 4T(n=2)+ n3.
4.3-2 A
T(n) = 7T(n=2)+n2, A0 | -
T0(n) = aT0(n=4)+n2. a
A0 , A?
4.3-3 , ,
T(n) = T(n=2)+ (1) ( , . . 1.3-5)
T(n) = (lgn).
4.3-4 , ( 3) -
: f,
", -
.

65. 4.4 4.1 65
? 4.4 4.1
4.1 -
,
.
. -
n, b -
. -
, -
. .
-
: -
, b,
,
.
, ,
, : T(n)
, T(n) = O(n) n,
, n. ( ,
T(n) = n n = 1 2 4 8 ::: T(n) = n2 n.)
4.4.1.
T(n) , ( )
b > 1 ( )
(4.5), . .
T(n) = aT(n=b)+ f(n):
T :
( 4.2), ( -
4.3) ( 4.4).
4.2. a > 1, b > 1 | , f(n) |
, b.
T(n) | , b
T(n) = aT(n=b)+ f(n) n > 1, T(1) > 0.
T(n) = (nlogb a) +
logb n;1X
j=0
ajf(n=bj): (4.6)

66. 66 4
. 4.3 T(n) = aT(n=b) + f(n)
a- logb n nlogb a. -
, (4.6).
.
,
T(n) = f(n)+ aT(n=b)
= f(n)+ af(n=b)+ a2T(n=b2)
= f(n)+ af(n=b)+ a2f(n=b2) + :::
+ alogb n;1f(n=blogb n;1) + alogb nT(1):
alogb n = nlogb a,
(nlogb a). ,
.
4.2
( . 4.3), a (
). f(n), a -
f(n=b), a2 f(n=b2) . .
j aj | f(n=bj). -
logb n ,
T(1) alogb n = nlogb a .

67. 4.4 4.1 67
(4.6) , :
j- ajf(n=bj),
logb n;1X
j=0
ajf(n=bj):
,
.
nlogb a 1, -
(nlogb a).
, -
.
, | ,
.
(4.6).
4.3. a > 1, b > 1 | , f(n) | -
, b. -
g(n),
g(n) =
logb n;1X
j=0
ajf(n=bj) (4.7)
( n, b).
1. f(n) = O(nlogb a;") " > 0,
g(n) = O(nlogb a).
2. f(n) = (nlogb a), g(n) = (nlogb a lgn).
3. af(n=b) 6 cf(n) c < 1
n > b, g(n) = (f(n)).
. 1. -
f(n) = n , = logb a;"
f (4.7) : (
= logb a)
g(n) = f(n) + af(n=b)+ a2f(n=b2) + :::+ ak;1f(n=bk;1)
= n + a(n=b) + a2(n=b2) + :::+ ak;1(n=bk;1) (4.8)
-
k = logb n a=b 1,
< logb a b < a. -
(
). O(ak) = O(nlogb a).
.

68. 68 4
2.
:
g(n) = f(n) + af(n=b)+ a2f(n=b2) + :::
= n + a(n=b) + a2(n=b2) + ::: (4.9)
= logb a -
1 . logb n, -
nlogb a logb n = (nlogb a lgn):
2 .
3. f ,
-
, c < 1.
c, -
( 1=(1 ; c))
( , -
). , g(n) = (f(n)) n,
b. .
, n b.
4.4. a > 1, b > 1 | , f(n) |
, b.
T(n) | , b
T(n) = aT(n=b)+ f(n) n > 1 T(1) > 0. :
1. f(n) = O(nlogb a;") " > 0, T(n) =
(nlogb a).
2. f(n) = (nlogb a), T(n) = (nlogb a)lgn.
3. f(n) = (nlogb a+") " > 0 af(n=b) 6
cf(n) c < 1 -
n, T(n) = (f(n)).
??????? ?
. 4.2 4.3.
T(n) = (nlogb a) + O(nlogb a) = (nlogb a):
T(n) = (nlogb a) + (nlogb a lgn) = (nlogb a lgn):

69. 4.4 4.1 69
T(n) = (nlogb a) + (f(n)) = (f(n)):
, f(n) = (nlogb a+")
" > 0 ,
( . . 4.4-3).
4.4 .
4.4.2.
n, -
b. n=b -
T(n) = aT(dn=be)+ f(n) (4.10)
T(n) = aT(bn=bc) + f(n): (4.11)
, .
, .
n n=b n=b2 ::: -
ni, :
ni =
(
n i = 0,
dni;1=be i > 0.
(4.12)
( ). -
| , .
, logb n -
, n ( b)
.
, dxe 6 x + 1 ,
n0 6 n
n1 6 n
b + 1
n2 6 n
b2 + 1
b + 1
n3 6 n
b3 + 1
b2 + 1
b + 1
::::::::::::::::::::::
1 + 1=b + 1=b2 + ::: 6 1=(b ; 1), i = blogb nc
ni 6 n=bi + b=(b;1) 6 b + b=(b;1) = O(1).

70. 70 4
(4.10),
T(n) = f(n0) + aT(n1)
= f(n0) + af(n1)+ a2T(n2)
6 f(n0) + af(n1)+ a2f(n2) + :::+
+ ablogb nc;1f(nblogb nc;1) + ablogb ncT(nblogb nc)
= (nlogb a) +
blogb nc;1X
j=0
ajf(nj): (4.13)
(4.6), n
b.
, .
, T(ndlogbne) .
, f(n) , . .
n, n0.
, n0.
.]
g(n) =
blogb nc;1X
j=0
ajf(nj): (4.14)
-
, ( -
), ( ) ( -
). -
, -
( ,
). , -
n=bi ni. -
, b=(b ; 1), -
. ,
f(nj) = O(nlogb a=aj) = O((n=bj)logb a), -
4.3 ( 2). , bj=n 6 1 j 6 blogb nc.
f(n) = O(nlogb a) , c

71. 4 71
nj
f(nj) 6 c n
bj + b
b;1
logb a
= c nlogb a
aj 1+ bj
n
b
b;1
logb a
6 c nlogb a
aj 1+ b
b;1
logb a
= O nlogb a
aj :
, c(1 + b=(b ; 1))logb a | -
. , 2 .
1 .
f(nj) = O(nlogb a;") ,
2, .
, n
.
, , nj
n=bj.
4.4-1? ni (4.12), b|
.
4.4-2? , f(n) = (nlogb a lgk n), k > 0,
(4.5) T(n) = (nlogb a lgk+1 n).
b.
4.4-3? , 3 -
: (af(n=b) 6 cf(n) -
c < 1) , " > 0,
f(n) = (nlogb a+").
4-1
-
( , T(n) | -
n 6 2):
. T(n) = 2T(n=2)+ n3.
. T(n) = T(9n=10)+ n.

72. 72 4
. T(n) = 16T(n=4)+ n2.
. T(n) = 7T(n=3)+ n2.
. T(n) = 7T(n=2)+ n2.
. T(n) = 2T(n=4)+
pn.
. T(n) = T(n;1) + n.
. T(n) = T(
pn) + 1.
4-2
A 1::n] 0 n, -
. , -
B 0::n] , A,
, ( -
O(n)). A
.
-
A: -
A.
, -
O(n).
4-3
, -
, |
. ,
, . -
. -
:
1. (1).
2. (N), N | .
3. , -
. (q ;p + 1),
A p::q].
.
( . 1.3-5). -
? ( -
, .)
. Merge-Sort -
1.3.1.
4-4
T(n) .
, T(n) | n 6 8.
. T(n) = 3T(n=2)+ nlgn.

73. 4 73
. T(n) = 3T(n=3+ 5)+ n=2.
. T(n) = 2T(n=2)+ n=lgn.
. T(n) = T(n ;1) + 1=n.
. T(n) = T(n ;1)+ lgn.
. T(n) =
pnT(
pn) + n.
4-5
T(n) n, -
b. -
n?
. T(n) h(n) | , -
( -
), T(n) 6 h(n) n, -
b > 1. , , h
: h(n) = O(h(n=b)). ,
T(n) = O(h(n)).
. T T(n) =
aT(n=b) + f(n) n > n0 a > 1, b > 1 f(n) -
. T(n) n 6 n0,
T(n0) 6 aT(n0=b)+f(n0): , T(n)
.
. 4.1
f(n).
4.4.
4-6
(2.13).
. -
(generating function), -
(formal power series)
F =
1X
i=0
Fizi = 0 + z + z2 + 2z3 + 3z4 + 5z5 + 8z6 + 13z7 + :::
. , F(z) = z + zF(z)+ z2F(z).
. ,
F(z) = z
1 ;z ;z2 = z
(1 ;'z)(1 ; b'z) = 1p
5
1
1 ;'z ; 1
1 ; b'z
' = 1 +
p
5
2 = 1 61803:::
b' = 1;
p
5
2 = ;0 61803:::

74. 74 4
. ,
F(z) =
1X
i=0
1p
5
('i ; b'i)zi:
. , Fi i > 0 'i=
p
5
. ( : jb'j < 1.)
. , Fi+2 > 'i i > 0.
4-7
n ,
, | . -
,
. -
,
.
A B
B A
B A
B A
B A
. , ,
, -
( ).
.
n , . ,
bn=2c , -
.
. . ,
(n) .
( .)
: -
n , -
. ,
, . ,
. -
O(n) . -
, -
. ,
, , . -
, -
:
.]

75. 4 75
(L. Fibonacci)
1202 . (A. De Moivre)
( . 4-6). -
-
, 26], -
( . 4.4-2). 121] 140]
,
. -
164].

76. 76 4
1257

77. 5
, , -
, , ,
. , , -
( ).
5.1.
(set) (members, elements).
x S, x 2 S ( -
x S ). x S, x =2 S.
-
. , S = f1 2 3g -
1 2 3 . 2
, 4 | , 2 2 S, 4 =2 S.
, -
. A B (are equal),
.
A = B. , f1 2 3 1g= f1 2 3g= f3 2 1g.
-
:
? (empty set),
.
Z (integers) Z =
f::: ;2 ;1 0 1 2 :::g
R ( )
(real numbers)
N (natural numbers)
N = f0 1 2 :::g1
A
1 1 | .

78. 78 5
B, ( x 2 A x 2 B), , A -
(subset) B A B. A
B, A
(proper subset) B A B. (
A B ,
.) A
A A. A B -
, A B B A. A, B C
A B B C A C. A
? A.
:
A ,
, B.
, fx : x 2
Z x=2 | g. x Z,
...
.
A B -
, -
(set operations):
(intersection) A B
A B = fx : x 2 A x 2 Bg:
(union) A B -
A B = fx : x 2 A x 2 Bg:
(di erence) A B -
AnB = fx : x 2 A x =2 Bg:
-
:
(empty set laws):
A ? = ? A ? = A:
(idempotency laws):
A A = A A A = A:
(commutative laws):
A B = B A A B = B A:

79. 79
??? n
. 5.1 , -
(5.2). A B C .
(associative laws):
A (B C) = (A B) C A (B C) = (A B) C:
(distributive laws):
A (B C) = (A B) (AC)
A (B C) = (A B) (A C)
(5.1)
(absorption laws):
A (A B) = A A (AB) = A
(DeMorgan's laws):
A n(B C) = (AnB) (AnC)
A n(B C) = (AnB) (AnC)
(5.2)
. 5.1 (5.1)
A, B C .
,
(universe). , ,
,
Z . U ,
(complement) A A =
U nA. A U :
A = A A A = ? A A = U:
(5.2) ,
A B U
A B = A B A B = AB:
A B (disjoint),
, . . AB = ?. ,
S = fSig
(partition) S,

80. 80 5
Si (are pairwise disjoint),
. . Si Sj = ? i 6= j,
S, . .
S =
Si2S
Si
, S S,
s 2 S
Si .
S (car-
dinality), (size), jSj.
,
. -
: j?j = 0. ( nite)
| (in nite) -
.
, , -
, (countably in nite)
, , -
(uncountable). Z ,
R .
A B
jA Bj = jAj+ jBj; jABj (5.3)
,
jA Bj 6 jAj+ jBj
A B , jA Bj = 0
: jA Bj = jAj+jBj. A
B, jAj 6 jBj.
n n- (n-
set) (sin-
gleton). k-
subset, k- ( - -
).
S
, S -
2S - (power set). ,
2fa bg = f? fag fbg fa bgg. S 2S -
2jSj .
a b (a b)
(a b) = fa fa bgg,
(a b) (b a).

