519.85(023) 22.18

052

, 004.1-60.

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/ . . . .: . , 2002. 341 : .

ISBN 5-94774-010-9 ,

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519.85(023) 22.18

:. (095)955-03-98, e-mail: lbz@aha.ru

-: (812)247-93-01, e-mail: dialect@sndlct.ioffe.rssi.ru

. ., 2002ISBN 5-94774-010-9 . , 2002

71. 9

1.1. 91.2. 21

2. 252.1. 252.2. 31

2.2.1. 312.2.2. 402.2.3. 462.2.4. 532.2.5. N

582.2.6. 612.2.7. 65

2.3. 683. 79

3.1. ( ) 793.2.

813.3. 963.4.

983.5. 1053.6. 1093.7. 113

4. 1414.1. 1414.2. 142

4.2.1. 1424.2.2. 143

4.3. 144

4.3.1. . 1444.3.2. 1454.3.3. . . . . 1484.3.4. . . 150

4.4. 1514.4.1. 1514.4.2. 1534.4.3. 154

4.5. 1574.5.1. 1574.5.2. 1584.5.3. 160

4.6. 1614.6.2. 1634.6.3. 1644.6.4. .

1664.7. 168

4.7.1. 1684.7.2.

1694.7.3. 1734.7.4. 1744.7.5. . . 175

4.8. 1804.8.1. 1804.8.2. . . . . 1814.8.3.

1854.9. , 186

4.9.1. 1864.9.2. . . . 1874.9.3. . . . 192

4.10. 195

4.10.1. 1954.10.2. 1974.10.3. 200

4.11. 2025. 222

5.1. 222

5.2. 2275.3. 2325.4. 2365.5. 2415.6. 2515.7. 259

6. 266

7. 3187.1. 3197.2. 3207.3.

( ) 328 340

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okulov@vspu.kirov.ru.

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30!= 265252859812191058636308480000000? -

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10 1.

Const MaxDig=1000;{* -.*}

Osn=10000;{*Ocoaue , - .*}

Type TLong=Array[0..MaxDig] Of Integer;{* - .*}

, .

23851674 (Osn) - 1000 ( ). - (- h) .

[0]30

11122233

[1]674

0

223

238385851516167674

[2]851

0

0002

23238385851

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1- 2- 3- 4- 5- 6- 7-

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()

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A[i] A[i+1], . . :

For i:=A[0] DownTo 1 Do BeginA[i+1] :=A[i+l] + (LongInt (A[i] ) *10) Div Osn;A[i] : = (LongInt (A[i] ) *10) Mod Osn;

End;

1. 11

23851674 6 - . -

ch 7. 7 - [1]. - . -

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i221

[1]516516160

[2]238380385

[3]022

ch7

[1] [0]. :

Procedure ReadLong(Var A:TLong);Var ch:Char;i:Integer;Begin

FillChar (A,SizeOf(A) ,0) ;RepeatRead (ch) ;Until ch In ['0'..'9'] {*

. *}While ch In ['0'..'9'] Do BeginFor i:=A[0] DownTo 1 Do Begin{*""

A[i] A[i+1]. *}

A[i+1]:=A[i+l]+(LongInt(A[i])*10) Div Osn;A[i] := (LongInt (A[i] ) *10) Mod Osn;

End;A[l] :=A[1]+Ord(ch)-Ord('0');{*

[1] . *}If [[0] +1]>0 Then Inc ([0]);{ * ,

. *}Read (ch);End;

End; .

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. , , 128400583274. 58, 0058, - . , .

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Procedure WriteLong(Const A:TLong) ;Var ls,s:String;

i: Integer;BeginStr(Osn Div 10,ls);Write(A[A[0]]);{* . *}For i:=A[0]-1 DownTo 1 Do BeginStr(A[i],s) ;While Length (s)

1. 13

i1234

A[i]945113028706

0

B[i]7461

5191213475

[1]6912691269126912

[2]1

135413541354

[3]00

78277827

[4]001

3476

=3476782713546912. , .

-

.

.

Procedure SumLongTwo(Const ,:TLong;Var C:TLong) ;Var i,k:Integer;Begin

FillChar (C,SizeOf (C) , 0) ;If A[0]>B[0] Then k:=A[0] Else k:=B[0];For i : = 1 To Do Begin [i+1]:=(C[i]+A[i]+B[i])

Div Osn;C[i]:=(C[i]+A[i]+B[i]) Mod Osn; {*

?*}End;If C[k+1]=0 Then C[0]:=k Else C[0]:=k+1;

End; .

(A=, , ==). = .

Function Eq(Const ,:TLong):Boolean;Var i:Integer;Begin

Eq:=False;If A[0]=B[0] Then Begin

i : =1 ;While (i .

Function More (,: TLong): Boolean;Var i:Integer;Begin

If A[0]

14 1.

Else If A[0]>B[0] Then More:=TrueElse Begin

i:=A[0];While (i>0) And (A[i]=B[i]) DoDec(i);If i=0 Then More:=False

Else If A[i]>B[i] Then More:=True ElseMore:=False;

End;End;

Eq More.Function Less(A,B:TLong):Boolean;{A

1. 1J5

While (i>sdvig) And (A [i] =B [i-sdvig] ) DoDec (i) ;

If i=sdvig Then Begin More:=0;{* . *}

For i:=1 To sdvig Do If A[i]>0 Then Exit;More:=2;{* , "" .*}

EndElse More:=Byte(A[i]0 Then C[0]:=A[0]+1Else [0]:=A[0];{*

.*}End;

End; -

, -

.

Procedure MulLong(Const ,: TLong; Var : TLong);{* "" "".*}Var i, j : Word;

dv: Longlnt;BeginFillChar(C

r SizeOf(C), 0) ;

For i:=1 To A[0] Do

16 1.

For j : = 1 B[0] Do Begindv:=LongInt (A[i])*B[j]+C[i+j-1];Inc (C[i+j], dv Div Osn);C[i+j-1]:=dv Mod Osn;

End;C[0] :=A[0]+B[0] ;While (C[0]>1) And (C[C[0]]=0) Do Dec(C[0]);

End;

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9 11 1 -, 10000 9 - .

Procedure Sub (Var A: TLong;Const : TLong; sp:Integer);

Var i , j : Integer;{ * sp, .*}

BeginFor i : = 1 To B[0] Do Begin Dec (A[i+sp], B[i]) ;

{* *} {*}j:=i; {*}While (A[j+sp]

1. 17

If A[i+sp]

18 1.

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. B*10, ( ) . -, 564, 63 . , -

, .

Down - , Up , Ost -.

Down05788

Up101010109

C=B*((Down+Up) Div2)315441504567504

Ost=564COstC>OstC>OstC>OstCOstC>OstC>OstC>OstC>OstC>OstC

1. 19

Function FindBin (Var Ost: TLong; Const : TLong;sp: Integer) : LongInt;

Var Down, Up: Word;C: TLong;

BeginDown:=0; Up:=Osn; {* .*}While Up-1>Down Do Begin {*

.

Up>Down. - . *}

Mul (B,(Up+Down) Div 2, );Case More (Ost, C, sp) Of

0: Down:=(Down+Up) Div 2;1: Up:= (Up+Down) Div 2;2: Begin Up:=(Up+Down) Div 2; Down:=Up; End;

End;End;Mul (B, (Up+Down) Div 2, C) ;If More(Ost,C,0) =0 Then Sub (Ost, C, sp)

{* . *}Else begin Sub(,Ost,sp); Ost:=C; End;

FindBin:=(Up+Down) Div 2;{* .*}End;

sp . , , 635 15. ? 63 15 , , . -

. 4, . . 635, 35. . . . 2 5. , ( ) 42, 5. , 10, 10000? - , ,

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Procedure MakeDel(Const ,: TLong;Var Res, Ost:TLong) ;

Var sp: Integer;Begin

Ost:=A;{* . *}sp:=A[0]-B[0];

20 1.

If More(,,sp)=1 Then Dec(sp);{* B*0sn>A, .*}

Res[0]:=sp+1;While sp>=0 Do Begin {*

.*}Res[sp+1]:=FindBin(Ost,B,sp);Dec(sp);End;End;

-

, () ( ). , .

F u n c t i o n Eq(Const , : TLong): B o o l e a n ;F u n c t i o n L e s s ( C o n s t A, B: TLong): B o o l e a n ;F u n c t i o n More(Const A, B: TLong): B o o l e a n ;F u n c t i o n L e s s _ E q ( C o n s t A, B: T L o n g ) : B o o l e a n ;F u n c t i o n More_Eq(Const A, B: TLong): B o o l e a n ;Function LongTurnShort (Const A: TLong; Var K:

LongInt): Boolean; {* LongInt.*}

Function LongModShort(Const A: TLong; Const K:Word): Word; {* Word.*}

Function HowDigits (Const A: TLong): Longlnt;{* .*}

Procedure Mul(Const A: TLong; : Word; Var :TLong); {* Word.*}

Procedure MulLong(Const A, B: TLong; Var C: TLong);Procedure SumLongTwo(Const A, B: TLong; Var C:

TLong);Procedure Sub(Var A: TLong; Const B: Tlong;Const

s p : I n t e g e r ) ;Procedure ShortTurnLong (K: LongInt; Var A: TLong);

{* LongInt .*}

Procedure LongDivShort(Const A: TLong; Const K:Word; Var B: TLong); {* Word.*}

1. 21

Procedure LongDivLong(Const , : TLong; Var ,ost: TLong);

Procedure SdvigLong(Var A: TLong; Const p: Word);{* .*}

1.2. 1.

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Const Basis=10;{* . *}Typept=^elem;

elem=Record data:Integer; next,pred:pt; End;Var head, tail :pt;

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2. 3. 4.

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574564 : 57 45 64. , - ( ).

2. 57 - ,

2 57, -

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3. 57

2=49: 57-49=8. -

, 845. 4. .

, , (2**10+)* 845 845. 5, (14*10+5)*5=725845. , 5 y=75.

5. 3. : 845-725=120, 12064.

6. 4.(2*y*10+)*

1. 23

11. 2, 8, 16.

12. .

13. , () 2 (). b :

b ,

b ,

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Function Sum (,: Integer) : Integer;Begin

If x=0 Then Sum:=yElse

If y=0 Then Sum:=xElse If (x Mod 2 =0) And (y Mod 2=0) Then

Sum:=Sum(x Div 2, Div 2)*2Else If (x Mod 2 =1) And (y Mod 2=1)

Then Sum:=Sum(x Div 2,y Div 2+1)*2Else Sum:=Sum(x Div 2, Div 2)*2+1;

End; (

) b. - 2 .

14. b , . ,

b=125, =37. :125*37250*18+125

24 1.

500*9+1251000*4+500+1252000*2+500+1254000*1+500+125. : 125*37=125+500+4000=4625 (

b ?). - :

,

,

:

Function RecMul (x,:Integer): Integer ;{* , Integer.*}

BeginIf =0 Then RecMul:=0Else If x Mod 2=0 Then RecMul:=RecMul (x Div

2,y)*2Else RecMul:=RecMul(x Div 2,y) *2+y;

End; ( -

) b. - 2 - .

