data structure part 1

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Array

. , . . ][L1 U1 , L2 U2 , Ln Un Array [L U] of items 1 + = U L ]1 + = [U1 L1 + 1][U2 L2 + 1][Un Ln n = (U L +1) n ) ( : ]002[ Float 0001 52 A . A[i] = (i L) n + 0011 = 0001 + 4 )0 52( = i .

2 1 3 0 2 1 5

5 6 4 2 1

2 3 Row Major 3 6 4 3 5 4

1.

Column Major 0 2 1 1 2 3 3 5 4 6 5 4

2.

A : Array [L1 U1 , L2 U2] of items ]1 + 2 = [U1 L1 + 1][U1 L = [(i L1) (U2 L2 + 1) + (j L2)] n + ] A[i , j = [(j L2) (U1 L1 + 1) + (i L1)] n + ] A[i , j


: , .2L1 U1 L2 U

]2 1 , 3 1[ : A

==

== C

]2[]3[A

2 5 1 6 3 4 5 = 1 + 2 2 = )1 2( + )1 + 1 2( )1 3( = ]2 , 3[ A 5 = 2 + 3 1 = )1 3( + )1 + 1 3( )1 2( = ]2 , 3[ A 1 = )1 2( + )1 + 1 2( )1 1( = ]2 , 1[ A 3 = )1 1( + )1 + 1 3( )1 2( = ]2 , 1[ A

: A [1 ... 100 , 1 ... 26] of integer 0001 ]6 , 06[ A ]4 , 02[ A .

8704 = 0001 + 2 )1 6( + )1 + 1 62( )1 06( = ]6 , 06[ A 8361 = 0001 + 2 )1 02( + )1 + 1 001( )1 4( = ]4 , 02[ A . )1 + n( n n . 2 1 6 7 )1 + 3 (3 6= = 0 2 5 2 0 0 4

0 = ]A [i , j 0 = ]A [i , j

>===== i > j >===== i < j


. . >===== 1 1 ( i 1) i +j 2 2 2 3 5 4 3 5 1 6 1

1 0 0 >===== 2 5 0 3 1 1

>=====

1 1

( j 1) j +i 2 2 6 3 2 4 7 5 5 6 4

1 6 7 >===== 0 2 5 0 0 4


, m n m n m n . .A m n + Bm n = C m n )for (i = 0 , i < m , + + i )for (j = 0 , j < n , + + j Cij = a ij + b ij

, A mL BLn .C mn = A m L B L ni k ki

3 2 1

4 5 0

1 3 2

2 1 1

43

2 1 0 1 = 1 1 3 5 23 = 24

)for (i = 0 , i < m , + + i )for (j = 0 , j < n , + + j { 0 = Cij )for (k = 0 , k < L , + + k Cij = a ik b kj + Cij }


: ]0, 1[ C . : for Trace. :0 = Cij Cij = a ik b kj + Cij 2 = 0 + 1 2 = Cij 2 = 2 + 0 5 = Cij 5 = 2 + 1 3 = Cij 8 = 5 + 3 1 = Cij i 1 j 0 k 0 1 2 3 L 4 Cij 0 2 2 5 8

.

2 A 1

2 5 T 3 A 7 0 5 2 0 4 2 1 1 1 2 5 0 2 3 1 3 8 4 5 8 3

1 0 7

A m n )for (i = 0 , i < m , + + i )for (j = 0 , j < n , + + j ]A [j] [i] = A [i] [j


Array A[n] , x Int search (A[n] , x) ; { int i = 1 ; while (i n ) return 1 // else return i // } 1 1 n 5 2 5 x 8 3 2 i 1 2 3 4 4 8 5 6 A[i] 1 5 2 8

x = A[4]

Int bsearch (A[n] , int x , int L , int U) { int i ; while {

L + U i= ; 2 if ( x < A[i] ) U = i 1 else if ( x > A[i] ) L = i + 1 else return i // i

} return 1 // }


x 8

i 5 2 4 3

L 1 3 4

U 01 4

]A[i 21 2 5 8

0P(x) = anxn + an1xn1 + . + a2x2 + a1x + a an 1an 2an . 1a a 2 + P(x) = 3x2 + 5x 3 + 3P(x) = 3x 0 3 3 3 0 3 5 0 5 2 3 = 5 +