81. 81
.
, , (a b) = (c d)
(a = c) (b = d).]
(cartesian product) A
B ,
A, | B. : A B.
A B = f(a b) : a 2 A b 2 Bg
, fa bg fa b cg = f(a a) (a b) (a c) (b a) (b b) (b c)g.
A B
:
jA Bj = jAj jBj: (5.4)
n A1 A2 ::: An
n- (n-tuples)
A1 A2 ::: An = f(a1 a2 ::: an) : ai 2 Ai i = 1 2 ::: ng
( (a b c) ((a b) c),
(a b c d) ((a b c) d) ).
:
jA1 A2 ::: Anj = jA1j jA2j ::: jAnj
An = A A ::: A
n A
An jAjn. , n-
n ( . . 5.3).
5.1-1 -
(5.1).
5.1-2 -
:
A1 A2 :::An = A1 A2 ::: An
A1 A2 ::: An = A1 A2 ::: An:

82. 82 5
5.1-3 (5.3), -
(principle of inclusion and exclusion):
jA1 A2 ::: Anj = jA1j+ jA2j+ :::+ jAnj
; jA1 A2j; jA1 A3j; :::
+ jA1 A2 A3j+ :::
+ (;1)n;1jA1 A2 ::: Anj
5.1-4 ,
.
5.1-5 , S n -
, - 2S 2n . ( -
, S 2n .)
5.1-6 n- , -
.
5.2.
(binary relation) R
A B -
A B. (a b) 2 R, aRb , a
R b.
A A A.
,
f(a b) : a b 2 N a < bg. n- -
(n-ary relation) A1 A2 ::: An
A1 A2 ::: An.
R A A (re ex-
ive),
aRa
a 2 A. , = 6 -
N, <
. R (symmetric),
aRb bRa
a b 2 A. ,
< 6 | . R
(transitive),
aRb bRc aRc

83. 83
a b c 2 A. , < , 6 =
, R = f(a b) : a b 2 N a = b; 1g |
, 3R4 4R5, 3R5.
, , -
, -
(equivalence relation). R |
A, (equiv-
alence class) a 2 A a] = fb 2 A : aRbg
, a. , -
,
a b , a + b . -
( R) -
. , a + a ,
a+ b = b+ a, R ,
a + b b + c | , a+ c = (a + b)+ (b+ c) ; 2b -
, R .
4 4] = f0 2 4 6 :::g, 3
3] = f1 3 5 7 :::g.
:
5.1 ( ).
A
A. , -
A
.
. , -
, , -
A. -
a 2 a], a.
, a] b] , -
. c | , aRc, bRc, cRb ( -
) aRb ( ). , b] a]:
x | b, bRx aRx.
, a] b] a] = b].
.
R A -
,
aRb bRa a = b:
, 6 -
, a 6 b b 6 a a = b. -
, -
(partial order),
-
(partially ordered set). ,

84. 84 5
-
, .
, ,
| x, yRx
y. -
: x (max-
imal), , . . xRy -
x = y. , -
( ),
(
).
(total order, linear or-
der), a b aRb, bRa
( | ).
, 6 -
, -
| ( ,
).
5.2-1 , -
Z , -
.
5.2-2 , n
a b (mod n) -
Z. ( , a b (mod n), q,
a;b = qn.)
?
5.2-3 ,
. ,
. ,
. , .
5.2-4 S | , R | -
S. , R ,
.
5.2-5 , -
, :
aRb bRa , aRa
. ?

85. 85
5.3.
A B. (function), -
A B, f A B, -
: a 2 A
b 2 B, (a b) 2 f. A -
(domain) B
, -
codomain.
, f -
A B. -
A
B, B -
A. , -
f = f(a b) : a 2 N b = a mod 2g
f : N ! f0 1g,
a b 2 f0 1g,
a mod 2. f(0) = 0, f(1) = 1, f(2) = 0
. ,
g = f(a b) : a b 2 N a + b g
, ( ) (1 3) (1 5),
, ,
.
(a b) f, -
, , b (value)
(argument) a, b = f(a). -
, , -
. ,
f: N ! N f(n) = 2n, , f = f(n 2n) :
n 2 Ng. f g: A ! B (equal),
f(a) = g(a) a 2 A. ( , -
f: A ! B1 g: A ! B2
B1 6= B2, f(a) = g(a) a!)
( nite sequence) n -
f,
f0 1 2 ::: n;1g.
, . . hf(0) f(1) ::: f(n;1)i. -
(in nite sequence) ,
N
. , , -
(2.13), h0 1 1 2 3 5 8 13 21 :::i.

86. 86 5
-
, -
. , f : A1 A2
::: An ! B, f(a1 a2 ::: an)
f((a1 a2 ::: an)). ai (ar-
gument) f,
n- (a1 a2 ::: an).
b = f(a) f : A ! B
a 2 A, b 2 B, b (image) a.
A0 A f(A0)
f(A0) = fb 2 B : b = f(a) a 2 A0g:
(range) f
, . . f(A). , -
f : N ! N, f(n) = 2n,
, -
f(N) = fm : m = 2n n 2 Ng.
f: A ! B (surjection), -
, B, . .
b 2 B a 2 A. -
, f : N ! N, f(n) = bn=2c, -
. f(n) = 2n ,
, f: N ! N, , -
-
. f : A ! B, ,
A B (onto B).
f: A ! B (injection), -
,
, . . f(a) 6= f(a0) a 6= a0. ,
f(n) = 2n N N, -
n
(n=2, n
). f(n) = bn=2c ,
( ) f(2) = f(3) = 1.
one-to-one function .
f: A ! B (bijection), -
. ,
f(n) = (;1)ndn=2e, ,

87. 87
N Z, :
0! 0
1!;1
2! 1
3!;2
4! 2
...
, Z
N. -
, Z -
N. -
(one-to-one correspon-
dence), -
A B. , -
A , (permutation) -
A.
f , (inverse)
f;1
f;1(b) = a , f(a) = b:
, f(n) = (;1)ndn=2e
f;1(m) =
(
2m m > 0,
;2m;1 m < 0.
5.3-1 A B | , f : A ! B | -
. ,
. f | , jAj 6 jBj
. f | , jAj > jBj.
5.3-2 f : N ! N,
f(x) = x+1? Z! Z,
.
5.3-3 .
( ,
.)
5.3-4? f: Z! Z Z.

88. 88 5
. 5.2 . ( ) -
(V E), V = f1 2 3 4 5 6g E =
f(1 2) (2 2) (2 4) (2 5) (4 1) (4 5) (5 4) (6 3)g. (2 2) -
. ( ) G = (V E), V = f1 2 3 4 5 6g
E = f(1 2) (1 5) (2 5) (3 6)g. 4 (
). ( ) ( ),
f1 2 3 6g.
5.4.
,
.
, -
,
. 23,
.
(directed graph)
(V E), V | , E | -
V, . . V V.
(digraph). -
V (vertex set)
(vertex
vertices). E (edge set) -
(edges). 5.2 ( ) -
f1 2 3 4 5 6g.
, | . ,
- (self-loops),
.
(undirected) G = (V E)
(V) (unordered) :
fu vg, u v 2 V u 6= v. -
(u v) fu vg
(u v) (v u)
. -
- ,
( ). . 5.2 ( ) -
f1 2 3 4 5 6g.

89. 89
-
( -
). (u v) ,
(incident from, leaves) u (incident to,
enters) v. , . 5.2 ( ) ,
2 ((2 2) (2 4) (2 5)) ,
((1 2) (2 2)). (u v) -
, (incident on vertices) u
v. , . 2.5 ( ) , 2
( (1 2) (2 5)).
G (u v), , v
u (is adjacent to u).
, -
. v
u , u ! v.
5.2 ( ) 5.2 ( ) 2 -
1, 1 2
( (2 1) ).
(degree) -
. , . 5.2 ( )
2 2. -
(out-degree), -
, (in-degree),
. -
(degree) . , -
2 . 5.2 ( ) 2,
3 5.
k (path of length k) u v -
hv0 v1 v2 ::: vki,
v0 = u, vk = v (vi;1 vi) 2 E i = 1 2 ::: k. -
, k k . (con-
tains) v0 v1 ::: vk (v0 v1) (v1 v2) ::: (vk;1 vk).
v0 , vk |
, v0 vk. u u0
p u u0, , u0 -
u p (u0 is reachable from u via p).
( ) u p
u0.
(simple), -
. , . 5.2 ( ) h1 2 5 4i -
3, h2 5 4 5i , -
.
(subpath) p = hv0 v1 ::: vki ,
, . .
hvi vi+1 ::: vji i j, -
0 6 i 6 j 6 k.

90. 90 5
(cycle) , -
-
. hv0 v1 ::: vki ,
( ),
. . v1 v2 ::: vk . -
1. , -
: k -
k (
k ). , . 5.2 ( ) h1 2 4 1i,
h2 4 1 2i h4 1 2 4i .
, h1 2 4 5 4 1i
. h2 2i, -
- (2 2). , -
- , (simple).
hv0 v1 ::: vki -
( ) , k > 3, v0 = vk
v1 v2 ::: vk . , . 5.2 ( ) -
h1 2 5 1i.
, , (acyclic).
(connected), -
-
. -
-
. -
(connected components) . , . 5.2 ( ) -
: f1 2 5g, f3 6g f4g. -
,
.
(strongly
connected), ( -
) . -
(strongly connected com-
ponents), -
u v v u .
, -
. . 5.2 ( )
: f1 2 4 5g, f3g f6g. ,
f3 6g , 3 -
6, .
G = (V E) G0 = (V0 E0)
(isomorphic),
f: V ! V0 ,
: (u v) 2 E
, (f(u) f(v)) 2 E0. , -
| ,
- . . 5.3 ( ) -

91. 91
. 5.3 ( ) . ( ) : -
4, | .
G G0 V = f1 2 3 4 5 6g
V 0 = fu v w x y zg. f : V ! V0, f(1) = u,
f(2) = v, f(3) = w, f(4) = x, f(5) = y, f(6) = z, -
. , . 5.3 ( ) ,
5 7 . , -
, ,
4, | .
G0 = (E0 V0) (subgraph) G =
(E V), E0 E V 0 V. G = (E V)
V0,
, ,
. . G0 = (E0 V0),
E0 = f(u v) 2 E : u v 2 V0g
G -
V0 (subgraph of G induced by V0).
. 5.2 ( ) f1 2 3 6g . 5.2 ( )
(1 2), (2 2), (6 3).
G
(directed version),
fu vg
(u v) (v u), . -
, -
(undirected version),
, - (u v)
(v u) fu vg.
(neighbor) u , -
( ) ,
v u , v u

92. 92 5
u v. v |
u v u .
.
(complete) , -
(
). (V E)
(bipartite), V -
V1 V2 , -
. -
(forest), -
( -
-
). - free
tree. (directed acyclic graph)
- dag ( ).
. ,
(multigraph),
, , -
, - . -
(hypergraph) ,
(hyperedges), -
, . -
-
.
5.4-1 , -
. . , -
,
(handshaking lemma): -
.
5.4-2 , k > 3 -
( ,
, , ).
5.4-3 , ( -
) u v,
u v. ,
, .
5.4-4 , (V E)
1: jEj >
jVj;1

93. 93
5.4-5 , -
( ) .
, -
, -
?
5.4-6 . 5.2 ( )
. 5.2 ( ).
5.4-7? -
, -
? ( :
,
.)
5.5.
, -
. -
. -
11.4 23.1
.
5.5.1.
5.4, ( ,
free tree) -
. ,
( ) , (forest) -
, ( -
).
. . 5.4 ( ) . 5.4 ( )
. 5.4 ( ) , .
5.4 ( ) , ,
.
-
.
5.2 ( ). G = (V E) | -
. :
1. G ( ).
2. G -
.
3. G , ,
.

94. 94 5
. 5.4 ( ) . ( ) . ( ) ,
, , .
. 5.5 u v
4. G jEj = jVj;1.
5. G jEj = jVj;1.
6. G , -
.
. (1) ) (2): ,
, , .
p1 p2 u
v ( . 5.5). ,
w | , x y |
, w p1 p2.
, ,
z. p0
w z ( x) p00 w z
( y). p0 p00 ( ),
z p1 w,
p2. ( p0
p00).
(2) ) (3) , , . (u v) | -
. ,
, u v | ,
.
(3) ) (4) , , jEj > jVj;1 ( . 5.4-
4). , jEj 6 jVj ; 1, .
n = 1 2 n;1 . G
n > 3 -
. G k > 2 -

95. 95
( , ).
(3). -
,
jVj; k 6 jVj; 2. , jVj; 1
.
(4) ) (5) jEj = jVj;1. ,
. ,
( ).
,
( ). ,
|
,
.
(5) ) (6) G jEj = jVj ; 1.
G k -
. ,
, k -
. , k = 1 .
( , (1) ) (2)),
.
(6) ) (1) , -
. , . -
, u v. ,
(u v) . -
(u v),
| , u
v, .
5.5.2. .
, (rooted tree), ,
( )
, (root). -
- nodes .
5.6 ( ) 12
7.
x | r.
r x ,
, (ancestors) x. y
x, x (descendant) y. -
.
x, x, -
(proper ancestors) (proper
descendants) x. : -

96. 96 5
= 4, 0, 1 . .
. 5.6 ( ) 4. :
( 7) , ( ,
1) , (
, 2) . .
. ( )
, ( 3),
.
-
, ,
.]
x ,
x, x . -
x (subtree rooted at x). ,
. 5.6 ( ) 8 8, 6, 5 9.
(y x) | x, y -
(parent) x, x y. -
(father) (son),
, - -
, | .
-
.]
, -
. , , -
siblings ,
, ( )
. , ,
(leaf, external node). , ,
(internal).
-
(degree). , , ,
, -
(
, ).
x (depth) -
x.

97. 97
. 5.7 . ( ) , -
. - ,
| - . ( ) ( , -
7 5 , ).
, . ( )
(a), ,
. -
.
(height) .
(ordered tree)
: -
( , ,
. .). . 5.6 -
, .
5.5.3. .
(binary tree) -
,
( ),
: , -
(root), , -
(left subtree) , , -
(right subtree) .
, , (emp-
ty). nil. ,
(left child)
(right child) .
, ,
(child is absent).
. 5.7 ( ).
-
,
2. , ,
1 |

98. 98 5
=3, 0, 1 . .
. 5.8 3 8 7
.
, -
. . 5.7 ( , ) ,
.
-
.
( , ). -
. 5.7 ( ).
: k-
(k-ary) k = 2. , -
(positional tree) ,
-
, . -
1, 2, 3 ,
( ) , -
(ith child is absent). k-
, k.
k- (complete k-ary tree) k-
,
k. ( -
.) . 5.8
3. ,
k- h.
0, k -
1, k2 k
kh h. ,
k- n logk n ( -
, ).
k- h
1 + k + k2 + :::+ kh;1 = kh ;1
k ; 1
( . (3.3)). , -
.

99. 5 99
5.5-1 ( ), -
A, B C. -
A, B C A. ( )
A, B C A.
A, B C A.
5.5-2 , n > 7 c n
, n
, n .
5.5-3 G = (V E) | ,
v0,
v 2 V . , -
G .
5.5-4 , -
2 .
5.5-5 , n -
blgnc.
5.5-6? (internal path length)
, 0
2, . -
. n | , i e
| . , e = i + 2n.
5.5-7? 2;d, d |
. ,
1 ( , Kraft inequality)
5.5-8? , L -
,
L=3 2L=3].
5-1
k- (V E) -
c: V ! f0 1 ::: k ; 1g, c(u) 6= c(v)
u v. (
.)
. , 2- .