15. - [2].

. t- numb 10*, 10*-|numb|. - [2, 23]. - i 9-i. - .

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Word, Integer, Char, Boolean . ,

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1. 1,2,5, -

1, 2, 3? : 6.2. ,

, , , 1, 2, 3,4? : 24.

3. -? : 120.

4. 8 , ? -: 40320.

N , N , N! (-)=1*2*3*...*N (PN=N!).

,

1 N. - . -

N N 1 N. 1 N. , 1 N.

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F G FG, - : FG(i)=F(G(i)). , F - ,

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FF-1=E. , -

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F : F=[1, 3], [2, 5, 4, 6]. , ifj.. (I) (-1)I. 6, - . , -

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1. 1, 2 1, 2, 5 ? : 6.

2. 1, 2 , , , ? : 12.

3. ? : 60.

4. 17 , .

? :4080.

5. 25 . - , , . -

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() MN. - : N -?

1. 1,2,5 ? : 3.

2. - , , , ? : 6.

28 2.

3. - ? : 10.

4. 17 ,

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2. 29

k. - .

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1. .: 9*10*10*10*10*10=900000.

2. 4, 33 , ,

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: 33*32*32*32=1081344.3. 16 (k).

, , 1 0 (N). 1 0 -? : 216=65536.

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x

, N2 , ..., Nk k-o

1. ? : 6 (, , , , ,).

2. ? : 180.

3. 2 - 3 ? : 10.

4. , 6 : 2 , 3 1 ? :60.

5. (, , ). -: bb, cabbac. - ? : 90.

30 2.

: N ==N1+N2+...+Nk .

. N - . k, ( - )?

. N=4. k=7. - . -

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. . (, , , ). - 7, - N-1 3, . . 10. -

3 10 , -

1. 30 . ? :

2. 3 - ? : 4 (N=2,

3. 13 52 , ? : 560 (N=4, k=13).

4. , - 2 ( , 36)? : 21 (N=6, k=2).

5. : 120 (N=4, k=7,

1101101101).. -

S, |S|=N ( ), k -,

2. 31

. S1 ,, ... . -

,

1. 1,2,5 - ? : 6.

2. , ,, ? : 6.

3. , ,

2? :4. 8

: , ?: 8!/(1!*3!*4!)=280.

. , -

2 6 ? :6!/(2!*2!*2!)=90.

S(N,k) - N- k . ,

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2.2 2.2.1.

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N. , -

:

123456

12624120720

32 2.

78910111213

...

30...

5040 ( Integer)40320362880362880039916800479001600 ( Longlnt)6227020800

265252859812191058636308480000000

.

Function Fac (k:LongInt): LongInt;Var r,i: LongLnt;Begin

r:=1;For i:=l To Do r:=r*i;Fa:=r;

End; N, 12.

N : ; ( 1).

.

N, -

. . ,

N. - -

.

1=1, P1[t]=

2. 33

()213231312321

()132312231321

(N=4). , , , -

.

123412431324134214231432

12342134132431242314

3214

213421432314234124132431

124321431423412324134213

312431423214324134123421

134231421432413234124312

412341324213423143124321

234132412431423134214321

. 11, 8, 5, 1, 7, 4, 10, 9, 6, 3, 2. ?

4 10. - , -

4, 6. 4 6 -, . . . -

-

:

34 2.

Procedure GetNext;Var i,j:Integer;Begin {* N, N-1,...,1

, . *}i : =N;While (i>l) And (P [i]

2. 35

:

i

tiyi

1110

281

351

411

570

643

7102

890

965

1035

1120

, 0=

36 2.

. ,

.

,

, -

.

N=4, - 0023. .

00232134

01231234

01221243

01211423

,

. y[i] - i , . ,

i ,

.

.

ConstNmax=12;Var P:Array[1. .Nmax] Of 0 . .Nmax; {*

.*}Y:Array[1..Nmax] Of 0..Nmax-1; {*

.*}D:Array[1..Nmax] Of-1..1; {*

"". *} -

Y D.Procedure First;

Var i:Integer;BeginFor i : = 1 To N Do BeginP[i]:=N-i+1; Y[i]:=0; D[i]:=1;

End;End;

2. 37

, .

D[i]=l Y[i]=i-1 D[i]=-l Y[i]=0 i - .

Function Ok:Integer; {* 1, , .*}

Var i:Integer;Begini:=N;While (i>1) And (((D[i]=1) And (Y[i]=i-1)) Or

((D[i]=-1) And (Y[i]=0))) DoDec(i);Ok: =i ;

End; :

BeginFirst;:=True;While pp Do BeginSolve (pp) ;If pp Then Print;End;End; .

Procedure Solve(Var q:Boolean);Var i, j, dj:Integer;Begini:=Ok; q:=(i>l); {*Ha ,

. *}If i>1 Then BeginY[i]:=Y[i]+D[i]; {* -

.*}For j:=i+1 To N Do D[j]:=-D[j]; {*

,

.*}j:=Who_i(i); {* ,

i.*}dj:=j+D[i]; {*

.*}Swap(P[j], P[dj]); {*.*}End;End;

38 2.

, -

, ,

i. .Function Who_i (i : Integer): Integer;Var j: Integer;Beginj:=N;While (j>0) And (P[j]i) Do Dec(j);Who_i :=j;

End; .

, , -

? -. . N 8 L, 37021. -. . -

7!, 5040 (1*******). 2 5040 (2*******). ,37021 Div 5040=7. , 8. L (37021 Mod 5040=1741) 1741. . , , .

,

4. , 1 2, : . , 4. 4, 5 7.

. N=

2. 39

.

Procedure GetPByNum(L:LongInt);Var Ws: Set Of Byte;

i,j,sc: Integer;Begin

Ws:=[]; {* , .*}

For i : = 1 N Do BeginSc:=L Div ResSm[N-i]; L:=L Mod ResSm[N-i]; j:=1;

While (sc0) Or (j In Ws) Do BeginIf Not (j In Ws) Then Dec(sc);Inc (j) ;

End;Ws:=Ws+[j]; {* . *}P[i]:=j;

End;End;

-

.

I

1

2

3

4

...

Sc

76543210210211021110

...

L370211741

301

61

13

...

J

12345678123123412

345

...

Ws

[]

[8]

[3,8]

[3,4,8]

[3,4,5,8]...

P********

8********

83******

834******

8345****...

40 2.

. , -

. .

N 8 53871462. :7!*6!*< 2- ,

3, 2>5!*< 3- ,

8, 5>4!*< 4- ,

7, 4>3!*< 5- ,

1, 0>2!*< 6- ,

4, 1>1!*< 7- ,

6, 1>,

7!*4+6!*2+5!*5+4!*4+3!*0+2!*1+1!*1=4*5040+2*720+5*120+4*24+0*6+1*2+1*1=22305.

Function GetNumByP: LongInt;Var ws:Set Of Byte;

i,j,sq:Integer; sc:LongInt;Beginws:=[]; {* .*} sc:=1;

{* .*}For i :=1 To N Do Begin

j:=1; sq:=0; {* .*}While j

2. 41

Function Plac(n,m: LongInt): LongInt;Var i,z: LongInt;Begin

z: =1 ;For i:=0 To m-1 Do z :=z* (n-i);Plac:=z;

End;

.

Function Plac (n,m:LongInt): LongInt;Begin

If m=0 Then Plac:=1Else Plac:=(n-m+1)*Plac (n,m-1);

End;

N=

42 2.

S:=S+[i]; {* .*}

A[t] :=i; {* .*}If t

2. 43

Procedure GetNext;Var i,j,k,q:Integer;

Free:Array[1..n] Of Boolean; {*

.*}Function FreeNext(t:Byte):Byte; {*

.*}BeginWhile (tn Then FreeNext:=0 {*

, .*}Else FreeNext:=t; {*

.*}End;BeginFor i:=1 To n Do Free [i]:=True; {*

.*}For i:=1 To m Do Free [A[i]]:=False; {*

.*}i:=m; {* .

,

.*}While (i>0) And (FreeNext(A[i])=0) Do Begin

{* , ,

.

.*}Free[A[i]]:=True;{* ,

i.*}Dec(i); {* .*}End;Free[A[i]]:=True; {*

.*}q:=FreeNext(A[i]+1); {*

, .*}Free[q]:=False; {* .*}A[i]:=q; {*3 .*}:=1; {* "" .*}For j:=i+1 To m Do Begin {*Co

.*}While (k

44 2.

If k>= Then k:=1 Else Inc(k);A[j]:=k; {*

.*}Free[k]:=False; {* .*}

End;End;

. N - L ( ).

5 3 , 60. - , , -

.

0.

012

345

123124125132134135

67

891011

142143145152153154

121314151617

213214215

231234235

1819202122

23

241243245251253254

242526272829

312314315321324325

3031323334

...

341

342345

351352

( ) 12.

12, . 32 1 2 3 4 5 ( - 0). 32 Div 12, 2, -, , 2- - , 3. 32 Mod 12 =8, 1 2 4 5. - 8 Div 3=2 , 4. 8 Mod 3=2, 1 2 5. - 2 Div 1 =2, 5.

Const n=5; m=3;Var A:Array[1. .] Of Integer;

L:LongInt; {* .*}Procedure GetPByNum(L:LongInt); {*B

Plac - .*}

Var i,j,q,t:LongInt;Free:Array[1..n] Of Byte; {*

.*}

2. 45

BeginFor i:=1 To n Do Free [i]:=i ; {*

.*}For i : = 1 To m Do Begin {*i -

.*}t:=Plac(n-i,m-i); {* ,

i.*}q:=L Div t; { *

.*}A[i]:=Free[q+1]; {*

.*}For j :=q+1 To n-i Do Free [j] :=Free [j+1];

{*, .*}

L:=L Mod t; {* ( ).*}

End;End;

. (- ).

. 345. , 3, 2(12*2=24). , 4, 2 24+3*2=30. , , 2. :12*2+3*2+1*2=32.

Function GetNumByP:LongInt; {* Plac .*}

Var L,i,j,num: LongInt;ws:Set Of Byte; {*

.*}Begin

ws: = [] ;L: =0 ;For i : = 1 To m Do Begin {*i -

.*}num:=0; {* .*}For j:=1To A[ i]-1 DoIf Not (j In ws) Then Inc (num); {*

j ws, ,

A[i].*}

46 2.

ws:=ws+[A[i]]; {* A[i] .*}

L:=L+num*Plac(n-i,m-i); {* ,

i.*}End;GetNumByP:=L;

End;

2.2.3. k- N- -

k - 1 N () - . , ,

[2, 6, 1, 3] - 1, 2, 3, 6. , - , k-- .

,

.

,

12341235

12361245

1246

1256

13451346

13561456

23452346

2356

2456

3456

. -

N k. . -

, -

-

, ,

, MaxInt. -

(. . 47). - . , -

-

N k, , -

.

CkN -. CkN .