2 + 001P(x) x

1 001

2 0 Stack

Stack . ) Last In First Owt (LIFO. , . top . , top top . top Stack top

. top n , top n .0 = Top Top = n === ===

push pop . ) Push (x x pop x .)x = pop pop (x


Stack : Array [1 .. n] of items int pop ( ) void push (int x) { { int x ; if (top = = n) if (top = 0) { { C out , = < , > , A[j { ; 1 = sw ; )]1 + swap (A[j] , A[j } ; if (sw = = 0) break } )(Insertion Sort

1 i . i . )For (i = 2 ; i < = n ; + + i { ; ]y = A[i ;1j=i ))]while (j > 0 & & (y < A[j { ; ]A[j + 1] = A[j ;1j=j } ; A[j + 1] = y }


05 1 05 05 04 01

06 2 06 04 05 04

04 3 04 06 06 05

02 4

01 5

03 6

i 2 3 4 5

n 6

y 06 04 02 01

06

j 1 2 1 0 3 2 1 0 4 3 2 1 0

n . : . 1 2 3 4 5 4 8 5 2 6 4 8 4 5 8 2 4 5 8 2 4 5 6 8

n 5

i 2 3 4

j 1 2 1 3 2 1 0 4 3

y 8 5 2 6

A


)(merge sort

n ) 2( 11 11 11 11 11 2 2 1 1 2 2 11 11 2 2 02 11 5 81 7 02 8 2 2 2 02 02 02 02 81 81 81 81 1 1 1 1 1 81 8 8 8 8 8 7 7 5 11 7 7 8 8 7 21 21 8 71 21 81 71 02 . 7 21 71 5 7 7 21 21 21 71 71 71 5 5 5 5 71

)Void mergsort (int L , int U { ; int i ) if ( L < U { ;2/)i=(L+U ; ) mergsort ( L , i ; ) mergsort ( i + 1 , U ; ) merg ( L , i , U } }


:

5 5 5 1 1

1 1 1 5 2

7 7 7 2 5

2 2 2 7 7 3=L 3=L 1=L 1=L 1=L 3=U 3=U 1=U 2=U 4=U 4=L 3=i 2=L 1=i 2=i 4=U 2=U

: merge sort . 1 5 2 3 3 1 4 4 4=L 3=L 3=L 4 4 5 2=L 1=L 1=L 1 3 4 1=L 4=U 4=U 4=U 2=U 1=U 2=U 4=U 1=i 2=i 3=i

A

) 4 , 1 ( Merge sort 1 3 2 5 3 1

: .


)(quick sort

, . , . . . ) 0(n Log n )2 0 (n . 1 21 )(Pivot 3 2 1 2 01 3 2 4 8 5 51 6 7 7 3 j 21 21 8 1 9 41 I 41 51

01 i

2 j 3

8 i 8

1 j 01

7 7

51 41

) Void quicksort (int L , int U { ; int i , j , pivot ) if ( L < U { ; ]i = L + 1 ; j = U ; pivot = A[L ) while ( i < j { ; + + while ( A [i] < pivot ) i ; - - while ( A[j] > pivot ) j ; )]if ( i < j ) swap ( A[i] , A[j } ; ) ]swap ( A[L] , A[j ; ) 1 quicksort ( L , j ; ) quicksort ( j + 1 , U } }


)(Link List

. )( . , . ) (node . . , . ) (Head ) (Header . 4 . 1- ) (node 2- ) (p 3- p . 4- p new node . ) Void insert ( int x , node * start { ; node * p , * q , * new node ; q = start next ; p = start ; new node = new ( node ) ; new node data = x ) 34 while ( q data < newnode 14424 datax

{ ;p=q ; q = q next } ; new node next = p next ; p next = new node }


: 51 :

)Start (p p Start 6 21 41

6 q q 6 21 41 02

21

41

02

03

(15) x

1- )(q2- q next p) p next ( 3- p .) void dellinklist ( int x , node * store { ; node * p , * q ; p = start ;q=p ) while ( p data ! = x { ;q=p ; p = p next } ; q next = p next ; )delete (p }