100. 100 5
. , -
G :
1. G .
2. G 2- .
3. G .
. d |
G. , G (d + 1)- .
. , G O(jVj) , G
O(
p
jVj)- .
5-2
. ( ,
,
.)
. n > 2
( ).
. 6 ,
, ,
.
.
,
.
. n n=2 ,
,
.
5-3
, -
, ,
-
.
. , n
,
3n=4 .
. , 3=4 ( ) ,
,
3n=4 .
. ,
n A B ,
jAj = bn=2c, jBj = dn=2e, ,
, O(lgn).

101. 5 101
(G. Boole) -
- ,
1854 . ( -
)
(G. Cantor) 1874{1895 . -
(G.W. Leibnitz) -
. -
. 1736 ,
(L. Euler) , -
.
94].

102. 6
-
. ,
.
, -
.
6.1 (
). 6.2 -
-
. 6.3 -
, . 6.4 -
. -
6.5
( ). 6.6 -
:
, -
.
6.1.
,
. , n-
n . -
. ,
( . . 5.1).
, -
, -
.
(rule of sum) , jA Bj = jAj + jBj
A B ( -
(5.3)).

103. 103
, , 26 + 10 = 36
, 26, 10.
(rule of product) , jA
Bj = jAj jBj A B, . (5.4). ,
28 4 ,
28 4 = 112 (
).
(string - ) -
-
S ( ). , 8 -
( ) 3:
000 001 010 011 100 101 110 111:
k k- (k-string). -
(substring) s0 s -
s. k-
(k-substring), k. , 010
3- 01101001 ( 4), 111
| .
k S
Sk, jSjk -
k. , 2k k. -
:
jSj jSj -
, | k , jSj jSj ::: jSj
(k ) .
(permutation) S -
, -
. ,
6 S = fa b cg :
abc acb bac bca cab cba:
n! n ,
n ,
n; 1 , n; 2 , . .

104. 104 6
,
,
. S n
k, n.
n k k,
S. ( | k-permutation.)
n(n; 1)(n;2) (n ; k + 1) = n!
(n; k)! (6.1)
n , n;1 -
, k- , -
n; k + 1 .
, 12 = 4 3
fa b c dg:
ab ac ad ba bc bd ca cb cd da db dc:
-
(
n = k).
(k-combinations) n k k-
- n- .
, fa b c dg 4 6 -
fa bg fa cg fa dg fb cg fb dg fc dg:
n k k!
( n k), k-
k! , -
. (6.1) ,
n k
n!
k!(n;k)!: (6.2)
k = 0 1, (
, 0! = 1).

105. 105
n k Ck
n
( )
;n
k :
Ck
n = n
k = n!
k!(n; k)!: (6.3)
k n ;k :
Ck
n = Cn;k
n : (6.4)
Ck
n (bi-
nomial coe cients), (binomial ex-
pansion):
(x + y)n =
nX
k=0
Ck
nxkyn;k (6.5)
( (x + y)n, , -
k x n ; k y,
k n, . . Ck
n).
x = y = 1
2n =
nX
k=0
Ck
n: (6.6)
( : 2n n -
: Ck
n k .)
(
).
-
. 1 6 k 6 n
Ck
n = n(n; 1) (n; k + 1)
k(k ;1) 1
= n
k
n;1
k ;1
n ;k + 1
1
> n
k
k
: (6.7)

106. 106 6
k! > (k
e)k, -
(2.12),
Ck
n = n(n ;1) (n;k + 1)
k(k ;1) 1
6 nk
k!
(6.8)
6 en
k
k
: (6.9)
0 6 k 6 n ( . . 6.1-12)
Ck
n 6 nn
kk(n;k)n;k (6.10)
00 = 1. k = n, 0 6 6
1,
C n
n 6 nn
( n) n((1; )n)(1; )n
= 1 1
1;
1; !n
(6.11)
= 2nH( ) (6.12)
H( ) = ; lg ;(1; )lg(1; ) (6.13)
( ) , - (bi-
nary) entropy function. 0lg0 = 0,
H(0) = H(1) = 0.
6.1-1 k- n? ( -
, ,
.) n?
6.1-2 (boolean function) n m -
| , ftrue falsegn
ftrue falsegm. -
n ?
n m ?
6.1-3 n ( )
? , -
, .

107. 107
6.1-4 -
f1 2 ::: 100g , ?
( .)
6.1-5
Ck
n = n
kCk;1
n;1 (6.14)
0 < k 6 n.
6.1-6
Ck
n = n
n; kCk
n;1
0 6 k < n.
6.1-7 k n, -
, . ,
,
Ck
n = Ck
n;1 + Ck;1
n;1:
6.1-8 6.1-7,
Ck
n n = 0 1 2 ::: 6 k 0 n -
(C0
0 , C0
1 C1
1 ,
).
(Pascal's triangle).
6.1-9
nX
i=1
i = C2
n+1:
6.1-10 , n > 0 Ck
n
( k 0 n) k =
bn=2c k = dn=2e ( n ,
| ).
6.1-11? , n > 0, j > 0, k > 0, j + k 6 n
Cj+k
n 6 Cj
nCk
n;j
, -
(6.3).
?
6.1-12? (6.10) k 6 n=2
, (6.4), k 6 n.

108. 108 6
6.1-13? , ,
Cn
2n = 22n
p n(1+ O(1=n)): (6.16)
6.1-14? H( ), , -
= 1=2. H(1=2)?
6.2.
-
.
S, -
(sample space), | -
(elementary events).
. -
, -
, 2, -
( ) ( ):
S = f g
(event) S.
,
, . . f g.
S ( ) -
(certain event), ? -
(null event). A B -
(mutually exclusive), AB = ?. -
s 2 S fsg S.
.
S. , -
S,
. - (
-
). ,
(
) .
(probability distribution) -
S P, -

109. 109
( -
, - probability axioms):
1. PA > 0 A.
2. PfSg = 1.
3. PfA Bg = PfAg+PfBg -
A B, , ,
Pf iAig =
X
i
PfAig:
( ) -
A1 A2 :::
PfAg A (probability of
the event A). , 2 -
, 1 -
.
.
Pf?g = 0. A B,
PfAg 6 PfBg. A S;A ( -
A), PfAg = 1;PfAg. A
B
PfA Bg = PfAg+ PfBg ;PfA Bg (6.17)
6 PfAg+ PfBg: (6.18)
,
1=4. -
Pf g = Pf g+ Pf g+ Pf g
= 3=4:
: , ,
Pf g = 1=4, -
1; 1=4 = 3=4.
6.2.1.
-
(discrete). -
PfAg =
X
s2A
Pfsg
A,
-
. S ,

110. 110 6
(uniform
probability distribution) S. -
, k
jSj, k=jSj. -
s 2 S .
-
( ipping a fair coin),
1=2. n , -
S = f gn,
2n . S -
n f g,
1=2n.
A = f k n ;k g
S jAj = Ck
n ,
Ck
n , k . , -
A PfAg = Ck
n=2n.
-
a b]. c d] a b]
Pf c d]g= d; c
b; a:
, ,
, , -
. , -
.
, 0, -
(c d] (c d) -
.
-
(continuous uniform probability distribu-
tion).
. , ,
. ,
?

111. 111
. -
, (
) 1=3.
(conditional probability) A B
PfAjBg
PfAjBg = PfABg
PfBg (6.19)
( , PfBg 6= 0). :
B -
, ,
A.
(independent),
PfABg = PfAgPfBg
PfBg 6= 0
PfAjBg = PfAg:
,
1=2, ( -
) | 1=4. -
,
. :
1=2, -
1=4.
.
, -
,
, ( . .
1=2).
A1 A2 ::: An
(pairwise independent),
PfAi Ajg = PfAigPfAjg
1 6 i < j 6 n.
A1 A2 An -
(mutually independent), Ai1 Ai2 ::: Aik
( 2 6 k 6 n 1 6 i1 < i2 < < ik 6 n)
PfAi1 Ai2 :::Aikg = PfAi1 gPfAi2 g PfAikg:

112. 112 6
| : , -
, -
,
.
(6.19) ,
A B, , -
PfA Bg = PfBgPfAjBg
= PfAgPfBjAg: (6.20)
PfAjBg,
PfAjBg = PfAgPfBjAg
PfBg (6.21)
(Bayes's theorem).
: B = (B A) (B A), B A
B A | ,
PfBg = PfB Ag+ PfB Ag
= PfAgPfBjAg+ PfAgPfBjAg:
(6.21),
:
PfAjBg = PfAgPfBjAg
PfAgPfBjAg+ PfAgPfBjAg:
.
: ,
.
, . ,
. ,
?
. -
A | , B |
. PfAjBg.
: PfAg = 1=2, PfBjAg = 1, PfAg = 1=2 PfBjAg = 1=4,
,
PfBjAg = (1=2) 1
(1=2) 1 + (1=2) (1=4) = 4=5:

113. 113
6.2-1 (Boole's inequality):
PfA1 A2 :::g 6 PfA1g+ PfA2g+ ::: (6.22)
A1 A2 :::.
6.2-2 , -
. ,
, ? (
.)
6.2-3 ( 1 10)
. ,
?
6.2-4? ,
p (0 < p < 1). -
, ,
. ( : -
, ,
.)
6.2-5? -
a=b, , -
? ( a b , 0 < a < b, -
lgb.)
6.2-6 ,
PfAjBg+ PfAjBg = 1:
6.2-7 , A1 A2 ::: An,
PfA1 A2 :::Ang =
= PfA1g PfA2jA1g PfA3jA1 A2g PfAnjA1 A2 An;1g:
6.2-8? n -
, k > 2
.
6.2-9? A B (condition-
ally independent) C,
PfABjCg = PfAjCg PfBjCg:

114. 114 6
( ) ,
,
.
6.2-10? ,
( , , ).
,
, , .
, . -
, ?
6.2-11?
X, Y Z, . -
. , ,
.
X -
Y Z, , ,
, , , ,
. -
X, Y . X , ,
1=2 (
, Z). ?
6.3.
(discrete random variable) X |
,
S . -
-
. ( X -
.)
, ,
.
X x -
X = x fs 2 S : X(s) = xg
PfX = xg =
X
fs2S:X(s)=xg
Pfsg:
f(x) = PfX = xg
(probability
density function) X.

115. 115
, PfX = xg > 0
P
x PfX = xg = 1.
-
. 36 , -
. , -
: Pfsg = 1=36. X,
, .
PfX = 3g = 5=36, X 3 5 -
( , (1 3), (2 3), (3 3), (3 2) (3 1)).
,
. X Y | -
,
f(x y) = PfX = x Y = yg
(joint probability density function) X Y. -
y
PfY = yg =
X
x
PfX = x Y = yg:
, x,
PfX = xg =
X
y
PfX = x Y = yg:
(6.19), -
PfX = xjY = yg = PfX = x Y = yg
PfY = yg :
(independent),
X = x Y = y
x y, , PfX = x Y = yg =
PfX = xgPfY = yg x y.
, -
,
, .
-
| (mean), -
(expected value, expectation). -
X
M X] =
X
x
xPfX = xg (6.23)

116. 116 6
,
. -
X , ,
.
-
3 2
. X -
,
M X] = 6 Pf2 g+ 1 Pf1 1 g;4 Pf2 g
= 6(1=4)+ 1(1=2); 4(1=4)
= 1:
-
:
M X + Y ] = M X]+ M Y ] (6.24)
M X] M Y] .
.
X | , g(x) | .
g(X) (
). (
)
M g(X)] =
X
x
g(x)PfX = xg:
g(x) = ax, a | ,
M aX] = aM X]: (6.25)
( ): X Y
a
M aX + Y ] = aM X]+ M Y]: (6.26)
X Y -
,
M XY] =
X
x
X
y
xyPfX = x Y = yg
=
X
x
X
y
xyPfX = xgPfY = yg
= (
X
x
xPfX = xg)(
X
y
yPfY = yg)
= M X]M Y]:

117. 117
, n -
X1 X2 ::: Xn, ,
M X1X2 Xn] = M X1]M X2] M Xn]: (6.27)
X -
(0 1 2 :::), -
:
M X] =
1X
i=0
iPfX = ig
=
1X
i=0
i(PfX > ig; PfX > i + 1g)
=
1X
i=1
PfX > ig: (6.28)
, PfX > ig i
i;1 (
PfX > 0g, ).
(variance) X
M X]
D X] = M (X ;M X])2]
= M X2 ;2XM X]+ M2 X]]
= M X2]; 2M XM X]]+ M2 X]
= M X2]; 2M2 X]+ M2 X]
= M X2]; M2 X]: (6.29)
M M2 X]] = M2 X] M XM X]] = M2 X] ,
M X] | ( )
(6.25), a = M X]. (6.29) :
M X2] = D X]+ M2 X] (6.30)
a
a2 :
D aX] = a2D X]:
X Y ,
D X + Y] = D X]+ D Y]:

118. 118 6
, n
X1 ::: Xn :
D
"
nX
i=1
Xi
#
=
nX
i=1
D Xi]: (6.31)
(standard deviation) -
X .
X
, ,
. 2.
6.3-1 .
?
-
?
6.3-2 A 1::n] n
-
. ,
? ,
?
6.3-3 .
1 6. -
. ,
.
,
.
?
6.3-4? X Y | . -
, f(X) g(Y) f
g.
6.3-5? X | -
M X].
(Markov's inequality)
PfX > tg 6 M X]=t (6.32)
t > 0.
.]
6.3-6? S | , -
X X0, X(s) > X0(s)

119. 119
s 2 S. , t,
PfX > tg > PfX0 > tg:
6.3-7 : -
?
6.3-8 , ,
0 1, D X] = M X]M 1;X].
6.3-9 (6.29), D aX] =
a2D X].
6.4.
|
(Bernouilli trials)
n , -
: (success), -
p, (failure), 1 ; p.
| -
| .
,
p ( q = 1;p).
? -
X | 1 2 :::,
PfX = kg = qk;1p (6.33)
( k, k ;1
, k- ). -
, (6.33),
(geometric distribution). . 6.1.
, p < 1, -
, (3.6):
M X] =
1X
k=1
kqk;1p = p
q
1X
k=0
kqk = p
q
q
(1;q)2 = 1=p: (6.34)
, 1=p ,
, , -
p.