Const MaxN=100;Var SmallSc:Array[0..MaxN,0..MaxN] Of LongInt;

{* ,

.*}Procedure FillSmallSc;Var i,j :Integer;BeginFillChar (SmallSc,SizeOf(SmallSc),0);For i:=0 To N Do SmallSc [i,0]:=1;For i:=1 To N DoFor j:=1 To k DoIf SmallSc[i-1,j]*1. 0+SmallSc[i-1,j-1]>

MaxLonglnt Then SmallSc[i,j]:=MaxLongIntElse SmallSc[i,j]:=SmallSc[i-1,j]+SmallSc

[i-1,j-1]; {* 1.0 ,

.

, ""

.*}End;

.

Type SmallSc=Array[0..MaxN] Of LongInt;Function Res(k:Integer): LongInt;Var A,B:SmallSc;

i,j:Integer;

48 2.

BeginFillChar (A,SizeOf (A),0) ;FillChar(B

rSizeOf(B),0) ;

A[0]:=1;A[1]=1;For i:=1 To N Do BeginB[0] :=1; B[1] :=1;For j : = 1 To k Do [j]:=A[j]+A[j-1]; {*

LongInt, ""

.

-

"" .*}:=;

End;Res:=A[k];

End; .

,

N=7 k=5. 21.

0

1

2

3

4

5

6

12345

12346

12347

12356

12357

12367

12456

7

8

9

10

11

12

13

12457

12467

12567

13456

13457

13467

13567

14

15

16

17

18

19

20

14567

23456

23457

23467

23567

24567

34567

1, 2, ..., k, N-k+1,N-k+2, ..., N. - . -

,

. , -

( ) - , .

Procedure GetNext;{*, ( ) .*}

Var i,j:Integer;Begini:=k;

2. 49

While (C[i]+k-i+1>N) DoDec(i); {*, .*}

Inc([i]); {* .*}For j : = i + 1 To k Do [j]:=[j-1]+1; {*

.*}End;

. L . (-) SmallSc, . . - ( ) N k. . , , 0 14 -, . -

( N k). L - L , , . L - .

Procedure GetWhByNum (L:LongInt);Var i,j,sc,ls:Integer;Begin

sc:=n;ls:=0; {* .*}For i:=1 To k Do Begin {*i -

; k-i - .*}

j:=1; {*sc-j - (), .*}

While L-SmallSc[sc-j,k-i]>=0 Do gi {*

SmallSc, L.*}

Dec (L,SmallSc[sc-j ,k-i]) ;Inc (j); {* While

,

SmallSc, .. ,

,

( ), L.*}

50 2.

End;[i]:=ls+j; {* .*}Inc(ls,j); {*

.*}Dec (sc,j); {* ,

.*}End;

End;

(N=7, k=5, L=17).

-

N=9, k=5, L=78. 23479. , .

2. 51

.

. ,

SmallSc. . SmallSc. - , . ,

3 (N=7, k=4). , -

1, 2 3. . , [0] 0 , - .

Function GetNumByWh:LongInt;Var sc:LongInt;

i, j; Integer;Begin

sc:=1;For i :=1 To DoFor j:=C[i-1]+1 To C[i]-1 Do Inc(sc, SmallSc[N-j,k-i]) ;

GetNumByWh:=sc ;End;

(N=9, k=5, C=023479).i1234

5

j1

2, j 4 3, j

568

SC

17171717578

SmallSc[N-j,k-i]70-

-

431

SC

71-

-

757879

(N=7, k=5, =023467).i12345

j1

3 2, j 4 3, j

5 7 6, j

SC

116161618

SmallSc[N-j,k-i]15-

-

2-

SC

16-

-

18-

, -

3 4 . 3 0, 4 1.

52 2.

. -

0 1. () N 0 1, [i]=1 , (N-i+1) - (). N=7, k=5. - , -

, .

0123456

0011111

0101111

011011101110110111101

0111110

1001111

12345

1234612356

1245613456

23456

12347

78910111213

1010111

10110111011101

1011110

1100111

11010111101101

12357

12457

13457

23457

1236712467

13467

14151617181920

1101110

1110011

11101011110110

1111001

11110101111100

2346712567

13567

23567

1456724567

34567

.

. C[i]=0 C[i+1]=1. - , .

, i i. ,

0, 1, 1 - .

.

Const MaxN=...;Type MyArray=Array[1..MaxN] Of 0. .1;Var Curr, Last: MyArray; {*

.*}i , j,Num, N, k: Integer;

Begin ...

For i : = 1 To N Do Begin Curr[i]:=0;Last[i];=0;End;For i:=1 To k Do Begin

Curr[N-i+1]:=1; Las t [ i ] :=1; End; {* .*}

While Not Eq(Curr,Last) Do Begini:=n-1;While (i>0) And Not ((Curr [i]=0) And

(Curr [i + 1] =1) ) Do Dec (i); {* 0 - 1 , ,

.*}Num:=0;

2. 53

For j : = i + 1 n Do Inc (Num,Curr [j] );{* .*}

Curr[i]:=1;For j : = i + 1 To n-Num+1 Do Curr[j]:=0; {*

.*}For j:=n-Num+2 To n Do Curr[j]:=1; {*3

.*}

End;End;

2.2.4. N. -

: N=al+a2++ak, k, a1 ..., ak - . ,

.

-

b1 ..., bk, . b1 ..., bk - N k .

, : - ; ,

; , - .

. N=8, -, .

1

2345678

111111112111111221111222112222311111321113221

910111213141516

3311332411114211422431445111

171819202122

5215361162718

N k P(N,k), P(N). , P(N) P(N,k) k. , (0)=(1)=1. .

54 2.

. N k N , .

. .

, , -

. P(i,j) P(i-j,k) k, k 0 j. -, j i i-j k . .

12345678

000000000

111111111

201122334

300112

.345

400011235

500001123

600000112

700000011

80000000

123

57111522

, (8,3)=(5,0)+(5,1)+(5,2)+(5,3). .

Const MaxN=100;Var N:LongInt;

SmallSc:Array[0..MaxN, 0..MaxN] Of LongInt;Procedure FillSmallSc;Var i, j , k:Integer;BeginFillChar(SmallSc,SizeOf(SmallSc),0);SmallSc[0

r0]:=1;

For i:=1 To N DoFor j : = 1 To i Do

2. 55

For k:=0 To j Do Inc (SmallSc [i, j ] ,SmallSc[i-j,k]);

SmallSc[0,0]:=0;End;

,

. -

.

Procedure Calc;Var i;Integer;

Res:LongInt;BeginRes:=SmallSc[N,1];For i:=2 To N Do Res:=Res+SmallSc [N , i ] ;WriteLn (Res:0:0);

End; . . -, -

.

.

Var Now:Array[0..MaxN] Of Integer; Now[0] , . . -

Now. - 8, 41111, 4211. Now[2]? , Now[1] Now[2], . . - Now[i-1]>Now[i], Now[i] , -

. -

? , 44, 5111. , , i 1. , , -

.

Procedure GetNext;Var i,j,sc:Integer;Begin

i:=Now[0]-1; {* .*}sc:=Now[i+1]-1;Now[i+1]:=0; {*B sc

,

.*}While (i>1) And (Now[i]=Now[i-1]) Do Beginsc:=sc+Now[i]; Now[i]:=0; Dec(i); {*

"".*}

56 2.

End;Inc (Now[i]); {*

.*}Now[0] :=i+sc; {* .*}For j :=1 To sc Do Now[j+i] :=1; {*3

,

.. .*}End;

-

( 8)? , . -.

, Now , Last . , , In.it. ....

While Not(Eq(Now,Last) ) Do Begin {* Eq "",

, ""

, -

While Now[0]1 Do ...*}GetNext;Print; {* .*}

End;

. (L) - (Now). - . L 0. .

sc:=N; i:=1; {*sc - , , i - .*}

While scOO Do Begin {* .*}j:=1; {*,

i .*}??????????? {*3 j

L, .*}Now[i]:=j; sc:=sc-j; Inc(i); {*

.*}End;

, ??? (), L, 0, -

2. 57

, . L 1. .

SmallSc[8,1]. L , L SmallSc[8,1], j , , 2111111., ? SmallSc sc , j, - L. sc, - j. .

Procedure GetWhByNum (L:LongInt);Var x,j,sc:Integer;Beginsc:=N; i:=1;While sc0 Do Begin j:=1;

While L>=SmallSc[sc,j] Do BeginDec(L,SmallSc[sc,j]); Inc(j);

End;Now[i]:=j; sc:=sc-j; Inc (i);

End;Now[0] :=i-1; {* .*}

End; . (Now) -

. . , -

, .

, 22211. , 2, . .

Function GetNumByWh:LongInt;Var i,jk,p:Integer;

sc:LongInt;Begin

sc:=1; {* .*}jk:=N; {* SmallSc.*}For i :=1 To Now[0] Do Begin {*

.*}For p:=0 To Now[i]-1 Do sc:=sc+SmallSc[jk,p];

{*3 SmallSc.*}

jk:=jk-Now[i]; {* .*}End;

58 2.

GetNumByWh:=sc;End;

(N=8, Now 22211).

SC1

2

3

44

jk8

6

4

21

i

1

2

3

45

01010100

SC

12

3

444

jk

6

4

210

, , 4.

2.2.5. N

. .

N=4. 8.

00000001

00100100

01011000

10011010

(- P(N))? N. .

P(N1). , - . -

0. - P(N-2). , : P(N)=P(N-1)+P(N-2), a , . -

, N, 1, N- . - LongInt (N=

2. 59

1597, 2584, 4181, 6765, 10946, 17711, 28657,46368, 75025, 121393, 196418, 317811, 514229,832040, 1346269, 2178309, 3524578, 5702887,9227465, 14930352, 24157817, 39088169,63245986, 102334155, 165580141, 267914296,433494437, 701408733, 1134903170, 1836311903);

.

Function GetSmall (N: Integer) : Longlnt;Begin

If N>NSmall Then GetSmall:=MaxLongIntElse GetSmall:=ResSm[N] ;

End; . . -

=6 ( , -).

0123456

000000000001000010

000100000101001000001001

78910111213

001010010000010001

010010010100010101

100000

14151617181920

100001

100010100100

100101

101000101001

101010

: - ,

..0010... ...0100... ..001010... 010000.... - . -

Now.Procedure GetNext;

Var i: Integer;Begin

i : =N;While (Now[i]=l) Or ( (i>l) And (Now[i-1] =1) )

Do BeginNow[i] :=0;Dec(i) ;

End;Now[i]:=2/

End;

60 2.

. -

. . L -, N1, () 0 1? : 1. !- - N1 - L. .

N=6, =15.

1522220

/1

_ _ _ 2 _ |3456

138532

1 '

Now (******)-j * * * * *

10****100***1000**10001*100010

Procedure GetWhByNum(L: Longlnt);Var i: Integer;BeginFor i:=1 To N DoIf L>=GetSmall (N-i) Then Begin Now[i]:=1;

L:=L-GetSmall(N-i)/EndElse Now[i]:=0;

End;

. -

-

. i 1,

N-i. , - 1 - .