: 3 .5 5 8 8 3 2 \ 2 \ p 5 8 3 q 5 8 2

: ) node * f ( int x , node * start { ; node * p ; p = start ; ) while ( p ; if ( p data ! = x && p ) p = p next ; else return p } ) void g ( node * start { ) if (start ! = Null { ; C out x ) return ; ( p Left ) right = p right ; ( p right ) Left = p Left ; ) delete ( p }

. . , . , .2 1 3 5 2 0 2 + 52x + 3x


)(Graph

G ) = (Vertex ) , = (Edge . . , , . ) .(Adjacement . n kn . 1 ) n (n )1 n (n 2 . B

A

C

F

D E

A B A B n . n . , . , , b .

a e c = c - b - e = a - b - e - a

d

= c - a - e - b


: . : . n 1 - n . : . : . : . .

1-

. n n n . : )( .

d a a a b c d e f g0 1 1 0 0 0 0

b

g f

b1 0 1 1 1 0 0

c1 1 0 0 1 0 0

d0 1 0 0 1 0 0

e0 1 1 1 0 1 0

f0 0 0 0 1 0 1

g0 0 0 0 0 1 0

c

e

77 w if i , j 1 = ] A [ i , j i j A [ i , j ] = 0 if : , .


3 a 1 c

b 9 01

7 8

d 7 g 2

a a b c d e f g0 3 1 0 0 0 0

b0 0 9 7 8 0 0

c0 0 0 0 0 0 0

d0 0 0 0 0 0 0

e0 0 01 7 0 0 0

f0 0 0 0 5 0 2

g0 0 0 0 0 0 0

e

5

f

a d a b =A c e a b c d e0 1 1 0 0

b1 0 1 0 0

c1 1 0 1 1

d0 0 1 0 1

e0 0 1 1 0

a d a b a b c d e0 1 0 0 0

b0 0 1 0 0

c1 0 0 0 1

d0 0 1 0 0

e0 0 0 1 0

c

e

.


: A 2 A 3 A .

a b c d e a b c d e0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 0

a b c d e a b c d e0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 0

a b c d e a b 2 = A c d e1 1 1 1 2 1 1 1 2 1 1 1 4 1 1 1 2 1 1 1 2 1 1 1 1

a b c d e a b c d e1 1 1 1 2 1 1 1 2 1 1 1 4 1 1 1 2 1 1 1 2 1 1 1 1

a b c d e a b c d e0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 03 = A

a b c d e a b c d e2 2 5 3 2 2 2 5 2 3 5 5 4 5 5 3 2 5 2 2 2 3 5 2 2

) 5 c - d ( :

c-d-c-d c-b-c-d c-e-c-d c-a-c-d c-d-e-d : 2 A . ] [ i , j Ak k i j . 2 n n . 2- . . . .


. n + e n + 2e. e n .

|n=|V |e=|E 1 2 3 4 5 6

2 1 2 3 3 2 2 \ \

3

5

\

3

5 6 1 \

6 1

\ \

1

4

6

5

.

)breadth first search (bfs )depth first search (dfs b a c e g d f a c e g b d

1. 2.

a-b-c-f-e-gd )(bfs

f


: .

, . . ) (bfs . , . ) ( . , ) bfs( . bfs . : .

b a d e

c a-b-f-c-e-d-g b a g f e d g

c

f

, . , . . , dfs .

b a d e

c a

a-b-c-d-f-e-g b

c

f

g

f e

d g


. e :

c a b e g d f

= e - b - c - d - f - g - a = e - c - b - a - d - f - g

Struct Linklist { int data ; struct Linklist *Next ; } typedef struct Linklist node ; struct LinkArray { int sw ; node *Link ; } typedef struct LinkArray pointer ; pointer *graphnodes ; graphnodes = New pointer [n] ; 1 2 3 4 5sw = 0

sw = 0

sw = 0

sw = 0

sw = 0


: . k Void bfs ( pointer graphnodes [ ] , int k ) { node *p ; graphnodes [ k ] sw = 1 ; C out

data structure part 1

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