120. 120 6
. 6.1 p = 1=3 -
q = 1 ;p. 1=p = 3.
,
D X] = q=p2: (6.35)
: ,
. 36 -
, 6 2 -
. p 8=36 = 2=9,
1=p = 9=2 = 4 5 ,
.
n
p q = 1;p. -
X | n .
0 1 ::: n,
PfX = kg = Ck
npk(1; p)n;k (6.36)
k = 0 1 ::: n Ck
n
n k ,
pkqn;k. (6.36) (bino-
mial distribution).

121. 121
. 6.2 b(k 15 1=3), n = 15 -
,
p = 1=3. np = 5.
b(k n p) = Ck
npk(1; p)n;k: (6.37)
6.2.
, -
(6.37) | k- (p+q)n. ,
p+ q = 1,
nX
k=0
b(k n p) = 1 (6.38)
( 2).
,
, (6.14)
(6.38). X | ,
b(k n p). q = 1 ;p. -

122. 122 6
,
M X] =
nX
k=0
kb(k n p)
=
nX
k=1
kCk
npkqn;k
= np
nX
k=1
Ck;1
n;1pk;1qn;k
= np
n;1X
k=0
Ck
n;1pkq(n;1);k
= np
n;1X
k=0
b(k n;1 p)
= np: (6.39)
: Xi | i- (
0 q 1 p).
M Xi] = p 1+q 0 = p. , X = X1 +:::+Xn,
(6.26)
M X] = M
"
nX
i=1
Xi
#
=
nX
i=1
M Xi] =
nX
i=1
p = np:
. (6.29) -
, D Xi] = M X2
i ] ; M2 Xi]. Xi
0 1, M X2
i ] = M Xi] = p, , ,
D Xi] = p ; p2 = pq: (6.40)
(6.31):
D X] = D
"
nX
i=1
Xi
#
=
nX
i=1
D Xi] =
nX
i=1
pq = npq: (6.41)
6.2 , b(k n p) k
, k np, -
. , -

123. 123
:
b(k n p)
b(k;1 n p) = Ck
npkqn;k
Ck;1n pk;1qn;k+1
= n!(k ;1)!(n; k + 1)!p
k!(n; k)!n!q
= (n;k + 1)p
kq (6.42)
= 1 + (n+ 1)p; k
kq :
1, (n + 1)p ; k ,
b(k n p) > b(k ; 1 n p) k < (n + 1)p ( ),
b(k n p) < b(k ; 1 n p) k > (n + 1)p ( ).
(n + 1)p | , :
(n+1)p (n+1)p;1 = np;q.
, k, np;
q < k < (n + 1)p.
-
.
6.1. n > 0, 0 < p < 1, q = 1 ; p, 0 6 k 6 n.
b(k n p) 6 np
k
k nq
n ;k
n;k
:
. (6.10),
b(k n p) = Ck
npkqn;k
6 n
k
k n
n ;k
n;k
pkqn;k
= np
k
k nq
n; k
n;k
:
6.4-1 2 .
6.4-2 6 -
3 3 ( )?
6.4-3 , b(k n p) = b(n; k n q), q = 1 ; p.
6.4-4 ,
b(k n p) 1=p2 npq, q = 1 ;p.

124. 124 6
6.4-5? ,
n 1=n
1=e. ,
1=e.
6.4-6? n ,
. , , -
, Cn
2n=4n. ( .
,
n 2n .)
nX
k=0
(Ck
n)2 = Cn
2n:
6.4-7? , 0 6 k 6 n,
b(k n 1=2)6 2nH(k=n);n
H(x) | (6.13).
6.4-8? n . pi | -
i- , X -
. p > pi
i = 1 2 ::: n. ,
PfX < kg 6
k;1X
i=0
b(i n p)
k = 1 2 ::: n.
6.4-9? X,
n p1 ::: pn. X0 |
,
p0
1 ::: p0
n. p0
i > pi i. ,
PfX0 > kg > PfX > kg:
k = 0 ::: n
( : , -
: ,
-
. 3.3-6.)
6.5.
-
!
k
, k ( -

125. 125
k ). ,
(tails) . -
,
(np) .
b(k n p). ( -
).
6.2. X | n
p.
PfX > kg =
nX
i=k
b(i n p) 6 Ck
npk:
. (6.15):
Ck+i
n 6 Ck
nCi
n;k:
,
PfX > kg =
nX
i=k
b(i n p)
=
n;kX
i=0
b(k + i n p)
=
n;kX
i=0
Ck+i
n pk+i(1 ;p)n;(k+i)
6
n;kX
i=0
Ck
nCi
n;kpk+i(1; p)n;(k+i)
= Ck
npk
n;kX
i=0
Ci
n;kpi(1; p)(n;k);i
= Ck
npk
n;kX
i=0
b(i n; k p)
= Ck
npk
Pn;k
i=0 b(i n;k p) = 1 (6.38).
,
6.3. X | n
p.
PfX 6 kg =
kX
i=0
b(i n p) 6 Cn;k
n (1; p)n;k = Ck
n(1 ;p)n;k:

126. 126 6
: -
, , .
6.4. X | n
p q =
1 ;p.
PfX < kg =
k;1X
i=0
b(i n p)< kq
np; kb(k n p)
0 < k < np.
.
Pk;1
i=0 b(i n p) -
( . . 3.2). i = 1 2 ::: k -
(6.42)
b(i;1 n p)
b(i n p)
= iq
(n;i + 1)p < i
n ; i
q
p 6 k
n; k
q
p :
x = k
n ; k
q
p < 1
, b(i n p) 0 6 i 6 k
, x. -
( i = 0 i = k ; 1) b(k n p),
x + x2 + x3 + ::: = x=(1; x):
k;1X
i=0
b(i n p)< x
1 ;xb(k n p) = kq
np ;kb(k n p):
k 6 np=2 kq=(np ; k) 1,
b(k n p)
. n -
. p = 1=2 k = n=4. 6.4 ,
n=4
n=4 . ( , -
r 6 n=4 r
r .) 6.4
-
, 6.1.
:

127. 127
6.5. X | n
p q =
1 ;p.
PfX > kg =
nX
i=k+1
b(i n p) < (n ;k)p
k ; np b(k n p)
np < k < n.
: -
.
6.6. n -
i- pi ( -
qi 1;pi). X
, = M X]. r > -
PfX ; > rg 6 e
r
r
:
. > 0 e x -
x,
PfX ; > rg = Pfe (X; ) > e rg
-
(6.32) ( ):
PfX ; > rg 6 M e (X; )]e; r: (6.43)
M e (X; )] . -
Xi, 1 i- -
0 .
X =
nX
i=1
Xi
X ; =
nX
i=1
(Xi ;pi):
, Xi .
e (Xi;pi)] ( . 6.3-4), -
(6.27) -
:
M e (X; )] = M
"
nY
i=1
e (Xi;pi)
#
=
nY
i=1
M e (Xi;pi)]:

128. 128 6
:
M e (Xi;pi)] = e (1;pi)pi + e (0;pi)qi
= pie qi + qie; pi
6 pie + 1 (6.44)
6 exp(pie )
exp(x) : exp(x) = ex.
( , > 0, qi 6 1, e qi 6 e e; pi 6 1,
(2.7).) ,
M e (X; )] 6
nY
i=1
exp(pie ) = exp( e )
=
Pn
i=1 pi. , (6.43)
PfX ; > rg 6 exp( e ; r): (6.45)
= ln(r= ) ( . . 6.5-6),
PfX ; > rg 6 exp( eln(r= ) ;rln(r= ))
= exp(r; rln(r= )) = er
(r= )r = e
r
r
:
r > ,
r e ,
.]
6.6 -
. = M X] = np,
6.7. X | n
p.
PfX ;np > rg =
nX
k=dnp+re
b(k n p) 6 npe
r
r
:
6.5-1? : n -
, n
4n ?

129. 129
6.5-2? ,
k;1X
i=0
Ci
nai < (a + 1)n k
na; k(a + 1)b(k n a=(a+ 1))
a > 0 0 < k < n.
6.5-3? , 0 < k < np, 0 < p < 1 q = 1 ; p,
k;1X
i=0
piqn;i < kq
np;k
np
k
k nq
n ; k
n;k
:
6.5-4? , 6.6 -
Pf ; X > rg 6 (n; )e
r
r
6.7
Pfnp ; X > rg 6 nqe
r
r
:
6.5-5? n -
i- pi ( qi
1 ; pi). X
, = M X]. , r > 0
PfX ; > rg 6 e;r2=2n:
6.5-6? , = ln(r= )
(6.45) .
6.6.
-
. : -
k , -
.
6.6.1.
(birthday paradox) -
: ,

130. 130 6
?
,
, .
, 365 k -
.
, .
, -
1=365.
( ) ,
, -
2=365 . , k
,
1 ; 1
365 1 ; 2
365 ::: 1 ; k ; 1
365 :
, n | , Ai |
(i+ 1)-
i . Bi = A1
A2 ::: Ai;1 i
.
Bk = Ak;1 Bk;1, (6.20)
PfBkg = PfBk;1gPfAk;1jBk;1g: (6.46)
: PfB1g = 1.
PfAk;1jBk;1g (n;k+1)=n,
n n ; (k ; 1) ( -
).
PfBkg = PfB1gPfA1jB1gPfA2jB2g PfAk;1jBk;1g
= 1 n; 1
n
n; 2
n ::: n; k + 1
n
= 1 1 ; 1
n 1 ; 2
n t::: 1 ; k ; 1
n
1 + x 6 ex (2.7) ,
PfBkg 6 e;1=ne;2=ne;(k;1)=n
= e;(1+2+3+:::+(k;1))=n
= e;k(k;1)=2n
6 1=2
;k(k ;1)=2n 6 ln(1=2). , k -
, 1=2 k(k;1) > 2nln2.

131. 131
, k > (1 +
p
1+ (8ln2)n)=2.
n = 365, k > 23. ,
23 , 1=2 -
. , -
669 , 31
.
, -
. (i j), -
, Xij
Xij =
(
1 i j ,
0 .
, -
, 1=n, -
(6.23)
M Xij] = 1 (1=n) + 0 (1; 1=n) = 1=n
i 6= k.
Xij 1 6 i < j 6 k -
,
C2
k = k(k ; 1)=2, k(k ; 1)=2n. -p
2n ,
1.
, n = 365 k = 28 , -
, (28 27)=(2 365) 1 0356.
38 .
, : (1) k
X =
P
Xij > 0 1=2, (2) k
X 1. -
( , PfX > 0g >
1=2, M X] > 1=2, X -
). ,
(
pn).
6.6.2.
b , 1 b.
:
. ,
1=b ( |

132. 132 6
). ,
.
? , -
, -
b(k n 1=b). n ,
n=b.
,
? -
1=b, 1=(1=b) = b.
, -
? -
. , , -
b. ni , -
, ,
i ; 1 i. ( , -
, , | ,
n1 = 1, n2 = 3.) -
n1 +n2 +:::+nb.
. -
i- , i;1 b
(b ; i + 1)=b. -
ni
b=(b;i+1). i
b(1=b+1=(b;1)+:::+1=2+1) = b(lnb+O(1)). ( . -
(3.5) .)
, blnb , -
.
6.6.3.
n . -
? -
, | (lgn).
,
O(lgn). Aik ,
k , i- . ,
PfAikg = 1=2k: (6.47)
k = 2dlgne k -
1=n2, ( i)
n, k ( - )
1=n.
:
n ( 1;1=n) -
2dlgne, d2lgne+n (1=n) = O(lgn).

133. 133
rdlgne
r (
2;rlgn = n;r, -
n n;r = n;(r;1).) , n = 1000
2dlgne = 20 -
1=n = 1=1000, 3dlgne = 30
1=n2 = 10;6.
: -
(lgn).
b(lgn)=2c. (6.47)
1=2b(lgn)=2c > 1=pn,
1;1=pn.
, b(lgn)=2c
. ( ,
.) 2n=lgn ;1.
,
, ,
(1; 1=
pn)2n=lgn;1 6 e;(2n=lgn;1)=pn
= O(e;lgn) = O(1=n)
, 1 + x 6 ex (2.7), ,
(2n=lgn;1)=pn > lgn ; O(1).
, 1 ; O(1=n)
blgn=2c = (lgn), -
(1 ; 1=n) (lgn) =
(lgn).
6.6-1 b
, .
,
?
6.6-2?
, ,
?
.
6.6-3? , -
,
?
6.6-4?
k- n- ?

134. 134 6
?
6.6-5? n n , ,
.
? ,
?
6.6-6? -
, , n
lgn; 2lglgn 1 ;1=n.
6-1
n
b .
. , ,
. , bn
.
. , -
. ,
(b+ n ; 1)!=(b; 1)! . ( : -
n b;1
.)
. , ,
. ,
Cn
b+n;1. ( : ,
( ).)
. , -
, Cn
b .
. ,
,
Cb;1
n;1.
6-2
-
A 1::n].
1 max ;1
2 for i 1 to n
3 do . A i] max.
4 if A i] > max
5 then max A i]
, -

135. 6 135
5. , A -
( -
).
. x i , -
?
. A i]
i, 5?
. 5
i?
. si | , 1 0
, 5 i- .
M si]?
. s = s1+s2+ +sn |
5 . , M s] = (lgn).
6-3
-
. n
.
, -
, .
.
k , -
. ,
k, . , -
. ,
-
( 1=e), k n=e.
6-4
t- 2t ; 1.
(probabilistic counting,
R. Morris) ,
.
-
ni ( i 0 2t;1). :
t- i, , -
( Increment)
ni. , n0 = 0.
Increment , -
i, . , i
1 1=(ni+1;ni), -
. ( i = 2t ; 1, .)
: i
ni+1 ;ni .