Function GetNumByWh: Longlnt;Var i: Integer;

sc: Longlnt;Beginsc:=l;For i:=l To N Do If Now[i]=l Then Inc

(sc, GetSmall (N-i));GetNumByWh :=s ;

End;

2. 61

2.2.6. . iV--

1 N , -

. -

: ; ;

.

. N 2N. , N . N=15 32768, Integer . N 100. - N -, : (- 1- , Mul) ( 1- , SayTLong).

N . . N , -.

Procedure Calc;Var i: Integer;

Tmp: TLong;Begin

i :=N; { * .

10000, 13 8192.*}Res[1]:=1; Res[0]:=1;{* 0 !*}While i>0 Do BeginIf i>13 Then Begin Mul(Res, 1 shl 13, Tmp);

i:=i-13; End {*, shl - , 1 13 , .. 213. 214?*}

Else Begin Mul (Res, 1 shl i, Tmp); i:=0; End;Res:=Tmp;

End;End;

. iV- 1 N N Now[l], Now[2],..., Now[N]

62 2.

( Now). Now[i], , , Ni+1 .

0123456789101112131415

()0000000100100011010001010110011110001001101010111100110111101111

(Now)0000000100110010011001110101010011001101111111101010101110011000

[][1]

[1,2][2]

[2,3][1,2,3]

[1,3][3]

[3,4][1,3,4]

[1,2,3,4][2,3,4]

[2,4][1,2,4]

[1,4][4]

-

0 2 4 -1. - ( ), , , -

2, , Now. . .,Now[l]:~B[l] i 2 N. , -

Now. , -

. Now .

Procedure TransferB;Var i: Integer;BeginFillChar (Now, SizeOf (Now) , 0) ;Now[l]:=B[1];For i:=2 To N Do Now[i] :=B[i] Xor B[i-1] ;

End;

2. 63

: [1]:=Now[1], i 2 N.

Procedure TransferToB;Var i , j : Integer;BeginFillChar(B, SizeOf(B), 0);B[l] :=Now[l] ;For i:=2 To N Do B[i] :=Now[i] Xor B[i-1] ;

End; ,

,

.

-

.

Procedure GetNext;Var i: Integer;Begin

i:=N+l;Repeat

Dec (i) ;B[i] :=B[i] Xor 1;

Until B[i]=l;Now[i]:=Now[i] Xor 1;

End; {* 1111 (N=4) ?*}

-.

,

, , -

. ,

2*N. - ( ) . ,

. , , -

, . -

0000 N=4 .

64 2.

0100:

(- 1100):

(- 1000):)

.

L L . -

L - .

Procedure GetWhByNum (L: Longlnt);Var i: Integer;Begin

For i:=l To N Do BeginB[N-i+l] :=L And 1; {*

L. *} L:=L shr 1; {*' L (

2) . *}End;

End; .

-

-

.

Function GetNumByWh: Longlnt;Var sc: Longlnt;

i: Integer;Begin

sc:=0;For i:=l To N Do sc:=sc*2 + B[i] ;GetNumByWh:=sc+l;

End;

2. 65

2.2.7. 2*N ,

i (l

66 2.

Now[i-1]: = ' ('; Now[i]: = ') ';{". *}Now:=Copy(Now,I,i);

sc:=0;{" . *}

For j:=l To i Do If Now[j ] = ' (' Then Inc(sc) ElseDec (sc) ;

While sc0 Do Begin{" . *}Now:=Now+r)';Dec(sc) ;

End;While Length (Now)

2. 67

. ScSmall[3,l]=2: ((), ()(. ScSmall[4,2]=3: (((),(()(, ()((. ScSmall[4,0]=2: ()(), (()). ScSmall[5.1]=5: ()()(,(())(, ((). (()(). ()(() :ScSmall[5,l]=ScSmall[4,0]+ScSmall[4,2]. - ScSmall[4,0] -, ScSmall[4,2] . - .

ScSmall[2*N,0].

01

2

34

567

8

-1

00

0

00

0

00

0

01

01

02

0

5014

1

01

02

0

5014

0

2

001

03

090

28

3

000

104

014

0

4

000

01

0

50

20

5

00

0

001

060

6

00

0

0

001

07

7

00

0

0

00

01

0

800

0

0

00

001

Longlnt.Const SmallN=37;Var ScSmall: Array[-1. .SmallN + 1, -1. .SmallN + 1]

Of Longlnt; -

.

Procedure FillSmallSc;Var i , j : Integer;BeginFillChar (ScSmall, SizeOf(ScSmall), 0) ;

ScSmall[0, 0]:=l;For i:=l To SmallN-1 DoFor j:=0 To SmallN Do ScSmall[i,j]:=ScSmall

[i~lrj-l]+ScSmall[i-l,j+l];

End;

SCSmall. 'Function GetSmallSc (N, Up: Longlnt): Longlnt;Begin

If (N

68 2.

If (N>SmallN) Or (Up>SmallN) Then GetSmallSc:=MaxLongIntElse GetSmallSc:=ScSmall[N, Up];

End; -

.

. -

Up, , - .

Procedure GetWhByNumfL: Longlnt);Var i, Up: Integer;

P: Longlnt;BeginNow:=rr; Up:=0;For i:=l To N*2 Do BeginP:=GetSmallSc(N*2-i, Up-1);If (L>=P) Then Begin Now:=Now+'('; Inc (Up);

L:=L-P; EndElse Begin Dec (Up); Now:=Now+')'; End;

End;End;Function GetNumByWh: Longlnt;Var sc: Longlnt;

i, Up: Integer;Beginsc:=l; Up:=0;For i:=l To N*2 DoIf Now[i]='(' Then Begin sc:=sc+GetSmallSc

(N*2-i,Up-l) ;Inc (Up) ;EndElse Dec(Up);GetNumByWh:=sc;

End;

2.3. 1. -

k 1, 2, ... N. 1, 1, ...,1, N, N, ..., N.

2. - k, i- i. 1, 1, ..., - 1, 2, ..., k.

2. 69

3. N - (, - , ) (, N=4):

4, 3+1, 2+2, 2+1+1, 1+1+1+1; 4, 2+2, 1+3, 1+1+2, 1+1+1+1; 1+1+1+1, 1+1+2, 1+3, 2+2, 4.

4. - 2*N, N N , ,

. .

.

5. , . - R(N,1).

Procedure R(trk:Integer);

Var i:Integer;Begin

If t= l Then Begin A[k]:=1;;End

Else Begin[]:=t;;For i:=l To t-1 Do Begin A[k]:=t-i;R(i,k+l);End;End;

End;

6. - . 2-5 . 2.2.1.

7. 1-5 . 2.2.1 -.

8. 1-5 . 2.2.2 -.

9. 1-5 . 2.2.3 -.

10. [18] - .

Const -4 ;Var A:Array[l..n] Of Integer;

i:Integer;Function Place (i ,m: Integer) -.Integer;Begin

If (m Mod 2=0) And (m>2) ThenIf i

70 2.

Else Place:=m-2Else Place:=m-l;

End;Procedure Perm (m:Integer);Var i,t:Integer;Begin

If m=l Then Begin For i:=l To n Do Write (A[i] :3) ;WriteLn;End

ElseFor i :=J To m Do Begin

Perm (m-1) ;If Km Then Begin t: =A[Place (i ,m) ] ;

AfPlace(i,m)]:=A[m];A[m]:=t;End;End;

End;BeginFor i:=l To n Do A[i]:=i;Perm (n);

End.11. [18] -

.

Const n=4;Var A:Array[1..] Of Integer;

i,j,p,l:Integer;BeginFor i:=l To n Do A[i]:=0;i:=0;Repeat

For 1:-1 To n Do Write (A[l] : 3) ;WriteLn ;i : =i +1 ;p: =1 ;j : =i ;While j Mod 2=0 Do Beginj:=j Div 2;p:=p+l;

End;If pn;End.

12. [18] - .

Const n=5;^;

Var A:Array 11. .] Of Integer;i,j,p:Integer;

2. 71

BeginFor i:-l To Do A[i] :=i;p:=k;While p>=l Do BeginFor i:=l To Do Write (A [i] : 3) ; WriteLn;If A[k]=n Then p:=p-l Else p:=k;If p>=l ThenFor i:=k DownTo p Do A[i] :=A[p] +i-p+l ;

End;End.

13. . - , -

: ( ) (). -

,

. ,

.

, -

-

N (1

72 2.

P(N-2). , : P(N)=P(N-l)+P(N-2), a , .

, N, - 1, iV- . Longlnt N

2. 73

: 1, 2 . . : - , -

( !) ,, , -

. , .** -

,

. ,

-

. 3 2 16 45. , () - .

, . , -

-

,

, . -

. -

.

, .

, , ( ) , , .

Const NMax=13;Type MyArray=Array[l..NMax] Of Word;Var A:MyArray;Function Check (Const P-.MyArray) :Boolean;

{* - ?*}

Begin

End;

Procedure Solver-Type pnt=*Ilem;

Ilem=Record{* . " ". .

- ykw, - ykr. *}

. , -

, , - .

74 2.

Next:pnt;Data:MyArray;End;

Var R,Q:MyArray;ykr,ykw, p:pnt;i -.Word;head:pnt;

Function NotQuerBoolean;{* - ?*}

Begin

End;BeginNew (ykw);.Data:=;.Next:=Nil;head:=ykw;{*' . *}

ykr:=head;While ykrOnil Do Begin {* . *}:=

.Data;{* . *}For i:=l To N-l Do Begin {*

. *}Q:=R;Swap(Q[i] ,Q[i+l]) ;If Check (Qj And NotQue Then Begin {*

,

. *}New ();.Data:=Q;pA.Next:=Nil;ykw*.Next:=p;ykw:=p;Print (Q);{^

. *}End;End;ykr:=.next;{*

. *}End;End;Begin {* . .

) . *}Solve;

End.

2. 75

15. N .

, -

-

. -

,

2 3 2 8 1. , ,

(, , , ). . , -

:

, -

N, . . in-

put.txt N (l

76 2.

(1

2. 77

tm:=tm*A[p+l] ;Inc (p) ;

End;GetMul:=tm;

End;

(, , . . ). - . 2N , 0 9, , N JV . - , . ,

[,], , , -

20 . =3, =1, 5=999999. 55251. - . , -

.

Function Rec (,Sum:Longlnt) :;{* k ~ Sum - .*}

Var i:Integer;Ans:Longlnt;

BeginIf k=2*N+l Then If Sum=0 Then Rec:=l Else Rec:=0Else Begin

Ans:=0;For i :=0 To 9 Do

If k

78 2.

17. N -. , -

( . . ).

. -

5. . (, ) , -

, -

. , N -, C3N, O(7V4).

3.

, -

*. ,

.

,

, .

, : , - : ( ) - (); - , ,

. -

-

. -

, , .

3.1. ( ) N U

v U2, ..., UN (N -

), =(1, 2, ..., aN), -

.

, , ,

,

-.

80 3.

-

. ,

-1 , A={av 2, ..., _1, ?, ..., ?), -

Else Begin

;For Do Backtrack (,i+1) ;

{*\\ - . *}End;

End; . -

.