136. 136 6
ni = i i > 0, .
, , , ni =
2i;1 i > 0 ni = Fi (i- , . . 2.2).
, n2t;1 , -
.
. ,
n Increment,
n.
. ,
n Increment, -
n0 n1 :::.
ni = 100i.
-
(B.Pascal) (P.de Fermat), -
1654 , (C.Huygens, 1657). -
(J.Bernoulli, 1713) (A.De Moivre, 1730). -
(P.S.de Laplace), (S.-D.Poisson) (C.F.Gauss).
. .
. . ( ). 1933 . . -
. -
40] 99]. -
(P.Erdos).
: 121], 140] ( -
) 28],
41], 57], 66], 171] ( -
) 30], 100], -
179] ( ).

138. II

139. ,
(sorting problem).
ha1 a2 ::: ani. -
ha0
1 a0
2 ::: a0
ni,
, : a0
1 6 a0
2 6 6
a0
n. ,
( , ).
.
(records),
, , -
. ,
, ( -
, , , , . .), -
-
.] , (
), (key), |
(satellite data).
: ,
( ) ,
. ( ,
, .)
, -
. ,
, ,
(
, ).

140. 140 II
1 , -
n (n2) . , -
,
(in place). ( ,
, -
).
,
.
-
.
(heapsort). |
, ( , -
, 7).
(nlgn)
.
8 , -
(quicksort). (n2),
| (nlgn),
( ,
).
( , , -
)
, . 9
,
(nlgn),
( ). , -
.
-
, . 9 -
. -
(counting sort) n
1 k O(n+k),
k. k = O(n),
. -
, ,
, n , -
d k- ,
O(d(n+k)). , d , k O(n),
O(n). ,
, , -
,
O(n) n .

141. II 141
i- ,
, (nlgn), -
( 9).
( -
),
i- (
). 10
: O(n2) -
O(n) , ,
O(n) .
7{10
. -
( ,
i- ) -
( 6). ,
i- (
) , .

142. 7
(heapsort). , -
O(nlgn) n , -
O(1) ( O(n) -
). ,
| -
.
, ( -
) -
. ,
( . . 7.5).
,
( , ).
( -
,
| , Lisp),
.
7.1.
(binary heap)
, -
, ( . 7.1).
.
i, bi=2c ( -
1 ), | 2i
2i+ 1. ,
A, length A]
heap-size A] ( ), heap-size A] 6 length A].
A 1] ::: A heap-size A]].

143. 143
. 7.1 ( ) ( ).
. .
Parent(i)
return bi=2c
Left(i)
return 2i
Right(i)
return 2i+ 1
A 1] .
Left
Parent
(Left, Parent) Right ,
.
, , -
(heap property): i,
( . . 2 6 i 6 heap-size A]),
A Parent(i)] > A i]: (7.1)
,
. , (
) ( -
).
(height) -
(
). -
, , . ,
, ( , , ),
. (lgn),
n | ( . . 7.1-2).
, -
, , O(lgn).

144. 144 7
.
:
Heapify -
(7.1). O(lgn).
Build-Heap ( -
) . O(n).
Heapsort , -
. O(nlgn).
Extract-Max ( ) Insert ( -
)
.
O(lgn).
7.1-1 h.
? ( .)
7.1-2 , n blgnc.
7.1-3 ,
.
7.1-4 ,
?
7.1-5 (
| ). ?
7.1-6 h23 17 14 6 13 10 1 5 7 12i?
7.2.
Heapify | . -
A i. , -
Left(i) Right(i) -
. i,
. :
i,
. ., A i]
.

145. 145
. 7.2 Heapify(A 2) heap-size A] = 10. ( ) -
. i = 2 .
, A 2] A 4]. ( )
4. -
Heapify(A 4) 4
A 4] $ A 9] ( ).
, Heapify(A 9) .
Heapify(A i)
1 l Left(i)
2 r Right(i)
3 if l 6 heap-size A] A l] > A i]
4 then largest l
5 else largest i
6 if r 6 heap-size A] A r] > A largest]
7 then largest r
8 if largest 6= i
9 then A i] $ A largest]
10 Heapify(A largest)
Heapify . 7.2. 3{7
largest
A i], A Left(i)] A Right(i)]. largest = i, A i]
, .
A i] A largest] ( -
(7.1) i, ,

146. 146 7
largest)
largest, .
Heapify. -
(1) , .
T(n) | , n -
. i n , -
Left(i) Right(i) 2n=3
( |
). ,
T(n) 6 T(2n=3)+ (1)
4.1 ( 2) , T(n) = O(lgn).
:
, O(lgn).
7.2-1 , . 7.2,
Heapify(A 3) h27 17 3 16 13 10 1 5 7 12 4 8 9 0i.
7.2-2 A i] , . -
Heapify(A i)?
7.2-3 i > heap-size A]=2.
Heapify(A i)?
7.2-4 Heapify, .
( -
.)
7.2-5 ,
Heapify n (lgn). ( : -
,
).
7.3.
A 1::n], -
, . -
Heapify, , -
. bn=2c + 1 ::: n
,
. ,
, Heapify.
,

147. 147
( ) -
.
Build-Heap(A)
1 heap-size A] length A]
2 for i blength A]=2c downto 1
3 do Heapify(A i)
Build-Heap . 7.3.
, Build-Heap -
O(nlgn). , Heapify O(n)
, O(lgn).
, .
, Heapify
, (
). h n -
dn=2h+1e ( . . 7.3-3),
blgnc ( . 7.1-2), Build-
Heap
blgncX
h=0
l n
2h+1
m
O(h) = O
0
@n
blgncX
h=0
h
2h
1
A (7.2)
x = 1=2 (3.6),
:
1X
h=0
h
2h = 1=2
(1 ;1=2)2 = 2:
, Build-Heap
O n
1X
h=0
h
2h
!
= O(n):
7.3-1 , . 7.3,
Build-Heap A = h5 3 17 10 84 19 6 22 9i.
7.3-2 Build-Heap , -
i blength A]=2c 1 ( )?
7.3-3 , n
dn=2h+1e h.

148. 148 7
. 7.3 Build-Heap.
Heapify 3.

149. 149
7.4.
.
Build-Heap, -
. : -
(A 1]).
c A n], 1 -
( -
Left(1) Right(1)
, Heapify).
-
. ,
.
Heapsort(A)
1 Build-Heap(A)
2 for i length A] downto 2
3 do A 1] $ A i]
4 heap-size A] heap-size A] ;1
5 Heapify(A 1)
. 7.4.
for ( 2).
Heapsort O(nlgn). -
, ( ) O(n),
n ;1 for O(lgn).
7.4-1 , . 7.4,
Heapsort A = h5 13 2 25 7 17 20 8 4i.
7.4-2 A -
. ?
?
7.4-3 , Heapsort -
(nlgn).
7.5.
| , ( . 8) -
. -

150. 150 7
. 7.4 Heapsort. -
Heapify.
.
.
|
.
(priority queue) | S,
. -
S hkey i, key | ,
(key).
| -
,
.
:
Insert(S x): x S

151. 151
Maximum(S):
Extract-Max(S): .
, , -
. -
, -
( Extract-Max). -
Insert.
| -
(event-driven simulation).
, , -
. ,
, .
Extract-Min (
), | Insert.
. -
.
, Maximum
(1). ,
, :
Heap-Extract-Max(A)
1 if heap-size A] < 1
2 then :
3 max A 1]
4 A 1] A heap-size A]]
5 heap-size A] heap-size A]; 1
6 Heapify(A 1)
7 return max
O(lgn) ( Heapify
).
,
( ), :
Heap-Insert(A key)
1 heap-size A] heap-size A]+ 1
2 i heap-size A]
3 while i > 1 A Parent(i)] < key
4 do A i] A Parent(i)]
5 i Parent(i)
6 A i] key
Heap-Insert . 7.5.
O(lgn), -
lgn ( i

152. 152 7
. 7.5 Heap-Insert.
15 ( ).
while ).
, n
O(lgn).
7.5-1 , . 7.5,
Heap-Insert(A 3) A = h15 13 9 5 12 8 7 4 0 6 2 1i.
7.5-2 , Heap-
Extract-Max .
7.5-3 , ( rst-in,
rst-out) . ( .
. 11.1).
7.5-4 Heap-Increase-Key(A i k) ( -
), A i] k,
k (A i] max(A i] k))
. | O(lgn).
7.5-5 Heap-Delete(A i) | -
i . O(lgn).

153. 7 153
7.5-6 ,
O(nlgk) k -
( n | ).
( : .)
7-1
,
Heap-Insert. -
:
Build-Heap0(A)
1 heap-size A] 1
2 for i 2 to length A]
3 do Heap-Insert(A A i])
. Build-Heap Build-Heap0 -
. ?
( .)
. , Build-Heap0 -
(nlgn) ( n | ).
7-2 d-
d- (d-ary heap), -
d .
. , Par-
ent, Left Right?
. d- n n
d?
. Extract-Max. -
( n d)?
. Insert. ?
. Heap-Increase-Key ( . 7.5-4). -
?
202]
.
Build-Heap 69].

154. 8
-
. n -
(n2),
: -
(nlgn), nlgn
. ,
-
.
8.1 -
.
8.2 ,
8.4. 8.3 -
, .
( -
) O(n2),
O(nlgn). ( , -
, ,
O(nlogn).) -
8.4,
.
8.1.
(quicksort), ,
( . . 1.3.1). -
A p::r] :
A ,
A p] ::: A q]
A q+1] ::: A r], q | p 6 q < r.
(partition).
A p::q] A q + 1::r].

155. 155
A p::r] .
, Quicksort
:
Quicksort(A p r)
1 if p < r
2 then q Partition(A p r)
3 Quicksort(A p q)
4 Quicksort(A q + 1 r)
Quicksort(A 1 length A]).
| Partition, -
A p::r] :
Partition(A p r)
1 x A p]
2 i p;1
3 j r + 1
4 while true
5 do repeat j j ;1
6 until A j] 6 x
7 repeat i i+ 1
8 until A i] > x
9 if i < j
10 then A i] $ A j]
11 else return j
Partition . 8.1. x =
A p] A p::q] -
, x, A q+1::r] | ,
x. , ,
x, (A p::i]), ,
x | (A j::r]). -
: i = p ; 1, j = r + 1.
while ( 5{8)
( ). -
A i] > x > A j]. A i]
A j] ,
.
i > j.
A p] ::: A j] A j + 1] ::: A r] -

156. 156 8
. 8.1 Partition.
, . ( ) -
, .
x = A p] = 5 . ( ) -
while ( 4{8). ( ) A i] A j] ( -
10). ( ) while. ( ) ( -
) while. i > j,
q = j. A j] ( )
, x = 5, A j] , x = 5.
. j.
,
. , , i j
p::r] . -
: , -
A p], , , A r]. ,
A r] | , -
i = j = r, q = j
| q < r, Quicksort
. Partition -
8-1.
Partition (n), n =
r ;p+ 1 ( . . 8.1-3).
8.1-1 , . 8.1,
Partition A = h13 19 9 5 12 8 7 4 11 2 6 21i.
8.1-2 A p::r] .
Partition?
8.1-3 , ,
Partition (n).

157. 157
8.1-4 ( -
), ?
8.2.
,
. -
, O(nlgn),
.
, O(n2), -
.
,
n;1 , | 1. ( 8.4.1,
.) -
, .
(n),
T(n)
T(n) = T(n ;1) + (n)
T(1) = (1),
T(n) = T(n;1) + (n) =
nX
k=1
(k) =
nX
k=1
k
!
= (n2)
( | , . -
(3.2)). . 8.2.
( . . 4.2.)
,
(n2), .
, , -
( , -
(n)).
, -
. -
,
T(n) = 2T(n=2)+ (n)
, 4.1 ( 2), T(n) = (nlgn). -
8.3.

158. 158 8
. 8.2 Quicksort ( -
Partition ,
). (n2
).
. 8.3 Quicksort ( -
). (nlgn).
8.4, ( -
)
. , , -
.
, ,
9 : 1.
T(n) = T(9n=10)+ T(n=10)+ n
( (n) n). -
8.4. n
, .

159. 159
. 8.4 , -
9 : 1. (nlgn).
log10=9 n = (lgn),
- (nlgn),
. , -
( )
- ,
(nlgn).
:
,
, -
. , , -
. (
.)
| ,
, . -
. 8.2-5, 80
9 : 1.
, -
,
( . 8.5( )). -
,
(lgn), -
, (lgn),
| (nlgn). , -
( ,

160. 160 8
. 8.5 ( ) : (n n;1 1) (n;1
). ( ) ,
.
).
8.2-1 , -
, Quicksort (nlgn).
8.2-2 . -
, Quicksort (n2).
8.2-3 ,
,
.
,
. , -
. ,
. , -
,
.
8.2-4 -
: 1; , 0 < 6 1=2. ,
;lgn=lg , -
;lgn=lg(1; ). ( .)
8.2-5? , 0 < 6 1=2
, -

161. 161
, : 1; , 1;2 .
1=2?
8.3.
,
. , ,
| -
. , , (
) ( . . 8.2-3).
,
-
.
, -
. ,
,
( O(n) | .
. 8.3-4).
,
(
).
(randomized), -
(random-number generator).
, Random -
: Random(a b)
a b. , Random(0 1) -
0 1 1=2. -
. ,
(b;a+ 1) -
. ( -
(pseudorandom-number generator) |
, ,
.)
-
: 13.4-4 -
, , ,
| , -
( ) .
,
, ,
, , , -
. ,

162. 162 8
( -
8.2, ).
, -
, Par-
tition. , A p::r]
A p] . -
,
-
.
. , -
O(nlgn),
, 8.4.
, , -
:
Randomized-Partition(A p r)
1 i Random(p r)
2 A p] $ A i]
3 return Partition(A p r)
Randomized-
Partition Partition:
Randomized-Quicksort(A p r)
1 if p < r
2 then q Randomized-Partition(A p r)
3 Randomized-Quicksort(A p q)
4 Randomized-Quicksort(A q + 1 r)
.
8.3-1 -
( ), -
?
8.3-2 Randomized-
Quicksort -
Random ? -
?
8.3-3? Random(a b),
, . . , 0 1.