, 2V, - I U^ *| U2\ *...*! UN\ . Ut , CN .

3. 81

3.2.

1 . -

. N*N N , -.

.

^2 ( 4,4*109 N=8). - , NN - (1,7*107 iV=8). , , (at, a2, ..., aN) , (1,2,..., N), N\ (4,0*104 N==8) . - ,

( 2V=8 2056 ). , -

N N*N. -

,

. .

, , -

. ,

: - ,

. , -

, af^N/2], - , a^N/2]. -, , , , 12 UJ^N1, .

.

.

Up: Array [2. .16] Of Boolean ;{ * . *}

Down:Array[-7.. 7] Of Boolean;{* . *}

Vr:Array[I..8] Of Boolean;{* . *}

X: Array [1. .8] Of Integer;{ * , . *}

82 3.

, -

( ).Procedure Hod(i,j:Integer); {* .*}Begin

X[d] :=j;Vr[j] : =False;Up [i+j] :=False;Down[i-j]:=False;

End;Procedure O_hod (i,j:Integer);{* .*}Begin

Vr [j]:=True;Up[i+j]:=True;Down[i-j]:=True;End;

Function D_hod(i,j:Integer):Boolean;{* (i,j). *}

BeginD_hod:=Vr[j] And Up[i+j] And Down [i-j] ;

End;

:

Procedure Solve(i:Integer;Var q:Boolean);Var j:Integer;

3. 83

Beginj:=0;RepeatInc (j);q:=False;{* . *}If D_hod (i,j) Then BeginHod(i,j);If i

84 3.

). , - N :

, -

; 90

; -

. ,

.

N, 1 N. , -, i-. -. , ,

,

. Siml, Sim2,Sim3 - , -

. ( ),

, ,

-

.

, True -, -

. ( ) .

Type TArray=Array [I. .N] Of 'Integer;

Procedure Siml(Var X:TArray);Var i:Integer;Begin

For i:=l To N Do X[i] : =N-X[i]+1 ;End;

Procedure Sim2 (Var X:TArray) ;Var i , r:Integer;Begin

For i:=l To N Div 2 Do Beginr:=X[i]; X[i]:=X[N-i+1];X[N-i+1]:=r;

End;End;

Procedure Sim3(Var X:TArray);

3. 85

Var YiTArray;i:Integer;

BeginFor i:=l To N Do Y[X[i]J:=i;X:=Y;

End;Function Cmp (X,Y:TArray) -.Boolean;Var i:Integer;Begini : =1 ;While (iN Then Cmp:=FalseElse If Y[i]

86 3.

-. -

. , - ,

.

: -

.

Program Ferz;Uses Crt;Const N=8;Var B:Array[1. .N] Of Integers-Procedure Rf (i:Integer) ;Var j,k,p,t:Integer;BeginFor j:=l To N Do BeginB[i]:=j;k:=l;p:=O;While (k

3. 87

( ). -

. , -

. -

.

.

Const N= ; M= ;Dx:Array[1..8] Of Integer=(~2,-1,1,2,2,1,-1-2);Dy:Array[1. .8] Of Integer=(1,2,2,1,-1,-2,-2,-1);Var A:Array[-l..N+2,-1..M+2] Of Integer;t:Integer;

Solve.Procedure Solve(,,q:Integer);Var z,i,j:Integer;BeginA[x,y]:=q;If q=N*M Then Inc (t)Else For z:=l To 8 Do Begin

i:=x+Dx[z];j :=y+Dy[z];If A[i,j]=O Then Solve (i , j ,q+l)

End;A[x,y]:=0;

End; :

For i:=-l To N+2 DoFor j:=-l To M+2 Do A[i,j]:=-l;{*3"" ,

If, . *}For i:=2 N Do' For j:=l To M Do A[i,j] : =0 /t:=0;For i:=1 To N DoFor j : =1 To M Do Solve (i,j ,1) ;

88 3.

WriteLn{' ',N,'*' ,,' - ' ,t) ;

-

N , ,

( !). ,

. ,

( 150 ). , ,

, ,

. -

(, N - ).

Solve .Procedure Solve(,,q:Integer);Var W:Array[1. .8] Of Integer;

xn,yn,i,j ,m,min -.Integer ;BeginA[x,y]:=q;If (q

3. 8!)

W[m] : =Maxint;{* . *}

End;Inc(i) ;End;EndElse Begin ;halt;{* ,

,

. *}End;[,]:=0;End;

3 . . , -

. .

, . -

-

.

-

.

, -

1891 . : ; ,

( -) .

. 1891 . .

Program Labirint;Const Nmax=...;Dx:Array[1. .4] Of Integer=(1,0,-1,0);Dy:Array[1. .4] Of Integer^(0,1,0,-1);Type MyArray=Array[0..Nmax+1,0..Nmax+1] Of Inte-

ger;Var A:MyArray;

xn,yn,xk,yk,N:Integer;Procedure Init;

Begin{* , .

. *};

90 3.

End;Procedure Print;

Begin{*' - .*};

End;Procedure Solve(,,k:Integer);{*k - ,

, - . *}Var i:Integer;BeginA[x

ry]:=k;

If (x=xk) And (y=yk) Then PrintElse For i:=l To 4 Do IfA[x+Dx[],y+Dy[i]]=O

Then Solve (x+Dx[i] ,y+Dy[i] ,k+l) ;A[x

ry]:=0;

End;Begin

Init;Solve(xn,yn,1);

End.

-

. -

( Dx, Dy) . , -

, , . , -

.

( - ) . . , , -

1, . -

, , -

. ,

, : - ; -

.

-

( , , , -

). -

( )

3. 91

(). . .

, . , -

,

.

Program Labirint;Const NMax=...;

MMax=...;Dx:Array[1..4] Of Integer^(-1,0,1,0) ;Dy-.Array [1. .4] Of Integer^ (0,1, 0 , -1) ;

Type MyArray=Array[O..NMax+1,0..MMax+1] Of Integer;Var A:MyArray;{* . *}

N,M: Integer;xn,yn,xk,:Integer;

Function Solve:Boolean;Type Och=Array[l . .*,1. .2] Of Integer;

Var tfi,j,ykr,ykw:Integer;:Och;{*. *}Y-.Boolean;

BeginA[xn,yn]:=1/ ykr:=0{*

. *};ykw:=l; {* .*}Y:=False;{* . *}[ykw,1]:=;0[ykw,2]:=;{*3

. *}While (ykr

92 3.

Inc(ykw) ;O[ykw,l] :=i+Dx[t] ;[ykw,2]:=j+Dy[t];{*'

. *}End;

End;Solve:=Y;{*' Solve. *}

End; .

BeginInit;{* ,

.

. *}Assign (Output,'Output.txt' );Rewrite(Output);

{*' ( ) .*}If Solve Then Print{* Print

. *}Else WriteLn (' '};

Close (Output);End.

, -

, , , t , .

4 ( ). N ( ). W. I wt vt(=1,2, ..., N). , W. i kt,

v1*k1+v2*k2+...+vN*kN

w1*k1+w2*k2+...+wN*kN=W, ki (O^k^lW/wJ), - .

,

N (-, ?). , :

Const MaxN=????;Var N,N:Integer;{^ ,

. *}Weight,Price:Array[1. .MaxN] Of Integer;{*Bec,

.*}

3. 93

Best,Now:Array[I..MaxN] Of Integer;{*

. *}MaxPrice:Longlnt;{^ . *}

, :

Procedure Solve(k,w:Integer;st:Longlnt); {*k - , w - ,

, st - . *}Var i:Integer;Begin

If (k>N) And (st>MaxPrice) Then Begin {*. *}

Best:=Now;MaxPrice:-st; EndElse If k

94 3.

:Array[1..,1..Max] Of Byte;{^,

,

,

.*}. (

).

@ .Way,BestWay:Array[1..Max] Of Byte;{*

. *}Nnew:Array[I..Max] Of Boolean;{*3

False ,

. *}BestCost:Integer;{* . *}

. v. .

1. (), - ,

(BestCost), - .

2. ( ), - -

.

, BestCost BestWay - .

3. v , - Count Way.

4. v . ,

v, Count, Cost, .

3. 95

5. v .

,

. -

.

,

.

, .

Procedure Solve(v,Count:Byte;Cost:Integer);{*v - ; Count - ; Cost - . *}Var i:Integer;Begin

If Cost>BestCost Then Exit;{*

. *}If Count=N Then BeginCost:=Cost+A[v,1]/Way[N]:=v;{*

.

. *}If Cost

96 3.

Nnew[v]:=False/Way[Count]:=v;{* v,

. *}For i:=1 N DoIf Nnew[B[v,i]] Then Solve (B[v,i],

Count+1,Cost+A[vrB[v,i]]); {* ,

,

v.*}Nnew[v] : =True; {^ v

. *}End;

Solve(l,l,0). - .

, -

( ). - 1 8 9 2 5 6 10 7 4 3 1, 158.

@321933224118151631

32@5158274235181734

1951@23354926343541

335823@333723464632

22273533@191023239

4142493719@24424210

183526231024@252514

15183446234225@132

161735462342251@32

31344132910143232@

.

, [1..50,1..50] . -

, 2500 - .

3.3. . , -

: -, .

. -

.

.

. -

3. 97

,

. .

, ,

21 . , 6: . 20 (3

). - . ( 21). - , 100 - 19 . . N, 8, 12870 . - 15 , 193050 12869 , - 200000 . 1000000 0.2 , ? , N 30, - 60!*(30!*30!). , - 101 7. , -, 60*1017. 1000000 3.2*1013 (). , .

. -

. , -

. ? X, Y . Z X 15 , Y 11 . - , . . 11. . . , -

, . -

.

,

. . . N, 300, 3*301*301, 1000000, -

98 3.

. , N=30 1 iV=300.

. -

. -

, -

. ,

-

. .

,

17 . -

.

3.4.

1 -. -

.

, -

, -

, -

-

.

-

.

> 1

3. 99

For j:=l i Do R[i,j ]=max(D[i,j ]+R[i-l,j ] ,D[,j]+R[i-l,j-l]);

-

.

R, 30. 2 .

N k. , kN. - , k. - , q q1 . , N. - N.

, N=5 (k*k)2*k. , Op[i]

i (Op[l]=0). , N- k, , -

.

.

,

h, N. - N, , .

:

opnl: =Min { : l

100 3.

Op[i] :=Min(Op[i] , Op [j ]+ [i-j ]+1) ;If i Mod j=0 Then Op[i] :=Min (Op[i] ,

Op[i Div j]+j-l) /End;

End;End;

3 - ( - ). -. -

(, , , ). ( N), , , , .

, .

-

.

-

-

. -

,

. -

W -: W[i,j]

W[i-l,j], W[i,j-1], , W[i-l,j-l]+l. - -

: ,

. , W[3,l] , . -

-

, .

, , -

.

. -

: .

.