163. 163
-
?
8.3-4? ,
(n)
A 1::n].
8.4.
-
8.2 . -
(
Quicksort, Randomized-Quicksort),
Randomized-Quicksort.
8.4.1.
8.2 , -
, (n2). -
, ( ) .
.
, O(n2),
( . . 4.1). T(n) |
n. ,
,
T(n) = max16q6n;1
(T(q)+ T(n;q)) + (n) (8.1)
( ).
, T(q) 6 cq2 c
q, n.
T(n) 6 max16q6n;1
(cq2+c(n;q)2)+ (n) = c max16q6n;1
(q2+(n;q)2)+ (n):
q2 + (n;q)2 -
1 6 q 6 n ; 1 ( q -
, , . . 8.4-2).
12 + (n;1)2 = n2 ;2(n ; 1).
T(n) 6 cn2 ;2c(n;1) + (n) 6 cn2
c ,
. , -
(n2).

164. 164 8
8.4.2.
8.2,
, , -
(lgn), | (nlgn). -
Randomized-
Quicksort, Parti-
tion,
( ).
, 3
Randomized-Partition Partition, -
A p]
A p::r]. ,
. -
, ,
, .
, , q, -
Partition, , -
, x = A p] (
(rank) x rank(x)).
n = r ; p + 1 | , , -
A p],
rank(x), 1 n, ( 1=n).
rank(x) > 1, , ,
rank(x);1 |
, x. rank(x) = 1,
(
i = j = p). , 1=n
2 3 ::: n;1 , 2=n |
.
Randomized-
Quicksort n T(n). ,
T(1) = (1).
Partition, (n),
q n ; q, q 2=n -
1 1=n | 2 ::: n ; 1.

165. 165
T(n) = 1
n
0
@T(1)+ T(n;1) +
n;1X
q=1
(T(q)+ T(n;q))
1
A + (n) (8.2)
( , q = 1, ).
T(1) = (1) T(n) = O(n2),
1
n(T(1)+ T(n; 1)) = 1
n( (1)+ O(n2)) = O(n):
T(1) T(n ; 1) (8.2)
(n).
T(n) = 1
n
n;1X
q=1
(T(q)+ T(n; q))+ (n): (8.3)
T(k), k = 1 ::: n;1,
, :
T(n) = 2
n
n;1X
k=1
T(k) + (n): (8.4)
(8.4) , .
, T(n) 6 anlgn + b, a > 0 b > 0
, .
n = 1 , a b. n > 1
T(n) = 2
n
n;1X
k=1
T(k)+ (n)
6 2
n
n;1X
k=1
(aklgk + b)+ (n)
= 2a
n
n;1X
k=1
klgk + 2b
n (n ;1) + (n):
, :
n;1X
k=1
klgk 6 1
2n2 lgn ; 1
8n2: (8.5)

166. 166 8
,
T(n) 6 2a
n
1
2
n2 lgn ; 1
8
n2 + 2b
n (n ;1) + (n)
6 anlgn; a
4n + 2b+ (n)
= anlgn+ b+ (n) + b; a
4n 6 anlgn+ b
a , a
4n (n)+b. ,
O(nlgn).
(8.5).
nlgn,
n;1X
k=1
klgk 6 n2 lgn:
|
1
2n2 lgn ; (n2).
lgk lgn, k
,
n;1X
k=1
klgk 6 lgn
n;1X
k=1
k = n(n ;1)
2 lgn 6 1
2n2 lgn:
, lgk lgn,
k 1
( k 6 n=2), (n=2)2=2 = n2=8.
,
n;1X
k=1
klgk =
dn=2e;1X
k=1
klgk +
n;1X
k=dn=2e
klgk
k < dn=2e lgk 6 lg(n=2) = lgn; 1.
n;1X
k=1
klgk 6 (lgn; 1)
dn=2e;1X
k=1
k + lgn
n;1X
k=dn=2e
k
= lgn
n;1X
k=1
k ;
dn=2e;1X
k=1
k 6 1
2
n(n; 1)lgn ; 1
2
n
2
;1 n
2
6 1
2n2 lgn ; 1
8n2

167. 167
n > 2. (8.5) .
-
(
) . . .
(1) : N -
.
: ,
, ( )
.
(2) ? , -
(
1 N),
( -
). ( , -
:
,
. .)
(3) , -
1 N | (
) . -
.
(4) i, j f1 ::: Ng p(i j) -
, -
. , p(i i+1)= 1, -
(
).
(5) , p(i i + 2) = 2=3. ,
,
i, i + 1 i + 2 i + 1 .
, p(i i + k) = 2=(k + 1) ( i i + k
, k + 1 i i + 1 ::: i + k
).
(6) P
m(i j) -
i j.
, m(i j) = p(i j). , -
:
X
16i<j6n
2
j ;i + 1:
(7) , 1 +
1=2 + 1=3 + ::: + 1=k = O(lgk), O(N lgN)
.
, ( )
.

168. 168 8
: (
): , ,
.]
8.4-1 , -
(nlgn).
8.4-2 , q2 + (n ; q)2 1 n ; 1]
.
8.4-3 ,
Randomized-Quicksort (nlgn).
8.4-4 -
, ( -
) -
. , , :
Randomized-Quicksort(A p r) , r;p+1 < k
( . . k ).
-
. , -
O(nk + nlg(n=k)). k?
8.4-5?
Z
xlnxdx = 1
2x2 lnx; 1
4x2
( )
( (8.5))
Pn;1
k=1 klgk.
8.4-6?
Randomized-Partition:
-
.
, ,
: 1 ; .
8-1
, Partition .
:

169. 8 169
. i j
p::r].
. j
r ( . . ).
. -
A p::j] A j + 1::r].
8-2
Partition, ,
. (N. Lomuto). -
A p::i] A i + 1::j],
x = A r], | x.
Lomuto-Partition(A p r)
1 x A r]
2 i p ;1
3 for j p to r
4 do if A j] 6 x
5 then i i+ 1
6 A i] $ A j]
7 if i < r
8 then return i
9 else return i; 1
. , Lomuto-Partition -
.
. Partition Lomuto-Partition -
? ( -
.)
. , Lomuto-Partition, -
Partition, (n), n |
.
. Quicksort Partition
Lomuto-Partition.
, ?
. Randomized-Lomuto-Partition,
A r] -
Lomuto-Partition. ,
, Randomized-Lomuto-
Partition q, , -
Randomized-Partition p + r ;q.
8-3
-
:

170. 170 8
Stooge-Sort(A i j)
1 if A i] > A j]
2 then A i] $ A j]
3 if i+ 1 > j
4 then return
5 k b(j ;i+ 1)=3c . .
6 Stooge-Sort(A i j ;k) . .
7 Stooge-Sort(A i+ k j) . .
8 Stooge-Sort(A i j ;k) . .
. , Stooge-Sort -
.
.
Stooge-Sort
.
. Stooge-Sort
( -
, , ).
?
8-4
Quicksort (
). -
, (
,
-
tail recursion):
Quicksort0(A p r)
1 while p < r
2 do . .
3 q Partition(A p r)
4 Quicksort0(A p q)
5 p q + 1
. , Quicksort0 -
.
, -
, -
. , -
, -
.
O(1), -
(stack depth) .
. , Quicksort0
(n).

171. 8 171
. Quicksort0 ,
(lgn) ( (nlgn)
).
8-5
Randomized-Quicksort ,
.
|
(median-of-3 method): -
.
,
A 1::n] n > 3. A0 1::n] -
( ).
pi = Pfx = A0 i]g, x | ,
.
. pi ( i = 2 3 ::: n ; 1) i n
( , p1 = pn = 0).
. (A0 b(n +
1)=2c]) , ? -
n ! 1?
. x -
, x = A0 i], n=3 6 i 6 2n=3.
, -
? ( : .)
. ,
(nlgn) (
nlgn).
98]. -
174] -
.
165].

172. 9
, -
n O(nlgn).
(merge sort) (heap sort)
, -
. -
O(nlgn) : -
n ,
(nlgn).
,
,
(comparison sort).
9.1 , -
n (nlgn) .
: -
, -
, .
9.2, 9.3 9.4 -
( ,
), . ,
(nlgn) , -
,
.
9.1.
, , -
, -
( ).
,
.

173. 173
. 9.1 , -
. -
3! = 6, 6 .
: . 9.1
(decision tree),
1.1.
n a1 ::: an. -
ai : aj, ,
(1 6 i j 6 n). -
h (1) (2) ::: (n)i, | -
n ( . . 6.1 ).
-
,
.
,
. .
( ) .
ai : aj. , ai 6 aj,
. ,
, ,
a (1) a (2) ::: a (n),
, .
n!
( -
).
|
.
.

174. 174 9
9.1. , -
n , (nlgn).
.
n ,
n!. h
2h , n! 6 2h. -
2 n! > (n=e)n,
(2.11), ,
h > nlgn ;nlge = (nlgn)
.
9.2. -
.
. O(nlgn) -
, .
9.1-1 -
?
9.1-2 lg(n!) =Pn
k=1 lgk , . 3.2.
9.1-3 , , -
,
n! n.
, 1=2 1=n?
1=2n?
9.1-4 ,
: - ai : aj
,
, : ai < aj,
ai = aj ai > aj. ,
n ,
(nlgn). , .
9.1-5 , -
n 2n;1 -
.
9.1-6 n ,
, k. -
. . (

175. 175
). -
, (nlgk) -
. ( : -
n=k k.)
9.2.
(counting sort) ,
n |
( -
k). k = O(n), -
O(n).
, x
, -
x, x
( , , 17
x, x
18). -
,
, -
.
C 1::k] k . -
A 1::n], -
B 1::n].
Counting-Sort(A B k)
1 for i 1 to k
2 do C i] 0
3 for j 1 to length A]
4 do C A j]] C A j]]+ 1
5 . C i] , i.
6 for i 2 to k
7 do C i] C i]+ C i ;1]
8 . C i] , i
9 for j length A] downto 1
10 do B C A j]]] A j]
11 C A j]] C A j]];1
. 9.2. ( 1{2) -
C i] A, i ( -
3{4), ,
C 1] C 2] ::: C k], | , i
( 6{7). , 9{11 -

176. 176 9
. 9.2 Counting-Sort, A 1::8],
, k = 6. ( ) A
C 3{4. ( )
C 6{7. ( { ) B -
C , 9{11.
,
. ( ) B .
A B. ,
n , -
A j] C A j]],
A A j] A -
, A j]
B C A j]] ( 11),
, A j],
.
.
1{2 6{7 O(k), 3{4
10{11 | O(n), , ,
O(k + n). k = O(n), O(n).
. 9.1
| ,
, .
,
(it is stable). ,
, -
, .
,
, ,
. , ,
, ht xi, t | 1 k,
x | , ,
.

177. 177
, -
.
.
, .
9.2-1 . 9.2.,
Counting-Sort k = 7 A = h7 1 3 1 2 4 5 7 2 4 3i.
9.2-2 , Counting-Sort -
.
9.2-3 9 Counting-Sort :
9 for j 1 to length A]
, . -
?
9.2-4
-
. Counting-Sort ,
, B
( A C). ( : A
-
?).
9.2-5 n 1 k. , -
,
O(1)
a b? .
O(n+ k).
9.3.
(radix sort) -
(
). -
. 80
12 . -
, -
( 80 ),
0{9 0{9 -
.
,

178. 178 9
329
457
657
839
436
720
355
)
720
355
436
457
657
329
839
"
)
720
329
436
839
355
457
657
"
)
329
355
436
457
657
720
839
"
. 9.3 -
. .
, (
, ).
,
10 , 0{9
.
( , | . .)?
, , |
. 10 ,
10 , |
,
( . 9.3-5).
, -
, 10 , -
. 10
: 0, -
1, . .
10 , , -
, . .
d- , d
. . 9.3 ,
, .
, , -
, : ,
, -
,
( , 10
).
, . , -
, .
,
: , | -
, | .

179. 179
, ,
: , -
, .
. -
, n- A
d , 1 | ,
d | .
Radix-Sort(A d)
1 for i 1 to d
2 do A
i
( . . 9.3-3). -
.
1 k, k -
, | . n
d 0 k;1 (n+k)
d , -
(dn+kd). d k = O(n),
.
-
, -
. :
64- . -
-
216,
, Counting-Sort B
216 ( -
) ,
lgn 20 . -
, , -
, ( , -
) , -
. , ,
.
9.3-1 . 9.3, , -
( ) COW, DOG, SEA,
RUG, ROW, MOB, BOX, TAB, BAR, EAR, TAR, DIG, BIG, TEA,
NOW, FOX.

180. 180 9
9.3-2
: , , -
, ? ,
-
.
?
9.3-3 , -
. -
?
9.3-4 , n
, n2, O(n).
9.3-5? -
, .
( ) d-
? -
?
9.4.
(bucket sort)
( ) . , -
: -
, ,
, -
0 1) ( -
. 6.2).
, | ( -
)
.]
, 0 1)
n , -
- (bucket), n
. -
0 1), , -
.
-
,
.
, n- A,
0 6 A i] < 1 i. -
B 0::n ; 1], ,
. , -

181. 181
. 9.4 Bucket-Sort. ( ) A 1::10].
( ) B 0::9] 5. i -
, i.
, B 0] ::: B 9].
. 11.2.
Bucket-Sort(A)
1 n length A]
2 for i 1 to n
3 do A i] B bnA i]c]
4 for i 0 to n; 1
5 do B i] ( )
6 B 0] B 1] ::: B n;1] ( )
. 9.4
10 .
, -
, A i] A j].
, ,
-
,
.
.
, , ( ) O(n).
O(n). ,
-
.
B i] ni (ni | ).
,
-
i O(M n2
i]),

182. 182 9
n;1X
i=0
O(M n2
i]) = O
n;1X
i=0
M n2
i]
!
: (9.1)
ni. -
, -
, ,
i, 1=n. ,
. 6.6.2 : n - , n - ,
p =
1=n. ni : -
, ni = k, Ck
npk(1 ; p)n;k,
M ni] = np = 1, D ni] = np(1;p) = 1;1=n.
(6.30) :
M n2
i ] = D ni]+ M2 ni] = 2 ; 1
n = (1):
(9.1), , -
O(n),
-
.
9.4-1 . 9.4, , -
Bucket-Sort A = h0:79 0:13 0:16 0:64 0:39
0:20 0:89 0:53 0:71 0:42i.
9.4-2 -
? ,
O(nlgn).
9.4-3? n
(xi yi), 1
( ,
- ). -
, -
(n).
( : , -
, ).
9.4-4? X | .
(probability distribution function) P(x) =

183. 9 183
PfX 6 xg. , -
n , -
P. P -
O(1). -
,
n?
9-1
,
n
, , (nlgn). ,
A,
TA | . -
, -
.
. TA ,
. , n! -
1=n!, .
. D(T) -
T ( , -
, | -
). T | k > 1 ,
LT RT . ,
D(T) = D(LT)+ D(RT)+ k.
. d(m) | D(T)
T m . , d(k) = min16i6k;1fd(i)+
d(k;i)+kg. ( : LT 1 k ;1
.)
. , k ilgi + (k ;
i)lg(k ; i) 1 k ; 1
i = k=2. , d(k) = (klgk).
. , D(TA) = (n!lg(n!)), ,
n TA,
, (nlgn).
B,
. ,
: -
.
Random(1 r) r , -
. (
r .)
. B.

184. 184 9
. -
T. , -
A, (
) T. ,
(
) (nlgn).
( . -
T, -
, T.)
9-2
. , -
, 0 1.
, -
O(1) (
,
).
. ( ) -
b- O(bn)? -
.
. n , -
1 k.
,
O(n + k), ( -
) O(k)? ( . k = 3.)
-
72]. -
123], , -
( ). -
- 23].
,
(1954 )
(H.H.Seward). , ,
. ,
1929
-
, (L.J.Comrie).
1956 , (E.J.Isaac)

185. 9 185
(R.C.Singleton) .

186. 10
:
n , i- ,
. -
i-
(order statistic). , (minimum) |
1, (maximum) | -
n. (median)
, ( )
. , n , |
i = (n + 1)=2, n ,
: i = n=2 i = n=2 + 1.
, , n,
i = b(n + 1)=2c i = d(n + 1)=2e. -
( ).
, ,
, ,
, , ,
.
(selection problem) -
:
: A n i,
1 6 i 6 n.
: x 2 A, i;1 -
A x.
O(nlgn): ,
i- . , ,
.
10.1 :
.
. 10.2
,
O(n)
(
). 10.3 ( -

187. 187
) -
, O(n) .
10.1.
, n
? n;1 : -
,
. , ,
A n.
Minimum(A)
1 min A 1]
2 for i 2 to length A]
3 do if min > A i]
4 then min A i]
5 return min
, .
? , . -
-
n , | ,
. , -
,
n;1 , Minimum -
.
, | -
4. -
6-2 , (lgn).
-
. , -
( ,
) , . -
.
, ,
n ; 1 ,
2n ; 2 , . , -
, 3dn=2e ; 2 . , -
, -
: ,

188. 188 10
, | .
-
( ,
).
10.1-1 , n
n + dlgne ; 2 . ( -
. .)
10.1-2? , -
n
d3n=2e; 2 . ( . -
: (1) ( -
, ) (2) ,
,
(3) , , -
(4) (
). a1 a2 a3 a4 |
. ?)
10.2.
,
, , ,
(n). -
Randomized-Select , -
.
8:
. ,
, , Randomized-Select |
. : -
(nlgn),
Randomized-Select (n).
Randomized-Select -
Randomized-Partition, 8.3,
( , , ) -
. Randomized-Select(A p r i)
i- A p::r].

189. 189
Randomized-Select(A p r i)
1 if p = r
2 then return A p]
3 q Randomized-Partition(A p r)
4 k q ; p + 1
5 if i 6 k
6 then return Randomized-Select(A p q i)
7 else return Randomized-Select(A q + 1 r i;k)
Randomized-Partition 3
A p::r] A p::q] A q +
1::r], A p::q]
A q+1::r]. 4
A p::q]. , -
i- A p::r]. i 6 k,
( A p::q]),
( ) Randomized-
Select 6. i > k, k
(A p::q]) ,
(i; k)- A q + 1::r] ( 7).
Randomized-Select ( -
Randomized-Partition)
(n2), : ,
, -
. -
,
.
. -
n -
( ). T(n) | -
. 8.4,
Randomized-Partition 1
2=n, i > 1 -
1=n i 2 n ; 1. , T(n)
n ( , T(n)
T0(n) = max16i6n T(i) T0). -
, , i-
, -

190. 190 10
. :
T(n) 6 1
n T(max(1 n; 1))+
n;1X
k=1
T(max(k n;k))
!
+ O(n)
6 1
n
0
@T(n;1) + 2
n;1X
k=dn=2e
T(k)
1
A + O(n)
= 2
n
n;1X
k=dn=2e
T(k) + O(n)
(
T(n; 1)=n
, T(n;1) = O(n2),
T(n; 1)=n O(n)).
, T(n) 6 cn, (
c, ). ,
T(j) 6 cj j < n. ,
, ,
T(n) 6 2
n
n;1X
k=dn=2e
ck + O(n)
6 2c
n
n
2
n=2 + 1 + n;1
2
+ O(n)
= 3cn
4
+ O(n) 6 cn
c , c=4 ,
O(n).
, , -
, .
10.2-1 Randomized-Select, -
.
10.2-2 , Randomized-
Select A =
h3 2 9 0 7 5 4 8 6 1i. ,
.
10.2-3 Randomized-Select -
, A ( ,
Randomized-Partition A p::r]

191. 191
A p::q] A q + 1::r] ,
A p::q] A q + 1::r])?
10.3.
( ) Select, -
O(n) -
. Randomized-Select,
-
, , ,
. Select ( -
) Partition ( .
. 8.1), , -
, , .
Select i- -
n > 1 :
1. n bn=5c 5
( ) , .
2. dn=5e (
).
3. ( ) Select, x, -
dn=5e .
4. Partition, -
. k | -
(
, , n ;k).
5. ( ) Select, i- -
, i 6 k, (i;k)-
, i > k.
Select. -
, x
( . . 10.1). ,
, x. ,
dn=5e , x,
: , x,
.
3 1
2
ln
5
m
;2 > 3n
10 ; 6:
, x, , -
3n=10 ; 6 , x. ,
Select, ,
7n=10+ 6.

192. 192 10
. 10.1 Select. ( n)
, | 5 , -
| . x.
. ,
x , x, x
, x. , x, .
T(n) | Select -
n . ,
O(n) ( O(n) -
O(1)),
T(dn=5e), , , | ,
T(b7n=10+6c) ( , ,
T(n) n). ,
T(n) 6 T(dn=5e) + T(b7n=10+ 6c) + O(n):
n (1=5 +
7=10 = 9=10) , -
, T(n) 6 cn c.
. , ,
T(m) 6 cm m < n,
T(n) 6 c(dn=5e)+ c(b7n=10+ 6c)+ O(n)
6 c(n=5+ 1) + c(7n=10+ 6) + O(n)
6 9cn=10+ 7c+ O(n) =
= cn;c(n=10;7) + O(n):
c cn
n > 70 ( , c(n=10 ; 6) ,
O(n)). ,
n > 70 ( , n
dn=5e b7n=10+ 6c n).
c ( ), , T(n)
cn n 6 70,
. , Select
( ).

193. 193
, Select Randomized-Select, -
9 -
, -
.
, -
, -
, (nlgn) ( -
9.1), ( 9-1).
10.3-1 Select ,
, -
? ,
.
10.3-2 x | Select (
n ). , n > 38 -
, x ( , x)
dn=4e.
10.3-3 , -
O(nlgn) .
10.3-4? i- -
. ,
-
, , .
10.3-5 - , -
.
, , -
.
10.3-6 k- (k-th quantiles) n
k ; 1 , -
: -
, k ( ,
) .
, O(nlgk) k-
.
10.3-7 , k
S k ,
. O(jSj).

194. 194 10
. 10.2 , -
?
10.3-8 X 1::n] Y 1::n] | .
, O(lgn) -
, .
10.3-9 ,
,
n . -
( , . 10.2).
-
, , -
, . ,
.
10-1 i
n i -
( ).
-
, n i
.
. i .
. i
Extract-Max.

195. 10 195
. 10.3 i-
( ),
i .
10-2
n x1 ::: xn, xi
( ) wi,
1. (weighted median)
xk,
X
xi<xk
wi 6 1
2
X
xi>xk
wi 6 1
2:
. , 1=n,
.
. n
O(nlgn) ?
. Select ( 10.3),
(n) ?
(post-o ce location problem)
. n p1 ::: pn n
w1 ::: wn p ( -
pi),
Pn
i=1 wid(p pi)
( d(a b) -
a b).
. , ( |
, d(a b) = ja;bj)
.
. ( |
), a =
(x1 y1) b = (x2 y2) L1- : d(a b) = jx1 ;
x2j+jy1;y2j ( Manhattan
distance, - ,
)
10-3 i- i
T(n) | Select,
n T(n) = (n),
n, , .
i n, i-
.

196. 196 10
. , i-
n , Ui(n) ,
Ui(n) =
(
T(n) n 6 2i,
n=2 + Ui(bn=2c) + T(2i+ 1) .
( : bn=2c i
.)
. , Ui(n) = n + O(T(2i)log(n=i)).
. , , i, Ui(n) = n+ O(lgn).
, 7,
i .]
. i = n=k, k > 2 , Ui(n) = n +
O(T(2n=k)lgk).
-
, , , 29].
97]. 70] -
, -
.

198. III

199. -
, . -
, -
( , -
dynamic) .
. , -
, -
, , -
. , , -
(dictionary). -
. , -
, 7 ,
( -
). , -
, .
| , -
.
(key), ,
| (satellite da-
ta), .
, -
, .
, -
, ( )
.
,
,

200. 200 III
( ,
)
( , -
, ).
, ,
, -
.
, -
,
.
,
.
(queries), -
, , (mod-
ifying operations). :
Search(S k) ( ). , S
k S -
k. S , -
nil.
Insert(S x) S , -
x ( ,
, x, ).
Delete(S x) S , -
x ( , x | ,
).
Minimum(S) S -
( , ).
Maximum(S) S -
.
Successor(S x) ( )
S, x (
). x |
, nil.
Predecessor(S x) ( ) -
, x ( x |
, nil).
Successor Predecessor -
,

201. III 201
. , -
Successor Predecessor , Minumum(S)
Successor,
. .
-
, . ,
14 , -
O(lgn),
n | .
III
11{15 ,
.
. , -
( )
7.
11
: , ,
. , -
,
. -
.
12 - , -
Insert, Delete Search.
- (n), , -
- ,
( ) O(1).
,
.
13 . -
.
(n),
-
O(lgn).
.
14 - .
: -
O(lgn).
-
( - )
19. -
.

202. 202 III
, .
15
- ,
( , -
).

203. 11
-
: , , , -
. , -
.
11.1.
| ,
, Delete,
, . ,
(stack) ,
:
| (last-in, rst-out | LIFO).
(queue), , , -
: -
| ( rst-in, rst-out, FIFO).
. ,
.
Push,
-
Pop (push , pop |
). ,
-
. |
| , .]
. 11.1 ,
n S 1::n].
top S], -
. S 1::top S]],

204. 204 11
. 11.1 S. -
. ( ) S 4 , | 9.
( ) Push(S 17) Push(S 3). ( )
, Pop(S) 3 (
). 3 - S,
| 17.
S 1] | ( ) S top S]] | -
, .
top S] = 0, (is empty). top S] = n,
(over ow), -
n.
|
| - ( - ?), -
under ow.
.
( , ,
) :
Stack-Empty(S)
1 if top S] = 0
2 then return true
3 else return false
Push(S x)
1 top S] top S]+ 1
2 S top S]] x
Pop(S)
1 if Stack-Empty(S)
2 then error under ow"
3 else top S] top S];1
4 return S top S]+ 1]
Push Pop . 11.1.
O(1).

205. 205
Enqueue, -
Dequeue. ( ,
Dequeue, .)
, : |
. ( , -
, - .)
, (head) (tail).
, , ,
, -
, , ,
.
. 11.2 , , -
n ; 1 , Q 1::n].
head Q] | , tail Q] | -
, -
. -
, head Q], head Q] + 1, ::: ,
tail Q];1 ( , : n -
1). head Q] = tail Q], .
head Q] = tail Q] = 1. ,
(under ow) head Q] = tail Q]+1,
, -
(over ow).
Enqueue Dequeue -
( 11.1-4
).
Enqueue(Q x)
1 Q tail Q]] x
2 if tail Q] = length Q]
3 then tail Q] 1
4 else tail Q] tail Q]+ 1
Dequeue(Q)
1 x Q head Q]]
2 if head Q] = length Q]
3 then head Q] 1
4 else head Q] head Q]+ 1
5 return x
Enqueue Dequeue . 11.2.