For i:=l To Length (SI) DoFor j:=l To Length (S2) Do BeginA[i,j] :=Max(A[i-l,j] ,A[i,j-l]) ;

3. 101

If SI [i]=S2 [j] Then A[i , j] :=Max (A[i , j] ,A[i-l,j-l]+l) ;

End;WriteLnf: ', A [Length (SI) , Length (S2) ]) ;

4 N-. - JV-, . N-2 -.

. -

.

.

S[k,l] -

A[k,l] - .

l=k+l k+2 S[k,l]=O,, l>k+2. - i, i - k+1 11, - -

A[k,l]. :S[k,l]~min{dMUHa + +S[k,i]+S[i,l]}. , i=k+l . i=ll.

5 . 1,2, .....pN W , , - W .

: N 5, Pj=5, 2=, 3=9, 4=11, 5=13, W=19. - (A:Array[l..N,0..W] Of Integer), ,

. , () , ( ).

. . , -

. 19 , 18.

12345

000000

100000

200000

300000

400000

555555

655555

757777

857777

957999

10 11 1257999

5791111

512121212

13 14 15 16 17 18 19512121212

512141414

512141414

512161616

512161616

512161818

512161818

( ), A[i,j] - Ni W=j. .

Procedure Solve;Var i,j:Integer;BeginFor i:=2 To N DoFor j:=1 To W Do BeginIf j-P[i]>=0 Then A[i,j]:= Max (A[i-l ,j] ,

A[i-l,j-P[i]]+P[i])Else A[i,j]:=A[i-l,j];

End;End;

, ,

-

. , -

, -

i /', P[i]Procedure Way(i,j:Integer);BeginIf (i=l) And (A[i,j]=O) Then ExitElse If i=l Then Begin Way (i ,j-P[i] ) ;

Write(P[i],' ');EndElse If A[i,j]OA[i-l,j] Then Begin

Way(i,j-P[i]) ;Write(P[i] ,' ') ;EndElse Way(i-l,j) ;

End; 6 (

). . N ( - ). W. i wi

102 3.

3. 103

vt (=1,2, ..., N). - , W. - , W. i kp - v1*k1+v2*k2++vN*kN

u>1*k1+w2*k2+...+wN*kN=W, kt (0

104 3.

) . ,

. -

.

.

Const MaxN=???;=???;

Type Thing=Record W,V:Nord; End;Var A:Array[1. .MaxN,0. .MaxK] Of Word;

P:Array[1..MaxN] of Thing;Old,NewA:Array[0..MaxK] Of Longlnt;N,W: Integers-

Procedure Solve;Var k,i,j:Integer;Begin

FillChar(Old,SizeOf(Old) ,0) ;For k:=l To N Do Begin { * . *}FillChar(NewA,SizeOf(NewA) ,0);For i:=0 To N Do {* . *}For j :=0 To i Div P[k] .W Do {*

-

.*}If j*P[k].V+01d[i-j*P[k].W]>=NewA[i]

Then BeginNewA[i] :=j*P[k] .V+Old [i-j *P [k] .N] ;A[k,i]:=j;{*3 j ?*}

End;Old:=NewA;

End;End;

.

Procedure OutWay(k,l:Integer);Begin

If k=0 Then ExitElse Begin

OutWay(k-l,l-A[k,l]*P[k].W) ;{*A . *}Write (A[k

r 1] ,' ');

End;End;

OutWay(N,W). - . .

3. 105

.

yL I , - i, i+1, ..., N j/ i=W j/.=0,i,...,W =2,3, ...,N. .

3.5.

. . .

.

(), ( ) - , ,

. ,

. -

( ). ,

-

, -

. -

., ., ., . . -

.// . 1965. 1. .13-22.

106 3.

, ,

.

. -

, -

, .

. :

24.

: - 2, 1. - 27. .

( ), 27. (1-3-2->4->5-6->1) , - . 27 . -

. ,

. -

. -

. .

3. 107

( 14) :

, .

. -

( ). -

, (1, 2)., , ,

. () ,

.

.

- ,

.

, 1+0=1. , , -

1. . -

. ,

. , . -

, (1,2) - (1,2). - , . . 15. .

() . , (2,1) , .

108 3.

. , , -

. , , -

(1,2), 14. ,

, .

-

(3,1) (3,1). - 14+5=19. -

, (2,3) - . -

.

-

. -

,

14+2=16. .

-

.

.

,

-

.

. , -

16 1246>531. -,

16, -

.

,

15. - 16 17. -. ,

16. -

( ). ? - , -

3. 109

-

. -

.

-

.

. 3.2.2 -

, -

. -

N? -,

iV

110 3.

Procedure Search (Var S:Number);Var i,j:2..N;Begin

S: = [2. .N] ;For i:=2 To N Div 2 DoIf i In S Then Beginj : =i +i ;Nhile j

3. 111

cnt:Integer;{* .*}ok:Boolean;{* - . *}

Procedure Init;Var i,j,k,1:Integer;Begin

For i:=2 To 6 DoFor j:=1 To 6 DoFor k:=l To 6 Do

For 1:=1 To 6 DoA[ (i-1) *216+(j-l) *36+(k-l) *6+l] ; =

Chr (i +Ord 0')) +Chr (j +Ord ('' 0')) ++Chr(k+Ord('0'))+Chr(1+Ord('0') ) ;

For i:=l To Do [i ] :=True;ent:=0;ok:=False;

End; . -

, .

32, 24. 1 0, - kr (1) bk(O).

11121314212223243132333441424344

TrueTrueTrueTrueTrueTrueTrueTrueTrueTrueTrueTrueTrueTrueTrueTrue

( )

FalseFalseFalseFalseFalse

False

False

( kr)False

False

False

False

False

FalseTrueFalseFalseFalseFalseFalseFalseFalseTrueFalseFalseTrueFalseTrueFalse

, 16 , - . ,

kr bk, :

112 3.

Function Pr (a,b:Post;kr,bk:Integer) -.Boolean;Var i,x:Integer;Begin{* "".*}

x:=0;For i:=l To 4 Do If a[i]=b[i] Then Inc(x) ;If xObk Then Begin Pr:=False;Exit;End;

{* "". *}x:=0;For i:=l To 4 Do If (a[i]ob[i]) And (Pos (b[i] ,a) 0)

Then Inc(x) ;If xOkr Then Begin Pr:=False;Exit;End;Pr:=True;

End; h.Procedure Hod(h:Post);Var i,kr

rbk:Integer;

BeginInc(cnt) ;Write(cnt:2,' . ' ,h,'-');ReadLn(kr,bk);If bk=4 Then Begin ok: =1'rue/;EndElse For i:=l To Pmax Do If B[i] And Not Pr(A[i]

r

h,kr,bk) Then B[i]:=False;End;

.

BeginClrScr;Init;Hod('1223');While Not ok Do Begin

\ "|Hod(h);

End;End.

:

cnt ? - .

? . , (-

3. 113

- True), - .

. -

.

3.7. 1. *. -

, .

, . ,

-

. ,

, , -

, . , -

, ,

.

, -

,

.

: 1 N (4

114 3.

, .

:

-

.

-

, ,

. , -

. .

,

, ,

, , -

.

( - ). -. . -

. : -, , , -

-

; -, ,

,

, -

; -, -

, ,

.

: . -

() .

, .

.

, .

3. 115

Const Nmax=150;Type Nint=0..Nmax+l;

Sset=Set of 0..Nmax;Var A:Array[Nint] Of ;

N:Integer;{* *.} .

Procedure Solve(k:Nint;Res,Rt:Sset); {*k- A; Res - , ; Rt - , ""

; min - ; mn - ; Rbest - ;Rwork - ; -Solved, [], [1..N]) .*}Var i:Nint;Begin

If Rt=[] Then Begin If nm

116 3.

i i , ,

i , .2. -

, ? . , -

9*9, 10*10 11*11.. ,

, ,

.

3. N*N(N

3. 117

, : ; ; ,

.

M*N (50). - 0 4 .

-

, ,

.

.

(0

118 3.

C[i] , i- (i+l)-u -, [] - . ,

, -

, -

, .

.

2* , 2** . - On, Ag: Array[1..M] Of Record wh, pr:Integer;End;. On[i], (wh) (jpr) i-ro , . i -: , . i - , +1, , - i=M. , On[i]=min{Z[i],C[i]+On[i+l].pr}. - On[i+lJ.pr? ; On[j]:=(j,Z[j]). On[j1],On[j-2], ..., On[j+1]. - Ag[i], - i On[i].pr Ag[i].pr .

10. (Array [1..N.1..M] Of Byte), 0 1, , 0.

. N=5, =6 (. ). (Array[l..N,l..M] Of Byte) . -

-

(B[i,j]=lA[i,j]) . , i=2..N j=2..M B[i,j]=O, A[i,j]=l, B[i,j]=Min(B[i-l,j],B[i,j-l],B[i-l,j-l])+1, A[i,j]=O. . -

3. , -

-

? ?

3. 119

Procedure Solve;Var i, j, , nx: Integer;Begin

sqa:=0;{*. *}For i:=1 To N DoFor j:=1 To M Do

If A[i, j]=0 Then B[i, j]:=B[i, j-l]+l ElseB[i, j]:=0;{*B , - . *}

For i:=1 N DoFor j :=1 To M Do Begin

nx:=B[i, j];For k:=i DownTo 1 Do Begin

nx:=Min(nx, B[k, j]);If nx* (i-k+1) >sqa Then sqa:=nx* (i-k+1) ;

End;End;

End;11. *. N*M -

2*1 (

120 3.

-

, .

.

,

, : 1, - 0. , -

00100011 (35). - 2' ( 0 2'-1), i .

,

. (Jk,j) - i, , - k. B[k,j] - .

B[O,j] - (i,j) .

, [0,0]=1 i, , -

, .

B[k,0]=0 k0, .

B[k,j] i. -, B[l,j1] I (0

3. 121

For j:=l 20 Do Begin {* . *}

For :- max Do {* . *}For l:=0 To max Do {* 1.*}If Can(k,l,i) Then [k,j] ;=S[k,j] +B[1 ,j-l] ;

{*' "" Can (k,l,i). *}

A[i,j]:=B[O,j];End;

End;End;

-

-

.

( ) - . -

, , -

I k, , .

-

.

. -, N ( ), - 0. -, =1 ^ 1. -, - . -,

2* - : A[2,t]=A[2,t-l]+A[2,t-2] ( ). -

-

Solve. . N - (t=8, y'=20), 3547073578562247994.

12. *. , .

. -

, ,

, , -

1993 .

122 3.

. -

, .

-

,

( ). -

. : - ; -, -

.

. -

,

.

West:

- -

, -

i, . . - 1..(1). ?

, ,

. Qt , - i i . , . . d[j] (jeQt) ,

.

:

, ? d , . d[i], /', . - : A:Array [I..max] Of Recordd, l:Byte; End;. , - . -

-

? . (-

3. 123

ray[ 1 ..max,I ..max] Of Record d,l:byte;End;). A[i,j].d ( i j) , : i 1( ) 1 / ( ). - N-ro N-&. , A[l,l].d=l uA[i,j]A[j,i]- Qt - , i. :

A[i,j] . d=max (A[k,j] .d+1) , keQi, koj , j1;A[i,i] .d=max (A[k,i] .d) , i>l keQi.13.

.

. , : New.Ar-[1..] Of Boolean; way, way_r:Array[l..2*max] Of Byte; -

(New[i]=True i -). - .

_.

. I.

.

Begin

FillChar(New,SizeOf(New) ,True);yk:=1;Way[yk]:=2;pp:=True;yk_max:=0;Solve (1) ;

End. .

Procedure Solve(i:Byte);Var j ,pn

rpv:Byte;

Begin

124 3.

If i=N Then pp:=False; {* . *}If (i=l) And Not pp Then Begin {*

. *}If yk>yk__max Then Begin {* .*} max:=yk ; way_r:=way;

End;pp:=True; {* . *}

End;(* , . *}

If pp Then Begin pn:=i+l; pv:=n; EndElse Begin pn:~l; pv:=i-l; End;

For j:=pn To pv DoIf West [i,j] And New[j] Then Begin { *B

j i, j . *}

Inc ();Way[yk]:=j;New[j]:=False; {* j . *}

Solve (j) ;New[j]:=True;Way[yk]:=0;Dec (yk); {*

j . *}If j=N Then pp: =True ; {*

(N) , . *}

End;End;14. . N . -

i . -

-

.

, . .

, , .

( 0). - i , , -

St.

, -

. .

, ,

, ,

-

.

3. 125

. N, , (l

126 3.

17. , , -

- -.

.

, -

(, 3 7, 5 6, 10 11), , - . ,

. , 7- - 11-, , 8-, . .

: N ,

I, m N=l*m (1

3. 127

18. N*M (2

128 3.

21. . - .

1*1*4 , .

. ( ) , .

22. 8*8 : -. -

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3. 139

[1,], Array[l..MaxN+l, O.MaxK] Of Longlnt. - -

.

Procedure Solve;Var i , j : Integer;

sum: Longlnt;Begin

sum:=0;For i : = N DownTo 1 Do Begin

sum:=sum + Pr[i] ;For j:=l To Do Begin

A[i, j]:=A[i, j-1];If i+j3. N2*(N2+l)/2. , N*(N2+1 )/2. : - , -

1, 2, 3, ..., N*N - .

. 9! N=3 8 , 161=20922789888000 - 7040 -

140 3.

. JV=4.

( - ). ( , -) , , -

/=4. iV .

4.

, -

.

( ), , , . -

-

, ,

.

4.1. V

G(V,E). V - N . - , . , -

, ; , , , .

, , . ,

, . -

. -

, (, v) . , -

, , -

.

, () -.

. -

.

: ; ; . : .

N*N.

142 4.

i j I j

R *2. .

4.2.

4.2.1. . -

v. , v. . . -

q , - q (), q , . , - , . ,

, ,

: Nnew : Array[1..N] Of Boole-an.

.

-

. -

-

.

,

-

,

.

.

Procedure Pg(v:Integer);{* Nnew . *}

Var j:Integer;BeginNnew[v]:=False; Write(v:3);For j:=l To N Do If (A[v,j ]0) And Nnew[j]

Then Pg (j) ;End;

.

4. 143

FillChar(Nnew,SizeOf(Nnew),True);For i:=l To N Do If Nnew[i] Then Pg(i) ;

-

. -

: ; Nnew . - St, yk.

Procedure (v:Integer);Var St:Array[l..N] Of Integer;

yk, t, j:Integer;pp:Boolean;

BeginFillChar(St,SizeOf (St),0); yk:=0;Inc(yk);St[yk]:=v;Nnew[v]:=False;While yk0 Do Begin {* . *}t:=St[yk];{* " "

. *}j:=2;:=False;Repeat

If (A[t,j] 0) And Nnew[j] Then pp:=TrueElse Inc (j ) ;

Until pp Or (j>=N); {* ,

, . *}If pp Then Begin

Inc (yk);Nnew[j]:=False;{*

. *}End

Else Dec (yk); {*"" . *}

End;End;

4.2.2. . ( )

, , .

, -

. -

.

144 4.

. -

.

-

.

-

.

.

Procedure Pw(v:Integer);Var :Array[1. .N] Of 0..N;{*. *}

ykl,yk2:Integer;{^ ,ykl - ; yk2 - . *}j -.Integer;

BeginFillChar(Og,SizeOf(Og) ,0) ;ykl:=0;yk2:=0;{* . *}Inc(ykl);Og[ykl]:=v;Nnew[v]:=False;{*B

- v. *}While yk2

4. 145

, ). , - .

.

.

-

JV NN'2.. .

(). - -

. -

,

, , Tree (Ar-ray[1..2,1..N] Of Integer).

. ,

.

.

4.3.2. . G=. .

.

. .

-

, .

G : - . G G_. G

146 4.

G , G. , -

,

.

G - :

Turn (Array[1..N] Of Integer) (Down) (Up) ;

Nnew; , , Tree; numb. :

FillChar(Nnew,SizeOf(Nnew),True);FillChar(Tree,SizeOf(Tree),0);Nnew[l]:=False;Turn[1]:=1; Down:=1;Up:=2; {*B

. *}numb:=0;

Procedure Solve(v,q:Integer) ;{*v - , , q - ,

. *}Var j:Integer;Begin

If Down>=Up Then Exit;j = q vWhile (j

4. 147

If numb=N-l Then Begin ; Exit End;{* , v,.

,

. *}If j=N+l Then Begin Inc(Down);Solve(Turn[Down],1);Dec(Down);

End;End;

-

. -

-

- . , .

.

. - , , . - 8 .

148 4.

4.3.3. . .

. G=. .

. (Array [ 1..3,1.-N* (N-1) Div 2] Of Integer). -. Q=,

. -

.

1. Q, N - .

2. G .

3. -

, -

Q, - -

: -

Q.

8 .

4. 149

4. 3 , Q N-1. .

3?

(Mark:Array[l..N] Of Integer). -

(Mark[i]=i i 1 N). - , , ,

. -

. , ,

Mark .

1234

-

[1,2,3,4,5][1,2,3,1,5][1,2,3,1,1][1,2,2,1,1][1,1,1,1,1]

.

Procedure Chang_Mark(l,m:Integer);{* Mark. *}

Var i,t:Integer;Begin

If m

150 4.

While (i

4. 151

.

Procedure Tree; { * - , . *}Var SM,SP:Set Of 1..N;

min,i,j,l,k,t:Integer;Beginmin:=maxint; SM:=[1..N];SP:=[];{*

. . *}For i:=l To N-l DoFor j:=i+l To N DoIf (A[i,j]

152 4.

, -

. -

v , , v. R :

R(v) - G, -

v. F(v) - G, -

1. 2() (()), . . - 2 . :

, , -

. ( ).

, -

R.Procedure Reach; {* R,

. -

,

. *}Var S,T:Set Of 1. .N;

i,j,l:Integer;BeginFillChar (R,SizeOf (R) ,0) ;For i:=l To N Do Begin {^

i . *}T: = [i] ;Repeat

S:=T;For 1:=1 To N DoIf 1 In S Then{ * ,

S. *}For j:=1 To N DoIf A[l

rj]=l Then T:=T+[j];

Until S=T; { * , ,

i .*}

4. 153

F o r j :=1 N DoIf j In T Then R[i,j]:=l;

End;End;

Q -:

.

1. , (v,u) (u,t) (v,t). Ch=(VJ) ' ,

, Gz . -

G.2. R(v) , v, a Q(u)

, . -

, -

.

4.4.2. . G , -

G i j. G , , G , . ,

i j i /

Q(v) G -, .

, v Q v R, . . Q=R', R* , - R.

A, R Q :

154 4.

j i. G - G. - .

-

. .

-

(1)={1,2,5} G, -, 1. : R(3)={1,2,3,4,5,6,7},

,

G. G*=(V*,E*) : -

G, (*, j*) G* , G (i,j), , I - , i*, a. j -, j * . G* G. -. G* . i . , -

-

, , -

.

. ,

G. - G .

. -

. * G* G - G*, . , G : - G, * G*, G.

4.4.3. , . -

, . -

, ,

. ,

. , ,

4. 155

. -

, . -

,

.

. .

4, 5 7.

. t - , , -

t, , (, ) t.

-

. .

Gv G2, G3, G4 G5 -

1, 2, 3. t, - G

v

Gx G2,

1. - G2 , -

1. G2 - ,

1. . G

x -

G3, G4 G5. - -

. G4 G5 3 G5 . , . . , -

G4, 2 . ,

, ,

, ,

156 4.

.

,

-

,

.

, -

. .

-

, -

(), .

.

?

Num. Num (1,2,3,4,5,6,7,9,8). (v,u), , - , Num[v] Num[u], - v , . Num , Lowpg - ,

, -

.

Num. - (v,u) :Lowpg[v]:=Min(Lowpg[v],Num[u]) v, . -

v (v,u) , ,

v (Lowpg[v]:= Min(Lowpg[v],Lowpg[u])). Lowpg :(1,1,1,2,4,4,4,9,8). - . (,), -, Lowpgfu] Num[v]. , - v . v - ,

, .

, .

Procedure Dvy(v,p:Integer);{* p - v. A, Num, Lowpg - .*}

4. 157

Var :Integer;Begin

Inc(nm);Num[v]:=nm;Lowpg[v]:=Num[v];For u:=l To N DoIf A[v,u]0 ThenIf Num[u]=0 Then Begin

;Dvy (u,v) ;Lowpglv] :=!Min (Lowpg[v] ,Lowpg[u] ) ; { *,

. *}If Lowpg[u]>=Num[v] Then ;

EndElse

If (up) And (Num[v]>Num[u]) Then Begin {*u v. *}

;Lowpg[v]:=Min(Lowpg[v],Num[u]);

End;End;

:

FillChar(Num,SizeOf(Num),0);FillChar (Lowpg,SizeOf (Lowpg) ,0) ;nm:=0;For v:=l To N Do If Num[v]=0 Then Dvy(v,0) ;

G , - .

. -

,

G. (-) ) ?

4.5. 4.5.1.

,

.

. G , -

.

158 4.

,

, , , ,

, -

. G, . - .

, . -

( ) . , , , -

, -

.

, . ,

( ) . , . ,

, -

.

Procedure Search(v:Integer);{*: - , CV - ;yk - . *}

Var j:Integer;BeginFor j:=1 To N DoIf A[v,j]0 Then Begin

A[v,j] :=0;A[j,v] :=0;Search (j)

End;Inc (yk);Cv[yk]:=v;

End;

-

Cv Se-arch.

4.5.2. ,

,

. -

.

.

, .

G. , . -

(backtracking). -, , . ,

k .

4. 159

, . ,

, -

.

(k+1) . , -

, . -

. () , .

,

, .

Procedure Gm(:Integer); {* - . : - ; St -

; Nnew - : . *}Var j,v:Integer;Beginv:=St [k-1]; {* . *}For j:=1 To N DoIf (A[v,j]0) Then {*

v j . *}If (k=N+l) And (j=l) Then ElseIf Nnew[j] Then Begin {^

. *}St[k]:=j;Nnew[j]:=False;Gm(k+1) ;Nnew[j]:=True;End;

End;

160 4.