206. 206 11
. 11.2 , Q 1::n]. - -
. ( ) 5 ( -
Q 7::11]). ( ) Enqueue(Q 17), En-
queue(Q 3) Enqueue(Q 5). ( ) De-
queue(Q) ( 15). -
6.
O(1).
11.1-1 . 11.1,
Push(S 4), Push(S 1), Push(S 3), Pop(S), Push(S 8) Pop(S)
, S 1::6].
.
11.1-2 A 1::n]
n? Push Pop
O(1).
11.1-3 . 11.2,
Enqueue(Q 4), Enqueue(Q 1), Enqueue(Q 3), Dequeue(Q),
Enqueue(Q 8) Dequeue(Q) , -
Q 1::5]. .
11.1-4 Enqueue Dequeue, -
under ow.
11.1-5
. -
, | . , -
(deque, double-ended queue |

207. 207
), -
. , -
O(1).
11.1-6 ,
. Enqueue Dequeue
?
11.1-7 , .
?
11.2.
( - linked list)
, -
, , ,
. -
, , -
( ).
, ,
, -
-
, .
, . 11.3, -
(doubly linked list) | ,
: key ( ) next ( ) prev (
previous | ). ,
. x | ,
next x] , prev x] |
. prev x] = nil, x -
: (head) . next x] = nil, x |
, , (tail).
,
, L -
head L] . head L] = nil, .
.
(singly linked) -
prev. (sorted)
, -
, | , -
(unsorted) . (circular list)
prev , next
.
,

208. 208 11
. 11.3 ( ) L 1 4 9 16. -
| -
( ).
next prev nil
( ) head L] . ( ) -
List-Insert(L x), key x] = 25,
25 , next
| 9. ( )
List-Delete(L x), x |
4.
.
List-Search(L k) L (
) , k.
, , nil,
. , , L |
. 11.3 , List-Search(L 4)
, List-Search(L 7) nil.
List-Search(L k)
1 x head L]
2 while x 6= nil and key x] 6= k
3 do x next x]
4 return x
n (
) (n) .
List-Insert x L,
( . 11.3 ).

209. 209
List-Insert(L x)
1 next x] head L]
2 if head L] 6= nil
3 then prev head L]] x
4 head L] x
5 prev x] nil
List-Insert O(1) (
).
List-Delete x L, -
. -
x. x,
-
List-Search.
List-Delete(L x)
1 if prev x] 6= nil
2 then next prev x]] next x]
3 else head L] next x]
4 if next x] 6= nil
5 then prev next x]] prev x]
. 11.3 .
List-Delete O(1) -
,
(n).
, -
List-Delete :
List-Delete0(L x)
1 next prev x]] next x]
2 prev next x]] prev x]
, L
nil L], next prev
. (
sentinel | ) . -
nil. -
: next nil L]] prev nil L]]
, prev next

210. 210 11
. 11.4 L, nil L] ( -
). head L] next nil L]]. ( ) .
( ) . 11.3 ( 9 | , 1 | ). ( )
List-Insert0(L x), key x] = 25. ( )
1. 4.
nil L] ( . 11.4).
next nil L]] | , head L]
. L , -
nil L] | .
List-Search nil nil L]
head L] next nil L]]:
List-Search0(L k)
1 x next nil L]]
2 while x 6= nil L] and key x] 6= k
3 do x next x]
4 return x
List-Delete0, -
. , :
List-Insert0(L x)
1 next x] next nil L]]
2 prev next nil L]]] x
3 next nil L]] x
4 prev x] nil L]
List-Insert0 List-Delete0
. 11.4.
, -
. (
, ),
.
. -
, -

211. 211
.
, .
11.2-1 ,
, O(1)?
.
11.2-2 .
Push Pop O(1).
11.2-3
. Enqueue Dequeue
O(1).
11.2-4 Insert, Delete
Search
. ?
11.2-5 Union ( ) -
(
). -
, O(1), -
.
11.2-6 ,
( ),
. ,
1 ( ).
?
11.2-7 ,
(n)
. (
) O(1).
11.2-8? -
, next prev -
np x]. , k- -
nil . np x]
np x] = next x] XOR prev x], XOR | -
2 ( ). ,
.
Search, Insert Delete? ,
O(1).

212. 212 11
. 11.5 . 11.3 ,
(key next prev). - .
,
. L |
.
11.3.
,
, .
,
.
, , -
( ). ,
. 11.5 ,
, . 11.3 : key ,
next prev | . -
(key x] next x] prev x]) -
x. x.
nil ,
( , 0 ;1). . 11.3
4 16 -
, 16 key 5,
4 | 2, next 5] = 2, prev 2] = 5.
key x]
: key x, key
x ( -
).
, -
. -
: -

213. 213
. 11.6 , . 11.3 11.5, -
A.
. key, next prev 0, 1 2. -
| . -
.
, ,
,
.
.
. - A. -
A j ::k].
| j -
0::k ; j] | . ,
, . 11.3 11.5,
, key, next prev 0, 1 2.
prev i], i | (= A),
, A i+ 2] ( . . 11.6).
-
( ),
, ,
, .
. ,
. -
| (garbage collector), ,
,
. , , -
. , -
,
.
, -

214. 214 11
. 11.7 Allocate-Object Free-Object. ( ) -
, . 11.5 ( - ) ( - ).
. ( )
Allocate-Object() ( 4), -
key 4] 25 List-Insert(L 4).
8 ( next 4]). ( ) -
List-Delete(L 5), Free-Object(5). 5 |
, 8.
, m,
n 6 m . n ; m ( ) -
(free).
, (free list).
next: , next i] -
, i. -
free.
L ( . 11.7). -
, 1 m]
L, .
: -
,
. -
-
Push Pop . -
Allocate-Object ( ) Free-Object ( -
) , free
( ) .

215. 215
. 11.8 L1 ( - ) L2 ( - ), -
( ) .
Allocate-Object()
1 if free = nil
2 then error
3 else x free
4 free next x]
5 return x
Free-Object(x)
1 next x] free
2 free x
n -
. , Allocate-
Object .
-
( . 11.8).
-
( O(1)).
, -
next ( ).
11.3-1 . 11.5, -
h13 4 8 19 5 11i -
,
. 11.6 , -
.
11.3-2 Allocate-Object Free-Object
, .
11.3-3 Allocate-Object Free-Object
prev?
11.3-4 ( , -
) -

216. 216 11
.
. Allocate-Object
Free-Object ,
1::m, m | . ( : -
.)
11.3-5 L m -
key, prev next n, -
.
Compactify-List ( ), -
L m m -
, (
). (m),
( )
O(1). .
11.4.
-
,
.
. -
, , -
.
.
, -
.
, , .
, .
. 11.9, -
T p, left right, -
, x -
. p x] = nil, x | x
, left x] right x] nil. T
root T] | . root T] = nil,
T .

217. 217
. 11.9 T. x
p x] ( ), left x] ( ) right x] ( ).
.
,
k, -
,
child1 child2 ::: childk, left right. -
, : -
, ( | -
) .
,
k,
k:
.
? ,
. :
, -
, (
).
.
,
, .
| (left-child, right-sibling representation)
. 11.10. - -
p root T]
. p,
:
1. left-child x] x

218. 218 11
. 11.10 T | .
x p x] ( ), left-child x] ( )
right-sibling x] ( ). .
2. right-sibling x] -
x ( )
x , -
left-child x] = nil. x |
, right-sibling x] = nil.
.
, 7
,
2. 22 ,
(
). -
.
11.4-1 ,
:

219. 11 219
key left right
1 12 7 3
2 15 8 nil
3 4 10 nil
4 10 5 9
5 2 nil nil
6 18 1 4
7 7 nil nil
8 14 6 2
9 21 nil nil
10 5 nil nil
| 6.
11.4-2
, -
.
11.4-3 ( -
), ? ( -
.)
11.4-4 , -
,
| .
11.4-5?
, ,
( , -
) O(1),
( ).
11.4-6? -
, (
, | )
.
11-1
( ) -
III ( . 194) , -
.

220. 220 11
, , ,
Search(L k)
Insert(L x)
Delete(L x)
Successor(L x)
Predecessor(L x)
Minimum(L)
Maximum(L)
11-2
(mergeable
heaps) ( ), -
: Make-Heap ( ),
Insert, Minimum, Extract-Min , , Union (
).
( -
) .
.
. .
. .
. , -
.
11-3
11.3-4 :
n- n -
. , -
( , key i] < key next i]] next i] 6= nil).
, -
o(n):
Compact-List-Search(L k)
1 i head L]
2 n length L]
3 while i 6= nil and key i] 6 k
4 do j Random(1 n)
5 if key i] < key j] and key j] < k
6 then i j
7 i next i]
8 if key i] = k
9 then return i
10 return nil

221. 11 221
4{6
. 4{6 -
j. key i] < key j] < k,
, ,
i j. ( , -
,
.)
. , ? -
, ( ) -
.
? , -
,
.
| -
.
,
, .
. t > 0
Xt , (
) i k t -
.
. , t > 0
Compact-List-Search O(t +
E Xt]).
. , E(Xt) 6
Pn
r=1(1;r=n)t. ( : -
(6.28)).
. ,
Pn;1
r=0 5 6 nt+1=(t+ 1).
. , E(Xt) 6 n=(t + 1), ,
.
e. , -
Compact-List-Search O(
pn).
| ,
5] 121].
90]
-
. 121] , 1947
(A.M.Turing) .

222. 222 11
, , , ,
. , -
. 1951
(G.M.Hopper) A-1, -
.
, -
IPL-II, 1956
, (A.Newell, J.C.Shaw, H.A.Simon). -
1957 IPL-III
.

223. 12 -
,
, -
. -
-
( ).
- ,
( (n)), .
- O(1).
-
. , -
,
-
O(1)
( , . . 12.1). -
,
, : -
, . 12.2 -
, 12.3 | -
.
.
, |
( O(1)
).
12.1.
,
.
U = f0 1 ::: m ; 1g ( m ).
, .
T 0::m ; 1],
(direct-address table).

224. 224 12 -
: universe of keys | -
, actual keys | , key | ,
satellite data | .
. 12.1 T -
. U = f0 1 ::: 9g. -
.
2, 3, 5 8 ( )
, ( - )
nil.
, , ( - slot position) -
U ( . 12.1: T k] |
, -
k k , T k] = nil).
:
Direct-Address-Search(T k)
return T k]
Direct-Address-Insert(T x)
T key x]] x
Direct-Address-Delete(T x)
T key x]] nil
O(1).
, T -
, . -
: -
. , ,
, .
12.1-1 -
,
. -

225. - 225
?
12.1-2 (bit vector) | (
). m
, m . , -
, , -
? -
O(1).
12.1-3 , -
,
? ,
O(1) ( ,
Delete
, ).
12.1-4? ,
, ,
.
- , ,
, . -
?
O(1), -
, O(1),
O(1). ( : , -
, -
, ).
12.2. -
: -
U , -
T jUj , . ,
jUj, .
, -
, - -
, . , -
(jKj), K | -
, - - O(1)
( , | ,
, -
).
k -

226. 226 12 -
: universe of keys | -
, actual keys |
. 12.2 -
- . - k2 k5 | -
.
k, -
h(k) - (hash table) T 0::m;1],
h: U ! f0 1 ::: m; 1g
| , - (hash function).
h(k) - (hash value) k.
. 12.2: m,
jUj, .
, , , -
. , ,
(collision). , :
- .
- , -
. jUj > m -
, - .
,
,
, - .
- , , -
.
- - , -
( to hash"
, ). ,
- -
,
- .
( , ) .
| | 12.4.

227. - 227
: universe of keys | -
, actual keys |
. 12.3 . T j]
- j. , h(k1) = h(k4)
h(k5) = h(k2) = h(k7).
(chaining) ,
, -
, - ( . 12.3). -
j , -
- j -
, j nil.
, :
Chained-Hash-Insert(T x)
x T h(key x])]
Chained-Hash-Search(T k)
k T h(k)]
Chained-Hash-Delete(T x)
x T h(key x])]
O(1).
( ). -
, O(1) | -
, ( -
, x
,
).

228. 228 12 -
-
.
T | - m , n
. (load factor) -
= n=m ( ,
). .
-
: - n ,
n,
(n), , -
- . ,
.
, -
- - .
, , -
12.3 ,
m
, .
(simple uniform hashing).
, k -
h(k), -
, k
T h(k)], -
.
: (
k ), |
.
12.1. T | - , -
. , -
. , ,
,
( - )
(1 + ).
. -
-
,
m ,
m . -
n=m = , -
, (1)
- .

229. - 229
12.2.
- (1 + ),
| .
.
, .
-
, -
,
.
: -
, ,
, -
(1 + ) ( : ,
)
:
-
, ,
,
, -
.
, -
, .
, n -
?
n ( ).
,
( . . 12.2-3),
, (
-
).
. i-
(1 + (i ; 1)=m)
( . ), -
,
n,
1
n
nX
i=1
1 + i ;1
m
!
= 1 + 1
nm
nX
i=1
(i; 1)
!
=
= 1+ 1
nm
(n;1)n
2 =
= 1+ 2 ; 1
2m = (1 + ):

230. 230 12 -
- -
, -
, ( -
) O(1). ,
n = O(m), = n=m = O(1) O(1 + ) = O(1).
- O(1) (
, . . 12.2-3), -
O(1) ( , ),
( -
) O(1).
12.2-1 h | - , -
n fk1 k2 ::: kng m
. ( -
, (i j), h(ki) = h(kj))?
12.2-2 - -
,
5 28 19 15 20 33 12 17 10 ( )?
9, - h(k) = k mod 9.
12.2-3 , -
( -
) O(1 + ),
.
12.2-4 ,
, -
.
, ,
, ?
12.2-5 - , -
- ( -
). ,
-
, , . ,
, -
O(1).
?
12.2-6 (
U) mn, m | - . ,

231. - 231
n - ,
-
(n).
12.3. -
, -
, - :
, .
- ?
- ( )
:
m - .
, -
P U ,
U ,
P. ,
X
k: h(k)=j
P(k) = 1
m j = 0 1 ::: m;1. (12.1)
, P , -
(
).
P . , ,
| , -
0 1). -
, - h(k) = bkmc (12.1).
-
, . , -
, -
.
( -
, pt pts). - ,
- .
- , -
-
, . ,
,
- -
.