.

St[l] :=l;Nnew[l] :=False; Gm(2) ;

4.5.3. (V,T) G=

N-1 , N G. , , . . -, -

.

-

G . N1 , G - -N+l , G.

,

.

, -

.

,

. -

( ) - St, ( Gnum) , -. , / , - v, .

.

:=0;:=0;For j : = 2 N Do Gnum[j ] :=0;

4. 161

.

Procedure Circl(v:integer);{*: - ; St -

,

; - St; Gnum -

(num.) , . *}

Var j:Integer;Begin

Inc(yk);St[yk]:=v;Inc (num);Gnum[v]:=num;For j :=1 To N DoIf A[v,j]0 ThenIf Gnum[j]=0 Then Circl[j] {* j

.*}Else

If (jSt[yk-l]) And (Gnum[j]

162 4.

. 1 4 3- 2- 6.

-

.

.

s1 =5, ,

,

1 5 . -

.

-

.

, . . -

, .

: - . D[s,t]. , D[s,t], s . D Array[1..N] Of Integer. , - D . - . S H ( - v, D[t]=D[v]+A[v,t]. v (, ). , , D[v]=D[u]+A[u,v], , - s. t, v, , .,., s - .

Procedure Way(s,t:Integer);{*D, A - . St - . *}

Var v,u:Integer;Procedure Print; (* St.*}Begin

End;Begin

;;v:=t;While vOs Do Begin:=;

;

4. 163

v:=u;End;

End;, D .

, . . -

D. - . -

.

, . ,

D[vJ, , , D[u]+A[u,v]

164 4.

Procedure Dist; (*, D, s, N - . *}Var i,u: Integer;

:Set Of 1. .N;Begin

For i:=l To N Do D[i] :=A[s, i] ;D[s]:=0;T: = [l..N]-[s];While TO[ ] Do Begin:=;T:=T-[u];For i:=1 To N DoIf i In T Then D[i] :=min (D[i]

rD[u] +A[u, i] ) ;

End;End;

4.6.3. G= ,

.

() D s .

, i /.

.

-

:

Numln, Numln[i] ,

-

i; Num, Num[i]

i;

4. 165

St, , - . -

; , .

. i, Nu-mln ( ), St, - ( Num), - Numln , I. , St . - -

.

123456

Numln[2,2,2,1,0,1][2,2,1,0,0,1][1,2,0,0,0,1][0,2,0,0,0,0][0,1,0,0,0,0][0,0,0,0,0,0][0,0,0,0,0,0]

Num[0,0,0,0,0,0][0,0,0,0,1,0][0,0,0,2,1,0][0,0,3,2,1,0][0,0,3,2,1,4][5,0,3,2,1,4][5,6,3,2,1,4]

St[5][4][3]

[6,1][1][2][ ]

_ |

Nm0123456

Procedure Change_Num; {*A, Num . *}

Var Numln,St:Array[1..N] Of Integer;i,j,,nm,yk:Integer;

BeginFillChar(Numln,SizeOf(Numln),0);For i:=1 To N DoFor j:=1 To N Do

If A[i,j]0 Then Inc(NumIn[j]) ;nm:=0;yk:=0;For i:=2 To N Do

If Numln[i]=0 Then Begin Inc (yk)/Stack[yk]:=i;End;

While yk0 Do Beginu:=Stack[yk];Dec[yk];Inc(nm);Num[u]:=nm;

For i:=1 To N DoIf A[u,i]0 Then BeginDec(Numln [i]) ;If Numln[i]=0 Then Begin

Inc(yk); Stack[yk]:=i;

166 4.

End;End;

End;End;

, G ( ) - ( ) . L - ,

i. , . .

Procedure Dist;{*D, - . *}Var i,j:Integer;Begin

D[l]:=0;For i:=2 To N Do D[i] :=!-;{*, Maxlnt .*}

For i:=2 N DoFor j:=1 To i-1 DoIf A[j ,i]Ooo Then D[i] : =Min (D[i] ,D[j] +A[j ,i] ) ;

End; , -

A[i,j] . - . -

Num. . -

?

4.6.4. .

G= (-[1 ..N,1 ..N] Of Integer). - D .

.

[l..m]. :

. -

[l..(m+l)].

4. 167

,

: i (+1) (+1) .

Procedure Dist/ {*Ar D -

. *}Var m,i,j :Integer;Begin

For i:=1 To N DoForj:=l To N Do D[i,j] :=A[i,j] ;

For i : = 2 To N Do D[i,i] :=0;For m:=l To N DoFor i:=2 To N Do

For j:=l To N Do D[i,j] :=Min{D[i,j] ,D[i,m]+D[m,j]};

End; -

D -.

D ( m Dist).

D. - ,

D. M[i,j] - i .

Dist . -, D[i,j] D[i,m]+D[m,j], D[i,jJ, M[i,j]. M[i,j] M[m,j]. .

, 3- - 2-. [3,2] 1, [3,1]. . [3,4] 3-. , :

168 4.

Procedure All_Way (i,j:Integer);{* i j . *}

BeginIf M[i,j]=i Then If i=j Write(i) Else Write (i, '~',j)Else Begin All_Way (i ,M[i,j ] ) ;All_Way (M[i ,j ] ,j) ;

End;End;

4.7. V G,

, , -

.

4.7.1. G=(V,E). -

G, , , . .

. , S, - V, , S - S , .

.

(1, 2), (3, 4, 5),(4, 7), (5, 6) . - -

, -

,

. Q - -

G,

G, S*, ,

. a[G]=3, a S* (3, 4, 5).

,

, (). ,

. -

G G', G' G.

4. 169

G' - ( ), ( ) G. - G' 4, - (2, 5, 7, 8), G ( ).

.2.

. -

-

. -

(k )., ,

.

.

Ss (Array[1..N] Of Integer), k .

( Print). -

, .

,

.

. -

. : Type Sset=Set Of 1..N Var A:Array[l..N] Of Sset.

, , -

i, .

.

. Gg

170 4.

-

.

k. - -

Ggl , , () . -

(Qp) (Qm). Qp Qm ft- - . k ,, i, Qp Qm - . ,

(k+1) Qp Qm - , i, - ( ),, , I. , .

Procedure Find(:Integer; Qp,Qm:Sset);Var Gg:Sset;

i:Byte;BeginIf (Qp=[ ]) And (Qm=[ ]) Then Begin Print () ;

Exit;End;;

1:=1;While i

4. 171

.

Sset.Function Number(A:Sset):Byte;Var i, ant:Byte;Begin

cnt:=0;For i;=1 To N Do

If i In A Then Inc(cnt) ;Number:=cnt;

End; , -

.

.

12

Qp[1-5][4,5]

[ ][ ]

Gg[1..5][4,5]

Ss(1)

(1,4)

2, 3 ,

4,5

.

If Qm [] Then

Else Gg:~Qp; ,

, -

Qp, Qp. Find.

3

2

[ ]

[5]

[ ]

[4]

[ ]

[5]

(1,4)

(1,5)

Find 4 Qp Qm

While Find - k, 2. - 5. Find -.

3

2

[ ]

[ ]

[ ]

[4,5]

[ ]

[ ]

(1,5)

While , Find (/c=1)

172 4.

. : - i Qp Qm. : - Gg. , i, - , -

Qp A[i]. , .

Qp: =Qp- [i ] ;Qm: =Qm+ [i ] ;Gg:=Qp*A[i] ; Find,

k=l.

1

1234

32

12

[2..5]

[2..5][3,5]

[5][ ]

[ ][5]

[3..5][5]

[1]

[1][ ][ ][ ]

[5][3]

[1.2][2]

[1-5]

[2..3][3,5]

[5][ ]

[ ][ ]

[2,3]

(2)(2,3)

(2,3,5)

(3)

Gg

,

Find, Gm

Gm. 1. /, - Qm, , A[j] Qp , -

A[j] ( ). 2. Qm, , - Qp ? :

/77 QP . - . ,

.

Begindelt:=W+1;For j:=2 To N Do

4. 173

If j In Qm ThenIf Number (A[j] *Qp)

174 4.

-

-

. -

(3, 5, 12, 14).

.

4.7.4. . -

*. -

G - *, -

.

, , -

.

, ,

() , . -

, ,

, .

. 1. - * , . .

, ,

.

2. *, . : /1=(1, 6,

4. 175

7, 8), /2=(1, 2, 5, 8), 73=(2, 3, 5), /4=(3, 4), U^(2, 3, 4, 5),/6=(5, 6), /7=(6, 7), !78=(7,8). , -, 5- , , 4- , - 5-. .

4.7.5. , ,

, . ,

.

*. ,

,

.

*,

. 1, : Type Pr = Array [L.MaxN, l..MaxN+l] Of Integer; VarBl:Pr; MaxN . ( ), ( Press).

1 : [12 3 . . N 0], .

,

* . -

. (i) 1. Bl[i,j]0, Bl[i,j], - *,

*. 1 -, 1, 1 , (Press). , N-1 1. - 1 :

176 4.

*, -

. Price(Array[L.MaxN] Of Integer) [15 13 43 8 9 10]. - 1

*. 1 :

(Blocs).Procedure Blocs;{^ , 1 -

. *}Procedure Sort;Begin

End;Procedure Press(i,j:Integer);{*

i, () j, .*}

Var k:Integer;Begink:=j;While Bl [i,k]0 Do Begin { *

.

0. *}Bl[i,k]:=Bl[i,k+l];Inc (k) ;End;End;

Var i,j,cnt:Integer;Begin

4. 177

FillChar (Bl,SizeOf (B1) ,0) ;For i:=l N Do Bl[1,i]:=i;{*,

. *}For i:=l To N-l Do Begin

j:=1;cnt:=0;While Bl[i,j]0 Do BeginIf A*[i,Bl[i,j]]=0 Then Begin{*

. *}Inc (cnt) ;Bl[i+1,cnt]:=B1[i,j];{*

. *}Press (i,j);Dec(j) ;

End;Inc(j);

End;End;Sort;

End; - * (*[2,7]=0, *[4,7]=1). -

-

-

. Bl , ( ) . - ,

, .

. -

, -

, , -

( . , . -,

, . . ,

( ), - . ,

,

,

() . -, . ,

- . , .

178 4.

Type Model=Array [I..MaxN] Of Boolean;Var Sbetter:Model;Pbetter:Integer;{*

. *}S:Model;P:Integer;{* . *}R:Model;{*R[i]=True - ,

i "" . *} () k -

( ) :Procedure Include (:Integer); {*

. *}Var j .-Integer;BeginP:=P+Price[];{* . *}S[]:=True;{* . *}For j:=1 N DoIf A*[j,]=1 Then R[j]:=True; {*,

"" .*}Ends-Procedure Exclude(:Integer);{*

. *}Var j:Integer;Beginp:=p-Pri[];S[k]:=False;For j:=1 To N DoIf (A*[j,k]=l) And R[j] Then R[j